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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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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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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Differential EquationsInternational Journal of
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Function Spaces
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 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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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
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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
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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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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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
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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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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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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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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of