Chaos, Solitons and Fractals 32 (2007) 80–94
www.elsevier.com/locate/chaos
Complex dynamic behaviors of a discrete-time
predator–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 of
period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richer
dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.
Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction
In population dynamics, there are two kinds of mathematical models: the continuous-time models described by differential equations or dynamical systems, and the discrete-time models described by difference equations, discrete
dynamical systems or iterative maps. The simplest continuous-time population model is the logistic differential equation
of a single species, first introduced by Verhulst [1] and later studied further by Pearl and Reed [2]:
x
x_ ¼ r0 x 1 ;
ð1:1Þ
k
where x(t) denotes the population of a single species at time t, k is the carrying capacity of the population, and r0 is the
intrinsic growth rate. Eq. (1.1) describes the growth rate of the population size of a single species. However, the population size of a single species may has a fixed interval between generations or possibly a fixed interval between measurements. For example, many species of insect have no overlap between successive generations, and thus their
population evolves in discrete-time steps. Such a population dynamics is described by a sequence {xn} that can be
modeled by the logistic difference equation
xn .
ð1:2Þ
xnþ1 ¼ xn þ r0 xn 1 k
We can see that (1.2) is time discretization of Eq. (1.1) by the forward Euler scheme with step one.
*
Corresponding author. Tel.: +86 21 54748556.
E-mail addresses: [email protected] (X. Liu), [email protected] (D. Xiao).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2005.10.081
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 dynamics of the population of a single species. As it is well known, the dynamics of (1.1) is trivial, i.e. every non-negative solution of (1.1) except the constant solution x 0 tends to the other constant solution x k as t ! 1 for all permissible
values of r0 and k. Hence, the population x(t) approaches the limit k as time evolves (cf. [3–5] and reference therein). On
the other hand, the dynamics of (1.2) is more complex. It is remarkable that for (1.2), period-doubling phenomenon and
the 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 for
chaotic behavior of discrete dynamical systems well beyond the discipline of mathematical biology. It is also remarkable
from a biological point of view that such a simple discrete model leads to unpredictable dynamic behaviors. This suggests 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 analyze 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–prey
model, which can be regarded as a coupling perturbation of (1.2) in R2 or time discretization of a Lotka–Volterra type
predator–prey system [15] by Euler method. About the discrete-time predator–prey models, an early work was done by
Beddington et al. [16]. From then, there has been a considerable amount of literatures on discrete-time predator–prey
models (e.g. see [17–21] and references therein). The basic forms of dynamics observed in their models are stable fixed
points, periodic orbits and some random motions.
We consider a Lotka–Volterra type predator–prey system [3,15]
x_ ¼ r0 xð1 kxÞ b0 xy;
ð1:3Þ
y_ ¼ ðd 0 þ cxÞy;
where x(t) and y(t) denote prey and predator densities respectively, b0x is the predator functional response, which represents the number of prey individuals consumed per unit area per unit time by an individual predator, c is the conversion efficiency of prey into predators, cxy is the predator numerical response, and d0 is the predator mortality rate. In
the absence of predator (i.e. y 0), this model reduces to (1.1).
Let us introduce scaled variables, X ¼ kx, Y ¼ bck0 y , and s ¼ kt , system (1.3) is reduced to
8
dX
>
>
¼ r0 kX ð1 X Þ k 2 cXY ;
<
ds
>
dY
>
:
¼ ðd 0 k þ k 2 cX ÞY .
ds
For the sake of simplicity, we rewrite the system above as
x_ ¼ rxð1 xÞ bxy;
y_ ¼ ðd 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 ! x þ d½rxð1 xÞ bxy;
ð1:5Þ
y ! y þ dðd þ bxÞy;
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 biology, 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 bifurcation by using center manifold theorem and bifurcation theory. Meanwhile, numerical simulations are presented not
only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors. These
results 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 R2þ . In Section 3, we show that there exist some values of parameters such that (1.5) undergoes the flip bifurcation and the Hopf bifurcation in the interior of R2þ . In Section 4, we present the numerical simulations, which not only illustrate our results with the theoretical analysis, but also exhibit the complex dynamical
behaviors 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 the
dynamical 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 R2þ of the (x, y) plane. We first discuss the existence of fixed points for (1.5), then study
the 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 x þ d½rxð1 xÞ bxy;
y ¼ y þ dðd 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 rðbdÞ
b2
Now we study the stability of these fixed points. Note that the local stability of a fixed point (x, y) is determined by
the 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 variational
matrix of (1.5) at a fixed point (x, y) is
1 þ rd 2rdx bdy
bdx
DV ðx; yÞ ¼
.
bdy
1 dd þ bdx
The characteristic equation of the variational matrix can be written as
k2 þ pðx; yÞk þ 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) = b2d2xy + (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 quadratic 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)
(ii)
(iii)
(iv)
(v)
jk1j < 1 and jk2j < 1 if and only if F(1) > 0 and C < 1;
jk1j < 1 and jk2j > 1 (or jk1j > 1 and jk2j < 1) if and only if F(1) < 0;
jk1j > 1 and jk2j > 1 if and only if F(1) > 0 and C > 1;
k1 = 1 and jk2j 5 1 if and only if F(1) = 0 and B 5 0, 2;
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 of
topological 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 fixed
point 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 < d2, O(0, 0) is a source if d > d2, and O(0, 0) is non-hyperbolic if
d ¼ d2.
We can see that when d ¼ d2, 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 of
d ¼ d2. 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*) from
Lemma 2.2.
Proposition 2.4. There exist at least four different topological types of A(1, 0) for all permissible values of parameters.
2 ;
(i) A(1, 0) is a sink if b < d and 0 < d < min 2r ; db
2 2 (ii) A(1, 0) is a source if b < d and d > max r ; db (or b > d and d > 2r );
2
;
(iii) A(1, 0) is not hyperbolic if either b = d, or d ¼ 2r or d ¼ bd
(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 the
following set:
2
2
F A ¼ ðb; d; r; dÞ : r ¼ ; b 6¼ d; d 6¼
; b > 0; d > 0; d > 0 .
d
bd
The fixed point A(1, 0) can undergo flip bifurcation when parameters vary in the small neighborhood of FA, since when
parameters 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 logistic model (1.2). Hence, in this case the predator becomes extinction and the prey undergoes the period-doubling
bifurcation 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:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rd rdðrd4bðbdÞÞ
;
(i.1) rd 4b(b d) P 0 and 0 < d <
rdðbdÞ
1
(i.2) rd 4b(b d) < 0 and 0 < d < bd
.
(ii) it is a source if one of the following conditions holds:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rdþ rdðrd4bðbdÞÞ
;
(ii.1) rd 4b(b d) P 0 and d >
rdðbdÞ
1
(ii.2) rd 4b(b d) < 0 and d > bd
.
(iii) it is not hyperbolic if one of the following conditions holds:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rd rdðrd4bðbdÞÞ
(iii.1) rd 4b(b d) P 0 and d ¼
;
rdðbdÞ
1
(iii.2) rd 4b(b d) < 0 and d ¼ bd
.
From Lemma 2.2, we can easily see that one of the eigenvalues of the positive fixed point B(x*, y*) is 1 and the
other 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 Proposition 2.5 as the following sets:
(
)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rd rdðrd 4bðb dÞÞ
F B1 ¼ ðb; d; r; dÞ : d ¼
; b > d > 0; rd > 4bðb dÞ; r > 0
rdðb dÞ
or
(
F B2 ¼
ðb; d; r; dÞ : d ¼
rd þ
)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rdðrd 4bðb dÞÞ
; b > d > 0; rd > 4bðb dÞ; r > 0 .
rdðb dÞ
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 be
written as the following set:
1
; b > d; 4bðb dÞ > rd; b > 0; d > 0; r > 0 .
H B ¼ ðb; d; r; dÞ : d ¼
bd
In the following section, we will study the flip bifurcation of the positive fixed point B(x*, y*) if parameters vary in the small
neighborhood of FB1 (or FB2), and the Hopf bifurcation of B(x*, y*) if parameters vary in the small neighborhood of HB.
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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 point
B(x*, y*) in this section. We choose parameter d as a bifurcation parameter to study the flip bifurcation and Hopf bifurcation 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. The
similar 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 described
by
x ! x þ d1 ½r1 xð1 xÞ b1 xy;
ð3:1Þ
y ! y þ d1 ðd 1 þ b1 xÞy.
Then map (3.1) has a unique positive fixed point B(x*, y*), whose eigenvalues are k1 = 1, k2 = 3 rx*d1 with jk2j 5 1
by Proposition 2.5, where x ¼ db11 ; y ¼ r1 ðbb1 2d 1 Þ.
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r1 d 1 r1 d 1 ðr1 d 1 4b1 ðb1 d 1 ÞÞ
Since (b1, d1, r1, d1) 2 FB1, d1 ¼
. Choosing d as a bifurcation parameter, we consider a perturr1 d 1 ðb1 d 1 Þ
bation of (3.1) as follows:
x ! x þ ðd1 þ d Þ½r1 xð1 xÞ b1 xy;
ð3:2Þ
y ! y þ ðd1 þ d Þðd 1 þ b1 xÞy;
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
0
1
a11 u þ a12 v þ a13 uv þ a14 u2 þ b0 ud
u
B
C
ð3:3Þ
!@
þb2 vd þ b3 uvd þ b4 u2 d
A;
v
a21 u þ v þ a23 uv þ c1 ud þ c2 uvd
where
a11 ¼ 1 þ d1 ðr1 2r1 x b1 y Þ; a12 ¼ b1 d1 x ; a13 ¼ b1 d1 ;
b0 ¼ r1 2r1 x b1 y ; b2 ¼ b1 x ; b3 ¼ b1 ; b4 ¼ r1 ;
a21 ¼ b1 d1 y ; a23 ¼ b1 d1 ; c1 ¼ b1 y ; c2 ¼ b1 .
a14 ¼ r1 d1 ;
We construct an invertible matrix
a12
a12
;
T ¼
1 a11 k2 a11
and use the translation
~x
u
¼T
~y
v
for (3.3), then the map (3.3) becomes
~x
~x
1 0
f ðu; v; d Þ
!
þ
;
~y
~y
0 k2
gðu; v; d Þ
where
½a13 ðk2 a11 Þ a12 a23 a14 ðk2 a11 Þ 2 ½b0 ðk2 a11 Þ a12 c1 uv þ
u þ
ud
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
b2 ðk2 a11 Þ ½b3 ðk2 a11 Þ a12 c2 b4 ðk2 a11 Þ 2 vd þ
uvd þ
ud;
þ
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
½a13 ð1 þ a11 Þ þ a12 a23 a14 ð1 þ a11 Þ 2 ½b0 ð1 þ a11 Þ þ a12 c1 uv þ
u þ
ud
gðu; v; d Þ ¼
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
b2 ð1 þ a11 Þ ½b3 ð1 þ a11 Þ þ a12 c2 b4 ð1 þ a11 Þ 2 vd þ
uvd þ
ud;
þ
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
a12 ðk2 þ 1Þ
uv ¼ a12 ð1 þ a11 Þ~x2 þ ½a12 ðk2 a11 Þ a12 ð1 þ a11 Þ~x~y þ a12 ðk2 a11 Þ~y 2 ;
u2 ¼ a212 ð~x2 þ 2~x~y þ ~y 2 Þ.
f ðu; v; d Þ ¼
ð3:4Þ
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 represented as follows:
W c ð0; 0Þ ¼ fð~x; ~y Þ : ~y ¼ a0 d þ a1~x2 þ a2~xd þ a3 d2 þ Oððj~xj þ jd jÞ3 Þg;
where Oððj~xj þ jd jÞ3 Þ is a function with order at least three in their variables ð~x; d Þ, and
a0 ¼ 0;
ð1 þ a11 Þ½a12 a14 a13 ð1 þ a11 Þ a12 a23 a1 ¼
;
1 k22
a2 ¼
b0 ð1 þ a11 Þ c1 a12
ð1 þ k2 Þ2
þ
b2 ð1 þ a11 Þ2
a12 ð1 þ k2 Þ2
;
a3 ¼ 0.
Therefore, we consider the map which is map (3.4) restricted to the center manifold Wc(0, 0)
f : ~x ! ~x þ h1~x2 þ h2~xd þ h3~x2 d þ h4~xd2 þ h5~x3 þ Oððj~xj þ jd jÞ4 Þ;
ð3:5Þ
where
1
fa12 a14 ðk2 a11 Þ ð1 þ a11 Þ½a13 ðk2 a11 Þ a12 a23 g;
k2 þ 1
1
½b0 a12 ðk2 a11 Þ a212 c1 b2 ðk2 a11 Þð1 þ a11 Þ;
¼
a12 ðk2 þ 1Þ
1
fa2 ðk2 1 2a11 Þ½a13 ðk2 a11 Þ a12 a23 þ 2a2 a12 a14 ðk2 a11 Þ þ a1 ½b0 ðk2 a11 Þ c1 a12 ¼
k2 þ 1
1
a1 b2 ðk2 a11 Þ2 ;
ð1 þ a11 Þ½b3 ðk2 c2 a12 Þ þ a12 b4 ðk2 a11 Þg þ
a12 ðk2 þ 1Þ
1
fa3 ðk2 1 2a11 Þ½a13 ðk2 a11 Þ a12 a23 þ 2a3 a12 a14 ðk2 a11 Þ
¼
k2 þ 1
a2 b2
þ a2 ½b0 ðk2 a11 Þ c1 a12 g þ
ðk2 a11 Þ2
a12 ðk2 þ 1Þ
a1
fðk2 1 2a11 Þ½a13 ðk2 a11 Þ a12 a23 þ 2a12 a14 ðk2 a11 Þg.
¼
k2 þ 1
h1 ¼
h2
h3
h4
h5
In order for map (3.5) to undergo a flip bifurcation, we require that two discriminatory quantities a1 and a2 are not
zero, where
2
of
1 of o2 f a1 ¼
þ
2 o~xd
2 od o~x
ð0;0Þ
and
a2 ¼
2 2 !
1 o3 f
1of
þ
6 o~x3
2 o~x2
.
ð0;0Þ
From a simple calculation, we obtain
a1 ¼
r1 d 1 ðb1 d 1 Þðd1 Þ2 4b1
6¼ 0
b1 d1 ð4 r1 x d1 Þ
and
a2 ¼ h5 þ
h21
¼
ðb1 d1 Þ2
ð1 þ k2 Þ2
(
)
2
d 1 r1 d1
d 1 r1 d1 ð2 þ d 1 d1 Þ2
4 d 1 d1 2 .
b1
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).
86
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
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).
In Section 4 we will give some values of parameters such that a2 5 0, thus the flip bifurcation occurs as d varies (see
Fig. 4.1).
Finally we discuss the Hopf bifurcation of B(x*, y*) if parameters vary in the small neighborhood of HB. Taking
parameters (b2, d2, r2, d2) arbitrarily from HB, we consider system (1.5) with parameters (b2, d2, r2, d2), which is described
by
x ! x þ d2 ½r2 xð1 xÞ b2 xy;
ð3:6Þ
y ! y þ d2 ðd 2 þ b2 xÞy.
Then map (3.6) has a unique positive fixed point B(x*, y*), where x ¼ db22 , y ¼ r2 ðbb2 2d 2 Þ.
2
Note that the characteristic equation associated with the linearization of the map (3.6) at B(x*, y*) is given by
k2 þ pk þ q ¼ 0;
where
d2 r2 d 2
;
b2
d2 r2 d 2 r2 d22 d 2 ðb2 d 2 Þ
q¼1
þ
.
b2
b2
p ¼ 2 þ
k with
Since parameters (b2, d2, r2, d2) 2 HB, the eigenvalues of B(x*, y*) are a pair of complex conjugate numbers k , and modulus 1 by Proposition 2.5, where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2 d 2 þ i r2 d 2 ð4b22 4b2 d 2 r2 d 2 Þ
p þ p2 4q
¼1þ
.
ð3:7Þ
k¼
2b2 ðb2 d 2 Þ
2
Now we consider a small perturbation of (3.6) by choosing the bifurcation parameter d as follows
x ! x þ ðd2 þ dÞ½r2 xð1 xÞ b2 xy;
y ! y þ ðd2 þ dÞðd 2 þ b2 xÞy;
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
u þ ðd2 þ dÞ½r2 uð1 uÞ 2x r2 u b2 x v b2 uðv þ y Þ
!
.
v
v þ ðd2 þ dÞb2 uðv þ y Þ
The characteristic equation associated with the linearization of the map (3.9) at (u, v) = (0, 0) is given by
k2 þ pðdÞk þ qðdÞ ¼ 0;
ð3:8Þ
ð3:9Þ
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
87
where
pðdÞ ¼ 2 þ
qðdÞ ¼ 1 r2 d 2 ðd2 þ dÞ
;
b2
r2 d 2 ðd2 þ dÞ r2 d 2 ðb2 d 2 Þðd2 þ dÞ2
þ
.
b2
b2
Correspondingly, when d varies in a small neighborhood of d = 0 the roots of the characteristic equation are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2 d 2 ðd2 þ dÞ þ iðd2 þ dÞ r2 d 2 ð4b22 4b2 d 2 r2 d 2 Þ
pðdÞ pðdÞ2 4qðdÞ
k1;2 ¼
¼1þ
2b2
2
and there have
jk1;2 j ¼ ðqðdÞÞ1=2 ;
l¼
djk1;2 j
r2 d 2
¼
> 0.
dd d¼0
2b2
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
1
2
(b2, d2, r2, d2) 2 HB. So r4b
> b2 d
> 0. Thus, p(0) 5 2, 2. We only need to require that p(0) 5 0, 1, which leads to
2 d2
2
1
jb
6¼ 2 ;
b2 d 2 r2 d 2
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 coordinate axes when d = 0 and (3.10) holds.
Next we study the normal form of (3.9) when d = 0.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
r2 d 2 ð4b22 4b2 d 2 r2 d 2 Þ
r2 d 2
Let a ¼ 1 2b2 ðb
;
b
¼
2b2 ðb2 d 2 Þ
2 d 2 Þ
0 1
T ¼
;
b a
and use the translation
~x
u
¼T
;
~y
v
the map (3.9) becomes
~x
a
!
~y
b
b
a
~x
þ
~y
!
f~ ð~x; ~y Þ
;
g~ð~x; ~y Þ
ð3:11Þ
where
ða þ 1Þd2
ad2 2
b2 ~y ðb~x þ a~y Þ þ
r2 ~y ;
f~ ð~x; ~y Þ ¼
b
b
2
g~ð~x; ~y Þ ¼ d2 ½b2 ~y ðb~x þ a~y Þ þ r2 ~y and
f~~x~x ¼ 0;
g~~x~x ¼ 0;
2ad2 ðb2 a þ b2 þ r2 Þ
f~ ~y~y ¼
; f~~x~y ¼ b2 ða þ 1Þd2 ; f~~x~x~x ¼ f~ ~y~y~y ¼ f~~x~y~y ¼ f~~x~x~y ¼ 0;
b
g~~y~y ¼ 2d2 ðr2 þ b2 aÞ; g~~x~y ¼ b2 d2 b; 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 not
zero [23]:
"
#
2
ð1 2kÞk
1
21 Þ;
n11 n20 kn11 k2 kn02 k2 þ Reðkn
a ¼ Re
ð3:12Þ
2
1k
88
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
where
1
n20 ¼ ½f~x~x f~y~y þ 2g~x~y þ iðg~x~x g~y~y 2f ~x~y Þ;
8
1
n11 ¼ ½f~x~x þ f~y~y þ iðg~x~x þ g~x~y Þ;
4
1
n02 ¼ ½f~x~x f~y~y þ 2g~x~y þ iðg~x~x g~y~y þ 2f ~x~y Þ;
8
1
n21 ¼ ½f~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~y Þ.
16
0.56
0.55
0.558
0.545
0.57
0.556
0.54
0.554
0.56
0.535
y
0.55
y
0.552
0.53
0.55
0.525
0.548
0.54
0.52
0.546
0.53
0.52
0.515
0.544
0.542
0.79
0.8 0.805 0.81 0.815 0.82 0.825 0.83 0.835 0.84 0.845 0.85
0.8
0.81
0.82
δ = 1.28
0.83
x
0.84
0.85
0.86
0.51
0.74 0.76 0.78
0.87
0.8
δ = 1.283
0.82 0.84 0.86 0.88
x
0.9
0.92
0.8
1.2
1.3
δ = 1.285
0.18
0.17
0.17
0.16
0.16
0.15
0.15
0.14
0.23
y
0.2
0.19
0.14
0.13
0.18
0.13
0.12
0.17
0.5
0.6
0.7
0.8
0.9
x
1
1.1
1.2
0.12
0.4
1.3
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0.11
0.4
1.3
0.5
0.6
0.7
x
δ = 1.316
δ = 1.348
0.16
0.18
0.15
0.17
x
0.9
1
1.1
δ = 1.3499
0.16
0.14
0.15
0.13
0.14
y
0.12
y
y
0.21
y
0.22
0.13
0.11
0.12
0.1
0.11
0.09
0.1
0.08
0.09
0.07
0.4
0.5
0.6
0.7
0.8
0.9
x
δ = 1.36
1
1.1
1.2
1.3
0.08
0.3
0.4
0.5
0.6
0.7
0.8
x
0.9
δ = 1.38
Fig. 4.2. Phase portraits for various values of d corresponding to Fig. 4.1(a).
1
1.1
1.2
1.3
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
89
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).
0.25
0.25
0.2
0.15
Maxmum Lypunov Exponent
Maximum Lyapunov Exponent
0.2
0.1
0.05
0
0.05
0.1
0.15
0.15
0.1
0.05
0
0.05
0.1
0.2
0.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.15
0.8
1
0.82
0.84
(a)
0.86
0.88
0.9
0.92
0.94
(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).
After some manipulation, we obtain
a¼
1
32ð1 aÞðb2 d 2 Þ2
h
i
8
3
<ð1 9a þ 14a2 þ 12a3 16a4 Þ 2ab22 þ 4a2 b22 þ 2a3 b22 þ ð1 a2 Þ2 b2 ðb2 r2 Þ þ 2ab2 r2 þ 4a2 b2 r2 þ 2a3 b2 r2 þ 2a2 r22
:
1 a2
)
i
h
3
1
2
3
2 2 2
2 2
1 a b2 þ a 1 a b2 ðb2 þ ab2 þ r2 Þ 2aðb2 r2 Þðb2 þ b2 a þ r2 Þ
þ 3 6a 12a þ 16a
3a2 ðb2 þ ab2 þ r2 Þ2
ð1 aÞ2 b22
16ð1 a2 Þðb2 d 2 Þ
32ðb2 d 2 Þ
2
2
ðb2 þ 2ab2 þ r2 Þ2
16ðb2 d 2 Þ2
.
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).
90
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
Fig. 4.5. Phase portraits for various values of d corresponding to Fig. 4.3(a).
In Section 4 we will choose some values of parameters to show the process of Hopf bifurcation for map (3.8) in
Fig. 4.5 by numerical simulation.
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
91
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) Local
amplification corresponding to (a).
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 using
numerical 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
5 point.
After calculation
for
the
positive
fixed
point
of
map
(1.5),
the
flip
bifurcation
emerges
from
the
fixed
point
; 5 at
6 9
pffiffiffiffiffi
pffiffiffiffiffi
d ¼ 10 76 with a1 = 1.55982, a2 = 0.307611 and ðb; d; r; dÞ ¼ ð0:6; 0:5; 2; 10 76Þ 2 F B1 . It shows the correctness
of Theorem 3.1.
pffiffiffiffiffi
From Fig. 4.1(a), we seepthat
ffiffiffiffiffi the fixed point is stable for d < 10 76; and loses its stability at the flip bifurcation
parameter value d ¼ 10 76, we also observe that there is a cascade of period-doubling. The maximum Lyapunov
exponents 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 period-2, 4, 8, 16 orbits. When d = 1.36, 1.38, we can see the chaotic sets. The maximum Lyapunov exponents corresponding 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
4 point.
After calculation for the positive fixed point
of
map
(1.5),
the
Hopf
bifurcation
emerges
from
the
fixed
point
; 18 at
7 49
d ¼ 23 with a = 1.15344 and ðb; d; r; dÞ ¼ 72 ; 2; 3; 23 2 HB . It shows the correctness of Theorem 3.2.
From Fig. 4.3(a), we observe that the fixed point 47 ; 18
of map (1.5) is stable for d < 23 ; loses its stability at d ¼ 23, and
49
an invariant circle appears when the parameter d exceeds 23. From Fig. 4.3, we see that there are period-doubling phenomenons. 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, the
non-chaotic region is bigger than the chaotic region (0.8217, 0.95). For d 2 (0.8217, 0.8515) some Lyapunov exponents
are bigger than 0, some are smaller than 0, so there exist stable fixed point or stable period windows in the chaotic
region. In general the positive Lyapunov exponent is considered to be one of the characteristics implying the existence
of 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
depicts the process of
4 18in Fig. 4.5, which clearly
2
.
When
d
exceeds
how a smooth invariant circle
bifurcates
from
the
stable
fixed
point
;
there
appears
a circle curve
7 49
3
enclosing the fixed point 47 ; 18
,
and
its
radius
becomes
larger
with
respect
to
the
growth
of
d.
When
d
increases
at cer49
tain values, for example, at d = 0.785, the circle disappears and a period-5 orbit appears, and some cascades of perioddoubling bifurcations lead to chaos. From Fig. 4.5 we observe that there are period-5, period-10, period-20, period-9
and 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 amplification in Fig. 4.6(b). The phase portraits
of various d corresponding to Fig. 4.6(a) are plotted in Fig. 4.7. From
Fig. 4.7, we see that the fixed point 12 ; 38 of map (1.5) is stable for d < 12 ; and loses its stability at d ¼ 12, an invariant
0.4
0.55
0.395
0.6
0.55
0.5
0.39
0.5
0.45
0.385
0.45
0.4
0.37
0.4
0.35
y
0.375
y
y
0.38
0.35
0.3
0.3
0.365
0.25
0.25
0.36
0.2
0.2
0.355
0.35
0.47
0.48
0.49
0.5
x
0.51
0.52
0.15
0.15
0.3
0.53
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
1
0.3
0.4
0.5
x
δ = 0.501
δ = 0.6
0.6
0.6
0.55
0.55
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.6
x
0.7
0.6
x
0.7
0.8
0.9
1
0.8
0.9
1
δ = 0.67
0.7
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.5
0.4
y
y
y
0.6
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
0.8
0.9
0.1
0.2
1
0.3
0.4
δ = 0.68
0.5
0.6
x
0.7
0.8
0.9
0
0.2
1
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0.4
0.5
0.6
0.7
x
0.8
δ = 0.722
0.5
y
0.6
y
0.7
y
0.7
0.3
0.4
δ = 0.72
0.7
0
0.2
0.3
δ = 0.71
0.9
1
1.1
1.2
0
0.2
0.3
0.4
0.5
0.6
0.7
x
0.8
0.9
1
1.1
1.2
0
0.2
0.3
0.4
0.5
δ = 0.723
Fig. 4.7. Phase portaits for various values of d corresponding to Fig. 4.6(a).
0.6
0.7
x
0.8
δ = 0.73
0.9
1
1.1
1.2
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
0.7
0.7
0.7
0.6
0.5
0.5
0.4
0.4
0.6
0.5
0.4
y
y
0.6
93
0.3
0.3
0.2
0.2
0.1
0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0
0.2
1.2
0.3
0.2
0.1
0.3
0.4
0.5
0.6
0.7
x
0.8
0.9
1
1.1
0
0.2
1.2
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0.5
0.6
0.7
x
0.8
δ = 0 .765
0.9
1
1.1
1.2
0.6
0.7
x
0.8
0.9
1
1.1
1.2
0.8
0.9
1
1.1
1.2
y
0.6
y
0.6
y
0.7
0.4
0.5
δ = 0 .76
0.7
0.3
0.4
δ = 0 .74
δ = 0 .735
0.7
0
0.2
0.3
0
0.2
0.3
0.4
0.5
0.6
0.7
x
0.8
0.9
δ = 0 .77
1
1.1
1.2
0
0.2
0.3
0.4
0.5
0.6
0.7
x
δ = 0 .8
Fig. 4.7 (continued)
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 all
orbits of the system in the interior of the first quadrant; otherwise, the system has a unique positive equilibrium and the
positive equilibrium attracts all orbits of the system in the interior of the first quadrant. Thus, system (1.3) has no limit
cycles for all parameter values. However, the discrete-time predator–prey model (1.5) has complex dynamics. In this
paper, we show that the unique positive fixed point of (1.5) can undergo flip bifurcation and Hopf bifurcation. Moreover, 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 predator
and prey can coexist in the stable period-n orbits and invariant cycle. These results reveal far richer dynamics of the
discrete model compared to the continuous model.
Acknowledgments
This work was supported by the National Natural Science Foundations of China (No. 10231020) and Program for
New Century Excellent Talents in Universities of China.
94
X. Liu, D. Xiao / Chaos, Solitons and Fractals 32 (2007) 80–94
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