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COMPLEX DYNAMIC BEHAVIORS OF A DISCRETE-TIME PREDATOR-PREY SYSTEM∗

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when some certain conditions are satisfied. 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-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.


Introduction
Predator-prey interaction is always a key issue in mathematical modelling of ecological processes. Considerable progress has been made since the famous Lotka-Volterra predator-prey model was proposed. In recent years, the study of the complex dynamics of the predator-prey models, including aspects as stability, periodic solutions, bifurcations and chaotic behavior, has drawn much attention to many excellent researchers [4-7, 11, 13-15, 17, 25].
The well known predator-prey model due to the mathematician Volterra [26] has the following form:   ẋ = ax − bxy, y = −cy + dxy, where x and y represent the number of a prey and a predator, respectively, and a, b, c, d are positive parameters. This model considers that, in the absence of predator the number of prey grows exponentially and in the absence of a prey population the number of predator decreases exponentially. This model was changed by Martelli [23] who considered some harvesting activities as: The author obtained the result that a moderate harvesting activity would favor the prey population. There are also many other predator-prey models of various types that have been extensively investigated and some of the relevant work may be found in [6,14,15,17]. These researches dealing with specific interactions have mainly focused on continuous predator-prey models with two variables. However, discrete-time models are widely used to understand the complex problem of the competition between two species. For example, the predator-prey models with age-structure for predator are studied in [19], in which they investigated the dynamical complexities including quasi-periodic attractors and strange attractors by using numerical analysis. Recently, bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type are investigated in [13]. They gave the conditions under which the system undergoes flip bifurcation and Hopf bifurcation by using the center manifold theorem and the bifurcation theory. The dynamical behaviors including the periodic solutions, bifurcations, chaos for the predator-prey systems as discrete-time models are also investigated in [4,5,7,11,25].
It is noteworthy to mention that in [8], Danca et al. considered the model (1.1) with the prey's growth be governed by a logistic map. The authors investigated local stability conditions of the fixed points in this model, determined the domains of the values of parameters for which the system has stationary states or chaotic behavior. The analysis indicated that the system undergoes Hopf bifurcation. But they only claimed numerical evidence of such bifurcation and chaotic behavior, and they did not theoretically prove the existence of them. Therefore, in this paper we will theoretically prove the existence of bifurcations. And we will also show the existence of Marotto's chaos.
In this paper, we consider the following discrete-time predator-prey system in [8] F : where x and y represent population densities of a prey and a predator, respectively, at time t, and r, b, d are positive parameters. Here, rx(1 − x) stands for the rate of increase of the prey population in the absence of predators, while the terms −bxy and dxy describe the predator-prey encounters which are favorable to predators and fatal to prey, where b and d are the predation parameters. Our main motivation in this paper is to investigate the system (1.1) in detail. Here, we derive the conditions of existence for flip bifurcation and Hopf bifurcation by using bifurcation theory and center manifold theorem [12,27]. The existence of chaos in the sense of Marotto [20,21] is proved by using both analytical and numerical methods. Numerical simulations, including bifurcation diagrams, phase portraits, maximum Lyapunov exponents and fractal dimension [1,2,16,24], are used to verify the theoretical analysis and to display new and interesting dynamical behaviors of the system (1.1). More specifically, this paper presents the period- 6,7,8,10,14,18,24,36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, the chaotic sets, nice chaotic behaviors which appear and disappear suddenly, interior crisis phenomena, and the new nice types of five and six coexisting chaotic attractors. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable fixed point. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results are interesting in mathematics and biology. Moreover, these results could be useful when the local and global stabilities in discrete-time predator-prey systems are concerned.
This paper is organized as follows: In section 2, we show the existence and stability of fixed points. In section 3, the sufficient conditions on the existence of codimension-one bifurcations, including flip bifurcation and Hopf bifurcations are obtained. In section 4, conditions on the existence of Marotto's chaos are given. In section 5, numerical simulation results are presented to support the theoretical analysis and they exhibit new and rich dynamical behaviors. In section 6, chaos is controlled to an unstable fixed point using the feedback control method. A brief conclusion is given in section 7.

Existence and Stability of Fixed Points
It is clear that the fixed points of system (1.1) satisfy the following equations: By a simple analysis, it is easy to see that the system (1.1) has one extinction fixed point (0, 0), one exclusion fixed point ( r−1 r , 0), and one coexistence(positive) fixed point (x * , y * ) = ( 1 d , rd−r−d bd ). Now we study the stability of these fixed points. For the fixed point (0, 0), the corresponding characteristic equation is λ 2 −rλ = 0 and its roots are λ 1 = 0, λ 2 = r that means (0, 0) is asymptotically stable when 0 < r < 1 and it is unstable when r > 1.
Below we will prove that when r = 3 the exclusion fixed point ( r−1 r , 0) is asymptotically stable and when d = r r−1 it is unstable by using center manifold theory. Let u = x − r−1 r and v = y in (1.1), we have Now we consider the first case, i.e., r = 3 and 0 < d < 3 2 . The system (2 We construct an invertible matrix then the map (2.2) can be written as: By the approximate computation for center manifold, we obtainα = 0 andβ = 0. Hence h(X) = 0, on the center manifold Y = 0, the new mapf is give bŷ Some computations show that the Schwarzian derivative of this map at X = 0 is S(f (0)) = −54 < 0. Hence, by [10], the exclusion fixed point ( r−1 r , 0) is asymptotically stable.
Next we consider the second case, i.e., 1 < r < 3 and d = r r−1 . The system We construct an invertible matrix and use the translation By the approximate computation for center manifold, we obtainα = − b 2 r(r−1) 2 and Computations show thatf (0) = 1 andf (0) = − 2b r−1 < 0. Hence, by [10], the exclusion fixed point ( r−1 r , 0) is unstable. More precisely, it is a semi-stable fixed point from the right. Therefore, ( r−1 r , 0) is asymptotically stable when 1 < r ≤ 3 and 0 < d < r r−1 . Finally, we consider the coexistence fixed point (x * , y * ) = ( 1 d , rd−r−d bd ) for d > r r−1 (r > 1). The Jacobian evaluated at the fixed point (x * , y * ) is given by and the characteristic equation of the Jacobian matrix J * can be written as: According to the Jury conditions [22], in order to find the asymptotically stable region of (x * , y * ), we need to find the region that satisfies the following conditions: , then from the relations P * (1) > 0, P * (−1) > 0 and detJ * < 1, we have that We now summarize the above analysis in the following result.
Proposition 2.1. For the predator-prey system (1.1), the following statements are true: (1) (0, 0) is asymptotically stable if 0 < r < 1; is asymptotically stable if and only if one of the following conditions holds:

Bifurcations
In this section, we mainly focus on the flip bifurcation and Hopf bifurcation of the coexistence fixed point (x * , y * ). We choose parameter d as a bifurcation parameter for analyzing the flip bifurcation and Hopf bifurcation of (x * , y * ) by using the center manifold theorem and bifurcation theory of [12,27]. First we consider the flip bifurcation of system (1.1).
If d * = 3r r+3 , then the eigenvalues of the fixed point (x * , y * ) are λ 1 = −1 and λ 2 = 6−r 3 . We require |λ 2 | = 1, thus r = 3 and r = 9. In addition, note that the existence of the coexistence fixed point is assured by the relation d > r r−1 , so, we get r > 3. Hence, we assume that r > 3 and r = 9 in the following discussion. Let we transform the fixed point (x * , y * ) of system (1.1) to the origin, and taked as a new dependent variable, then system then the map (3.1) can be written as: By the center manifold theorem, the stability of (X, Y ) = (0, 0) near µ = 0 can be determined by studying a one-parameter family of maps on a center manifold, which can be written as: by the approximate computation for center manifold, we obtain Therefore, the map restricted to the center manifold is given bỹ , If map (3.2) undergoes a flip bifurcation, then it must satisfy the following conditions By a simple calculation, we obtain It is easy to check that if 3 < r < 9, then |λ 2 | < 1 and α 2 < 0. From the above analysis, we have the following result. r+3 . Moreover, if 3 < r < 9, then period-2 points that bifurcate from this fixed point are unstable.
Next we give the conditions of existence for Hopf bifurcation. The characteristic equation associated with the linearized system (1.1) at the fixed point (x * (d), y * (d)) is given by The eigenvalues of the characteristic equation (3.3) are given as We get q(d * ) = 1 and Under the conditions of (3.4) and (3.5), we have In addition, if p(d * ) = 0, 1, which leads to r = 5 and r = 7, then we obtain that λ n 1,2 (d * ) = 1(n = 1, 2, 3, 4). where

XY.
Notice that (3.7) is exactly in the form on the center manifold, in which the coefficient k [12] is given by Thus, some complicated calculation gives From the above analysis, we have the theorem stated as follows.

Existence of Marotto's chaos
In this section, we rigorously prove that the system (1.1) possesses a chaotic behavior in the sense of Marotto's definition (see [21], [20]) and present the conditions for the existence of chaotic phenomena.
According to the definition of snap-back repeller, one also needs to find one point z 1 (x 1 , y 1 ) ∈ U r (z 0 ) such that z 1 = z 0 , F M (z 1 ) = z 0 and |DF M (z 1 )| = 0 for some positive integer M , where map F is defined by (1.1).
In fact, by calculating the inverse iterations of the fixed point z 0 twice, we have and Now a map F 2 has been constructed to map the point z 1 (x 1 , y 1 ) to the fixed point z 0 (x * , y * ) after two iterations if there are solutions different from z 0 for Eqs. From (4.2), one can get By a simple calculation, we obtain that the equation (4.3) has one real root which is different from x * denoted as Substituting x 2 into (4.2), we get
Obviously, if the conditions in Lemma 4.1 and the relations (4.9) are satisfied, then z 0 is a snap-back repeller in U r (z 0 ). Thus, the following theorem is established.

Numerical simulations
In this section, numerical simulations are given, including bifurcation diagrams, Lyapunov exponents (M L), fractal dimension (F D) and phase portraits, to illustrate the above theoretical analysis and to show new and more complex dynamic behaviors in the system (1.1).
The fractal dimension [1,2,16,24] is defined by using Lyapunov exponents as follows: with L 1 , L 2 , . . . , L n being Lyapunov exponents, where j is the largest integer such that L i < 0. Our model is a two-dimensional map which has the fractal dimension in the form

Numerical Simulations for Marotto's Chaos
In this subsection, numerical simulations are shown for verifying the conditions in Theorem 4.1. Fig. 2(a) for r = 4, b = 2 and d ∈ (3.1, 3.32) with initial value (0.3, 0.8). The maximum Lyapunov exponents corresponding to (a) are computed as shown in Fig.  2(b). The Marotto's chaotic attractor is given in Fig. 2(c) for d = 3.25, which verifies Theorem 4.1. Case 2. By Example 4.2, the bifurcation diagram in (d, x) plane is displayed in Fig. 3(a) for r = 4, b = 2 and d ∈ (3.35, 3.45) with initial value (0.3, 0.8). The maximum Lyapunov exponents corresponding to (a) are computed as shown in Fig.  3(b). The Marotto's chaotic attractor is given in Fig. 3(c) for d = 3.4. This shows the correctness of Theorem 4.1.

Further Numerical Simulations for the Map (1.1)
In this subsection, new and complex dynamical behaviors are investigated as the parameters vary. The bifurcation diagrams in the two-dimensional plane are considered in the following four cases:  The diagrams show that there is a stable fixed point for d ∈ (2.4, 2.625) and the fixed point loses its stability as d increases. Hopf bifurcation occurs at d ∼ 2.625, and invariant circle appears as d increases and the invariant circle becomes to period-14 orbits at d ∼ 2.788 suddenly. As the growth of d, quasi-periodic orbits appear, and the system undergoes period-doubling bifurcation to chaos with period-24 windows and interior crisis at d ∼ 2.91 and period-10 windows and interior crisis at d ∼ 2.96, respectively. Then, the chaotic behavior disappears at d ∼ 3.18 suddenly. The phase portraits for various values of d are shown in Figs. 7(a)-7(i), which clearly depict how a smooth invariant circle bifurcates from the stable fixed point and an invariant circle to chaotic attractors. From Fig. 7 we observe that there are period-20, period-24, quasi-periodic orbits, six-coexisting chaotic attractors and attracting chaotic sets. For case (iv). The bifurcation diagram of map (1.1) in (r, x) plane for d = 3.5 and b = 2 with initial value (0.28, 0.33) is shown in Fig. 8(a) Lyapunov exponents are bigger than 0, some are smaller than 0, which implies that there exist stable fixed points or stable period windows in the chaotic region. The diagrams show that there is a stable fixed point for r ∈ (1.5, 2.33) and the fixed point loses its stability as r increases. Hopf bifurcation occurs at r ∼ 2.33, and invariant circle appears as r increases and the invariant circle suddenly becomes to period-7 orbits at r ∼ 2.719 and period-13 orbits at r ∼ 3.11. Furthermore, as the growth of r, we can observe the period-6, 7, 13, 14, 50 windows within the chaotic regions and boundary crisis at r = 4.0. The phase portraits for various values of r are revealed in Fig. 9.

Chaos control
In this section, we apply the state feedback control method [3,9,18] to stabilize chaotic orbits at an unstable fixed point of system (1.1). Consider the following controlled form of system (1.1): with the following feedback control law as the control force: where k 1 and k 2 are the feedback gains, (x * , y * ) is the coexistence fixed point of system (1.1).  The Jacobian matrix J of the controlled system (6.1) evaluated at the fixed point (x * , y * ) is given by and the characteristic equation of the Jacobian matrix J(x * , y * ) is λ 2 − (r + (d − 2r)x * − by * − k 1 )λ + d((r − k 1 )x * − 2rx * 2 + k 2 y * ) = 0.

Conclusions
In this paper, we investigated the complex dynamic behaviors of the predator-prey system (1.1) as a discrete-time dynamical system, and showed that the unique coexistence fixed point of system (1.1) can undergo flip bifurcation and Hopf bifurcation. Moreover, system (1.1) displays much more interesting dynamical behaviors, which include orbits of period-6, 7, 8, 10, 14, 18, 24, 36, 50, invariant cycles, quasi-periodic orbits and chaotic sets, in particular, Marotto's chaotic attractors. They all imply that the predator and prey can coexist at period-n oscillatory balance behaviors or a oscillatory balance behavior, but the predator-prey system are unstable if chaotic behavior occurs, and particularly, the predator will ultimately tend to extinct, or tend to a stable fixed point if the prey is in chaotic. These results show far richer dynamics of the discrete-time model. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.