Applied Mathematics and Computation 217 (2011) 5449–5457

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Stability and bifurcation analysis for a discrete-time modelof Lotka–Volterra type with delay

Wei Han ⇑, Maoxing LiuDepartment of Mathematics, North University of China, Taiyuan, Shanxi 030051, PR China

a r t i c l e i n f o

Keywords:Lotka–Volterra modelDelayDiscrete-timeBifurcation

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.12.014

⇑ Corresponding author.E-mail address: sh_hanweiwei1@126.com (W. H

a b s t r a c t

A discrete model of Lotka–Volterra type with delay is considered, and a bifurcation analysisis undertaken for the model. We derive the precise conditions ensuring the asymptotic sta-bility of the positive equilibrium, with respect to two characteristic parameters of the sys-tem. It is shown that for certain values of these parameters, fold or Neimark–Sackerbifurcations occur, but codimension 2 (fold-Neimark–Sacker, double Neimark–Sackerand resonance 1:1) bifurcations may also be present. The direction and the stability ofthe Neimark–Sacker bifurcations are investigated by applying the center manifold theoremand the normal form theory.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In recent years, the study on population models in various fields of mathematical biology has become a very popular topicsince the pioneering theoretical works by [1] and [2] in which the well-known Lotka–Volterra predator–prey system de-scribed by the ordinary differential equations was proposed and studied. The dynamical behaviors of population models gov-erned by difference equations has been studied by a number of papers (see [10–17]).

The continuous Lotka–Volterra population model have been studied very deeply (for details, see [8,9]). On the one hand,based on the classical Lotka–Volterra predator–prey model, many population models reflecting various interactions betweentwo species have been proposed. For example, when there are two different species in a certain environment and they live bycompeting the resources, if the past history has the effect on the dynamical behaviors of system and the species will migratetowards regions of lower population density to add the possibility of survival, under the assumption that in the absence ofone of the species, the growth of the other species will be governed by the delayed logistic diffusion equation and the dimen-sion of space will be taken as one (see [8]). Yan and Zhang [9] studied the following delayed Lotka–Volterra two species com-petition diffusion system with a single discrete delay and subject to homogeneous Dirichlet boundary conditions

@uðt;xÞ@t ¼ d1

@2uðt;xÞ@x2 þ uðt; xÞ r1 � a11uðt � s11; xÞ � a12vðt � s12; xÞ½ �;

@vðt;xÞ@t ¼ d2

@2vðt;xÞ@x2 þ vðt; xÞ r2 � a21uðt � s21; xÞ � a22vðt � s22; xÞ½ �;

8<: ð1:1Þ

where uðt; xÞ and vðt; xÞ represent the densities of two species at time t and space x, respectively. They investigated the direc-tion of Hopf bifurcation resulting from the increase of delay and showed that the system under consideration can undergo asupercritical Hopf bifurcation near the spatially in homogeneous positive stationary solution when the delay crosses througha sequence of critical values by applying the implicit function theorem.

. All rights reserved.

an).

5450 W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457

On the other hand, the impulsive Lotka–Volterra population models have also been studied extensively, for example, Guoand Chen [17] presented a kind of time-limited pest control of a Lotka–Volterra model with impulsive harvest, described bythe initial and boundary value problem of impulsive differential equation. They obtained the conditions under which themodel has a solution by the comparison principle by a series of the upper solutions and the conditions under which the mod-el has no solution are also given by a series of the lower solutions.

In the discrete Lotka–Volterra population model, there are also very many results about it (see [3,13]). In [3], Wang and Luconsidered the global stability of discrete model for the following Lotka–Volterra model

xiðkþ 1Þ ¼ xiðkÞ exp riðkÞ �Xn

j¼1

aijðkÞxjðkÞ( )

; i ¼ 1;2; . . . ;n; ð1:2Þ

where xiðkÞ is the density of population i at kth generation, riðkÞ is the growth rate of population i at kth generation, aijðkÞmeasures the intensity of intraspecific competition or interspecific action of species. Zou et al. [13] discussed the followingLotka–Volterra equation

u0nl ¼ unðun�1 � unþ1Þ þ unðun�2 � unþ2Þ;unð0Þ ¼ n:

�ð1:3Þ

They generalized the homotopy analysis method to solve (1.3). At the same time, Bifurcation analysis especially in discrete-time models have been studied very extensively. In [4], Kaslik and Balint studied the following discrete-time Hopfield neuralnetwork of two neurons with a single delay

xnþ1 ¼ axn þ T11g1ðxn�kÞ þ T12g2ðyn�kÞ;ynþ1 ¼ ayn þ T21g1ðxn�kÞ þ T22g2ðyn�kÞ:

�ð1:4Þ

The authors obtained the bifurcation analysis for the above system by applying the center manifold theorem and thenormal form theory. Yan and Zhang [5] studied the following delayed Lotka–Volterra predator–prey system with a singledelay

_xðtÞ ¼ xðtÞ r1 � a11xðt � sÞ � a12yðt � sÞ½ �;_yðtÞ ¼ yðtÞ �r2 þ a21xðt � sÞ � a22yðt � sÞ½ �:

�ð1:5Þ

By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system ofthe original system at the positive equilibrium, they studied the linear stability of the system Hopf bifurcations. In [6,7],Zhang and Zheng also obtained the Hopf bifurcation analysis by using the normal form method and the center manifoldtheorem.

Motivated by the above recent papers, in this paper, we are interested in the bifurcation analysis and the direction and thestability of the Neimark–Sacker bifurcations for the following Lotka–Volterra model

xkþ1 ¼ xk exp r1 � a11xk�s � a12yk�sf g;ykþ1 ¼ yk exp r2 � a21xk�s � a22yk�sf g;

�8k P s; ð1:6Þ

where xk is the density of first population at kth generation, yk denotes the density of second population at kth generation, ri

is the growth rate of population i, aij stands for the intensity of intraspecific competition or interspecific action of species.To prove the main results in this paper, we transform the system (1.6) into the following equivalent system of 2sþ 2

equations without delays

xð0Þkþ1 ¼ xð0Þk exp r1 � a11xðsÞk � a12yðsÞk

n o;

y0kþ1 ¼ yð0Þk exp r2 � a21xðsÞk � a22yðsÞk

n o;

xðjÞkþ1 ¼ xðj�1Þk ; yj

kþ1 ¼ yðj�1Þk ; j ¼ 1;2; . . . ; s;

8>>><>>>:

8k P s; ð1:7Þ

where

xð0Þk ¼ xk; yð0Þk ¼ yk:

To date no paper has appeared in the literature which discusses the bifurcation analysis for the discrete model of Lotka–Volterra type with delay. This paper attempts to fill this gap in the literature. Our work focuses on the stability and bifur-cation analysis and the direction analysis of the Neimark–Sacker bifurcations by applying the center manifold theoremand the normal form theory. The method of the paper is similar to the work of Kaslik and Balint [4].

The rest of this paper is as follows: We state some basic preliminaries and prove several lemmas in Section 2. Thestability of the positive equilibrium and bifurcation analysis will be discussed in Sections 3 and 4, respectively. Section 5is devoted to the direction and stability of the Neimark–Sacker bifurcations, the main tool being the center manifoldtheorem and the normal form theory. At the end of the paper, some numerical simulations to demonstrate our theoret-ical results are given.

W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457 5451

2. Preliminaries and some lemmas

To guarantee that system (1.6) has always a positive equilibrium, throughout this paper, we assume that the coefficientsof system (1.6) satisfies the following condition

ðHÞ r1a22 � r2a12 > 0; r2a11 � r1a21 > 0:

Clearly, under the hypothesis ðHÞ, system (1.6) has a unique positive equilibrium E�ðx�; y�Þ, where

x� ¼ r1a22 � r2a12

a11a22 � a12a21; y� ¼ r2a11 � r1a21

a11a22 � a12a21:

For convenience, we copy (1.7) in the following equation

xð0Þkþ1 ¼ xð0Þk exp r1 � a11xðsÞk � a12yðsÞk

n o;

y0kþ1 ¼ yð0Þk exp r2 � a21xðsÞk � a22yðsÞk

n o;

xðjÞkþ1 ¼ xðj�1Þk ; yj

kþ1 ¼ yðj�1Þk ; j ¼ 1;2; . . . ; s;

8>>><>>>:

8k P s; ð2:1Þ

where

xð0Þk ¼ xk; yð0Þk ¼ yk:

The Jacobian matrix of system (2.1) at the point xð0Þk ; yð0Þk ; xð1Þk ; yð1Þk ; . . . ; xðsÞk ; yðsÞk

� �2 R2sþ2 is

J ¼

A 0 0 � � � 0 B

E2 0 0 � � � 0 00 E2 0 � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � E2 0

0BBBBBBB@

1CCCCCCCAð2sþ2Þ�ð2sþ2Þ

;

where

A ¼exp r1 � a11xðsÞk � a12yðsÞk

n o0

0 exp r2 � a21xðsÞk � a22yðsÞk

n o0B@

1CA;

B ¼�a11xð0Þk exp r1 � a11xðsÞk � a12yðsÞk

n o�a12xð0Þk exp r1 � a11xðsÞk � a12yðsÞk

n o�a21yð0Þk exp r2 � a21xðsÞk � a22yðsÞk

n o�a22yð0Þk exp r2 � a21xðsÞk � a22yðsÞk

n o0B@

1CA;

E2 and 0 are the identity and the null matrices, respectively.Then we obtain the Jacobian matrix of system (2.1) at the positive equilibrium ðx�; y�; x�; y�; . . . ; x�; y�Þ 2 R2sþ2

J0 ¼

E2 0 0 � � � 0 B0

E2 0 0 � � � 0 00 E2 0 � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � E2 0

0BBBBBBB@

1CCCCCCCAð2sþ2Þ�ð2sþ2Þ

; ð2:2Þ

where

B0 ¼�a11x� �a21x�

�a21y� �a22y�

� �;

E2 and 0 are the identity and the null matrices, respectively. We obtain the following characteristic equation of J0

k2sðk� 1Þ2 � 2aksðk� 1Þ þ b ¼ 0; ð2:3Þ

where 2a ¼ trðB0Þ and b ¼ detðB0Þ. Therefore, the eigenvalues of the matrix J0 are the solutions of the following two equations

ksðk� 1Þ ¼ gi; i ¼ 1;2; ð2:4Þ

where g1 and g2 are the eigenvalues of the matrix B0.We will find the set of the parameters ða; bÞ 2 R2 for which the positive equilibrium is asymptotically stable. And a bifur-

cation analysis will be undertaken along the boundary of this set by using the techniques from [19].

5452 W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457

To prove our main results in this paper, we will employ following useful lemmas.

Lemma 2.1. (see [18]). The polynomial PðkÞ ¼ ksþ1 � aks þ b, where a; b 2 R is of Schur type (i.e. all its eigenvalues are inside theunit circle) if and only if one of the following conditions hold

1. if basþ16 0 then jaj þ jbj < 1;

2. if basþ1> 0 then jaj 6 sþ1

s and jaj � 1 < jbj < ða2 þ 1� 2jaj cos /Þ12,

where / 2 0; psþ1

� �is the solution of the equation jaj sin s/ ¼ sinðsþ 1Þ/.

Let us denote by /s 2 0; psþ1

� �the unique solution of the equation sin s/ ¼ sinðsþ 1Þ/ and let cs ¼ �ð2� 2 cos /sÞ

12 < 0. We

will denote by PiðkÞ ¼ ksþ1 � ks � gi; i ¼ 1;2, where g1 and g2 are the eigenvalues of the matrix B0. We have the following

Lemma 2.2. If b 6 a2, the positive equilibrium of (2.1) is asymptotically stable if and only if g1 and g2 belong to the intervalðcs;0Þ.

Proof. By b 6 a2, the eigenvalues g1;2 ¼ a�ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � b

pof the matrix B0 are real. Based on (2.4), the positive equilibrium of (2.1)

is asymptotically stable if and only if both polynomials PiðkÞ ¼ kkþ1 � kk � gi; i ¼ 1;2 are Schur polynomials. Lemma 2.1 pro-vides that this is true if and only if cs < gi < 0; i ¼ 1;2. h

Lemma 2.3. If b > a2, the matrix J0 has eigenvalues on the unit circle if and only if 0 6 b 6 4 and

a ¼ cos ðsþ 1Þ arccos2� b

2

� �� cos s arccos

2� b2

� �: ð2:5Þ

More, the only eigenvalues of J0 of modulus 1 are e�ih where h ¼ arccos 2�b2 .

Proof. In the case b > a2, the eigenvalues g1 and g2 of the matrix B0 are complex, that is g1 ¼ g2 ¼ aþ iffiffiffiffiffiffiffiffiffiffiffiffiffiffib� a2

p.

The eigenvalues of J0 are the roots of the polynomials PiðkÞ ¼ ksþ1 � ks � gi; i ¼ 1;2. Let us underline that k is a root ofP1ðkÞ if and only if �k is a root of P2ðkÞ, as g1 ¼ g2. Suppose that the polynomial P1ðkÞ has a root on the unit circle, k ¼ eih withh 2 ½0;p�. Therefore, eishðeih � 1Þ ¼ g1 and jeih � 1j2 ¼ jg1j

2 ¼ b. Hence, 0 6 b 6 4 and h ¼ arccos 2�b2 . On the other hand, we

also have that Reðeishðeih � 1ÞÞ ¼ Reðg1Þ ¼ a which leads us to relation (2.5). h

Lemma 2.4. The function gs : ½0;4� ! R defined by

gsðbÞ ¼ cos ðsþ 1Þ arccos2� b

2

� �� cos s arccos

2� b2

� �ð2:6Þ

is a polynomial function of degree ðsþ 1Þ, it is strictly decreasing on the interval ½0; c2s � and gsð½0; c2

s �Þ ¼ ½cs; 0�. More, jgsðbÞj 6ffiffiffibp

for any b 2 ½0;4�, specifically, we have

g1ðbÞ ¼12ðb2 � 3bÞ; g2ðbÞ ¼

12ð5b2 � 5b� b3Þ:

Let hs : ½cs;0� ! ½0; c2s � be the inverse of the restriction of the function gs to the interval ½0; c2

s �, i.e. hs ¼ ðgsj½0;c2s �Þ�1, where gs is the

function defined in the expression (2.6).

Lemma 2.5. (see [6,7]). Suppose that S � Ris a compact and connected set, Pðk;aÞ ¼ km þ p1ðaÞkm�1 þ p2ðaÞkm�2 þ � � � pmðaÞ iscontinuous on C � S. Then, as the parameter a varies, the sum of the order of the zeros of Pðk;aÞ out of the unit circle, i.e. cardfk 2 C : Pðk;aÞ ¼ 0; jkj > 1gð Þ, can change only if a zero appears on or crossed the unit circle.

3. The Stability of the positive equilibrium

In [3], we know that equation (1.6) has a positive equilibrium ðx�; y�Þ under the hypothesis ðHÞ. We have the followingresult on the stability of the positive equilibrium.

Theorem 3.1. The positive equilibrium of (1.6) is asymptotically stable if and only if one of the following inequalities hold

1: b 6 a2; b > 2csa� c2s ; b > 0; cs < a < 0: ð3:1Þ

2

2: b > a ; b < hsðaÞ; cs < a < 0: ð3:2Þ

(see Fig. 1.)

W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457 5453

Proof

1. It can be easily verified that the eigenvalues g1 and g2 of the matrix B0 belongs to the interval ðcs;0Þ if and only if the half-trace a and the determinant b of the matrix B0 verify the set of inequalities (3.1). Hence, by Lemma 2.2, it follows that thepositive equilibrium is asymptotically stable if and only if inequalities (3.1) hold.

2. The statements follows directly from Lemmas 2.3, 2.4, 2.5 and from the results obtained in the case when b 6 a2, and weomit it here. The proof is complete. h

4. Bifurcation analysis

In this section, we will give the bifurcation analysis of our model.

Theorem 4.1. a; b; cs are defined as in the Section 2. If b 6 a2, then the following hold

1. If a 2 cs;cs2

� , and b ¼ 2csa� c2

s , then in this case, a Neimark–Sacker bifurcation occurs in system (1.6), i.e. a unique closedinvariant curve bifurcates from the positive equilibrium near b ¼ 2csa� c2

s .2. If a 2 cs

2 ;0�

, and b ¼ 0, then in this case, system (1.6) has a fold bifurcation at the positive equilibrium.3. If a ¼ cs

2 , and b ¼ 0, then in this case, a fold-Neimark–Sacker bifurcation occurs at the positive equilibrium in system (1.6).

Proof

1. Assume that a 2 cs;cs2

� and let b ¼ b� ¼ 2csa� c2

s , the eigenvalues of the matrix B0 are g1 ¼ cs and g2 ¼ 2a� cs. Asg2 2 ðcs;0Þ, by Lemma 2.1, the roots of the polynomial P2ðkÞ ¼ kkþ1 � kk � g2 are all inside the unit circle. The polynomial

P1ðkÞ ¼ kkþ1 � kk � g1 has a simple pair of roots of eigenvalues inside the unit circle and two eigenvalues of modulus 1. More,

as /s 2 0; psþ1

� �, we have that e�i/s is not a root of order 1, 2, 3, or 4 of the unity, therefore, the first non-degeneracy condition

for the Neimark–Sacker bifurcation holds [19]. The second non-degeneracy condition [19] that we still need to verify isdjkjdb jb¼b�–0, where kðbÞ is the root of the equation kkþ1 � kk � g1ðbÞ ¼ 0, with g1ðb

�Þ ¼ csand kðb�Þ ¼ ei/s . Indeed, we have

Figequ

djkj2

db¼ �k

dkdbþ k

d�kdb¼ dg1

db

�k

ks�1½ðsþ 1Þk� s�� k

�ks�1½ðsþ 1Þ�k� s�

!;

and therefore

djkj2

dbjb¼b� ¼

ðsþ 1Þ cosððsþ 1Þ/sÞ � s cosðs/sÞða� csÞððsþ 1Þ2 � 2sðsþ 1Þ cos /s þ s2Þ

:

NS-NS 1:1

Fold-NS

0 ; fold

2

);(h NS

22 cc

NS

. 1. The stability domain in the ða; bÞ plain (region inclosed by the black lines and curve) and bifurcations with loss of stability for the positiveilibrium of (1.6).

5454 W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457

The denominator of the above fraction is strictly positive, so the sign of djkj2db jb¼b� is given by the numerator, which can be

proven to be strictly negative for any s 2 N. We give the proof of the strict negativity of the numerator as follows. Since/s 2 0; p

sþ1

� �is the unique solution of the equation sin s/ ¼ sinðsþ 1Þ/, and we note that 0 < s/s < ðsþ 1Þ/s < p, so we

get that p=2 < ðsþ 1Þ/s < p; 0 < s/s < p=2, and s/s þ ðsþ 1Þ/s ¼ p, we conclude that ðsþ 1Þ cosððsþ 1Þ/sÞ�s cosðs/sÞ < 0. Hence, a Neimark–Sacker bifurcation occurs at the origin in system (1.6) at b� ¼ 2csa� c2

s .2. Assume that a 2 cs

2 ;0�

and b ¼ 0, the eigenvalues of the matrix B0 are g1 ¼ 0 and g2 ¼ 2a. As g2 2 ðcs;0Þ, by Lemma 2.1,the roots of the polynomial P2ðkÞ ¼ ksþ1 � ks � g2 are all inside the unit circle. The polynomial P1ðkÞ ¼ ksþ1 � ks � g1 has asimple root k ¼ 1 on the unit circle, and all its other roots are inside the unit circle. Therefore, a fold bifurcation occurs atthe origin in system (1.6).

3. Assume that a ¼ cs2 and b ¼ 0, the eigenvalues of the matrix B0 are g1 ¼ 0 and g2 ¼ cs. Therefore, the matrix J0 has exactly

three eigenvalues on the unit circle: a simple root 1 and a simple pair of roots e�i/s , and all its other eigenvalues are insidethe unit circle. This means that a fold-Neimark–Sacker bifurcation occurs at the origin in system (1.6). h

Theorem 4.2. a; b; cs are defined as in the Section 2. If b > a2, then the following hold

1. If a 2 ðcs;0Þ, and b ¼ hsðaÞ, then in this case, system (1.6) has a Neimark–Sacker bifurcation at the positive equilibrium. That is,system (1.6) has a unique closed invariant curve bifurcating from the positive equilibrium near b ¼ hsðaÞ.

2. If a ¼ cs, and b ¼ c2s , then in this case, the positive equilibrium of (1.6) is a double Neimark–Sacker bifurcation point.

3. If a ¼ 0 and b ¼ 0, then in this case, the system (1.6) has a strong 1:1 resonant bifurcation at the positive equilibrium.

Proof.

1. We note that if a 2 ðcs;0Þ and b ¼ b� ¼ hsðaÞ, so b� ¼ hsðaÞ 2 ð0; c2sÞ and therefore, 0 < arccos 2�b�

2 < /s <p

sþ1. Hence, e�ih

with h ¼ arccos 2�b�

2 are not roots of order p; p ¼ 1;2;3;4 of the unity, therefore, the first non-degeneracy condition forthe Neimark–Sacker bifurcation holds [19]. We will need to check that the second non-degeneracy condition [19], i.e.djkjdb jb¼b� – 0, where kðbÞ is the root of the equation ksþ1 � ks � gðbÞ ¼ 0, with gðbÞ ¼ aþ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib� a2

pand kðb�Þ ¼ eih. Indeed,

we have

djkj2

db¼ �k

dkdbþ k

d�kdb¼ dg

db

�k

ks�1½ðsþ 1Þk� s�� k

�ks�1½ðsþ 1Þ�k� s�

!;

and therefore

djkj2

dbjb¼b� ¼

ðsþ 1Þ sinððsþ 1ÞhÞ � s sinðshÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib� � a2

pððsþ 1Þ2 � 2sðsþ 1Þ cos /s þ s2Þ

> 0:

The denominator of the above fraction is strictly positive, so the sign of djkj2db jb¼b� is given by the numerator, which can be

proven to be strictly positive for any s 2 N and h 2 ð0;/sÞ. We give the proof of the strict positivity of the numerator asfollowsSince /s 2 ð0; p

sþ1Þ is the unique solution of the equation sin s/ ¼ sinðsþ 1Þ/, and we note that 0 < s/s < ðsþ 1Þ/s <

p;p=2 < ðsþ 1Þ/s < p; 0 < s/s < p=2, and s/s þ ðsþ 1Þ/s ¼ p, thus /s ¼ p=ð2sþ 1Þ, and 0 < sh < sp=ð2sþ 1Þ <p=2; 0 < ðsþ 1Þh < ðsþ 1Þp=ð2sþ 1Þ < p; ðsþ 1Þhþ sh < p. We discuss in two cases respectively: One case isðsþ 1Þh 6 p=2, we have sh < p=2, so by the monotonicity, sinððsþ 1ÞhÞ > sinðshÞ, and thus ðsþ 1Þ sinððsþ 1ÞhÞ�s sinðshÞ > 0. The other case is ðsþ 1Þh > p=2, we have sh < p=2;p� ðsþ 1Þh 2 ð0;p=2Þ, and p� ðsþ 1Þh > sh, sosinððsþ 1ÞhÞ ¼ sinðp� ðsþ 1ÞhÞ > sinðshÞ, and thus ðsþ 1Þ sinððsþ 1ÞhÞ � s sinðshÞ > 0. In conclusion. we obtainðsþ 1Þ sinððsþ 1ÞhÞ � s sinðshÞ > sinðshÞ > 2sh=p > 0.Hence, a Neimark–Sacker bifurcation occurs at the origin in system (1.6) at b� ¼ hsðaÞ.

2. If a ¼ cs and b ¼ c2s , then g1 ¼ g2 ¼ cs are the eigenvalues of the matrix B0. Therefore, e�i/s are roots of both polynomials P1

and P2. Hence the matrix J0 has a double pair of eigenvalues on the unit circle. The positive equilibrium of (1.6) is a doubleNeimark–Sacker bifurcation point.

3. If a ¼ 0 and b ¼ 0, then g1 ¼ g2 ¼ 0 are the eigenvalues of the matrix B0. In this case, 1 is a root of both polynomials P1 andP2. Therefore, the matrix J0 has a double 1 eigenvalue, meaning that the system (1.6) has a strong 1:1 resonant bifurcationat the origin. h

5. Direction and stability of the Neimark–Sacker bifurcations

Let the function F : R2sþ2 ! R2sþ2 be given by:

W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457 5455

Fðxð0Þ; yð0Þ; xð1Þ; yð1Þ; . . . ; xðsÞ; yðsÞÞ ¼

x0 expfr1 � a11xs � a12ysgy0 expfr2 � a21xs � a22ysg

xð0Þ

yð0Þ

. . .

xðs�1Þ

yðs�1Þ

0BBBBBBBBBBB@

1CCCCCCCCCCCA; 8xðjÞ; yðjÞ 2 R; j ¼ 0;1; . . . ; s:

The function F is the right hand side of system (2.1). It is clear that J0 ¼ DFð0Þ. We will denote J00 ¼ D2Fð0Þ and J000 ¼ D3Fð0Þ.According to Theorems 4.1 and 4.2, Neimark–Sacker bifurcations occur at the origin for system (1.6) if one of the follow-

ing cases holds:

(i) a 2 cs;cs2

� and b ¼ 2csa� c2

s . In this case, matrix J0 has a simple pair of eigenvalues e�i/s , on the unit circle. More,k ¼ ei/s satisfies

ksþ1 � ks ¼ g;

where g ¼ cs is an eigenvalue of the matrix B0.

(ii) a 2 ðcs;0Þ and b ¼ hsðaÞ. In this case, matrix J0 has a simple pair of eigenvalues e�ih, with h ¼ arccos 2�hsðaÞ

2 on the unitcircle. More, k ¼ eih satisfies

ksþ1 � ks ¼ g;

where g ¼ aþ iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihsðaÞ � a

pis an eigenvalue of matrix B0.

In both cases, the restriction of system (2.1) to its two dimensional center manifold at the critical parameter values can betransformed into the normal form written in complex coordinates [19]

x#kx 1þ 12

djxj2� �

þ Oðjxj4Þ; x 2 C;

where

d ¼ �k p; J000 q; q; �qð Þ þ 2J00 q; ðE� J0Þ�1J00ðq; �qÞ� �

þ J00 �q; ðk2E� J0Þ�1J00ðq; qÞ� �D E

;

where J0q ¼ kq; J0T p ¼ �kp and hp; qi ¼ 1 (with hp; qi ¼ �pT q). Direct computations result in the following theorems [19,20]

Theorem 5.1. The vectors q and p of C2sþ2 which verify

J0q ¼ kq; J0T p ¼ �kp and hp; qi ¼ 1; ð5:1Þ

are given by

q ¼ ðu1; �ku1; �k2u1; . . . ; �ksu1ÞT ; p ¼ ðu2; �gksu2; �gks�1u2; . . . ; �gku2ÞT ; ð5:2Þ

where u1; u2 2 C2 are given by

B0u1 ¼ gu1; B0u2 ¼ �gu2; hu1; u2i ¼1

1þ sg�ksþ1:

Theorem 5.2. The direction and stability of the Neimark-Sacher bifurcation is determined by the sign of ReðdÞ. If ReðdÞ < 0 thenthe bifurcation is supercritical, i.e. the closed invariant curve bifurcating from the origin is asymptotically stable. If ReðdÞ > 0, thebifurcation is subcritical, i.e. the closed invariant curve bifurcating from the origin is unstable.

6. Some applications

In this section, we present some simple examples to explain our results.

Example 6.1. Consider the following model

xkþ1 ¼ xk expf1� 4xk�1 � 0:5yk�1g;ykþ1 ¼ yk expf0:9� xk�1 � 2yk�1g;

�8k P 1; ð6:1Þ

where we have

Fig. 2.to the p

5456 W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457

r1 ¼ 1; r2 ¼ 0:9; s ¼ 1;

and r1a22 � r2a12 ¼ 1� 2� 0:9� 0:5 ¼ 1:55 > 0; r2a11 � r1a21 ¼ 0:9� 4� 1� 1 ¼ 2:6 > 0, so the condition ðHÞ hold, system(6.1) has a unique positive equilibrium E�ðx�; y�Þ, where

x� ¼ r1a22 � r2a12

a11a22 � a12a21¼ 0:2067; y� ¼ r2a11 � r1a21

a11a22 � a12a21¼ 0:3467:

We compute that / ¼ p3 ; cos / ¼ 1

2 ; cs ¼ �1 and a ¼ 12 ð�4� 0:2067� 2� 0:3467Þ ¼ �0:7601; b ¼ 4� 0:2067� 2�

0:3467� 0:5� 0:2067� 0:3467 ¼ 0:5375. From the expression of gsðbÞ,

gsðbÞ ¼ cos ðsþ 1Þ arccos2� b

2

� �� cos s arccos

2� b2

� �¼ cos 2 arccos

2� b2

� �� cos arccos

2� b2

� �

¼ 12½ð2� bÞ2 � ð2� bÞ � 2�;

we get hsðaÞ ¼ 0:6457.So 0:5375 6 ð�0:7601Þ2 ¼ 0:5778;0:5375 > 2� ð�1Þ � ð�0:7601Þ � 12 ¼ 0:5202; b > 0, and �1 < �0:7601 < 0. then the

condition 1 of Theorem 3.1 is satisfied, so the positive equilibrium E�ð0:2067;0:3467Þ of (6.1) is asymptotically stable (seeFig. 2).

Example 6.2. Consider the following model

xkþ1 ¼ xk expf1:0365� 4:1xk�1 � 0:8yk�1g;ykþ1 ¼ yk expf0:9� 1:5xk�1 � 3yk�1g;

�8k P 1; ð6:2Þ

where we have

r1 ¼ 1:0365; r2 ¼ 0:9; s ¼ 1;

and r1a22 � r2a12 ¼ 1:0365� 3� 0:9� 0:8 ¼ 2:3895 > 0; r2a11 � r1a21 ¼ 0:9� 4:1� 1:0365� 1:5 ¼ 2:1353 > 0, so the con-dition ðHÞ hold, system (6.2) has a unique positive equilibrium E�ðx�; y�Þ, where

x� ¼ r1a22 � r2a12

a11a22 � a12a21¼ 0:2153; y� ¼ r2a11 � r1a21

a11a22 � a12a21¼ 0:1924:

We compute that / ¼ p3 ; cos / ¼ 1

2 ; cs ¼ �1 and a ¼ �0:7299; b ¼ 0:4597; 2csa� c2s ¼ �2a� 1 ¼ 0:4597.

So 0:4597 6 ð�0:7299Þ2 ¼ 0:5328; a ¼ �0:7299 2 �1;� 12

� ; b ¼ 0:4597 ¼ 2csa� c2

s . Then the condition 1 of Theorem4.1 is satisfied, so the positive equilibrium E�ð0:2067;0:3467Þ of (6.2) is unstable, and in this case, a Neimark–Sackerbifurcation occurs in system (6.2), i.e. an asymptotically stable closed invariant curve is present. The trajectory of (6.2)converges to the asymptotically stable invariant curve (see Fig. 3).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x(k)

y(k)

For r1 ¼ 1; r2 ¼ 0:9; a11 ¼ 4; a12 ¼ 0:5; a21 ¼ 1; a22 ¼ 2, the positive equilibrium of (6.1) is asymptotically stable. The trajectory of (6.1) convergesositive equilibrium ðx�; y�Þ ¼ ð0:2067;0:3467Þ.

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x(k)

y(k)

Fig. 3. For r1 ¼ 1:5; r2 ¼ 0:9; a11 ¼ 4; a12 ¼ 0:5; a21 ¼ 1; a22 ¼ 10, the the positive equilibrium ðx�; y�Þ ¼ ð0:3684;0:0532Þ of (6.2) is unstable and anasymptotically stable closed invariant curve is present. The trajectory of (6.2) converges to the asymptotically stable invariant curve.

W. Han, M. Liu / Applied Mathematics and Computation 217 (2011) 5449–5457 5457

Acknowledgments

The authors thank the Editor and the anonymous referees for their very valuable comments and helpful suggestions,which have been very useful for improving this paper.

This research is supported by the Youth Science Foundation of Shanxi Province (2010021001-2), the National SciencesFoundation of China (10901145), the Top Young Academic Leaders of Higher Learning Institutions of Shanxi, the NationalNatural Science Foundation of China (under Grant No. 60771026), the Programme for New Century Excellent Talents in Uni-versity (NCET050271), and the Special Scientific Research Foundation for the Subjects of Doctors in University(20060110005).

References

[1] A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, New York, 1925.[2] V. Volterra, Variazionie fluttuazioni del numero d’ individui in specie animali conviventi, Mem. Acad. Licei. 2 (1926) 31–113.[3] W.D. Wang, Z.Y. Lu, Global stability of discrete models of Lotka–Volterra type, Nonlinear Anal. 35 (1999) 1019–1030.[4] E. Kaslik, St. Balint, Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network, Chaos Soliton Fractal 34 (2007) 1245–

1253.[5] X.P. Yan, C.H. Zhang, Hopf bifurcation in a delayed Lotka–Volterra predator-prey system, Nonlinear Anal. Real World Appl. 9 (2008) 114–127.[6] C.R. Zhang, B.D. Zheng, Hopf bifurcation in numerical approximation of a n-dimension neural network model with multi-delays, Chaos Soliton Fractal

25 (2005) 129–146.[7] C.R. Zhang, B.D. Zheng, Stability and bifurcation of a two-dimension discrete neural network model with multi-delays, Chaos Soliton Fractal 31 (2007)

1232–1242.[8] M. Delgado, M. Montenegro, A. Suarez, A Lotka–Volterra symbiotic model with cross-diffusion, J. Differ. Equations 246 (2009) 2131–2149.[9] X.P. Yan, C.H. Zhang, Direction of Hopf bifurcation in a delayed Lotka–Volterra competition diffusion system, Nonlinear Anal. Real World Appl. 10

(2009) 2758–2773.[10] R.Q. Shi, L.S. Chen, Staged-structured Lotka–Volterra predator-prey models for pest management, Appl. Math. Comput. 203 (2008) 258–265.[11] L. Wan, Q.H. Zhou, Stochastic Lotka–Volterra model with infinite delay, Stat. Probabil. Lett. 79 (2009) 698–706.[12] N. Fang, X.X. Chen, Permanence of a discrete multispecies Lotka–Volterra competition predator-prey system with delays, Nonlinear Anal. Real World

Appl. 9 (2008) 2185–2195.[13] L. Zou, Z. Zong, G.H. Dong, Generalizing homotopy analysis method to solve Lotka–Volterra equation, Comput. Math. Appl. 56 (2008) 2289–2293.[14] F.D. Chen, Permanence in a discrete Lotka–Volterracompetition model with deviating arguments, Nonlinear Anal. Real World Appl. 9 (2008) 2150–

2155.[15] J.C.F. Wong, The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions, J.

Comput. Appl. Math. 223 (2009) 421–437.[16] H. Fang, Y.F. Xiao, Existence of multiple periodic solutions for delay Lotka–Volterra competition patch systems with harvesting, Appl. Math. Model. 33

(2009) 1086–1096.[17] H.J. Guo, L.S. Chen, Time-limited pest control of a Lotka–Volterra model with impulsive harvest, Nonlinear Anal. Real World Appl. 10 (2009) 840–848.[18] S. Kuruklis, The aymptotic stability of xnþ1 � axn þ bxn�k ¼ 0, J. Math. Anal. Appl. 188 (3) (1994) 719–731.[19] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, second ed., Springer-Verlag, NewYork, 1998.[20] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 2003.