Chaos, Solitons and Fractals 32 (2007) 80–94

www.elsevier.com/locate/chaos

Complex dynamic behaviors of a discrete-timepredator–prey system

Xiaoli Liu *, Dongmei Xiao

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China

Accepted 20 October 2005

Abstract

The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant R2þ. It is shown that

the system undergoes flip bifurcation and Hopf bifurcation in the interior of R2þ by using center manifold theorem and

bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis,but also to exhibit the complex dynamical behaviors, such as the period-5, 6, 9, 10, 14, 18, 20, 25 orbits, cascade ofperiod-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richerdynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically com-puted to confirm further the complexity of the dynamical behaviors.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

In population dynamics, there are two kinds of mathematical models: the continuous-time models described by dif-ferential equations or dynamical systems, and the discrete-time models described by difference equations, discretedynamical systems or iterative maps. The simplest continuous-time population model is the logistic differential equationof a single species, first introduced by Verhulst [1] and later studied further by Pearl and Reed [2]:

0960-0doi:10

* CoE-m

_x ¼ r0x 1� xk

� �; ð1:1Þ

where x(t) denotes the population of a single species at time t, k is the carrying capacity of the population, and r0 is theintrinsic growth rate. Eq. (1.1) describes the growth rate of the population size of a single species. However, the pop-ulation size of a single species may has a fixed interval between generations or possibly a fixed interval between mea-surements. For example, many species of insect have no overlap between successive generations, and thus theirpopulation evolves in discrete-time steps. Such a population dynamics is described by a sequence {xn} that can bemodeled by the logistic difference equation

xnþ1 ¼ xn þ r0xn 1� xn

k

� �. ð1:2Þ

We can see that (1.2) is time discretization of Eq. (1.1) by the forward Euler scheme with step one.

779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved..1016/j.chaos.2005.10.081

rresponding author. Tel.: +86 21 54748556.ail addresses: xiaoliliu@sjtu.edu.cn (X. Liu), xiaodm@sjtu.edu.cn (D. Xiao).

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 81

One expects the deterministic models may provide a useful way of gaining sufficient understanding about the dynam-ics of the population of a single species. As it is well known, the dynamics of (1.1) is trivial, i.e. every non-negative solu-tion of (1.1) except the constant solution x � 0 tends to the other constant solution x � k as t!1 for all permissiblevalues of r0 and k. Hence, the population x(t) approaches the limit k as time evolves (cf. [3–5] and reference therein). Onthe other hand, the dynamics of (1.2) is more complex. It is remarkable that for (1.2), period-doubling phenomenon andthe onset of chaos in the sense of Li–Yorke [6] occur for some values of r0 [7,8]. Now Eq. (1.2) becomes a prototype forchaotic behavior of discrete dynamical systems well beyond the discipline of mathematical biology. It is also remarkablefrom a biological point of view that such a simple discrete model leads to unpredictable dynamic behaviors. This sug-gests the possibility that the governing laws of ecological systems may be relatively simple and therefore discoverable.May [9,10] have clearly documented the rich array of dynamic behavior possible in simple discrete-time models. Hence,the discrete version has been an important subject of study in diverse phenomenology or as an object interesting to ana-lyze by itself from the mathematical point of view [11–14].

The purpose of this paper is to analyze qualitatively the dynamical complexity of a discrete-time predator–preymodel, which can be regarded as a coupling perturbation of (1.2) in R2 or time discretization of a Lotka–Volterra typepredator–prey system [15] by Euler method. About the discrete-time predator–prey models, an early work was done byBeddington et al. [16]. From then, there has been a considerable amount of literatures on discrete-time predator–preymodels (e.g. see [17–21] and references therein). The basic forms of dynamics observed in their models are stable fixedpoints, periodic orbits and some random motions.

We consider a Lotka–Volterra type predator–prey system [3,15]

_x ¼ r0xð1� xkÞ � b0xy;

_y ¼ ð�d0 þ cxÞy;

�ð1:3Þ

where x(t) and y(t) denote prey and predator densities respectively, b0x is the predator functional response, which rep-resents the number of prey individuals consumed per unit area per unit time by an individual predator, c is the conver-sion efficiency of prey into predators, cxy is the predator numerical response, and d0 is the predator mortality rate. Inthe absence of predator (i.e. y � 0), this model reduces to (1.1).

Let us introduce scaled variables, X ¼ xk, Y ¼ b0y

ck , and s ¼ tk, system (1.3) is reduced to

dXds¼ r0kX ð1� X Þ � k2cXY ;

dYds¼ ð�d0k þ k2cX ÞY .

8>><>>:

For the sake of simplicity, we rewrite the system above as

_x ¼ rxð1� xÞ � bxy;

_y ¼ ð�d þ bxÞy;

�ð1:4Þ

where r, b and d are positive parameters, r = r0k, b = k2c and d = d0k. Applying the forward Euler scheme to system(1.4), we obtain the discrete-time predator–prey system as follows:

x! xþ d½rxð1� xÞ � bxy�;y ! y þ dð�d þ bxÞy;

�ð1:5Þ

where d is the step size.It is clear that system (1.5) can be regarded as a coupling perturbation of (1.2) in R2. From the point of view of biol-

ogy, we will focus on the dynamical behaviors of (1.5) in the closed first quadrant R2þ, and show that the dynamics of the

discrete time model (1.5) can produce a much richer set of patterns than those discovered in continuous-time model(1.3). More precisely, in this paper, we rigorously prove that (1.5) undergoes the flip bifurcation and the Hopf bifurca-tion by using center manifold theorem and bifurcation theory. Meanwhile, numerical simulations are presented notonly to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors. Theseresults reveal far richer dynamics of the discrete model compared with the continuous model.

This paper is organized as follows. In Section 2, we discuss the existence and stability of fixed points for system (1.5)in the closed first quadrant R2

þ. In Section 3, we show that there exist some values of parameters such that (1.5) under-goes the flip bifurcation and the Hopf bifurcation in the interior of R2

þ. In Section 4, we present the numerical simula-tions, which not only illustrate our results with the theoretical analysis, but also exhibit the complex dynamicalbehaviors such as the period-5, 6, 9, 10, 14, 18, 20, 25 orbits, cascade of period-doubling bifurcation in period-2, 4,

82 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

8, quasi-periodic orbits and the chaotic sets. The Lyapunov exponents are computed numerically to confirm further thedynamical behaviors. A brief discussion is given in Section 5.

2. The existence and stability of fixed points

In this section and through out the paper, from the point of view of biology, we consider the discrete-time model(1.5) in the closed first quadrant R2

þ of the (x,y) plane. We first discuss the existence of fixed points for (1.5), then studythe stability of the fixed point by the eigenvalues for the variational matrix of (1.5) at the fixed point.

It is clear that the fixed points of (1.5) satisfy the following equations:

x ¼ xþ d½rxð1� xÞ � bxy�;y ¼ y þ dð�d þ bxÞy.

�

By a simple computation, it is straightforward to obtain the following results:

Lemma 2.1

(i) For all parameter values, (1.5) has two fixed points, O(0,0) and A(1,0);

(ii) if b > d, then (1.5) has, additionally, a unique positive fixed point, B(x*, y*), where x� ¼ db ; y� ¼ rðb�dÞ

b2 .

Now we study the stability of these fixed points. Note that the local stability of a fixed point (x,y) is determined bythe modules of eigenvalues of the characteristic equation at the fixed point.

Let the vector function (x + d(rx(1 � x) � bxy),y + d(�d + bx)y)T be denoted by V(x,y). Then the variationalmatrix of (1.5) at a fixed point (x,y) is

DV ðx; yÞ ¼1þ rd� 2rdx� bdy �bdx

bdy 1� ddþ bdx

� �.

The characteristic equation of the variational matrix can be written as

k2 þ pðx; yÞkþ qðx; yÞ ¼ 0; ð2:1Þ

which is a quadratic equation with one variable, p(x,y) = �2 + dd � rd + (2rd � bd)x + bdy, and q(x,y) = b2d2-

xy + (1 + rd � 2rdx � bdy)(1 � dd + bdx).In order to study the modulus of eigenvalues of the characteristic equation (2.1) at the positive fixed point B(x*,y*),

we first give the following lemma, which can be easily proved by the relations between roots and coefficients of the qua-dratic equation.

Lemma 2.2. Let F(k) = k2 + Bk + C. Suppose that F(1) > 0, k1 and k2 are two roots of F(k) = 0. Then

(i) jk1j < 1 and jk2j < 1 if and only if F(�1) > 0 and C < 1;

(ii) jk1j < 1 and jk2j > 1 (or jk1j > 1 and jk2j < 1) if and only if F(�1) < 0;

(iii) jk1j > 1 and jk2j > 1 if and only if F(�1) > 0 and C > 1;

(iv) k1 = �1 and jk2j5 1 if and only if F(�1) = 0 and B 5 0,2;

(v) k1 and k2 are complex and jk1j = jk2j = 1 if and only if B2 � 4C < 0 and C = 1.

Let k1 and k2 be two roots of (2.1), which called eigenvalues of the fixed point (x,y). We recall some definitions oftopological types for a fixed point (x,y). (x,y) is called a sink if jk1j < 1 and jk2j < 1. A sink is locally asymptotic stable.(x,y) is called a source if jk1j > 1 and jk2j > 1. A source is locally unstable. (x,y) is called a saddle if jk1j > 1 and jk2j < 1(or jk1j < 1 and jk2j > 1). And (x,y) is called non-hyperbolic if either jk1j = 1 or jk2j = 1.

Substituting the coordinates of the fixed point O(0,0) for (x,y) of (2.1) and computing the eigenvalues of the fixedpoint O(0,0) straightforward, we can obtain the following proposition.

Proposition 2.3. The fixed point O(0,0) is a saddle if 0 < d < 2d, O(0,0) is a source if d > 2

d, and O(0,0) is non-hyperbolic if

d ¼ 2d.

We can see that when d ¼ 2d, one of the eigenvalues of the fixed point O(0,0) is �1 and the other is not one with

module. Thus, the flip (or period-doubling) bifurcation may occur when parameters vary in the neighborhood ofd ¼ 2

d. However, one can see the flip bifurcation can not occur for the original parameters of (1.5) by computation,and O(0,0) is degenerate with higher codimension.

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 83

The following propositions show the local dynamics of fixed point A(1,0) and positive fixed point B(x*,y*) fromLemma 2.2.

Proposition 2.4. There exist at least four different topological types of A(1,0) for all permissible values of parameters.

(i) A(1,0) is a sink if b < d and 0 < d < min 2r ;

2d�b

� �;

(ii) A(1,0) is a source if b < d and d > max 2r ;

2d�b

� �(or b > d and d > 2

r);

(iii) A(1,0) is not hyperbolic if either b = d, or d ¼ 2r or d ¼ 2

b�d;

(iv) A(1,0) is a saddle for the other values of parameters except those values in (i)–(iii).

We can easily see that one of the eigenvalues of fixed point A(1,0) is �1 and the other is neither 1 nor �1 if the term(iii) of Proposition 2.4 holds. And the conditions in the term (iii) of Proposition 2.4 imply all parameters locate in thefollowing set:

F A ¼ ðb; d; r; dÞ : r ¼ 2

d; b 6¼ d; d 6¼ 2

b� d; b > 0; d > 0; d > 0

� .

The fixed point A(1,0) can undergo flip bifurcation when parameters vary in the small neighborhood of FA, since whenparameters are in FA a center manifold of (1.5) at A(1,0) is y = 0 and (1.5) restricted to this center manifold is the lo-gistic model (1.2). Hence, in this case the predator becomes extinction and the prey undergoes the period-doublingbifurcation to chaos in the sense of Li-Yorke by choosing bifurcation parameter r.

Proposition 2.5. When b > d, system (1.5) has a unique positive fixed point B(x*, y*) and

(i) it is a sink if one of the following conditions holds:

(i.1) rd � 4b(b � d) P 0 and 0 < d <rd�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirdðrd�4bðb�dÞÞp

rdðb�dÞ ;

(i.2) rd � 4b(b � d) < 0 and 0 < d < 1b�d .

(ii) it is a source if one of the following conditions holds:

(ii.1) rd � 4b(b � d) P 0 and d >rdþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirdðrd�4bðb�dÞÞp

rdðb�dÞ ;

(ii.2) rd � 4b(b � d) < 0 and d > 1b�d.

(iii) it is not hyperbolic if one of the following conditions holds:

(iii.1) rd � 4b(b � d) P 0 and d ¼ rd�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirdðrd�4bðb�dÞÞp

rdðb�dÞ ;(iii.2) rd � 4b(b � d) < 0 and d ¼ 1

b�d .

From Lemma 2.2, we can easily see that one of the eigenvalues of the positive fixed point B(x*,y*) is �1 and theother is neither 1 nor �1 if the term (iii.1) of Proposition 2.5 holds. We rewrite the conditions in the term (iii.1) of Prop-osition 2.5 as the following sets:

F B1 ¼ ðb; d; r; dÞ : d ¼ rd �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirdðrd � 4bðb� dÞÞ

prdðb� dÞ ; b > d > 0; rd > 4bðb� dÞ; r > 0

( )

or

F B2 ¼ ðb; d; r; dÞ : d ¼ rd þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirdðrd � 4bðb� dÞÞ

prdðb� dÞ ; b > d > 0; rd > 4bðb� dÞ; r > 0

( ).

When the term (iii.2) of Proposition 2.5 holds, we can obtain that the eigenvalues of the positive fixed point B(x*,y*)are a pair of conjugate complex numbers with module one. The conditions in the term (iii.2) of Proposition 2.5 can bewritten as the following set:

H B ¼ ðb; d; r; dÞ : d ¼ 1

b� d; b > d; 4bðb� dÞ > rd; b > 0; d > 0; r > 0

� .

In the following section, we will study the flip bifurcation of the positive fixed point B(x*,y*) if parameters vary in the smallneighborhood of FB1 (or FB2), and the Hopf bifurcation of B(x*,y*) if parameters vary in the small neighborhood of HB.

84 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

3. Flip bifurcation and Hopf bifurcation

Based on the analysis in Section 2, we discuss the flip bifurcation and Hopf bifurcation of the positive fixed pointB(x*,y*) in this section. We choose parameter d as a bifurcation parameter to study the flip bifurcation and Hopf bifur-cation of B(x*,y*) by using center manifold theorem and bifurcation theory in [22,23].

We first discuss the flip bifurcation of (1.5) at B(x*,y*) when parameters vary in the small neighborhood of FB1. Thesimilar arguments can be applied to the other case FB2.

Taking parameters (b1,d1, r1,d1) arbitrarily from FB1, we consider system (1.5) with (b1,d1, r1,d1), which is describedby

x! xþ d1½r1xð1� xÞ � b1xy�;y ! y þ d1ð�d1 þ b1xÞy.

�ð3:1Þ

Then map (3.1) has a unique positive fixed point B(x*,y*), whose eigenvalues are k1 = �1, k2 = 3 � rx*d1 with jk2j5 1by Proposition 2.5, where x� ¼ d1

b1; y� ¼ r1ðb1�d1Þ

b21

.

Since (b1,d1, r1,d1) 2 FB1, d1 ¼r1d1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir1d1ðr1d1�4b1ðb1�d1ÞÞp

r1d1ðb1�d1Þ . Choosing d� as a bifurcation parameter, we consider a pertur-bation of (3.1) as follows:

x! xþ ðd1 þ d�Þ½r1xð1� xÞ � b1xy�;y ! y þ ðd1 þ d�Þð�d1 þ b1xÞy;

�ð3:2Þ

where jd*j � 1, which is a small perturbation parameter.Let u = x � x* and v = y � y*. Then we transform the fixed point B(x*,y*) of map (3.2) into the origin. We have

u

v

� �!

a11uþ a12vþ a13uvþ a14u2 þ b0ud�

þb2vd� þ b3uvd� þ b4u2d�

a21uþ vþ a23uvþ c1ud� þ c2uvd�

0B@

1CA; ð3:3Þ

where

a11 ¼ 1þ d1ðr1 � 2r1x� � b1y�Þ; a12 ¼ �b1d1x�; a13 ¼ �b1d1; a14 ¼ �r1d1;

b0 ¼ r1 � 2r1x� � b1y�; b2 ¼ �b1x�; b3 ¼ �b1; b4 ¼ �r1;

a21 ¼ b1d1y�; a23 ¼ b1d1; c1 ¼ b1y�; c2 ¼ b1.

We construct an invertible matrix

T ¼a12 a12

�1� a11 k2 � a11

� �;

and use the translation

u

v

� �¼ T

~x

~y

� �

for (3.3), then the map (3.3) becomes

~x

~y

� �!

�1 0

0 k2

� �~x

~y

� �þ

f ðu; v; d�Þgðu; v; d�Þ

� �; ð3:4Þ

where

f ðu; v; d�Þ ¼ ½a13ðk2 � a11Þ � a12a23�a12ðk2 þ 1Þ uvþ a14ðk2 � a11Þ

a12ðk2 þ 1Þ u2 þ ½b0ðk2 � a11Þ � a12c1�a12ðk2 þ 1Þ ud�

þ b2ðk2 � a11Þa12ðk2 þ 1Þ vd� þ ½b3ðk2 � a11Þ � a12c2�

a12ðk2 þ 1Þ uvd� þ b4ðk2 � a11Þa12ðk2 þ 1Þ u2d�;

gðu; v; d�Þ ¼ ½a13ð1þ a11Þ þ a12a23�a12ðk2 þ 1Þ uvþ a14ð1þ a11Þ

a12ðk2 þ 1Þ u2 þ ½b0ð1þ a11Þ þ a12c1�a12ðk2 þ 1Þ ud�

þ b2ð1þ a11Þa12ðk2 þ 1Þ vd

� þ ½b3ð1þ a11Þ þ a12c2�a12ðk2 þ 1Þ uvd� þ b4ð1þ a11Þ

a12ðk2 þ 1Þ u2d�;

uv ¼ �a12ð1þ a11Þ~x2 þ ½a12ðk2 � a11Þ � a12ð1þ a11Þ�~x~y þ a12ðk2 � a11Þ~y2;

u2 ¼ a212ð~x2 þ 2~x~y þ ~y2Þ.

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 85

Next, we determine the center manifold Wc(0,0) of (3.4) at the fixed point (0,0) in a small neighborhood of d* = 0.From the center manifold theorem, we know there exists a center manifold Wc(0,0), which can be approximately rep-resented as follows:

W cð0; 0Þ ¼ fð~x; ~yÞ : ~y ¼ a0d� þ a1~x

2 þ a2~xd� þ a3d

�2 þ Oððj~xj þ jd�jÞ3Þg;

where Oððj~xj þ jd�jÞ3Þ is a function with order at least three in their variables ð~x; d�Þ, and

a0 ¼ 0;

a1 ¼ð1þ a11Þ½a12a14 � a13ð1þ a11Þ � a12a23�

1� k22

;

a2 ¼�b0ð1þ a11Þ � c1a12

ð1þ k2Þ2þ b2ð1þ a11Þ2

a12ð1þ k2Þ2;

a3 ¼ 0.

Therefore, we consider the map which is map (3.4) restricted to the center manifold Wc(0,0)

f : ~x! �~xþ h1~x2 þ h2~xd

� þ h3~x2d� þ h4~xd

�2 þ h5~x3 þ Oððj~xj þ jd�jÞ4Þ; ð3:5Þ

where

h1 ¼1

k2 þ 1fa12a14ðk2 � a11Þ � ð1þ a11Þ½a13ðk2 � a11Þ � a12a23�g;

h2 ¼1

a12ðk2 þ 1Þ ½b0a12ðk2 � a11Þ � a212c1 � b2ðk2 � a11Þð1þ a11Þ�;

h3 ¼1

k2 þ 1fa2ðk2 � 1� 2a11Þ½a13ðk2 � a11Þ � a12a23� þ 2a2a12a14ðk2 � a11Þ þ a1½b0ðk2 � a11Þ � c1a12�

� ð1þ a11Þ½b3ðk2 � c2a12Þ� þ a12b4ðk2 � a11Þg þ1

a12ðk2 þ 1Þ a1b2ðk2 � a11Þ2;

h4 ¼1

k2 þ 1fa3ðk2 � 1� 2a11Þ½a13ðk2 � a11Þ � a12a23� þ 2a3a12a14ðk2 � a11Þ

þ a2½b0ðk2 � a11Þ � c1a12�g þa2b2

a12ðk2 þ 1Þ ðk2 � a11Þ2

h5 ¼a1

k2 þ 1fðk2 � 1� 2a11Þ½a13ðk2 � a11Þ � a12a23� þ 2a12a14ðk2 � a11Þg.

In order for map (3.5) to undergo a flip bifurcation, we require that two discriminatory quantities a1 and a2 are notzero, where

a1 ¼o2fo~xd�

þ 1

2

ofod�

o2f

o~x2

� �����ð0;0Þ

and

a2 ¼1

6

o3f

o~x3þ 1

2

o2f

o~x2

� �2 !�����

ð0;0Þ

.

From a simple calculation, we obtain

a1 ¼r1d1ðb1 � d1Þðd1Þ2 � 4b1

b1d1ð4� r1x�d1Þ6¼ 0

and

a2 ¼ h5 þ h21 ¼

ðb1d1Þ2

ð1þ k2Þ24� d1d1 2� d1r1d1

b1

� �� 2

� d1r1d1ð2þ d1d1Þ2

b1

( ).

From the above analysis and the theorem in [22], we have the following theorem.

Theorem 3.1. If a2 5 0, then map (3.2) undergoes a flip bifurcation at the fixed point B(x*, y*) when the parameter d*

varies in the small neighborhood of the origin . Moreover, if a2 > 0 (resp., a2 < 0), then the period-2 points that bifurcate

from B(x*, y*) are stable (resp., unstable).

Fig. 4.1. (a) Bifurcation diagram of map (1.5) with d covering [1.26,1.4], r = 2, b = 0.6, d = 0.5, the initial value is (0.83,0.55). (b)Maximum Lyapunov exponents corresponding to (a).

86 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

In Section 4 we will give some values of parameters such that a2 5 0, thus the flip bifurcation occurs as d varies (seeFig. 4.1).

Finally we discuss the Hopf bifurcation of B(x*,y*) if parameters vary in the small neighborhood of HB. Takingparameters (b2,d2, r2,d2) arbitrarily from HB, we consider system (1.5) with parameters (b2,d2, r2,d2), which is describedby

x! xþ d2½r2xð1� xÞ � b2xy�;y ! y þ d2ð�d2 þ b2xÞy.

�ð3:6Þ

Then map (3.6) has a unique positive fixed point B(x*,y*), where x� ¼ d2

b2, y� ¼ r2ðb2�d2Þ

b22

.

Note that the characteristic equation associated with the linearization of the map (3.6) at B(x*,y*) is given by

k2 þ pkþ q ¼ 0;

where

p ¼ �2þ d2r2d2

b2

;

q ¼ 1� d2r2d2

b2

þ r2d22d2ðb2 � d2Þ

b2

.

Since parameters (b2,d2, r2,d2) 2 HB, the eigenvalues of B(x*,y*) are a pair of complex conjugate numbers k , and �k withmodulus 1 by Proposition 2.5, where

k ¼ �p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 � 4q

p2

¼ 1þ�r2d2 þ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2d2ð4b2

2 � 4b2d2 � r2d2Þq

2b2ðb2 � d2Þ. ð3:7Þ

Now we consider a small perturbation of (3.6) by choosing the bifurcation parameter d as follows

x! xþ ðd2 þ dÞ½r2xð1� xÞ � b2xy�;y ! y þ ðd2 þ dÞð�d2 þ b2xÞy;

�ð3:8Þ

where jdj � 1 , which is a small parameter.Moving B(x*,y*) to the origin, let u = x � x* and let v = y � y* we have

u

v

� �!

uþ ðd2 þ dÞ½r2uð1� uÞ � 2x�r2u� b2x�v� b2uðvþ y�Þ�vþ ðd2 þ dÞb2uðvþ y�Þ

� �. ð3:9Þ

The characteristic equation associated with the linearization of the map (3.9) at (u,v) = (0,0) is given by

k2 þ pðdÞkþ qðdÞ ¼ 0;

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 87

where

pðdÞ ¼ �2þ r2d2ðd2 þ dÞb2

;

qðdÞ ¼ 1� r2d2ðd2 þ dÞb2

þ r2d2ðb2 � d2Þðd2 þ dÞ2

b2

.

Correspondingly, when d varies in a small neighborhood of d = 0 the roots of the characteristic equation are

k1;2 ¼�pðdÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðdÞ2 � 4qðdÞ

q2

¼ 1þ�r2d2ðd2 þ dÞ þ iðd2 þ dÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2d2ð4b2

2 � 4b2d2 � r2d2Þq

2b2

and there have

jk1;2j ¼ ðqðdÞÞ1=2; l ¼ djk1;2jdd

����d¼0

¼ r2d2

2b2

> 0.

In addition, it is required that when d = 0, km1;2 6¼ 1; m ¼ 1; 2; 3; 4; which is equivalent to p(0) 5 �2,0,1,2. Note that

(b2,d2, r2,d2) 2 HB. So 4b2

r2d2> 1

b2�d2> 0. Thus, p(0) 5 �2,2. We only need to require that p(0) 5 0,1, which leads to

1

b2 � d2

6¼ jb2

r2d2

; j ¼ 2; 3. ð3:10Þ

Therefore, the eigenvalues k1,2 of fixed point (0,0) of (3.9) do not lay in the intersection of the unit circle with the coor-dinate axes when d = 0 and (3.10) holds.

Next we study the normal form of (3.9) when d = 0.

Let a ¼ 1� r2d2

2b2ðb2�d2Þ; b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2d2ð4b2

2�4b2d2�r2d2Þp

2b2ðb2�d2Þ

T ¼0 1

b a

� �;

and use the translation

u

v

� �¼ T

~x

~y

� �;

the map (3.9) becomes

~x

~y

� �!

a �b

b a

� �~x

~y

� �þ

~f ð~x;~yÞ~gð~x;~yÞ

!; ð3:11Þ

where

~f ð~x; ~yÞ ¼ ðaþ 1Þd2

bb2~yðb~xþ a~yÞ þ ad2

br2~y

2;

~gð~x; ~yÞ ¼ �d2½b2~yðb~xþ a~yÞ þ r2~y2�

and

~f ~x~x ¼ 0; ~f ~y~y ¼2ad2ðb2aþ b2 þ r2Þ

b; ~f ~x~y ¼ b2ðaþ 1Þd2; ~f ~x~x~x ¼ ~f ~y~y~y ¼ ~f ~x~y~y ¼ ~f ~x~x~y ¼ 0;

~g~x~x ¼ 0; ~g~y~y ¼ �2d2ðr2 þ b2aÞ; ~g~x~y ¼ �b2d2b; ~g~x~x~x ¼ ~g~y~y~y ¼ ~g~x~y~y ¼ ~g~x~x~y ¼ 0.

In order for map (3.11) to undergo Hopf bifurcation, we require that the following discriminatory quantity a is notzero [23]:

a ¼ �Reð1� 2�kÞ�k2

1� kn11n20

" #� 1

2kn11k2 � kn02k2 þ Reð�kn21Þ; ð3:12Þ

88 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

where

0.52

0.53

0.54

0.55

0.56

0.57

0.17

0.18

0.19

0.2

0.21

0.22

0.23

y

y

n20 ¼1

8½f~x~x � f~y~y þ 2g~x~y þ iðg~x~x � g~y~y � 2f ~x~yÞ�;

n11 ¼1

4½f~x~x þ f~y~y þ iðg~x~x þ g~x~yÞ�;

n02 ¼1

8½f~x~x � f~y~y þ 2g~x~y þ iðg~x~x � g~y~y þ 2f ~x~yÞ�;

n21 ¼1

16½f~x~x~x þ f~x~y~y þ g~x~x~y þ g~y~y~y þ iðg~x~x~x þ g~x~y~y � f~x~x~y � f~y~y~yÞ�.

0.8 0.805 0.81 0.815 0.82 0.825 0.83 0.835 0.84 0.845 0.85 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.870.542

0.544

0.546

0.548

0.55

0.552

0.554

0.556

0.558

0.56

x

y

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.920.51

0.515

0.52

0.525

0.53

0.535

0.54

0.545

0.55

x

y

δ = 1.28 δ = 1.283 δ = 1.285

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3x

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.12

0.13

0.14

0.15

0.16

0.17

0.18

x

y

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.11

0.12

0.13

0.14

0.15

0.16

0.17

x

y

δ = 1.316 δ = 1.348 δ = 1.3499

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

x0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

x

y

δ = 1.36 δ = 1.38

Fig. 4.2. Phase portraits for various values of d corresponding to Fig. 4.1(a).

Fig. 4.3. (a) Bifurcation diagram of map (1.5) with d covering [0.5,0.95], r = 3, b = 3.5, d = 2, the initial value is (0.57143,0.36735). (b)Local amplification corresponding to (a).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

Max

imum

Lya

puno

v E

xpon

ent

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.940.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

Max

mum

Lyp

unov

Exp

onen

t

(a) (b)

Fig. 4.4. (a) Maximum Lyapunov exponent with d covering [0,0.95], r = 3, b = 3.5, d = 2, the initial value is (0.57143,0.36735). (b)Local amplification corresponding to (a).

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 89

After some manipulation, we obtain

a¼ 1

32ð1�aÞðb2�d2Þ2

��1�9aþ14a2þ12a3�16a4ð Þ 2ab2

2þ4a2b22þ2a3b2

2þ 1�a2ð Þ32b2ðb2� r2Þþ2ab2r2þ4a2b2r2þ2a3b2r2þ2a2r2

2

h i1�a2

8<:

þ 3�6a�12a2þ16a3� �

1�a2� �3

2b22þa 1�a2

� �12b2ðb2þab2þ r2Þ�2aðb2� r2Þðb2þb2aþ r2Þ

h i)

� 3a2ðb2þab2þ r2Þ2

16ð1�a2Þðb2�d2Þ2� ð1�aÞ2b2

2

32ðb2�d2Þ2�ðb2þ2ab2þ r2Þ2

16ðb2�d2Þ2.

From above analysis and the theorem in [23], we have the following theorem.

Theorem 3.2. If the condition (3.10) holds and a 5 0, then map (3.8) undergoes Hopf bifurcation at the fixed point

B(x*, y*) when the parameter d varies in the small neighborhood of the origin. Moreover, if a < 0 (resp., a > 0), then an

attracting (resp., repelling) invariant closed curve bifurcates from the fixed point for d > 0 (resp., d < 0).

Fig. 4.5. Phase portraits for various values of d corresponding to Fig. 4.3(a).

90 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

In Section 4 we will choose some values of parameters to show the process of Hopf bifurcation for map (3.8) inFig. 4.5 by numerical simulation.

Fig. 4.6. (a) Bifurcation diagram of map (1.5) with d covering [0.45,0.8], r = 3, b = 4, d = 2, the initial value is (0.501,0.374). (b) Localamplification corresponding to (a).

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 91

4. Numerical simulations

In this section, we present the bifurcation diagrams, phase portraits and Maximum Lyapunov exponents for system(1.5) to confirm the above theoretical analysis and show the new interesting complex dynamical behaviors by usingnumerical simulations. The bifurcation parameters are considered in the following three cases:

(1) Varying d in range 1.26 < d 6 1.4, and fixing b = 0.6, d = 0.5, r = 2.(2) Varying d in range 0.5 < d 6 0.95, and fixing b = 3.5, d = 2, r = 3.(3) Varying d in range 0.5 < d 6 0.8, and fixing b = 4, d = 2, r = 3.

For case (1). b = 0.6, d = 0.5, r = 2, based on Lemma 2.1, we know the map (1.5) has only one positive fixed point.After calculation for the positive fixed point of map (1.5), the flip bifurcation emerges from the fixed point 5

6; 5

9

� �at

d ¼ 10�ffiffiffiffiffi76p

with a1 = �1.55982, a2 = 0.307611 and ðb; d; r; dÞ ¼ ð0:6; 0:5; 2; 10�ffiffiffiffiffi76pÞ 2 F B1. It shows the correctness

of Theorem 3.1.From Fig. 4.1(a), we see that the fixed point is stable for d < 10�

ffiffiffiffiffi76p

; and loses its stability at the flip bifurcationparameter value d ¼ 10�

ffiffiffiffiffi76p

, we also observe that there is a cascade of period-doubling. The maximum Lyapunovexponents corresponding to Fig. 4.1(a) are computed in Fig. 4.1(b).

The phase portraits which are associated with Fig. 4.1(a) are disposed in Fig. 4.2. For d 2 (1.28,1.35), there are per-iod-2, 4, 8, 16 orbits. When d = 1.36, 1.38, we can see the chaotic sets. The maximum Lyapunov exponents correspond-ing to d = 1.36, 1.38 are larger than 0 that confirm the existence of the chaotic sets.

For case (2). b = 3.5, d = 2, r = 3, according to Lemma 2.1, we know the map (1.5) has only one positive fixed point.After calculation for the positive fixed point of map (1.5), the Hopf bifurcation emerges from the fixed point 4

7; 18

49

� �at

d ¼ 23

with a = �1.15344 and ðb; d; r; dÞ ¼ 72; 2; 3; 2

3

� �2 H B. It shows the correctness of Theorem 3.2.

From Fig. 4.3(a), we observe that the fixed point 47; 18

49

� �of map (1.5) is stable for d < 2

3; loses its stability at d ¼ 2

3, and

an invariant circle appears when the parameter d exceeds 23. From Fig. 4.3, we see that there are period-doubling phe-

nomenons. Fig. 4.3(b) is the local amplification for d 2 [0.78,0.873].The maximum Lyapunov exponents corresponding to Fig. 4.3(a) are calculated and plotted in Fig. 4.4 where we can

easily see that that the maximum Lyapunov exponents are negative for the parameter d 2 (0.5,0.8217), that is to say, thenon-chaotic region is bigger than the chaotic region (0.8217,0.95). For d 2 (0.8217,0.8515) some Lyapunov exponentsare bigger than 0, some are smaller than 0, so there exist stable fixed point or stable period windows in the chaoticregion. In general the positive Lyapunov exponent is considered to be one of the characteristics implying the existenceof chaos.

92 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

The phase portraits which are associated with Fig. 4.3(a) are disposed in Fig. 4.5, which clearly depicts the process ofhow a smooth invariant circle bifurcates from the stable fixed point 4

7; 18

49

� �. When d exceeds 2

3there appears a circle curve

enclosing the fixed point 47; 18

49

� �, and its radius becomes larger with respect to the growth of d. When d increases at cer-

tain values, for example, at d = 0.785, the circle disappears and a period-5 orbit appears, and some cascades of period-doubling bifurcations lead to chaos. From Fig. 4.5 we observe that there are period-5, period-10, period-20, period-9and quasi-periodic orbits.

For case (3). b = 4, d = 2, r = 3 and d is varying. We draw the bifurcation diagram in Fig. 4.6(a) with local ampli-fication in Fig. 4.6(b). The phase portraits of various d corresponding to Fig. 4.6(a) are plotted in Fig. 4.7. FromFig. 4.7, we see that the fixed point 1

2; 3

8

� �of map (1.5) is stable for d < 1

2; and loses its stability at d ¼ 1

2, an invariant

0.47 0.48 0.49 0.5 0.51 0.52 0.530.35

0.355

0.36

0.365

0.37

0.375

0.38

0.385

0.39

0.395

0.4

x

y

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

x

y

δ δ δ

δ δ δ

δ δ δ

= 0.501 = 0.6 = 0.67

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

= 0.68 = 0.71 = 0.72

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

= 0.722 = 0.723 = 0.73

Fig. 4.7. Phase portaits for various values of d corresponding to Fig. 4.6(a).

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

= 0 .735 δ = 0 .74 δ = 0 .76

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

δ

δ

= 0 .765 δ = 0 .77 δ = 0 .8

Fig. 4.7 (continued)

X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94 93

circle appears when the parameter d exceeds 12. There is an invariant circle for more large regions of d 2 (0.5,0.66). when

d increases, there are: period-6 orbits, period-25 orbits and attracting chaotic sets.

5. Discussion

It is well known that the dynamics of system (1.3) is trivial in the first quadrant for all parameters. More precisely,for some parameter values the system has no positive equilibria and the one of boundary equilibria, (k, 0), attracts allorbits of the system in the interior of the first quadrant; otherwise, the system has a unique positive equilibrium and thepositive equilibrium attracts all orbits of the system in the interior of the first quadrant. Thus, system (1.3) has no limitcycles for all parameter values. However, the discrete-time predator–prey model (1.5) has complex dynamics. In thispaper, we show that the unique positive fixed point of (1.5) can undergo flip bifurcation and Hopf bifurcation. More-over, system (1.5) displays much interesting dynamical behaviors, including period-5, 6, 9, 10, 14, 18, 20, 25 orbits,invariant cycle, cascade of period-doubling, quasi-periodic orbits and the chaotic sets, which implies that the predatorand prey can coexist in the stable period-n orbits and invariant cycle. These results reveal far richer dynamics of thediscrete model compared to the continuous model.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (No. 10231020) and Program forNew Century Excellent Talents in Universities of China.

94 X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94

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