Research ArticleNonlinear Dynamics of a Toxin-Phytoplankton-ZooplanktonSystem with Self- and Cross-Diffusion

Pengfei Wang12 Min Zhao23 Hengguo Yu23 Chuanjun Dai23

Nan Wang12 and Beibei Wang12

1School of Mathematics and Information Science Wenzhou University Wenzhou Zhejiang 325035 China2School of Life and Environmental Science Wenzhou University Wenzhou Zhejiang 325035 China3Zhejiang Provincial Key Laboratory for Water Environment and Marine Biological Resources ProtectionWenzhou University Wenzhou Zhejiang 325035 China

Correspondence should be addressed to Min Zhao zmcntomcom

Received 27 November 2015 Accepted 28 March 2016

Academic Editor Amit Chakraborty

Copyright copy 2016 Pengfei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nonlinear system describing the interaction between toxin-producing phytoplankton and zooplankton was investigatedanalytically and numerically where the system was represented by a couple of reaction-diffusion equations We analyzed the effectof self- and cross-diffusion on the system Some conditions for the local and global stability of the equilibrium were obtainedbased on the theoretical analysis Furthermore we found that the equilibrium lost its stability via Turing instability and patternsformation then occurred In particular the analysis indicated that cross-diffusion can play an important role in pattern formationSubsequently we performed a series of numerical simulations to further study the dynamics of the system which demonstratedthe rich dynamics induced by diffusion in the system In addition the numerical simulations indicated that the direction of cross-diffusion can influence the spatial distribution of the population and the population density The numerical results agreed with thetheoretical analysis We hope that these results will prove useful in the study of toxic plankton systems

1 Introduction

In marine ecosystems most aquatic life relies on planktonwhich comprises phytoplankton and zooplankton Phyto-plankton comprises most of the primary energy sources inaquatic food webs and it accounts for a large proportion ofthe worldrsquos fixed production Phytoplankton is consumed byzooplankton which provides food for fish and other aquaticanimals In fact the phytoplankton can also render very use-ful service by producing a huge amount of oxygen for otheranimals after absorbing carbon dioxide from environments[1] Thus plankton forms the basis of all aquatic food chainsand it has an essential role in the study of marine ecology [23] However the biomass of phytoplankton population maybe of rapid increase or almost equally rapid decrease and thisphenomenon of rapid change in phytoplankton population iscalled ldquobloomrdquo at some fixed time Because some phytoplank-ton can produce toxin and the accumulation of high biomass

some of these blooms are known as ldquoharmful algal bloomsrdquo[4] which can have toxic effects onmarine ecosystems or evenhuman health thereby causing great socioeconomic damageThus research into bloom dynamics is widespread with aspecial emphasis on harmful algal blooms

In the past two decades there have been major increasesin harmful plankton blooms in aquatic ecosystems [5ndash7]Studies have shown that there are at least eight differentmodes and mechanisms that allow harmful phytoplanktonspecies to cause mortality physiological impairment orother negative in situ effects [8] It is well known that thetoxin-producing phytoplankton has important impacts onthe growth of the zooplankton and thus studies of marineplankton are ubiquitous and significant while the dynamicbehavior of interacting species in the marine is also amajor topic [9] However researchers have paid less atten-tion towards toxin-producing plankton blooms in recentyears [10 11] although some have recognized the role

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 4893451 11 pageshttpdxdoiorg10115520164893451

2 Discrete Dynamics in Nature and Society

of toxin-producing phytoplankton in reducing the grazingpressure due to zooplankton [12ndash15] In [14] a three-speciesmodel comprised a toxin-producing phytoplankton and zoo-plankton and fish population was studied by Upadhyay andChattopadhyay who employed Holling type I II and IIIfunctional responses to describe the liberation of the toxinwhere they concluded that the toxin-producing phytoplank-tonmade a stabilizing contribution to aquatic systems In [15]the dynamical behaviors of toxin-producing phytoplanktonand zooplankton were investigated where the phytoplanktonwas divided into two groups that is susceptible and infectedphytoplankton Of course given the global increase in harm-ful plankton blooms in aquatic ecosystems the study of theeffects of toxic phytoplankton has begun to become a growingconcern in recent years [16ndash19]

In marine ecosystems the diffusion phenomenon ofpopulation exists widely which can impact the distributionof population Many factors can influence the diffusion suchas turbulence and foraging Actually it is indicated that thediffusion induced by mixing play an important role in thepopulation dynamics of the phytoplankton [20] And climatemodels predict that global warming will increase the stabi-lity of vertical stratification [21] and reduce vertical mixing[22] which means that climate change will affect the popu-lation dynamics Hence in order to understand how climatechanges affect the population dynamics the study of thediffusion mechanism of population becomes much moreimportant In this aspect the reaction-diffusion equationmaybe a useful tool to study the spatiotemporal dynamics inducedby diffusion [23 24]

The dynamics of interacting populations with self- andcross-diffusion have been studied widely [25ndash27] In par-ticular Dubey et al [25] analyzed a mathematical modelof a predator-prey interaction with self- and cross-diffusionand obtained the criteria for local stability instability andglobal stability Another study [26] considered a nutrient-plankton model of an aquatic environment in the context ofphytoplankton blooms to analyze the diffusion-driven insta-bility and stability as well as cross-diffusion of zooplanktonunder the influence of phytoplankton in the spatial modelThe results [26] indicated the influence of cross-diffusion ofzooplankton and thus we can consider the effects of morecomplex cross-diffusion in marine ecosystems In the previ-ous work some researchers have studied the dynamical pro-perties of the toxic plankton system with diffusion [28ndash30]Jang et al studied a mathematic model which described thephytoplankton-zooplankton interactions with toxin-produc-ing phytoplankton the study revealed that passive diffusionof both populations can simplify the dynamics of the interac-tions and exhibit plankton patchiness [28] In [29] authorsstudied a delay-diffusion model of marine plankton eco-system which exhibited cyclic nature of blooms In [30] Royconsidered a nontoxic phytoplankton-toxin-phytoplankton-zooplankton model which is described by a reaction-diffusion equation The study demonstrated that spatialmovements of planktonic systems in the presence of TPPgenerate and maintain inhomogeneous biomass distributionof competing phytoplankton as well as grazer zooplanktonThese works have studied the influence of diffusion on toxin

plankton system and obtained some good results Howeverthe influence of cross-diffusion on the toxin plankton ecosys-temwas seldom considered In order to study the influence ofcross-diffusion in the toxic plankton ecosystem we proposea toxin-phytoplankton-zooplankton system with self- andcross-diffusion

2 Model and Linear Stability Analysis

21 Model Analysis In this study we consider a toxin-phytoplankton-zooplankton system with Holling type IIresponse as follows

119889119875

119889119905= 1199061119875 minus 1199031119875 minus 11990621198752minus

1205731119875119885

119870 + 119875

119889119885

119889119905=

1205732119875119885

119870 + 119875minus V1119885 minus V21198852minus

1205733119875119885

119870 + 119875

(1)

where 119875 = 119875(119905) is the density of the phytoplankton popula-tion and 119885 = 119885(119905) is the density of the zooplankton popu-lation at time 119905 119906

1is the phytoplankton natural birth rate

1199031is the phytoplankton natural mortality rate and thus

1199061minus 1199031is the intrinsic growth rate of the phytoplankton 119906

2

denotes phytoplankton intraspecies competition119875119885(119870+119875)

is Holling type II response 119870 is the half-saturation constantfor a Holling type II functional response 120573

1is the rate of

predation for zooplankton V1is the zooplanktonrsquos natural

mortality rate V2is the zooplankton intraspecies competition

1205732is the rate of biomass consumption by zooplankton during

its growth and 1205733denotes the rate of toxin liberation by the

toxin-producing phytoplankton populationWemake a basicassumption that 120573

2gt 1205733 The parameters 119906

1 1199031 1199062119870 1205731 1205732

V1 V2 and 120573

3are all positive constants

Considering the relationship between the climate and thediffusion of species and the fact on the existence of diffusionin population system (1) is developed into a spatial systemwith diffusion We expect to explore the effect of climatechange phytoplankton population by studying the spatialdynamics of the diffusion system The spatial system can bedescribed as

120597119875

120597119905= 1199061119875 minus 1199031119875 minus 11990621198752minus

1205731119875119885

119870 + 119875+ 11986311Δ119875 + 119863

12Δ119885

120597119885

120597119905=

1205732119875119885

119870 + 119875minus V1119885 minus V21198852minus

1205733119875119885

119870 + 119875+ 11986321Δ119875

+ 11986322Δ119885

(2)

where 11986311and 119863

22are the constant diffusion coefficients for

the phytoplankton and zooplankton respectively and 11986312

and 11986321

are the cross-diffusion coefficients for the phyto-plankton and zooplankton respectively Biologically cross-diffusion implies countertransport and it means that the preyexercised a self-defense mechanism to protect against attackby a predator [10] different from the self-diffusion and thevalues of 119863

12and 119863

21may be positive or negative We take

the zero-flux boundary condition of the plankton populations119875(119904 119905) and 119885(119904 119905) in 119904 isin Ω 119905 gt 0 as follows

120597119875

120597119899=

120597119885

120597119899= 0 119904 isin 120597Ω 119905 gt 0 (3)

Discrete Dynamics in Nature and Society 3

and the initial conditions are

119875 (0 119904) = 1198750(119904) gt 0

119885 (0 119904) = 1198850(119904) gt 0

119904 isin Ω

(4)

where Ω sube 1198772 is a bounded spatial domain with smooth

boundary 120597ΩIn addition we take

119863 = (

11986311

11986312

11986321

11986322

) (5)

as the diffusive matrix The determinant of 119863 is det(119863) =

1198631111986322

minus 1198631211986321 We assume that the self-diffusion is

stronger than cross-diffusion thus diffusive matrix 119863 ispositive definite

Let 119891(119875 119885) and 119892(119875 119885) be the right hand sides of theequations in (1) It is obvious that the equilibria 119864

0= (0 0)

and 1198641= ((1199061minus 1199031)1199062 0) always exist

By 119891(119875 119885) = 0 we can obtain the vertical isoclines1198971 119885 = minus(119906

21205731)(1198752+ (119870 minus (119906

1minus 1199031)1199062)119875 minus (119906

1minus 1199031)1198701199062)

This is a downward parabola so the axis of symmetry is 119875 =

(12)((1199061minus 1199031)1199062minus 119870) and the fellowship with the 119910-axis

is 119885 = (1199061minus 1199031)1198701205731 By 119892(119875 119885) = 0 we can obtain the

horizontal isoclines 1198972 119885 = (120573

2minus 1205733)119875V2(119870 + 119875) minus V

1V2

It is easy to show that the line 119875 = minus119870 is the asymptote of1198972 We can prove that 119897

2increases monotonically when 119875 isin

(minus119870 +infin) and the fellowship with the 119910-axis is 119885 = minusV1V2

Based on the analysis above we define

119865 (119875) = minus1199062

1205731

(1198752+ (119870 minus

1199061minus 1199031

1199062

)119875 minus1199061minus 1199031

1199062

119870)

119866 (119875) =(1205732minus 1205733) 119875

V2(119870 + 119875)

minusV1

V2

(6)

and the following conclusions can be established

Lemma 1 (1) If (1199061minus1199031)1199062gt 119870 119906

2((1199061minus1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus119870)V

2((1199061minus 1199031)1199062+119870) minus V

1V2 then

there exists one positive equilibrium point 1198641198901

= (1198751198901 1198851198901) in

system (1) where 1198751198901

lt (12)((1199061minus 1199031)1199062minus 119870)

(2) If (1199061minus 1199031)1199062lt 119870 lt ((119906

1minus 1199031)1199062V1)(1205732minus 1205733minus V1)

then there exists one positive equilibrium point1198641198902

= (1198751198902 1198851198902)

in system (1) it is obvious that 1198751198902

gt (12)((1199061minus 1199031)1199062minus 119870)

(3) If (1199061minus 1199031)1199062

gt 119870 and there at least exists 1198750

isin

(0 (12)((1199061minus 1199031)1199062minus 119870)) to guarantee 119865(119875

0) lt 119866(119875

0) but

119865((12)((1199061minus 1199031)1199062minus119870)) gt 119866((12)((119906

1minus 1199031)1199062minus119870)) then

there exist three positive equilibria 11986411989031

= (11987511989031

11988511989031

) 11986411989032

=

(11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) in system (1) where 11987511989031

lt

11987511989032

lt (12)((1199061minus1199031)1199062minus119870) 119875

11989033gt (12)((119906

1minus1199031)1199062minus119870)

22 Linear Stability Analysis For 1198640= (0 0) and 119864

1= ((1199061minus

1199031)1199062 0) we can find that 119864

0= (0 0) is always unstable and

that1198641= ((1199061minus1199031)1199062 0) is locally asymptotically stablewhen

V1gt (1205732minus 1205731)(1199061minus 1199031)(1199062119870 + 119906

1minus 1199031)

Now let 119864 = (119875 119885) be an arbitrary positive equilibriumand thus 119864 = (119875 119885) satisfies the equations 119891(119875 119885) = 0 and119892(119875 119885) = 0 Then the Jacobian matrix for 119864 = (119875 119885) is givenby

119869119864= (

11986911

11986912

11986921

11986922

)

= (

119875[1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] minus

1205731119875

119870 + 119875

(1205732minus 1205733)119870119885

(119870 + 119875)2

minusV2119885

)

(7)

Note that

tr (119869119864) = 119875 [

1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] minus V2119885

det (119869119864) = 119875 [

1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] V2119885

+(1205732minus 1205733)119870119885

(119870 + 119875)2

1205731119875

119870 + 119875

(8)

From 119869119864 we can find that if 119875 gt (12)((119906

1minus 1199031)1199062minus119870) then

11986911

lt 0 and thus tr(119869119864) lt 0 and det(119869

119864) gt 0 in this case the

equilibrium119864 = (119875 119885) is always locally asymptotically stableBut if 119875 lt (12)((119906

1minus 1199031)1199062minus 119870) then 119869

11gt 0

Next we analyze the stability and instability of system(2) Let 0 = 119906

0lt 1199061

lt 1199062

lt sdot sdot sdot lt infin (lim119894rarrinfin

119906119894= infin)

be the eigenvalues of minusnabla2 on Ω with the zero-flux boundary

condition Set

119883 fl 119880 = (119875 119885) isin [1198621(Ω)]2

120597119875

120597119899=

120597119885

120597119899= 0 (9)

and consider 119883 = ⨁infin

119894=0119883119894 where 119883

119894is the eigenspace that

corresponds to 119906119894

In the condition where (1199061

minus 1199031)1199062

lt 119870 lt ((1199061

minus

1199031)1199062V1)(1205732

minus 1205733

minus V1) we obtain the following conclu-

sions

Lemma 2 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is always locally asymptotically

stable

Theorem 3 If 11986311

= 0 11986322

= 0 11986312

= 0 and 11986321

= 0 thenthe equilibrium 119864

1198902= (1198751198902 1198851198902) is uniformly asymptotically

stable when 11986312

gt 0 11986321

lt 0

Proof The linearization of system (2) at the positive equilib-rium 119864

1198902= (1198751198902 1198851198902) is

4 Discrete Dynamics in Nature and Society

120597

120597119905(119875

119885) = Γ(

119875

119885) + (

1198911(119875 minus 119875

1198902 119885 minus 119885

1198902)

1198912(119875 minus 119875

1198902 119885 minus 119885

1198902)

)

Γ = (

1198751198902

[1

119870 + 1198751198902

(1199061minus 1199031minus 11990621198751198902) minus 1199062] + 119863

11Δ minus

12057311198751198902

119870 + 1198751198902

+ 11986312Δ

(1205732minus 1205733)1198701198851198902

(119870 + 1198751198902)2

+ 11986321Δ minusV

21198851198902

+ 11986322Δ

) = (

Γ11

+ 11986311Δ Γ12

+ 11986312Δ

Γ21

+ 11986321Δ Γ22

+ 11986322Δ

)

(10)

where 119891119899(1205761 1205762) = 119874(120576

2

1+ 1205762

2) (119899 = 1 2)

For each 119894 (119894 = 0 1 2 ) 119883119894is invariant under the

operator Γ and 120582119894is an eigenvalue of Γ on 119883

119894if and only if

120582119894is an eigenvalue of the matrix

119867119894= (

Γ11

minus 11986311119906119894

Γ12

minus 11986312119906119894

Γ21

minus 11986321119906119894

Γ22

minus 11986322119906119894

) (11)

Note that

tr (119867119894) = Γ11

+ Γ22

minus (11986311

+ 11986322) 119906119894

det (119867119894) = Γ11Γ22

minus Γ12Γ21

+ 11986311119863221199062

119894

minus 119906119894(11986322Γ11

+ 11986311Γ22) minus 11986312119863211199062

119894

+ (Γ1211986321

+ Γ2111986312) 119906119894

(12)

When 11986312

gt 0 11986321

lt 0 then tr(119867119894) lt 0 det(119867

119894) gt 0 and

the two eigenvalues 120582+

119894and 120582

minus

119894have negative real parts For

any 119894 ge 0 we find the followingIf (tr(119867

119894))2minus 4 det(119867

119894) le 0 then Re(120582plusmn

119894) = (12) tr(119867

119894) le

(12)(Γ11

+ Γ22) lt 0 and if (tr(119867

119894))2minus 4 det(119867

119894) gt 0 since

tr(119867119894) lt 0 and det(119867

119894) gt 0 then

Re (120582minus119894) =

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

2

le1

2tr (119867119894) le

1

2(Γ11

+ Γ22) lt 0

Re (120582+119894) =

tr (119867119894) + radic(tr (119867

119894))2

minus 4 det (119867119894)

2

=2 det (119867

119894)

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

ledet (119867

119894)

tr (119867119894)

lt 119862

(13)

where 119862 is independent of 119894Thus a negative constant 119862 exists which is independent

of 119894 such that Re(120582plusmn119894) lt 119862 for any 119894 By referring to [31] we

can prove that the positive equilibrium 1198641198902

= (1198751198902 1198851198902) of

system (1) is uniformly asymptotically stable when 11986312

gt 011986321

lt 0

Theorem 4 (1) If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870

(2) If11986311

= 011986322

= 011986312

= 011986321

= 0 then the positiveequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022 where

11986011

= 11986311

119875

1198752

11986022

= 11986322

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

11986012

= minus11986312

119875

1198752minus 11986321

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

(14)

Proof (1) Define a Lyapunov function

1198811(119875 119885) = int

119875

119875

119883 minus 119875

119883119889119883

+1205731(119870 + 119875)

(1205732minus 1205733)119870

int

119885

119885

119884 minus 119885

119884119889119884

(15)

We note that 1198811(119875 119885) is nonnegative and 119881

1(119875 119885) = 0 if and

only if (119875(119905) 119885(119905)) = (119875 119885)Furthermore

1198891198811

119889119905=

119875 minus 119875

119875

119889119875

119889119905+

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885 minus 119885

119885

119889119885

119889119905 (16)

By substituting the expressions for 119889119875119889119905 and 119889119885119889119905 fromsystem (1) we obtain

1198891198811

119889119905= (119875 minus 119875)(119906

1minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875 + 119885)

+1205731(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)

sdot ((1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885)

(17)

Discrete Dynamics in Nature and Society 5

Using the condition that

1199061minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875= 0

(1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885 = 0

(18)

(17) can be simplified as

1198891198811

119889119905= minus(119906

2minus

1205731119885

(119870 + 119875) (119870 + 119875)) (119875 minus 119875)

2

minus1205731V2(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)2

(19)

minus(1205731V2(119870+119875)(120573

2minus1205733)119870)(119885 minus 119885)

2

lt 0 and minus(1199062minus1205731119885(119870+

119875)(119870 + 119875))(119875 minus 119875)2lt 0 always occurs when 119906

2gt 1205731119885(119870 +

119875)119870Therefore 119889119881

1119889119905 lt 0 if 119906

2gt 1205731119885(119870 + 119875)119870 holds

Proof (2) Then by referring to [32 33] we choose thefollowing Lyapunov function

1198812= ∬Ω

1198811(119875 119885) 119889Ω (20)

and by differentiating 1198812with respect to time 119905 along the

solutions of system (2) we can obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω + ∬

Ω

((11986311Δ119875 + 119863

12Δ119885)

1205971198811

120597119875

+ (11986321Δ119875 + 119863

22Δ119885)

1205971198811

120597119885)119889Ω

(21)

Using Greenrsquos first identity in the plane we obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω minus 119863

11∬Ω

12059721198811

1205971198752|nabla119875|2119889Ω

minus 11986322

∬Ω

12059721198811

1205971198852|nabla119885|2119889Ω

minus 11986321

∬Ω

12059721198811

1205971198852nabla119875nabla119885119889Ω

minus 11986312

∬Ω

12059721198811

1205971198752nabla119875nabla119885119889Ω

(22)

where

12059721198811

1205971198752=

119875

1198752gt 0

12059721198811

1205971198852=

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852gt 0

(23)

Based on the analysis above

1199062gt

1205731119885

(119870 + 119875)119870 (24)

In the case of increasing 11986311 11986322

to sufficiently largevalues and if 119863

12 11986321meet the conditions

1198602

12lt 41198601111986022

(25)

then 1198891198811119889119905 lt 0 and thus 119889119881

2119889119905 lt 0 Therefore 119864

1198902=

(1198751198902 1198851198902) of system (2) is globally asymptotically stable when

1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022

Under the condition of1199061minus 1199031

1199062

gt 119870

1199062((1199061minus 1199031) 1199062+ 119870)2

41205731

lt(1205732minus 1205733) ((1199061minus 1199031) 1199062minus 119870)

V2((1199061minus 1199031) 1199062+ 119870)

minusV1

V2

(26)

we have the following conclusions

Lemma 5 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198901= (1198751198901 1198851198901) is locally asymptotically stable

when

119865 (119875lowast) minus 119866 (119875

lowast) gt 0

(119875lowast=

(1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1)2

+ 81199062V1119870

41199062

)

(27)

and the equilibrium 1198641198901

= (1198751198901 1198851198901) is unstable when 119865(119875

lowast) minus

119866(119875lowast) lt 0

Proof Define

119867(119875) = 119865 (119875) minus 119866 (119875)

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(28)

under the condition of Lemma 1 we can find that 1198671015840(119875) lt

0 (0 lt 119875 lt ((1199061minus 1199031)1199062minus 119870)2)

6 Discrete Dynamics in Nature and Society

Thus

(1205732minus 1205733)119870

V2(119870 + 119875)

2lt minus

1199062

1205731

[2119875 + (119870 minus1199061minus 1199031

1199062

)]

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(29)

and combined with det(1198641198901) we can find that det(119864

1198901) gt 0

and

tr (1198641198901)

=minus211990621198752

1198901+ (1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) 1198751198901

+ V1119870

119870 + 1198751198901

(30)

Hence if 119865(119875lowast) minus 119866(119875

lowast) gt 0 then 119875

1198901gt 119875lowast and in this case

tr(1198641198901) lt 0 thus 119864

1198901= (1198751198901 1198851198901) is stable If 119865(119875

lowast)minus119866(119875

lowast) lt

0 then 1198751198901

lt 119875lowast and in this case tr(119864

1198901) gt 0 thus 119864

1198901=

(1198751198901 1198851198901) is unstable

By analyzing Lemma 5 we can find that 119865(119875lowast) minus119866(119875

lowast) =

0 is the value where the plankton system populations bifur-cate into periodic oscillation By 119865(119875

lowast) = 119866(119875

lowast) then we can

obtain

1205731= 120573 =

minus1199062V2(119870 + 119875

lowast) ((119875lowast)2

+ (119870 minus (1199061minus 1199031) 1199062) 119875lowastminus ((1199061minus 1199031) 1199062)119870)

(1205732minus 1205733minus V1) 119875lowast minus V

1119870

(31)

which is a bifurcation value of the parameter 1205732 Now the

characteristic equation of the 1198641198901reduces to

1205822+ det (119864

1198901) = 0 (32)

where det(1198641198901) gt 0 and the equation has purely imaginary

rootsWe take 120582 = 119886+119887119894 If the stability change occurs at 1205731=

120573 it is obvious that 119886(120573) = 0 119887(120573) = 0 and using 119886(120573) = 0119887(120573) = 0 we get (119889119886119889120573

1)|1205731=120573

= ((12)119889 tr(1198641198901)1198891205731)|1205731=120573

gt

0 Thus the transversality condition for a Hopf bifurcation issatisfied

Theorem 6 If (1199061minus 1199031)1199062gt 119870 119906

2((1199061minus 1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus 119870)V

2((1199061minus 1199031)1199062+ 119870) minus V

1V2 and

119865(119875lowast) minus119866(119875

lowast) gt 0 then the criterion for the Turing instability

of system (2) satisfies the following condition (1198861111986322

+1198862211986311

minus

1198631211988621

minus 1198632111988612)2gt 4 det(119863) det(119869

1198641198901

) Consider

(1198691198641198901

= (11988611

11988612

11988621

11988622

) 119894119904 119905ℎ119890 119869119886119888119900119887119894119886119899 119898119886119905119903119894119909 119886119905 1198641198901

= (1198751198901 1198851198901))

(33)

Proof The linearized form of system (2) corresponding to theequilibrium 119864

1198901(1198751198901 1198851198901) is given by

120597119901

120597119905= 11988611119901 + 11988612119911 + 119863

11

1205972119901

1205971199042+ 11986312

1205972119911

1205971199042

120597119911

120597119905= 11988621119901 + 11988622119911 + 119863

21

1205972119901

1205971199042+ 11986322

1205972119911

1205971199042

(34)

where 119875 = 1198751198901

+ 119901 119885 = 1198851198901

+ 119911 and (119901 119911) aresmall perturbations in (119875 119885) about the equilibrium 119864

1198901=

(1198751198901 1198851198901) We expand the solution of system (34) into a

Fourier series

(119901

119911) = sum

119896

(1198881

119896

1198882

119896

)119890120582119896119905+119894119896119904

(35)

where 119896 is the wave number of the solution By combining(34) and (35) we obtain

120582119896(1198881

119896

1198882

119896

) = (11988611

minus 119896211986311

11988612

minus 119896211986312

11988621

minus 119896211986321

11988622

minus 119896211986322

)(

1198881

119896

1198882

119896

) (36)

Hence we can obtain the characteristic equation as follows

1205822minus tr119896120582119896+ Δ119896= 0 (37)

where

tr119896= 11988611

+ 11988622

minus 1198962(11986311

+ 11986322)

Δ119896= det (119869

1198641198901

)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321)

(38)

In order to allow the system to achieve Turing instabilityat least one of the following conditionsmust be satisfied tr

119896gt

0 and Δ119896lt 0 However it is obvious that tr

119896lt 0 Thus only

the condition that Δ119896

lt 0 exists can give rise to instabilityTherefore a necessary condition for the system to be unstableis that

det (1198691198641198901

) minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321) lt 0

(39)

We define

119865 (1198962) = 1198964(1198631111986322

minus 1198631211986321)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ det (1198691198641198901

)

(40)

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

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Differential EquationsInternational Journal of

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Complex AnalysisJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

of toxin-producing phytoplankton in reducing the grazingpressure due to zooplankton [12ndash15] In [14] a three-speciesmodel comprised a toxin-producing phytoplankton and zoo-plankton and fish population was studied by Upadhyay andChattopadhyay who employed Holling type I II and IIIfunctional responses to describe the liberation of the toxinwhere they concluded that the toxin-producing phytoplank-tonmade a stabilizing contribution to aquatic systems In [15]the dynamical behaviors of toxin-producing phytoplanktonand zooplankton were investigated where the phytoplanktonwas divided into two groups that is susceptible and infectedphytoplankton Of course given the global increase in harm-ful plankton blooms in aquatic ecosystems the study of theeffects of toxic phytoplankton has begun to become a growingconcern in recent years [16ndash19]

In marine ecosystems the diffusion phenomenon ofpopulation exists widely which can impact the distributionof population Many factors can influence the diffusion suchas turbulence and foraging Actually it is indicated that thediffusion induced by mixing play an important role in thepopulation dynamics of the phytoplankton [20] And climatemodels predict that global warming will increase the stabi-lity of vertical stratification [21] and reduce vertical mixing[22] which means that climate change will affect the popu-lation dynamics Hence in order to understand how climatechanges affect the population dynamics the study of thediffusion mechanism of population becomes much moreimportant In this aspect the reaction-diffusion equationmaybe a useful tool to study the spatiotemporal dynamics inducedby diffusion [23 24]

The dynamics of interacting populations with self- andcross-diffusion have been studied widely [25ndash27] In par-ticular Dubey et al [25] analyzed a mathematical modelof a predator-prey interaction with self- and cross-diffusionand obtained the criteria for local stability instability andglobal stability Another study [26] considered a nutrient-plankton model of an aquatic environment in the context ofphytoplankton blooms to analyze the diffusion-driven insta-bility and stability as well as cross-diffusion of zooplanktonunder the influence of phytoplankton in the spatial modelThe results [26] indicated the influence of cross-diffusion ofzooplankton and thus we can consider the effects of morecomplex cross-diffusion in marine ecosystems In the previ-ous work some researchers have studied the dynamical pro-perties of the toxic plankton system with diffusion [28ndash30]Jang et al studied a mathematic model which described thephytoplankton-zooplankton interactions with toxin-produc-ing phytoplankton the study revealed that passive diffusionof both populations can simplify the dynamics of the interac-tions and exhibit plankton patchiness [28] In [29] authorsstudied a delay-diffusion model of marine plankton eco-system which exhibited cyclic nature of blooms In [30] Royconsidered a nontoxic phytoplankton-toxin-phytoplankton-zooplankton model which is described by a reaction-diffusion equation The study demonstrated that spatialmovements of planktonic systems in the presence of TPPgenerate and maintain inhomogeneous biomass distributionof competing phytoplankton as well as grazer zooplanktonThese works have studied the influence of diffusion on toxin

plankton system and obtained some good results Howeverthe influence of cross-diffusion on the toxin plankton ecosys-temwas seldom considered In order to study the influence ofcross-diffusion in the toxic plankton ecosystem we proposea toxin-phytoplankton-zooplankton system with self- andcross-diffusion

2 Model and Linear Stability Analysis

21 Model Analysis In this study we consider a toxin-phytoplankton-zooplankton system with Holling type IIresponse as follows

119889119875

119889119905= 1199061119875 minus 1199031119875 minus 11990621198752minus

1205731119875119885

119870 + 119875

119889119885

119889119905=

1205732119875119885

119870 + 119875minus V1119885 minus V21198852minus

1205733119875119885

119870 + 119875

(1)

where 119875 = 119875(119905) is the density of the phytoplankton popula-tion and 119885 = 119885(119905) is the density of the zooplankton popu-lation at time 119905 119906

1is the phytoplankton natural birth rate

1199031is the phytoplankton natural mortality rate and thus

1199061minus 1199031is the intrinsic growth rate of the phytoplankton 119906

2

denotes phytoplankton intraspecies competition119875119885(119870+119875)

is Holling type II response 119870 is the half-saturation constantfor a Holling type II functional response 120573

1is the rate of

predation for zooplankton V1is the zooplanktonrsquos natural

mortality rate V2is the zooplankton intraspecies competition

1205732is the rate of biomass consumption by zooplankton during

its growth and 1205733denotes the rate of toxin liberation by the

toxin-producing phytoplankton populationWemake a basicassumption that 120573

2gt 1205733 The parameters 119906

1 1199031 1199062119870 1205731 1205732

V1 V2 and 120573

3are all positive constants

Considering the relationship between the climate and thediffusion of species and the fact on the existence of diffusionin population system (1) is developed into a spatial systemwith diffusion We expect to explore the effect of climatechange phytoplankton population by studying the spatialdynamics of the diffusion system The spatial system can bedescribed as

120597119875

120597119905= 1199061119875 minus 1199031119875 minus 11990621198752minus

1205731119875119885

119870 + 119875+ 11986311Δ119875 + 119863

12Δ119885

120597119885

120597119905=

1205732119875119885

119870 + 119875minus V1119885 minus V21198852minus

1205733119875119885

119870 + 119875+ 11986321Δ119875

+ 11986322Δ119885

(2)

where 11986311and 119863

22are the constant diffusion coefficients for

the phytoplankton and zooplankton respectively and 11986312

and 11986321

are the cross-diffusion coefficients for the phyto-plankton and zooplankton respectively Biologically cross-diffusion implies countertransport and it means that the preyexercised a self-defense mechanism to protect against attackby a predator [10] different from the self-diffusion and thevalues of 119863

12and 119863

21may be positive or negative We take

the zero-flux boundary condition of the plankton populations119875(119904 119905) and 119885(119904 119905) in 119904 isin Ω 119905 gt 0 as follows

120597119875

120597119899=

120597119885

120597119899= 0 119904 isin 120597Ω 119905 gt 0 (3)

Discrete Dynamics in Nature and Society 3

and the initial conditions are

119875 (0 119904) = 1198750(119904) gt 0

119885 (0 119904) = 1198850(119904) gt 0

119904 isin Ω

(4)

where Ω sube 1198772 is a bounded spatial domain with smooth

boundary 120597ΩIn addition we take

119863 = (

11986311

11986312

11986321

11986322

) (5)

as the diffusive matrix The determinant of 119863 is det(119863) =

1198631111986322

minus 1198631211986321 We assume that the self-diffusion is

stronger than cross-diffusion thus diffusive matrix 119863 ispositive definite

Let 119891(119875 119885) and 119892(119875 119885) be the right hand sides of theequations in (1) It is obvious that the equilibria 119864

0= (0 0)

and 1198641= ((1199061minus 1199031)1199062 0) always exist

By 119891(119875 119885) = 0 we can obtain the vertical isoclines1198971 119885 = minus(119906

21205731)(1198752+ (119870 minus (119906

1minus 1199031)1199062)119875 minus (119906

1minus 1199031)1198701199062)

This is a downward parabola so the axis of symmetry is 119875 =

(12)((1199061minus 1199031)1199062minus 119870) and the fellowship with the 119910-axis

is 119885 = (1199061minus 1199031)1198701205731 By 119892(119875 119885) = 0 we can obtain the

horizontal isoclines 1198972 119885 = (120573

2minus 1205733)119875V2(119870 + 119875) minus V

1V2

It is easy to show that the line 119875 = minus119870 is the asymptote of1198972 We can prove that 119897

2increases monotonically when 119875 isin

(minus119870 +infin) and the fellowship with the 119910-axis is 119885 = minusV1V2

Based on the analysis above we define

119865 (119875) = minus1199062

1205731

(1198752+ (119870 minus

1199061minus 1199031

1199062

)119875 minus1199061minus 1199031

1199062

119870)

119866 (119875) =(1205732minus 1205733) 119875

V2(119870 + 119875)

minusV1

V2

(6)

and the following conclusions can be established

Lemma 1 (1) If (1199061minus1199031)1199062gt 119870 119906

2((1199061minus1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus119870)V

2((1199061minus 1199031)1199062+119870) minus V

1V2 then

there exists one positive equilibrium point 1198641198901

= (1198751198901 1198851198901) in

system (1) where 1198751198901

lt (12)((1199061minus 1199031)1199062minus 119870)

(2) If (1199061minus 1199031)1199062lt 119870 lt ((119906

1minus 1199031)1199062V1)(1205732minus 1205733minus V1)

then there exists one positive equilibrium point1198641198902

= (1198751198902 1198851198902)

in system (1) it is obvious that 1198751198902

gt (12)((1199061minus 1199031)1199062minus 119870)

(3) If (1199061minus 1199031)1199062

gt 119870 and there at least exists 1198750

isin

(0 (12)((1199061minus 1199031)1199062minus 119870)) to guarantee 119865(119875

0) lt 119866(119875

0) but

119865((12)((1199061minus 1199031)1199062minus119870)) gt 119866((12)((119906

1minus 1199031)1199062minus119870)) then

there exist three positive equilibria 11986411989031

= (11987511989031

11988511989031

) 11986411989032

=

(11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) in system (1) where 11987511989031

lt

11987511989032

lt (12)((1199061minus1199031)1199062minus119870) 119875

11989033gt (12)((119906

1minus1199031)1199062minus119870)

22 Linear Stability Analysis For 1198640= (0 0) and 119864

1= ((1199061minus

1199031)1199062 0) we can find that 119864

0= (0 0) is always unstable and

that1198641= ((1199061minus1199031)1199062 0) is locally asymptotically stablewhen

V1gt (1205732minus 1205731)(1199061minus 1199031)(1199062119870 + 119906

1minus 1199031)

Now let 119864 = (119875 119885) be an arbitrary positive equilibriumand thus 119864 = (119875 119885) satisfies the equations 119891(119875 119885) = 0 and119892(119875 119885) = 0 Then the Jacobian matrix for 119864 = (119875 119885) is givenby

119869119864= (

11986911

11986912

11986921

11986922

)

= (

119875[1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] minus

1205731119875

119870 + 119875

(1205732minus 1205733)119870119885

(119870 + 119875)2

minusV2119885

)

(7)

Note that

tr (119869119864) = 119875 [

1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] minus V2119885

det (119869119864) = 119875 [

1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] V2119885

+(1205732minus 1205733)119870119885

(119870 + 119875)2

1205731119875

119870 + 119875

(8)

From 119869119864 we can find that if 119875 gt (12)((119906

1minus 1199031)1199062minus119870) then

11986911

lt 0 and thus tr(119869119864) lt 0 and det(119869

119864) gt 0 in this case the

equilibrium119864 = (119875 119885) is always locally asymptotically stableBut if 119875 lt (12)((119906

1minus 1199031)1199062minus 119870) then 119869

11gt 0

Next we analyze the stability and instability of system(2) Let 0 = 119906

0lt 1199061

lt 1199062

lt sdot sdot sdot lt infin (lim119894rarrinfin

119906119894= infin)

be the eigenvalues of minusnabla2 on Ω with the zero-flux boundary

condition Set

119883 fl 119880 = (119875 119885) isin [1198621(Ω)]2

120597119875

120597119899=

120597119885

120597119899= 0 (9)

and consider 119883 = ⨁infin

119894=0119883119894 where 119883

119894is the eigenspace that

corresponds to 119906119894

In the condition where (1199061

minus 1199031)1199062

lt 119870 lt ((1199061

minus

1199031)1199062V1)(1205732

minus 1205733

minus V1) we obtain the following conclu-

sions

Lemma 2 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is always locally asymptotically

stable

Theorem 3 If 11986311

= 0 11986322

= 0 11986312

= 0 and 11986321

= 0 thenthe equilibrium 119864

1198902= (1198751198902 1198851198902) is uniformly asymptotically

stable when 11986312

gt 0 11986321

lt 0

Proof The linearization of system (2) at the positive equilib-rium 119864

1198902= (1198751198902 1198851198902) is

4 Discrete Dynamics in Nature and Society

120597

120597119905(119875

119885) = Γ(

119875

119885) + (

1198911(119875 minus 119875

1198902 119885 minus 119885

1198902)

1198912(119875 minus 119875

1198902 119885 minus 119885

1198902)

)

Γ = (

1198751198902

[1

119870 + 1198751198902

(1199061minus 1199031minus 11990621198751198902) minus 1199062] + 119863

11Δ minus

12057311198751198902

119870 + 1198751198902

+ 11986312Δ

(1205732minus 1205733)1198701198851198902

(119870 + 1198751198902)2

+ 11986321Δ minusV

21198851198902

+ 11986322Δ

) = (

Γ11

+ 11986311Δ Γ12

+ 11986312Δ

Γ21

+ 11986321Δ Γ22

+ 11986322Δ

)

(10)

where 119891119899(1205761 1205762) = 119874(120576

2

1+ 1205762

2) (119899 = 1 2)

For each 119894 (119894 = 0 1 2 ) 119883119894is invariant under the

operator Γ and 120582119894is an eigenvalue of Γ on 119883

119894if and only if

120582119894is an eigenvalue of the matrix

119867119894= (

Γ11

minus 11986311119906119894

Γ12

minus 11986312119906119894

Γ21

minus 11986321119906119894

Γ22

minus 11986322119906119894

) (11)

Note that

tr (119867119894) = Γ11

+ Γ22

minus (11986311

+ 11986322) 119906119894

det (119867119894) = Γ11Γ22

minus Γ12Γ21

+ 11986311119863221199062

119894

minus 119906119894(11986322Γ11

+ 11986311Γ22) minus 11986312119863211199062

119894

+ (Γ1211986321

+ Γ2111986312) 119906119894

(12)

When 11986312

gt 0 11986321

lt 0 then tr(119867119894) lt 0 det(119867

119894) gt 0 and

the two eigenvalues 120582+

119894and 120582

minus

119894have negative real parts For

any 119894 ge 0 we find the followingIf (tr(119867

119894))2minus 4 det(119867

119894) le 0 then Re(120582plusmn

119894) = (12) tr(119867

119894) le

(12)(Γ11

+ Γ22) lt 0 and if (tr(119867

119894))2minus 4 det(119867

119894) gt 0 since

tr(119867119894) lt 0 and det(119867

119894) gt 0 then

Re (120582minus119894) =

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

2

le1

2tr (119867119894) le

1

2(Γ11

+ Γ22) lt 0

Re (120582+119894) =

tr (119867119894) + radic(tr (119867

119894))2

minus 4 det (119867119894)

2

=2 det (119867

119894)

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

ledet (119867

119894)

tr (119867119894)

lt 119862

(13)

where 119862 is independent of 119894Thus a negative constant 119862 exists which is independent

of 119894 such that Re(120582plusmn119894) lt 119862 for any 119894 By referring to [31] we

can prove that the positive equilibrium 1198641198902

= (1198751198902 1198851198902) of

system (1) is uniformly asymptotically stable when 11986312

gt 011986321

lt 0

Theorem 4 (1) If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870

(2) If11986311

= 011986322

= 011986312

= 011986321

= 0 then the positiveequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022 where

11986011

= 11986311

119875

1198752

11986022

= 11986322

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

11986012

= minus11986312

119875

1198752minus 11986321

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

(14)

Proof (1) Define a Lyapunov function

1198811(119875 119885) = int

119875

119875

119883 minus 119875

119883119889119883

+1205731(119870 + 119875)

(1205732minus 1205733)119870

int

119885

119885

119884 minus 119885

119884119889119884

(15)

We note that 1198811(119875 119885) is nonnegative and 119881

1(119875 119885) = 0 if and

only if (119875(119905) 119885(119905)) = (119875 119885)Furthermore

1198891198811

119889119905=

119875 minus 119875

119875

119889119875

119889119905+

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885 minus 119885

119885

119889119885

119889119905 (16)

By substituting the expressions for 119889119875119889119905 and 119889119885119889119905 fromsystem (1) we obtain

1198891198811

119889119905= (119875 minus 119875)(119906

1minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875 + 119885)

+1205731(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)

sdot ((1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885)

(17)

Discrete Dynamics in Nature and Society 5

Using the condition that

1199061minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875= 0

(1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885 = 0

(18)

(17) can be simplified as

1198891198811

119889119905= minus(119906

2minus

1205731119885

(119870 + 119875) (119870 + 119875)) (119875 minus 119875)

2

minus1205731V2(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)2

(19)

minus(1205731V2(119870+119875)(120573

2minus1205733)119870)(119885 minus 119885)

2

lt 0 and minus(1199062minus1205731119885(119870+

119875)(119870 + 119875))(119875 minus 119875)2lt 0 always occurs when 119906

2gt 1205731119885(119870 +

119875)119870Therefore 119889119881

1119889119905 lt 0 if 119906

2gt 1205731119885(119870 + 119875)119870 holds

Proof (2) Then by referring to [32 33] we choose thefollowing Lyapunov function

1198812= ∬Ω

1198811(119875 119885) 119889Ω (20)

and by differentiating 1198812with respect to time 119905 along the

solutions of system (2) we can obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω + ∬

Ω

((11986311Δ119875 + 119863

12Δ119885)

1205971198811

120597119875

+ (11986321Δ119875 + 119863

22Δ119885)

1205971198811

120597119885)119889Ω

(21)

Using Greenrsquos first identity in the plane we obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω minus 119863

11∬Ω

12059721198811

1205971198752|nabla119875|2119889Ω

minus 11986322

∬Ω

12059721198811

1205971198852|nabla119885|2119889Ω

minus 11986321

∬Ω

12059721198811

1205971198852nabla119875nabla119885119889Ω

minus 11986312

∬Ω

12059721198811

1205971198752nabla119875nabla119885119889Ω

(22)

where

12059721198811

1205971198752=

119875

1198752gt 0

12059721198811

1205971198852=

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852gt 0

(23)

Based on the analysis above

1199062gt

1205731119885

(119870 + 119875)119870 (24)

In the case of increasing 11986311 11986322

to sufficiently largevalues and if 119863

12 11986321meet the conditions

1198602

12lt 41198601111986022

(25)

then 1198891198811119889119905 lt 0 and thus 119889119881

2119889119905 lt 0 Therefore 119864

1198902=

(1198751198902 1198851198902) of system (2) is globally asymptotically stable when

1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022

Under the condition of1199061minus 1199031

1199062

gt 119870

1199062((1199061minus 1199031) 1199062+ 119870)2

41205731

lt(1205732minus 1205733) ((1199061minus 1199031) 1199062minus 119870)

V2((1199061minus 1199031) 1199062+ 119870)

minusV1

V2

(26)

we have the following conclusions

Lemma 5 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198901= (1198751198901 1198851198901) is locally asymptotically stable

when

119865 (119875lowast) minus 119866 (119875

lowast) gt 0

(119875lowast=

(1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1)2

+ 81199062V1119870

41199062

)

(27)

and the equilibrium 1198641198901

= (1198751198901 1198851198901) is unstable when 119865(119875

lowast) minus

119866(119875lowast) lt 0

Proof Define

119867(119875) = 119865 (119875) minus 119866 (119875)

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(28)

under the condition of Lemma 1 we can find that 1198671015840(119875) lt

0 (0 lt 119875 lt ((1199061minus 1199031)1199062minus 119870)2)

6 Discrete Dynamics in Nature and Society

Thus

(1205732minus 1205733)119870

V2(119870 + 119875)

2lt minus

1199062

1205731

[2119875 + (119870 minus1199061minus 1199031

1199062

)]

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(29)

and combined with det(1198641198901) we can find that det(119864

1198901) gt 0

and

tr (1198641198901)

=minus211990621198752

1198901+ (1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) 1198751198901

+ V1119870

119870 + 1198751198901

(30)

Hence if 119865(119875lowast) minus 119866(119875

lowast) gt 0 then 119875

1198901gt 119875lowast and in this case

tr(1198641198901) lt 0 thus 119864

1198901= (1198751198901 1198851198901) is stable If 119865(119875

lowast)minus119866(119875

lowast) lt

0 then 1198751198901

lt 119875lowast and in this case tr(119864

1198901) gt 0 thus 119864

1198901=

(1198751198901 1198851198901) is unstable

By analyzing Lemma 5 we can find that 119865(119875lowast) minus119866(119875

lowast) =

0 is the value where the plankton system populations bifur-cate into periodic oscillation By 119865(119875

lowast) = 119866(119875

lowast) then we can

obtain

1205731= 120573 =

minus1199062V2(119870 + 119875

lowast) ((119875lowast)2

+ (119870 minus (1199061minus 1199031) 1199062) 119875lowastminus ((1199061minus 1199031) 1199062)119870)

(1205732minus 1205733minus V1) 119875lowast minus V

1119870

(31)

which is a bifurcation value of the parameter 1205732 Now the

characteristic equation of the 1198641198901reduces to

1205822+ det (119864

1198901) = 0 (32)

where det(1198641198901) gt 0 and the equation has purely imaginary

rootsWe take 120582 = 119886+119887119894 If the stability change occurs at 1205731=

120573 it is obvious that 119886(120573) = 0 119887(120573) = 0 and using 119886(120573) = 0119887(120573) = 0 we get (119889119886119889120573

1)|1205731=120573

= ((12)119889 tr(1198641198901)1198891205731)|1205731=120573

gt

0 Thus the transversality condition for a Hopf bifurcation issatisfied

Theorem 6 If (1199061minus 1199031)1199062gt 119870 119906

2((1199061minus 1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus 119870)V

2((1199061minus 1199031)1199062+ 119870) minus V

1V2 and

119865(119875lowast) minus119866(119875

lowast) gt 0 then the criterion for the Turing instability

of system (2) satisfies the following condition (1198861111986322

+1198862211986311

minus

1198631211988621

minus 1198632111988612)2gt 4 det(119863) det(119869

1198641198901

) Consider

(1198691198641198901

= (11988611

11988612

11988621

11988622

) 119894119904 119905ℎ119890 119869119886119888119900119887119894119886119899 119898119886119905119903119894119909 119886119905 1198641198901

= (1198751198901 1198851198901))

(33)

Proof The linearized form of system (2) corresponding to theequilibrium 119864

1198901(1198751198901 1198851198901) is given by

120597119901

120597119905= 11988611119901 + 11988612119911 + 119863

11

1205972119901

1205971199042+ 11986312

1205972119911

1205971199042

120597119911

120597119905= 11988621119901 + 11988622119911 + 119863

21

1205972119901

1205971199042+ 11986322

1205972119911

1205971199042

(34)

where 119875 = 1198751198901

+ 119901 119885 = 1198851198901

+ 119911 and (119901 119911) aresmall perturbations in (119875 119885) about the equilibrium 119864

1198901=

(1198751198901 1198851198901) We expand the solution of system (34) into a

Fourier series

(119901

119911) = sum

119896

(1198881

119896

1198882

119896

)119890120582119896119905+119894119896119904

(35)

where 119896 is the wave number of the solution By combining(34) and (35) we obtain

120582119896(1198881

119896

1198882

119896

) = (11988611

minus 119896211986311

11988612

minus 119896211986312

11988621

minus 119896211986321

11988622

minus 119896211986322

)(

1198881

119896

1198882

119896

) (36)

Hence we can obtain the characteristic equation as follows

1205822minus tr119896120582119896+ Δ119896= 0 (37)

where

tr119896= 11988611

+ 11988622

minus 1198962(11986311

+ 11986322)

Δ119896= det (119869

1198641198901

)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321)

(38)

In order to allow the system to achieve Turing instabilityat least one of the following conditionsmust be satisfied tr

119896gt

0 and Δ119896lt 0 However it is obvious that tr

119896lt 0 Thus only

the condition that Δ119896

lt 0 exists can give rise to instabilityTherefore a necessary condition for the system to be unstableis that

det (1198691198641198901

) minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321) lt 0

(39)

We define

119865 (1198962) = 1198964(1198631111986322

minus 1198631211986321)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ det (1198691198641198901

)

(40)

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

and the initial conditions are

119875 (0 119904) = 1198750(119904) gt 0

119885 (0 119904) = 1198850(119904) gt 0

119904 isin Ω

(4)

where Ω sube 1198772 is a bounded spatial domain with smooth

boundary 120597ΩIn addition we take

119863 = (

11986311

11986312

11986321

11986322

) (5)

as the diffusive matrix The determinant of 119863 is det(119863) =

1198631111986322

minus 1198631211986321 We assume that the self-diffusion is

stronger than cross-diffusion thus diffusive matrix 119863 ispositive definite

Let 119891(119875 119885) and 119892(119875 119885) be the right hand sides of theequations in (1) It is obvious that the equilibria 119864

0= (0 0)

and 1198641= ((1199061minus 1199031)1199062 0) always exist

By 119891(119875 119885) = 0 we can obtain the vertical isoclines1198971 119885 = minus(119906

21205731)(1198752+ (119870 minus (119906

1minus 1199031)1199062)119875 minus (119906

1minus 1199031)1198701199062)

This is a downward parabola so the axis of symmetry is 119875 =

(12)((1199061minus 1199031)1199062minus 119870) and the fellowship with the 119910-axis

is 119885 = (1199061minus 1199031)1198701205731 By 119892(119875 119885) = 0 we can obtain the

horizontal isoclines 1198972 119885 = (120573

2minus 1205733)119875V2(119870 + 119875) minus V

1V2

It is easy to show that the line 119875 = minus119870 is the asymptote of1198972 We can prove that 119897

2increases monotonically when 119875 isin

(minus119870 +infin) and the fellowship with the 119910-axis is 119885 = minusV1V2

Based on the analysis above we define

119865 (119875) = minus1199062

1205731

(1198752+ (119870 minus

1199061minus 1199031

1199062

)119875 minus1199061minus 1199031

1199062

119870)

119866 (119875) =(1205732minus 1205733) 119875

V2(119870 + 119875)

minusV1

V2

(6)

and the following conclusions can be established

Lemma 1 (1) If (1199061minus1199031)1199062gt 119870 119906

2((1199061minus1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus119870)V

2((1199061minus 1199031)1199062+119870) minus V

1V2 then

there exists one positive equilibrium point 1198641198901

= (1198751198901 1198851198901) in

system (1) where 1198751198901

lt (12)((1199061minus 1199031)1199062minus 119870)

(2) If (1199061minus 1199031)1199062lt 119870 lt ((119906

1minus 1199031)1199062V1)(1205732minus 1205733minus V1)

then there exists one positive equilibrium point1198641198902

= (1198751198902 1198851198902)

in system (1) it is obvious that 1198751198902

gt (12)((1199061minus 1199031)1199062minus 119870)

(3) If (1199061minus 1199031)1199062

gt 119870 and there at least exists 1198750

isin

(0 (12)((1199061minus 1199031)1199062minus 119870)) to guarantee 119865(119875

0) lt 119866(119875

0) but

119865((12)((1199061minus 1199031)1199062minus119870)) gt 119866((12)((119906

1minus 1199031)1199062minus119870)) then

there exist three positive equilibria 11986411989031

= (11987511989031

11988511989031

) 11986411989032

=

(11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) in system (1) where 11987511989031

lt

11987511989032

lt (12)((1199061minus1199031)1199062minus119870) 119875

11989033gt (12)((119906

1minus1199031)1199062minus119870)

22 Linear Stability Analysis For 1198640= (0 0) and 119864

1= ((1199061minus

1199031)1199062 0) we can find that 119864

0= (0 0) is always unstable and

that1198641= ((1199061minus1199031)1199062 0) is locally asymptotically stablewhen

V1gt (1205732minus 1205731)(1199061minus 1199031)(1199062119870 + 119906

1minus 1199031)

Now let 119864 = (119875 119885) be an arbitrary positive equilibriumand thus 119864 = (119875 119885) satisfies the equations 119891(119875 119885) = 0 and119892(119875 119885) = 0 Then the Jacobian matrix for 119864 = (119875 119885) is givenby

119869119864= (

11986911

11986912

11986921

11986922

)

= (

119875[1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] minus

1205731119875

119870 + 119875

(1205732minus 1205733)119870119885

(119870 + 119875)2

minusV2119885

)

(7)

Note that

tr (119869119864) = 119875 [

1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] minus V2119885

det (119869119864) = 119875 [

1

119870 + 119875(1199061minus 1199031minus 1199062119875) minus 119906

2] V2119885

+(1205732minus 1205733)119870119885

(119870 + 119875)2

1205731119875

119870 + 119875

(8)

From 119869119864 we can find that if 119875 gt (12)((119906

1minus 1199031)1199062minus119870) then

11986911

lt 0 and thus tr(119869119864) lt 0 and det(119869

119864) gt 0 in this case the

equilibrium119864 = (119875 119885) is always locally asymptotically stableBut if 119875 lt (12)((119906

1minus 1199031)1199062minus 119870) then 119869

11gt 0

Next we analyze the stability and instability of system(2) Let 0 = 119906

0lt 1199061

lt 1199062

lt sdot sdot sdot lt infin (lim119894rarrinfin

119906119894= infin)

be the eigenvalues of minusnabla2 on Ω with the zero-flux boundary

condition Set

119883 fl 119880 = (119875 119885) isin [1198621(Ω)]2

120597119875

120597119899=

120597119885

120597119899= 0 (9)

and consider 119883 = ⨁infin

119894=0119883119894 where 119883

119894is the eigenspace that

corresponds to 119906119894

In the condition where (1199061

minus 1199031)1199062

lt 119870 lt ((1199061

minus

1199031)1199062V1)(1205732

minus 1205733

minus V1) we obtain the following conclu-

sions

Lemma 2 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is always locally asymptotically

stable

Theorem 3 If 11986311

= 0 11986322

= 0 11986312

= 0 and 11986321

= 0 thenthe equilibrium 119864

1198902= (1198751198902 1198851198902) is uniformly asymptotically

stable when 11986312

gt 0 11986321

lt 0

Proof The linearization of system (2) at the positive equilib-rium 119864

1198902= (1198751198902 1198851198902) is

4 Discrete Dynamics in Nature and Society

120597

120597119905(119875

119885) = Γ(

119875

119885) + (

1198911(119875 minus 119875

1198902 119885 minus 119885

1198902)

1198912(119875 minus 119875

1198902 119885 minus 119885

1198902)

)

Γ = (

1198751198902

[1

119870 + 1198751198902

(1199061minus 1199031minus 11990621198751198902) minus 1199062] + 119863

11Δ minus

12057311198751198902

119870 + 1198751198902

+ 11986312Δ

(1205732minus 1205733)1198701198851198902

(119870 + 1198751198902)2

+ 11986321Δ minusV

21198851198902

+ 11986322Δ

) = (

Γ11

+ 11986311Δ Γ12

+ 11986312Δ

Γ21

+ 11986321Δ Γ22

+ 11986322Δ

)

(10)

where 119891119899(1205761 1205762) = 119874(120576

2

1+ 1205762

2) (119899 = 1 2)

For each 119894 (119894 = 0 1 2 ) 119883119894is invariant under the

operator Γ and 120582119894is an eigenvalue of Γ on 119883

119894if and only if

120582119894is an eigenvalue of the matrix

119867119894= (

Γ11

minus 11986311119906119894

Γ12

minus 11986312119906119894

Γ21

minus 11986321119906119894

Γ22

minus 11986322119906119894

) (11)

Note that

tr (119867119894) = Γ11

+ Γ22

minus (11986311

+ 11986322) 119906119894

det (119867119894) = Γ11Γ22

minus Γ12Γ21

+ 11986311119863221199062

119894

minus 119906119894(11986322Γ11

+ 11986311Γ22) minus 11986312119863211199062

119894

+ (Γ1211986321

+ Γ2111986312) 119906119894

(12)

When 11986312

gt 0 11986321

lt 0 then tr(119867119894) lt 0 det(119867

119894) gt 0 and

the two eigenvalues 120582+

119894and 120582

minus

119894have negative real parts For

any 119894 ge 0 we find the followingIf (tr(119867

119894))2minus 4 det(119867

119894) le 0 then Re(120582plusmn

119894) = (12) tr(119867

119894) le

(12)(Γ11

+ Γ22) lt 0 and if (tr(119867

119894))2minus 4 det(119867

119894) gt 0 since

tr(119867119894) lt 0 and det(119867

119894) gt 0 then

Re (120582minus119894) =

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

2

le1

2tr (119867119894) le

1

2(Γ11

+ Γ22) lt 0

Re (120582+119894) =

tr (119867119894) + radic(tr (119867

119894))2

minus 4 det (119867119894)

2

=2 det (119867

119894)

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

ledet (119867

119894)

tr (119867119894)

lt 119862

(13)

where 119862 is independent of 119894Thus a negative constant 119862 exists which is independent

of 119894 such that Re(120582plusmn119894) lt 119862 for any 119894 By referring to [31] we

can prove that the positive equilibrium 1198641198902

= (1198751198902 1198851198902) of

system (1) is uniformly asymptotically stable when 11986312

gt 011986321

lt 0

Theorem 4 (1) If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870

(2) If11986311

= 011986322

= 011986312

= 011986321

= 0 then the positiveequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022 where

11986011

= 11986311

119875

1198752

11986022

= 11986322

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

11986012

= minus11986312

119875

1198752minus 11986321

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

(14)

Proof (1) Define a Lyapunov function

1198811(119875 119885) = int

119875

119875

119883 minus 119875

119883119889119883

+1205731(119870 + 119875)

(1205732minus 1205733)119870

int

119885

119885

119884 minus 119885

119884119889119884

(15)

We note that 1198811(119875 119885) is nonnegative and 119881

1(119875 119885) = 0 if and

only if (119875(119905) 119885(119905)) = (119875 119885)Furthermore

1198891198811

119889119905=

119875 minus 119875

119875

119889119875

119889119905+

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885 minus 119885

119885

119889119885

119889119905 (16)

By substituting the expressions for 119889119875119889119905 and 119889119885119889119905 fromsystem (1) we obtain

1198891198811

119889119905= (119875 minus 119875)(119906

1minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875 + 119885)

+1205731(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)

sdot ((1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885)

(17)

Discrete Dynamics in Nature and Society 5

Using the condition that

1199061minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875= 0

(1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885 = 0

(18)

(17) can be simplified as

1198891198811

119889119905= minus(119906

2minus

1205731119885

(119870 + 119875) (119870 + 119875)) (119875 minus 119875)

2

minus1205731V2(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)2

(19)

minus(1205731V2(119870+119875)(120573

2minus1205733)119870)(119885 minus 119885)

2

lt 0 and minus(1199062minus1205731119885(119870+

119875)(119870 + 119875))(119875 minus 119875)2lt 0 always occurs when 119906

2gt 1205731119885(119870 +

119875)119870Therefore 119889119881

1119889119905 lt 0 if 119906

2gt 1205731119885(119870 + 119875)119870 holds

Proof (2) Then by referring to [32 33] we choose thefollowing Lyapunov function

1198812= ∬Ω

1198811(119875 119885) 119889Ω (20)

and by differentiating 1198812with respect to time 119905 along the

solutions of system (2) we can obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω + ∬

Ω

((11986311Δ119875 + 119863

12Δ119885)

1205971198811

120597119875

+ (11986321Δ119875 + 119863

22Δ119885)

1205971198811

120597119885)119889Ω

(21)

Using Greenrsquos first identity in the plane we obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω minus 119863

11∬Ω

12059721198811

1205971198752|nabla119875|2119889Ω

minus 11986322

∬Ω

12059721198811

1205971198852|nabla119885|2119889Ω

minus 11986321

∬Ω

12059721198811

1205971198852nabla119875nabla119885119889Ω

minus 11986312

∬Ω

12059721198811

1205971198752nabla119875nabla119885119889Ω

(22)

where

12059721198811

1205971198752=

119875

1198752gt 0

12059721198811

1205971198852=

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852gt 0

(23)

Based on the analysis above

1199062gt

1205731119885

(119870 + 119875)119870 (24)

In the case of increasing 11986311 11986322

to sufficiently largevalues and if 119863

12 11986321meet the conditions

1198602

12lt 41198601111986022

(25)

then 1198891198811119889119905 lt 0 and thus 119889119881

2119889119905 lt 0 Therefore 119864

1198902=

(1198751198902 1198851198902) of system (2) is globally asymptotically stable when

1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022

Under the condition of1199061minus 1199031

1199062

gt 119870

1199062((1199061minus 1199031) 1199062+ 119870)2

41205731

lt(1205732minus 1205733) ((1199061minus 1199031) 1199062minus 119870)

V2((1199061minus 1199031) 1199062+ 119870)

minusV1

V2

(26)

we have the following conclusions

Lemma 5 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198901= (1198751198901 1198851198901) is locally asymptotically stable

when

119865 (119875lowast) minus 119866 (119875

lowast) gt 0

(119875lowast=

(1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1)2

+ 81199062V1119870

41199062

)

(27)

and the equilibrium 1198641198901

= (1198751198901 1198851198901) is unstable when 119865(119875

lowast) minus

119866(119875lowast) lt 0

Proof Define

119867(119875) = 119865 (119875) minus 119866 (119875)

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(28)

under the condition of Lemma 1 we can find that 1198671015840(119875) lt

0 (0 lt 119875 lt ((1199061minus 1199031)1199062minus 119870)2)

6 Discrete Dynamics in Nature and Society

Thus

(1205732minus 1205733)119870

V2(119870 + 119875)

2lt minus

1199062

1205731

[2119875 + (119870 minus1199061minus 1199031

1199062

)]

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(29)

and combined with det(1198641198901) we can find that det(119864

1198901) gt 0

and

tr (1198641198901)

=minus211990621198752

1198901+ (1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) 1198751198901

+ V1119870

119870 + 1198751198901

(30)

Hence if 119865(119875lowast) minus 119866(119875

lowast) gt 0 then 119875

1198901gt 119875lowast and in this case

tr(1198641198901) lt 0 thus 119864

1198901= (1198751198901 1198851198901) is stable If 119865(119875

lowast)minus119866(119875

lowast) lt

0 then 1198751198901

lt 119875lowast and in this case tr(119864

1198901) gt 0 thus 119864

1198901=

(1198751198901 1198851198901) is unstable

By analyzing Lemma 5 we can find that 119865(119875lowast) minus119866(119875

lowast) =

0 is the value where the plankton system populations bifur-cate into periodic oscillation By 119865(119875

lowast) = 119866(119875

lowast) then we can

obtain

1205731= 120573 =

minus1199062V2(119870 + 119875

lowast) ((119875lowast)2

+ (119870 minus (1199061minus 1199031) 1199062) 119875lowastminus ((1199061minus 1199031) 1199062)119870)

(1205732minus 1205733minus V1) 119875lowast minus V

1119870

(31)

which is a bifurcation value of the parameter 1205732 Now the

characteristic equation of the 1198641198901reduces to

1205822+ det (119864

1198901) = 0 (32)

where det(1198641198901) gt 0 and the equation has purely imaginary

rootsWe take 120582 = 119886+119887119894 If the stability change occurs at 1205731=

120573 it is obvious that 119886(120573) = 0 119887(120573) = 0 and using 119886(120573) = 0119887(120573) = 0 we get (119889119886119889120573

1)|1205731=120573

= ((12)119889 tr(1198641198901)1198891205731)|1205731=120573

gt

0 Thus the transversality condition for a Hopf bifurcation issatisfied

Theorem 6 If (1199061minus 1199031)1199062gt 119870 119906

2((1199061minus 1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus 119870)V

2((1199061minus 1199031)1199062+ 119870) minus V

1V2 and

119865(119875lowast) minus119866(119875

lowast) gt 0 then the criterion for the Turing instability

of system (2) satisfies the following condition (1198861111986322

+1198862211986311

minus

1198631211988621

minus 1198632111988612)2gt 4 det(119863) det(119869

1198641198901

) Consider

(1198691198641198901

= (11988611

11988612

11988621

11988622

) 119894119904 119905ℎ119890 119869119886119888119900119887119894119886119899 119898119886119905119903119894119909 119886119905 1198641198901

= (1198751198901 1198851198901))

(33)

Proof The linearized form of system (2) corresponding to theequilibrium 119864

1198901(1198751198901 1198851198901) is given by

120597119901

120597119905= 11988611119901 + 11988612119911 + 119863

11

1205972119901

1205971199042+ 11986312

1205972119911

1205971199042

120597119911

120597119905= 11988621119901 + 11988622119911 + 119863

21

1205972119901

1205971199042+ 11986322

1205972119911

1205971199042

(34)

where 119875 = 1198751198901

+ 119901 119885 = 1198851198901

+ 119911 and (119901 119911) aresmall perturbations in (119875 119885) about the equilibrium 119864

1198901=

(1198751198901 1198851198901) We expand the solution of system (34) into a

Fourier series

(119901

119911) = sum

119896

(1198881

119896

1198882

119896

)119890120582119896119905+119894119896119904

(35)

where 119896 is the wave number of the solution By combining(34) and (35) we obtain

120582119896(1198881

119896

1198882

119896

) = (11988611

minus 119896211986311

11988612

minus 119896211986312

11988621

minus 119896211986321

11988622

minus 119896211986322

)(

1198881

119896

1198882

119896

) (36)

Hence we can obtain the characteristic equation as follows

1205822minus tr119896120582119896+ Δ119896= 0 (37)

where

tr119896= 11988611

+ 11988622

minus 1198962(11986311

+ 11986322)

Δ119896= det (119869

1198641198901

)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321)

(38)

In order to allow the system to achieve Turing instabilityat least one of the following conditionsmust be satisfied tr

119896gt

0 and Δ119896lt 0 However it is obvious that tr

119896lt 0 Thus only

the condition that Δ119896

lt 0 exists can give rise to instabilityTherefore a necessary condition for the system to be unstableis that

det (1198691198641198901

) minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321) lt 0

(39)

We define

119865 (1198962) = 1198964(1198631111986322

minus 1198631211986321)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ det (1198691198641198901

)

(40)

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

120597

120597119905(119875

119885) = Γ(

119875

119885) + (

1198911(119875 minus 119875

1198902 119885 minus 119885

1198902)

1198912(119875 minus 119875

1198902 119885 minus 119885

1198902)

)

Γ = (

1198751198902

[1

119870 + 1198751198902

(1199061minus 1199031minus 11990621198751198902) minus 1199062] + 119863

11Δ minus

12057311198751198902

119870 + 1198751198902

+ 11986312Δ

(1205732minus 1205733)1198701198851198902

(119870 + 1198751198902)2

+ 11986321Δ minusV

21198851198902

+ 11986322Δ

) = (

Γ11

+ 11986311Δ Γ12

+ 11986312Δ

Γ21

+ 11986321Δ Γ22

+ 11986322Δ

)

(10)

where 119891119899(1205761 1205762) = 119874(120576

2

1+ 1205762

2) (119899 = 1 2)

For each 119894 (119894 = 0 1 2 ) 119883119894is invariant under the

operator Γ and 120582119894is an eigenvalue of Γ on 119883

119894if and only if

120582119894is an eigenvalue of the matrix

119867119894= (

Γ11

minus 11986311119906119894

Γ12

minus 11986312119906119894

Γ21

minus 11986321119906119894

Γ22

minus 11986322119906119894

) (11)

Note that

tr (119867119894) = Γ11

+ Γ22

minus (11986311

+ 11986322) 119906119894

det (119867119894) = Γ11Γ22

minus Γ12Γ21

+ 11986311119863221199062

119894

minus 119906119894(11986322Γ11

+ 11986311Γ22) minus 11986312119863211199062

119894

+ (Γ1211986321

+ Γ2111986312) 119906119894

(12)

When 11986312

gt 0 11986321

lt 0 then tr(119867119894) lt 0 det(119867

119894) gt 0 and

the two eigenvalues 120582+

119894and 120582

minus

119894have negative real parts For

any 119894 ge 0 we find the followingIf (tr(119867

119894))2minus 4 det(119867

119894) le 0 then Re(120582plusmn

119894) = (12) tr(119867

119894) le

(12)(Γ11

+ Γ22) lt 0 and if (tr(119867

119894))2minus 4 det(119867

119894) gt 0 since

tr(119867119894) lt 0 and det(119867

119894) gt 0 then

Re (120582minus119894) =

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

2

le1

2tr (119867119894) le

1

2(Γ11

+ Γ22) lt 0

Re (120582+119894) =

tr (119867119894) + radic(tr (119867

119894))2

minus 4 det (119867119894)

2

=2 det (119867

119894)

tr (119867119894) minus radic(tr (119867

119894))2

minus 4 det (119867119894)

ledet (119867

119894)

tr (119867119894)

lt 119862

(13)

where 119862 is independent of 119894Thus a negative constant 119862 exists which is independent

of 119894 such that Re(120582plusmn119894) lt 119862 for any 119894 By referring to [31] we

can prove that the positive equilibrium 1198641198902

= (1198751198902 1198851198902) of

system (1) is uniformly asymptotically stable when 11986312

gt 011986321

lt 0

Theorem 4 (1) If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870

(2) If11986311

= 011986322

= 011986312

= 011986321

= 0 then the positiveequilibrium 119864

1198902= (1198751198902 1198851198902) is globally asymptotically stable

when 1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022 where

11986011

= 11986311

119875

1198752

11986022

= 11986322

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

11986012

= minus11986312

119875

1198752minus 11986321

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852

(14)

Proof (1) Define a Lyapunov function

1198811(119875 119885) = int

119875

119875

119883 minus 119875

119883119889119883

+1205731(119870 + 119875)

(1205732minus 1205733)119870

int

119885

119885

119884 minus 119885

119884119889119884

(15)

We note that 1198811(119875 119885) is nonnegative and 119881

1(119875 119885) = 0 if and

only if (119875(119905) 119885(119905)) = (119875 119885)Furthermore

1198891198811

119889119905=

119875 minus 119875

119875

119889119875

119889119905+

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885 minus 119885

119885

119889119885

119889119905 (16)

By substituting the expressions for 119889119875119889119905 and 119889119885119889119905 fromsystem (1) we obtain

1198891198811

119889119905= (119875 minus 119875)(119906

1minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875 + 119885)

+1205731(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)

sdot ((1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885)

(17)

Discrete Dynamics in Nature and Society 5

Using the condition that

1199061minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875= 0

(1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885 = 0

(18)

(17) can be simplified as

1198891198811

119889119905= minus(119906

2minus

1205731119885

(119870 + 119875) (119870 + 119875)) (119875 minus 119875)

2

minus1205731V2(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)2

(19)

minus(1205731V2(119870+119875)(120573

2minus1205733)119870)(119885 minus 119885)

2

lt 0 and minus(1199062minus1205731119885(119870+

119875)(119870 + 119875))(119875 minus 119875)2lt 0 always occurs when 119906

2gt 1205731119885(119870 +

119875)119870Therefore 119889119881

1119889119905 lt 0 if 119906

2gt 1205731119885(119870 + 119875)119870 holds

Proof (2) Then by referring to [32 33] we choose thefollowing Lyapunov function

1198812= ∬Ω

1198811(119875 119885) 119889Ω (20)

and by differentiating 1198812with respect to time 119905 along the

solutions of system (2) we can obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω + ∬

Ω

((11986311Δ119875 + 119863

12Δ119885)

1205971198811

120597119875

+ (11986321Δ119875 + 119863

22Δ119885)

1205971198811

120597119885)119889Ω

(21)

Using Greenrsquos first identity in the plane we obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω minus 119863

11∬Ω

12059721198811

1205971198752|nabla119875|2119889Ω

minus 11986322

∬Ω

12059721198811

1205971198852|nabla119885|2119889Ω

minus 11986321

∬Ω

12059721198811

1205971198852nabla119875nabla119885119889Ω

minus 11986312

∬Ω

12059721198811

1205971198752nabla119875nabla119885119889Ω

(22)

where

12059721198811

1205971198752=

119875

1198752gt 0

12059721198811

1205971198852=

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852gt 0

(23)

Based on the analysis above

1199062gt

1205731119885

(119870 + 119875)119870 (24)

In the case of increasing 11986311 11986322

to sufficiently largevalues and if 119863

12 11986321meet the conditions

1198602

12lt 41198601111986022

(25)

then 1198891198811119889119905 lt 0 and thus 119889119881

2119889119905 lt 0 Therefore 119864

1198902=

(1198751198902 1198851198902) of system (2) is globally asymptotically stable when

1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022

Under the condition of1199061minus 1199031

1199062

gt 119870

1199062((1199061minus 1199031) 1199062+ 119870)2

41205731

lt(1205732minus 1205733) ((1199061minus 1199031) 1199062minus 119870)

V2((1199061minus 1199031) 1199062+ 119870)

minusV1

V2

(26)

we have the following conclusions

Lemma 5 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198901= (1198751198901 1198851198901) is locally asymptotically stable

when

119865 (119875lowast) minus 119866 (119875

lowast) gt 0

(119875lowast=

(1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1)2

+ 81199062V1119870

41199062

)

(27)

and the equilibrium 1198641198901

= (1198751198901 1198851198901) is unstable when 119865(119875

lowast) minus

119866(119875lowast) lt 0

Proof Define

119867(119875) = 119865 (119875) minus 119866 (119875)

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(28)

under the condition of Lemma 1 we can find that 1198671015840(119875) lt

0 (0 lt 119875 lt ((1199061minus 1199031)1199062minus 119870)2)

6 Discrete Dynamics in Nature and Society

Thus

(1205732minus 1205733)119870

V2(119870 + 119875)

2lt minus

1199062

1205731

[2119875 + (119870 minus1199061minus 1199031

1199062

)]

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(29)

and combined with det(1198641198901) we can find that det(119864

1198901) gt 0

and

tr (1198641198901)

=minus211990621198752

1198901+ (1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) 1198751198901

+ V1119870

119870 + 1198751198901

(30)

Hence if 119865(119875lowast) minus 119866(119875

lowast) gt 0 then 119875

1198901gt 119875lowast and in this case

tr(1198641198901) lt 0 thus 119864

1198901= (1198751198901 1198851198901) is stable If 119865(119875

lowast)minus119866(119875

lowast) lt

0 then 1198751198901

lt 119875lowast and in this case tr(119864

1198901) gt 0 thus 119864

1198901=

(1198751198901 1198851198901) is unstable

By analyzing Lemma 5 we can find that 119865(119875lowast) minus119866(119875

lowast) =

0 is the value where the plankton system populations bifur-cate into periodic oscillation By 119865(119875

lowast) = 119866(119875

lowast) then we can

obtain

1205731= 120573 =

minus1199062V2(119870 + 119875

lowast) ((119875lowast)2

+ (119870 minus (1199061minus 1199031) 1199062) 119875lowastminus ((1199061minus 1199031) 1199062)119870)

(1205732minus 1205733minus V1) 119875lowast minus V

1119870

(31)

which is a bifurcation value of the parameter 1205732 Now the

characteristic equation of the 1198641198901reduces to

1205822+ det (119864

1198901) = 0 (32)

where det(1198641198901) gt 0 and the equation has purely imaginary

rootsWe take 120582 = 119886+119887119894 If the stability change occurs at 1205731=

120573 it is obvious that 119886(120573) = 0 119887(120573) = 0 and using 119886(120573) = 0119887(120573) = 0 we get (119889119886119889120573

1)|1205731=120573

= ((12)119889 tr(1198641198901)1198891205731)|1205731=120573

gt

0 Thus the transversality condition for a Hopf bifurcation issatisfied

Theorem 6 If (1199061minus 1199031)1199062gt 119870 119906

2((1199061minus 1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus 119870)V

2((1199061minus 1199031)1199062+ 119870) minus V

1V2 and

119865(119875lowast) minus119866(119875

lowast) gt 0 then the criterion for the Turing instability

of system (2) satisfies the following condition (1198861111986322

+1198862211986311

minus

1198631211988621

minus 1198632111988612)2gt 4 det(119863) det(119869

1198641198901

) Consider

(1198691198641198901

= (11988611

11988612

11988621

11988622

) 119894119904 119905ℎ119890 119869119886119888119900119887119894119886119899 119898119886119905119903119894119909 119886119905 1198641198901

= (1198751198901 1198851198901))

(33)

Proof The linearized form of system (2) corresponding to theequilibrium 119864

1198901(1198751198901 1198851198901) is given by

120597119901

120597119905= 11988611119901 + 11988612119911 + 119863

11

1205972119901

1205971199042+ 11986312

1205972119911

1205971199042

120597119911

120597119905= 11988621119901 + 11988622119911 + 119863

21

1205972119901

1205971199042+ 11986322

1205972119911

1205971199042

(34)

where 119875 = 1198751198901

+ 119901 119885 = 1198851198901

+ 119911 and (119901 119911) aresmall perturbations in (119875 119885) about the equilibrium 119864

1198901=

(1198751198901 1198851198901) We expand the solution of system (34) into a

Fourier series

(119901

119911) = sum

119896

(1198881

119896

1198882

119896

)119890120582119896119905+119894119896119904

(35)

where 119896 is the wave number of the solution By combining(34) and (35) we obtain

120582119896(1198881

119896

1198882

119896

) = (11988611

minus 119896211986311

11988612

minus 119896211986312

11988621

minus 119896211986321

11988622

minus 119896211986322

)(

1198881

119896

1198882

119896

) (36)

Hence we can obtain the characteristic equation as follows

1205822minus tr119896120582119896+ Δ119896= 0 (37)

where

tr119896= 11988611

+ 11988622

minus 1198962(11986311

+ 11986322)

Δ119896= det (119869

1198641198901

)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321)

(38)

In order to allow the system to achieve Turing instabilityat least one of the following conditionsmust be satisfied tr

119896gt

0 and Δ119896lt 0 However it is obvious that tr

119896lt 0 Thus only

the condition that Δ119896

lt 0 exists can give rise to instabilityTherefore a necessary condition for the system to be unstableis that

det (1198691198641198901

) minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321) lt 0

(39)

We define

119865 (1198962) = 1198964(1198631111986322

minus 1198631211986321)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ det (1198691198641198901

)

(40)

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Using the condition that

1199061minus 1199031minus 1199062119875 minus

1205731119885

119870 + 119875= 0

(1205732minus 1205733) 119875

119870 + 119875minus V1minus V2119885 = 0

(18)

(17) can be simplified as

1198891198811

119889119905= minus(119906

2minus

1205731119885

(119870 + 119875) (119870 + 119875)) (119875 minus 119875)

2

minus1205731V2(119870 + 119875)

(1205732minus 1205733)119870

(119885 minus 119885)2

(19)

minus(1205731V2(119870+119875)(120573

2minus1205733)119870)(119885 minus 119885)

2

lt 0 and minus(1199062minus1205731119885(119870+

119875)(119870 + 119875))(119875 minus 119875)2lt 0 always occurs when 119906

2gt 1205731119885(119870 +

119875)119870Therefore 119889119881

1119889119905 lt 0 if 119906

2gt 1205731119885(119870 + 119875)119870 holds

Proof (2) Then by referring to [32 33] we choose thefollowing Lyapunov function

1198812= ∬Ω

1198811(119875 119885) 119889Ω (20)

and by differentiating 1198812with respect to time 119905 along the

solutions of system (2) we can obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω + ∬

Ω

((11986311Δ119875 + 119863

12Δ119885)

1205971198811

120597119875

+ (11986321Δ119875 + 119863

22Δ119885)

1205971198811

120597119885)119889Ω

(21)

Using Greenrsquos first identity in the plane we obtain

1198891198812

119889119905= ∬Ω

1198891198811

119889119905119889Ω minus 119863

11∬Ω

12059721198811

1205971198752|nabla119875|2119889Ω

minus 11986322

∬Ω

12059721198811

1205971198852|nabla119885|2119889Ω

minus 11986321

∬Ω

12059721198811

1205971198852nabla119875nabla119885119889Ω

minus 11986312

∬Ω

12059721198811

1205971198752nabla119875nabla119885119889Ω

(22)

where

12059721198811

1205971198752=

119875

1198752gt 0

12059721198811

1205971198852=

1205731(119870 + 119875)

(1205732minus 1205733)119870

119885

1198852gt 0

(23)

Based on the analysis above

1199062gt

1205731119885

(119870 + 119875)119870 (24)

In the case of increasing 11986311 11986322

to sufficiently largevalues and if 119863

12 11986321meet the conditions

1198602

12lt 41198601111986022

(25)

then 1198891198811119889119905 lt 0 and thus 119889119881

2119889119905 lt 0 Therefore 119864

1198902=

(1198751198902 1198851198902) of system (2) is globally asymptotically stable when

1199062gt 1205731119885(119870 + 119875)119870 and 119860

2

12lt 41198601111986022

Under the condition of1199061minus 1199031

1199062

gt 119870

1199062((1199061minus 1199031) 1199062+ 119870)2

41205731

lt(1205732minus 1205733) ((1199061minus 1199031) 1199062minus 119870)

V2((1199061minus 1199031) 1199062+ 119870)

minusV1

V2

(26)

we have the following conclusions

Lemma 5 If 11986311

= 11986312

= 11986321

= 11986322

= 0 then theequilibrium 119864

1198901= (1198751198901 1198851198901) is locally asymptotically stable

when

119865 (119875lowast) minus 119866 (119875

lowast) gt 0

(119875lowast=

(1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1)2

+ 81199062V1119870

41199062

)

(27)

and the equilibrium 1198641198901

= (1198751198901 1198851198901) is unstable when 119865(119875

lowast) minus

119866(119875lowast) lt 0

Proof Define

119867(119875) = 119865 (119875) minus 119866 (119875)

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(28)

under the condition of Lemma 1 we can find that 1198671015840(119875) lt

0 (0 lt 119875 lt ((1199061minus 1199031)1199062minus 119870)2)

6 Discrete Dynamics in Nature and Society

Thus

(1205732minus 1205733)119870

V2(119870 + 119875)

2lt minus

1199062

1205731

[2119875 + (119870 minus1199061minus 1199031

1199062

)]

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(29)

and combined with det(1198641198901) we can find that det(119864

1198901) gt 0

and

tr (1198641198901)

=minus211990621198752

1198901+ (1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) 1198751198901

+ V1119870

119870 + 1198751198901

(30)

Hence if 119865(119875lowast) minus 119866(119875

lowast) gt 0 then 119875

1198901gt 119875lowast and in this case

tr(1198641198901) lt 0 thus 119864

1198901= (1198751198901 1198851198901) is stable If 119865(119875

lowast)minus119866(119875

lowast) lt

0 then 1198751198901

lt 119875lowast and in this case tr(119864

1198901) gt 0 thus 119864

1198901=

(1198751198901 1198851198901) is unstable

By analyzing Lemma 5 we can find that 119865(119875lowast) minus119866(119875

lowast) =

0 is the value where the plankton system populations bifur-cate into periodic oscillation By 119865(119875

lowast) = 119866(119875

lowast) then we can

obtain

1205731= 120573 =

minus1199062V2(119870 + 119875

lowast) ((119875lowast)2

+ (119870 minus (1199061minus 1199031) 1199062) 119875lowastminus ((1199061minus 1199031) 1199062)119870)

(1205732minus 1205733minus V1) 119875lowast minus V

1119870

(31)

which is a bifurcation value of the parameter 1205732 Now the

characteristic equation of the 1198641198901reduces to

1205822+ det (119864

1198901) = 0 (32)

where det(1198641198901) gt 0 and the equation has purely imaginary

rootsWe take 120582 = 119886+119887119894 If the stability change occurs at 1205731=

120573 it is obvious that 119886(120573) = 0 119887(120573) = 0 and using 119886(120573) = 0119887(120573) = 0 we get (119889119886119889120573

1)|1205731=120573

= ((12)119889 tr(1198641198901)1198891205731)|1205731=120573

gt

0 Thus the transversality condition for a Hopf bifurcation issatisfied

Theorem 6 If (1199061minus 1199031)1199062gt 119870 119906

2((1199061minus 1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus 119870)V

2((1199061minus 1199031)1199062+ 119870) minus V

1V2 and

119865(119875lowast) minus119866(119875

lowast) gt 0 then the criterion for the Turing instability

of system (2) satisfies the following condition (1198861111986322

+1198862211986311

minus

1198631211988621

minus 1198632111988612)2gt 4 det(119863) det(119869

1198641198901

) Consider

(1198691198641198901

= (11988611

11988612

11988621

11988622

) 119894119904 119905ℎ119890 119869119886119888119900119887119894119886119899 119898119886119905119903119894119909 119886119905 1198641198901

= (1198751198901 1198851198901))

(33)

Proof The linearized form of system (2) corresponding to theequilibrium 119864

1198901(1198751198901 1198851198901) is given by

120597119901

120597119905= 11988611119901 + 11988612119911 + 119863

11

1205972119901

1205971199042+ 11986312

1205972119911

1205971199042

120597119911

120597119905= 11988621119901 + 11988622119911 + 119863

21

1205972119901

1205971199042+ 11986322

1205972119911

1205971199042

(34)

where 119875 = 1198751198901

+ 119901 119885 = 1198851198901

+ 119911 and (119901 119911) aresmall perturbations in (119875 119885) about the equilibrium 119864

1198901=

(1198751198901 1198851198901) We expand the solution of system (34) into a

Fourier series

(119901

119911) = sum

119896

(1198881

119896

1198882

119896

)119890120582119896119905+119894119896119904

(35)

where 119896 is the wave number of the solution By combining(34) and (35) we obtain

120582119896(1198881

119896

1198882

119896

) = (11988611

minus 119896211986311

11988612

minus 119896211986312

11988621

minus 119896211986321

11988622

minus 119896211986322

)(

1198881

119896

1198882

119896

) (36)

Hence we can obtain the characteristic equation as follows

1205822minus tr119896120582119896+ Δ119896= 0 (37)

where

tr119896= 11988611

+ 11988622

minus 1198962(11986311

+ 11986322)

Δ119896= det (119869

1198641198901

)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321)

(38)

In order to allow the system to achieve Turing instabilityat least one of the following conditionsmust be satisfied tr

119896gt

0 and Δ119896lt 0 However it is obvious that tr

119896lt 0 Thus only

the condition that Δ119896

lt 0 exists can give rise to instabilityTherefore a necessary condition for the system to be unstableis that

det (1198691198641198901

) minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321) lt 0

(39)

We define

119865 (1198962) = 1198964(1198631111986322

minus 1198631211986321)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ det (1198691198641198901

)

(40)

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Thus

(1205732minus 1205733)119870

V2(119870 + 119875)

2lt minus

1199062

1205731

[2119875 + (119870 minus1199061minus 1199031

1199062

)]

(0 lt 119875 lt(1199061minus 1199031) 1199062minus 119870

2)

(29)

and combined with det(1198641198901) we can find that det(119864

1198901) gt 0

and

tr (1198641198901)

=minus211990621198752

1198901+ (1199061minus 1199031minus 1199062119870 minus 120573

2+ 1205733+ V1) 1198751198901

+ V1119870

119870 + 1198751198901

(30)

Hence if 119865(119875lowast) minus 119866(119875

lowast) gt 0 then 119875

1198901gt 119875lowast and in this case

tr(1198641198901) lt 0 thus 119864

1198901= (1198751198901 1198851198901) is stable If 119865(119875

lowast)minus119866(119875

lowast) lt

0 then 1198751198901

lt 119875lowast and in this case tr(119864

1198901) gt 0 thus 119864

1198901=

(1198751198901 1198851198901) is unstable

By analyzing Lemma 5 we can find that 119865(119875lowast) minus119866(119875

lowast) =

0 is the value where the plankton system populations bifur-cate into periodic oscillation By 119865(119875

lowast) = 119866(119875

lowast) then we can

obtain

1205731= 120573 =

minus1199062V2(119870 + 119875

lowast) ((119875lowast)2

+ (119870 minus (1199061minus 1199031) 1199062) 119875lowastminus ((1199061minus 1199031) 1199062)119870)

(1205732minus 1205733minus V1) 119875lowast minus V

1119870

(31)

which is a bifurcation value of the parameter 1205732 Now the

characteristic equation of the 1198641198901reduces to

1205822+ det (119864

1198901) = 0 (32)

where det(1198641198901) gt 0 and the equation has purely imaginary

rootsWe take 120582 = 119886+119887119894 If the stability change occurs at 1205731=

120573 it is obvious that 119886(120573) = 0 119887(120573) = 0 and using 119886(120573) = 0119887(120573) = 0 we get (119889119886119889120573

1)|1205731=120573

= ((12)119889 tr(1198641198901)1198891205731)|1205731=120573

gt

0 Thus the transversality condition for a Hopf bifurcation issatisfied

Theorem 6 If (1199061minus 1199031)1199062gt 119870 119906

2((1199061minus 1199031)1199062+119870)241205731lt

(1205732minus 1205733)((1199061minus 1199031)1199062minus 119870)V

2((1199061minus 1199031)1199062+ 119870) minus V

1V2 and

119865(119875lowast) minus119866(119875

lowast) gt 0 then the criterion for the Turing instability

of system (2) satisfies the following condition (1198861111986322

+1198862211986311

minus

1198631211988621

minus 1198632111988612)2gt 4 det(119863) det(119869

1198641198901

) Consider

(1198691198641198901

= (11988611

11988612

11988621

11988622

) 119894119904 119905ℎ119890 119869119886119888119900119887119894119886119899 119898119886119905119903119894119909 119886119905 1198641198901

= (1198751198901 1198851198901))

(33)

Proof The linearized form of system (2) corresponding to theequilibrium 119864

1198901(1198751198901 1198851198901) is given by

120597119901

120597119905= 11988611119901 + 11988612119911 + 119863

11

1205972119901

1205971199042+ 11986312

1205972119911

1205971199042

120597119911

120597119905= 11988621119901 + 11988622119911 + 119863

21

1205972119901

1205971199042+ 11986322

1205972119911

1205971199042

(34)

where 119875 = 1198751198901

+ 119901 119885 = 1198851198901

+ 119911 and (119901 119911) aresmall perturbations in (119875 119885) about the equilibrium 119864

1198901=

(1198751198901 1198851198901) We expand the solution of system (34) into a

Fourier series

(119901

119911) = sum

119896

(1198881

119896

1198882

119896

)119890120582119896119905+119894119896119904

(35)

where 119896 is the wave number of the solution By combining(34) and (35) we obtain

120582119896(1198881

119896

1198882

119896

) = (11988611

minus 119896211986311

11988612

minus 119896211986312

11988621

minus 119896211986321

11988622

minus 119896211986322

)(

1198881

119896

1198882

119896

) (36)

Hence we can obtain the characteristic equation as follows

1205822minus tr119896120582119896+ Δ119896= 0 (37)

where

tr119896= 11988611

+ 11988622

minus 1198962(11986311

+ 11986322)

Δ119896= det (119869

1198641198901

)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321)

(38)

In order to allow the system to achieve Turing instabilityat least one of the following conditionsmust be satisfied tr

119896gt

0 and Δ119896lt 0 However it is obvious that tr

119896lt 0 Thus only

the condition that Δ119896

lt 0 exists can give rise to instabilityTherefore a necessary condition for the system to be unstableis that

det (1198691198641198901

) minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ 1198964(1198631111986322

minus 1198631211986321) lt 0

(39)

We define

119865 (1198962) = 1198964(1198631111986322

minus 1198631211986321)

minus 1198962(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)

+ det (1198691198641198901

)

(40)

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

If we let 1198962min be the corresponding value of 1198962 for the mini-

mum value of 119865(1198962) then

1198962

min =1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612

2 (1198631111986322

minus 1198631211986321)

gt 0 (41)

Thus the corresponding minimum value of 119865(1198962) is

119865 (1198962

min)

= det (1198691198641198901

)

minus(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

4 det (119863)

(42)

In addition if we let 119865(1198962

min) lt 0 the sufficient condition forthe system to be unstable reduces to

(1198861111986322

+ 1198862211986311

minus 1198631211988621

minus 1198632111988612)2

gt 4 det (119863) det (1198691198641198901

)

(43)

Noting If there exists three positive equilibria 11986411989031

=

(11987511989031

11988511989031

) 11986411989032

= (11987511989032

11988511989032

) and 11986411989033

= (11987511989033

11988511989033

) insystem (1) where 119875

11989031lt 11987511989032

lt (12)((1199061minus 1199031)1199062minus 119870)

11987511989033

gt (12)((1199061minus 1199031)1199062minus 119870) then a bistable system exists

when

11987511989031

gt(1199061minus 1199031minus 1199062119870 minus 120573

1+ V1) + radic(119906

1minus 1199031minus 1199062119870 minus 120573

1+ V1)2

+ 81199062V1119870

41199062

(44)

We take 1199061= 09 119903

1= 02 119906

2= 014 120573

1= 12 120573

2= 07

1205733= 01 V

1= 005 V

2= 05 Table 1 show that the condition

for Lemma 1(3) is achievable

3 Numerical Results

31 Numerical Analysis In this section system (2) is analyzedusing the numerical technique to show its dynamic complex-ity However the existence of a positive equilibrium and itsstability in system (1) are given first before this Table 2 showsthe existence of the equilibrium and its stability for differentvalues of the parameter119870 while the other parameters remainfixed 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2=

02428 1205733= 01 V

1= 009 and V

2= 0013Table 2 indicates

that the positive equilibrium is unstable when 119870 = 01 but itis locally asymptotically stable when 119870 = 0205 and globalasymptotically stable when 119870 = 06

The parameters are taken as follows 1199061

= 07 1199031

= 021199062= 0382 120573

1= 021 120573

2= 02428 120573

3= 01 V

1= 009 and

V2= 0013Figure 2 presents a series of one-dimensional numerical

solutions of system (2) Figure 2(a) shows that the solution ofsystem (2) tends to a positive equilibriumwhere119863

21= 11986312

=

0 which implies that positive equilibrium is stable Thismeans that the self-diffusion does not lead to the occurrenceof instability In fact as shown in Figure 1 the self-diffusiondoes not cause instability in system (2) when 119863

21= 11986312

=

0 However instability may occur when the cross-diffusioncoefficients satisfy some conditions

Therefore different values of 11986312

and 11986321

are employedto illustrate the instability induced by cross-diffusionFigure 2(b) shows the phytoplankton density where thevalues of 119863

12and 119863

21are negative whereas the values

of 11986312

and 11986321

are positive in Figure 2(d) It is obviousthat the spatial distributions of phytoplankton in Figures2(b) and 2(d) are markedly different In ecology positive

cross-diffusion indicates that one species tends to move inthe direction with a lower concentration of another specieswhereas negative cross-diffusion denotes that the populationtends to move in the direction with a higher concentration ofanother species [30] In Figures 2(c) and 2(f) 119863

12= 0 and

11986321

= 05minus15 respectively whichmeans that only the influ-ence of zooplankton cross-diffusion is considered in the sys-tem The spatial distributions of phytoplankton are not obvi-ously different in Figures 2(c) and 2(f) but the amplitudesof the phytoplankton density differ which means that thedirection of zooplankton cross-diffusion can affect the phy-toplankton density In Figure 2(e) coefficient 119863

21is positive

and11986312is negative and it is not difficult to see that the spatial

distribution of phytoplankton differs significantly from theother panels shown in Figure 2 These results demonstratethat cross-diffusion plays a significant role in the stability ofthe system but they also show the effects of the various typesof cross-diffusion that a plankton system may encounter

32 Pattern Formation In order to illustrate the spatialdistribution of phytoplankton more clearly we show thepatterns formed by system (2) in a two-dimensional space inFigure 3

Figure 3 shows two different spatial distributions of phy-toplankton Figure 3(a) shows the initial spatial distributionsof phytoplankton Figures 3(b) and 3(c) illustrate differenttypes of pattern formation for different values of 119863

12and

11986321

at 119905 = 1000 When 11986321

= 04 11986312

= 01 Figure 3(b)shows that a spotted pattern and the spots appear to becluttered with an irregular shape However when 119863

21=

091 11986312

= minus05 a circular spotted pattern appears andthe distribution of the spots is more regular as shown inFigure 3(c) As discussed in Section 31 cross-diffusion willlead to instability in the system where both positive cross-diffusion and negative cross-diffusion play important rolesin the stability of the system The spatial distributions of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

minus1 0 1 2

D21

I

minus3

minus2

minus1

0

1

2

3

D12

Figure 1 Bifurcation diagramwith11986312and119863

21 where the parameters values are taken as119870 = 0205 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732= 02428 120573

3= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 The gray line represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

= 0the blue curve represents det(119863) det(119869

1198641198901) = 0 and the red curve represents 119863

1111988622

+ 1198632211988611

minus 1198631211988621

minus 1198632111988612

minus 2radicdet(119863) det(1198691198641198901

) = 0thus 119868 shows the zone where the instability induced by diffusion occurs The remainder of this paper is organized as follows In Section 2 wedescribe and analyze the toxin-phytoplankton-zooplankton system and we derive the criteria for the stability and instability of a system withcross-diffusion In Section 3 we present the results of numerical simulations and discuss their implications Finally we give our conclusionsin Section 4

Table 1 The condition for three positive equilibria exists in system

119870 Equilibrium tr (119869) det (119869) Stability

05(07441 06177) lt0 gt0 Locally stable

(1 07) lt0 Unstable(22559 08823) lt0 gt0 Locally stable

phytoplankton shown in Figures 2(b) and 2(c) agree well withthe results given in the previous section

4 Conclusion and Discussion

The system is simple because it is only an abstract descriptionof a marine plankton system However the system exhibitedsome rich features of the marine plankton systemThe resultsshow that the spatial distribution of phytoplankton is rela-tively uniform and stable under some conditions if we onlyconsider the self-diffusion in system (2) When the influenceof cross-diffusion is considered however the distribution ofphytoplanktonwill change significantly as shown in Figure 2Our results demonstrate that the cross-diffusion can greatlyaffect the dynamic behavior of plankton system

In Section 2 we showed that complex equilibria existin the system and that a bistable system exists when theequilibrium satisfies certain conditions The results shownin Table 1 verified these conclusions We also found thecondition forHopf bifurcation InTable 2 we take parametersvalues of 119906

1= 07 119903

1= 02 119906

2= 0382 120573

1= 021

1205732

= 02428 1205733

= 01 V1

= 009 and V2

= 0013 and

Table 2 The positive equilibrium and its stability

119870 Equilibrium tr (119869) det (119869) Stability01 (02205 06346) gt0 gt0 Unstable0205 (05408 10420) lt0 gt0 Locally stable06 (11915 03826) lt0 gt0 Global stable

we illustrated the changes in stability by varying the half-saturation constant for Holling type II functional response insystem (1) Although the system is simple the behavior of thesystem was found to be very rich

In contrast to previous studies of marine plankton sys-tems diffusion system (2) considered the mutual effectsof self-diffusion and cross-diffusion Furthermore variouseffects of the saturation of cross-diffusion were consideredin our study and we analyzed the influence of differences inreal saturation In the paper theoretical proof of the stabilityof the system we considered positive or negative or zerocross-diffusion and we found that the conditions for systeminstability were caused by cross-diffusion In our numericalanalysis we focused on the influence of cross-diffusion onthe system Figure 2 shows the influence of cross-diffusionon the system where we considered the complexity of theactual situation and the results are illustrated for differentvalues of 119863

12and 119863

21 In Figures 2(a)ndash2(f) the results of

our theoretical derivation are verified and they also reflectthe complex form of the system under the influence of cross-diffusion Turing first proposed the reaction-diffusion equa-tion and described the concept of Turing instability in 1952

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

06

05

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(a)

0

40

20

0x

t

Phytop

lank

ton

1

05

0

1250

2500

(b)

09

02

40

20

0 0

1000

2000

x t

Phytop

lank

ton

(c)

40

20

0 0

1000

2000

x

t

09

02

Phytop

lank

ton

(d)

40

20

0 0

1000

2000

x t

Phytop

lank

ton

09

01

(e)

40

20

0 0

1000

2000

xt

1

05

01

Phytop

lank

ton

(f)

Figure 2 Numerical stability of system (2) with the following parameters 119870 = 0205 1199061= 07 119903

1= 02 119906

2= 0382 120573

1= 021 120573

2= 02428

1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (a) 119863

21= 0 119863

12= 0 (b) 119863

21= minus005 119863

12= minus15 (c) 119863

21= 05 119863

12= 0 (d)

11986321

= 05 11986312

= 01 (e) 11986321

= 1 11986312

= minus05 (f) 11986321

= minus15 11986312

= 0

[34] and pattern formation has now become an importantcomponent when studying the reaction-diffusion equationThus based on Figure 2 we analyzed the patterns formed tofurther illustrate the instability of the system as in Figure 3

In addition it is known that climate change can affectthe diffusion of population in marine Our studies show thatthe diffusion of phytoplankton and zooplankton can affectthe spatial distribution of population under some conditionsHence climate change may have a significant effect on thepopulation in marine We hope that this finding can be

further studied and proved to be useful in the study of toxicplankton systems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31370381)

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

0539 054 0541 0542(a)

02 04 06(b)

02 04 06(c)

Figure 3 Patterns obtained with system (2) using the following parameters 119870 = 0205 1199061

= 07 1199031

= 02 1199062

= 0382 1205731

= 021 1205732

=

02428 1205733= 01 V

1= 009 V

2= 0013 119863

11= 004 and 119863

22= 35 (b) 119863

21= 04 119863

12= 01 (c) 119863

21= 091 119863

12= minus05 Time steps (a) 0

(b) 1000 and (c) 1000

References

[1] J Duinker and G Wefer ldquoDas CO2-problem und die rolle des

ozeansrdquo Naturwissenschaften vol 81 no 6 pp 237ndash242 1994[2] T Saha and M Bandyopadhyay ldquoDynamical analysis of toxin

producing phytoplankton-zooplankton interactionsrdquoNonlinearAnalysis Real World Applications vol 10 no 1 pp 314ndash3322009

[3] Y F Lv Y Z Pei S J Gao and C G Li ldquoHarvesting of aphytoplankton-zooplankton modelrdquo Nonlinear Analysis RealWorld Applications vol 11 no 5 pp 3608ndash3619 2010

[4] T J Smayda ldquoWhat is a bloom A commentaryrdquo Limnology andOceanography vol 42 no 5 pp 1132ndash1136 1997

[5] T J Smayda ldquoNovel and nuisance phytoplankton blooms inthe sea evidence for a global epidemicrdquo in Toxic MarinePhytoplankton E Graneli B Sundstrom L Edler and D MAnderson Eds pp 29ndash40 Elsevier New York NY USA 1990

[6] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[7] G M Hallegraeff ldquoA review of harmful algal blooms and theirapparent global increaserdquo Phycologia vol 32 no 2 pp 79ndash991993

[8] R R Sarkar S Pal and J Chattopadhyay ldquoRole of twotoxin-producing plankton and their effect on phytoplankton-zooplankton systemmdasha mathematical study supported byexperimental findingsrdquo BioSystems vol 80 no 1 pp 11ndash232005

[9] F Rao ldquoSpatiotemporal dynamics in a reaction-diffusiontoxic-phytoplankton-zooplankton modelrdquo Journal of StatisticalMechanics Theory and Experiment vol 2013 pp 114ndash124 2013

[10] S R Jang J Baglama and J Rick ldquoNutrient-phytoplankton-zooplankton models with a toxinrdquoMathematical and ComputerModelling vol 43 no 1-2 pp 105ndash118 2006

[11] S Chaudhuri S Roy and J Chattopadhyay ldquoPhytoplankton-zooplankton dynamics in the lsquopresencersquoor lsquoabsencersquo of toxicphytoplanktonrdquo Applied Mathematics and Computation vol225 pp 102ndash116 2013

[12] S Chakraborty S Roy and J Chattopadhyay ldquoNutrient-limitedtoxin production and the dynamics of two phytoplankton inculturemedia amathematicalmodelrdquoEcologicalModelling vol213 no 2 pp 191ndash201 2008

[13] S Chakraborty S Bhattacharya U Feudel and J Chattopad-hyay ldquoThe role of avoidance by zooplankton for survival anddominance of toxic phytoplanktonrdquo Ecological Complexity vol11 no 3 pp 144ndash153 2012

[14] R K Upadhyay and J Chattopadhyay ldquoChaos to order role oftoxin producing phytoplankton in aquatic systemsrdquo NonlinearAnalysis Modelling and Control vol 4 no 4 pp 383ndash396 2005

[15] S Gakkhar and K Negi ldquoA mathematical model for viral infec-tion in toxin producing phytoplankton and zooplankton sys-temrdquo Applied Mathematics and Computation vol 179 no 1 pp301ndash313 2006

[16] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[17] Y P Wang M Zhao C J Dai and X H Pan ldquoNonlineardynamics of a nutrient-plankton modelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 451757 10 pages 2014

[18] S Khare O P Misra and J Dhar ldquoRole of toxin producingphytoplankton on a plankton ecosystemrdquo Nonlinear AnalysisHybrid Systems vol 4 no 3 pp 496ndash502 2010

[19] W Zhang and M Zhao ldquoDynamical complexity of a spatialphytoplankton-zooplanktonmodel with an alternative prey andrefuge effectrdquo Journal of Applied Mathematics vol 2013 ArticleID 608073 10 pages 2013

[20] J Huisman N N Pham Thi D M Karl and B Sommei-jer ldquoReduced mixing generates oscillations and chaos in theoceanic deep chlorophyll maximumrdquoNature vol 439 no 7074pp 322ndash325 2006

[21] J L Sarmiento T M C Hughes S Ronald and S ManabeldquoSimulated response of the ocean carbon cycle to anthropogenicclimate warmingrdquo Nature vol 393 pp 245ndash249 1998

[22] L Bopp P Monfray O Aumont et al ldquoPotential impact of cli-mate change onmarine export productionrdquoGlobal Biogeochem-ical Cycles vol 15 no 1 pp 81ndash99 2001

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

[23] Z Yue andW J Wang ldquoQualitative analysis of a diffusive ratio-dependent Holling-tanner predator-prey model with smithgrowthrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 267173 9 pages 2013

[24] Y X Huang and O Diekmann ldquoInterspecific influence onmobility and Turing instabilityrdquo Bulletin of Mathematical Biol-ogy vol 65 no 1 pp 143ndash156 2003

[25] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modelling vol141 no 1ndash3 pp 67ndash76 2001

[26] B Mukhopadhyay and R Bhattacharyya ldquoModelling phyto-plankton allelopathy in a nutrient-plankton model with spatialheterogeneityrdquo Ecological Modelling vol 198 no 1-2 pp 163ndash173 2006

[27] W Wang Y Lin L Zhang F Rao and Y Tan ldquoComplex pat-terns in a predator-prey model with self and cross-diffusionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 2006ndash2015 2011

[28] S Jang J Baglama and L Wu ldquoDynamics of phytoplankton-zooplankton systems with toxin producing phytoplanktonrdquoApplied Mathematics and Computation vol 227 pp 717ndash7402014

[29] B Mukhopadhyay and R Bhattacharyya ldquoA delay-diffusionmodel of marine plankton ecosystem exhibiting cyclic natureof bloomsrdquo Journal of Biological Physics vol 31 no 1 pp 3ndash222005

[30] S Roy ldquoSpatial interaction among nontoxic phytoplanktontoxic phytoplankton and zooplankton emergence in space andtimerdquo Journal of Biological Physics vol 34 no 5 pp 459ndash4742008

[31] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer New York NY USA 1981

[32] B Dubey and J Hussain ldquoModelling the interaction of two bio-logical species in a polluted environmentrdquo Journal ofMathemat-ical Analysis and Applications vol 246 no 1 pp 58ndash79 2000

[33] B Dubey N Kumari and R K Upadhyay ldquoSpatiotemporal pat-tern formation in a diffusive predator-prey system an analyticalapproachrdquo Journal of Applied Mathematics and Computing vol31 no 1 pp 413ndash432 2009

[34] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society Part B vol 52 no 1-2pp 37ndash72 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of