Neimark-Sacker bifurcation of a two-dimensional discrete-time predator-prey model

In this paper, we study the dynamics and bifurcation of a two-dimensional discrete-time predator-prey model in the closed first quadrant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}_+^2$$\end{document}R+2. The existence and local stability of the unique positive equilibrium of the model are analyzed algebraically. It is shown that the model can undergo a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium and an invariant circle will appear. Some numerical simulations are presented to illustrate our theocratical results and numerically it is shown that the unique positive equilibrium of the system is globally asymptotically stable.

but there are many articles that discuss the dynamical behavior of discrete-time models for exploring the possibility of bifurcation and chaos phenomena (Hu et al. 2011;Sen et al. 2012;Chen and Changming 2008;Gakkhar and Singh 2012;Jing and Yang 2006;Zhang et al. 2010;Smith 1968).
We consider the following discrete predator-prey model described by difference equations which was proposed by Smith et al. (1968): where x n and y n denotes the numbers of prey and predator respectively. Moreover the parameters α, β and the initial conditions x 0 , y 0 are positive real numbers.
The organization of the paper is as follows: In Sect. "Existence of equilibria and local stability", we discuss the existence and local stability of equilibria for system (1) in R 2 + . This also include the specific parametric condition for the existence of Neimark-Sacker bifurcation of the system (1). In Sect. "Neimark-Sacker bifurcation", we study the Neimark-Sacker bifurcation by choosing α as a bifurcation parameter. In Sect. "Numerical simulations", numerical simulations are presented to verify theocratical discussion. Finally a brief conclusion is given in Sect. "Conclusion".

Existence of equilibria and local stability
In this section, we will study the existence and stability of equilibria of system (1) in the close first quadrant R 2 + . So, we can summarized the results about the existence of equilibria of system (1) as follows: Lemma 2.1 (i) System (1) has a unique equilibrium O(0, 0) if α < 1 1−β and β < 1; (ii) System (1) has two equilibria O(0, 0) and A(β, α(1 − β) − 1) if α > 1 1−β and β < 1 . More precisely, system (1) has a unique positive equilibrium A(β, α(1 − β) − 1) if α > 1 1−β and β < 1. Now we will study the dynamics of system (1) about these equilibria. The Jacobian matrix of linearized system of (1) about the equilibrium (x, y) is The characteristic equation of the Jacobian matrix J of linearized system of (1) about the unique positive equilibrium A(β, α(1 − β) − 1) is given by where Moreover the eigenvalues of the Jacobian matrix of linearized system of (1) about the unique positive equilibrium A(β, α(1 − β) − 1) is given by where Now we will state the topological classification of these equilibria as follows: Lemma 2.2 (i) For the equilibrium point O(0, 0), following topological classification holds: (1), following topological classification holds:

Lemma 2.3 For the unique positive equilibrium
is a sink if one of the following parametric conditions holds: is a source if one of the following parametric conditions holds: is non-hyperbolic if one of the following parametric conditions holds: From Lemmas 2.2 and 2.3, we summarize the local dynamics of system (1) as follows: Theorem 2.4 (i) If α < 1 1−β and β < 1, then system (1) has a unique equilibrium O(0, 0), which is locally asymptotically stable; (ii) If α > 1 1−β and β < 1, then system (1) has two equilibria O(0, 0) and In the following section, we will study the Neimark-Sacker bifurcation about the unique positive equilibrium A(β, α(1 − β) − 1) by using bifurcation theory (Guckenheimer and Holmes 1983; Kuznetson 2004).

Numerical simulations
In this section, we will give some numerical simulations for the system (1) to support our theoretical results. If we choose β = 0.23, then from non-hyperbolic condition (iii.2) of Lemma 2.3, the value of bifurcation parameter is α = 1.85185. In theoretical point of view, the unique positive equilibrium is stable if α < 1.85185, loss its stability at α = 1.85185 and an attracting invariant close curves appear from the positive equilibrium when α > 1.85185. From subfigures a and b of Fig. 1 it is clear that if α = 1.48 < 1.85185, then unique positive equilibrium is locally stable and corresponding to Fig. 1a, b one can easily seen from Fig. 2a that it is an attractor. So, Fig. 1 shows the local stability of system (1) whereas Fig. 2 shows that the unique positive equilibrium of system (1) is globally asymptotically stable. Figure 3 shows that for different choices of parameters when α > 1.85185, then unique positive equilibrium is unstable and meanwhile an attracting invariant closed curve bifurcates from the positive equilibrium, as in Fig. 3a-i.
(11) ξ 02 = 1 8 F X n X n −F Y n Y n + 2Ḡ X n Y n + ι Ḡ X n X n −Ḡ Y n Y n + 2F X n Y n | (0,0) , 16 F X n X n X n +F X n Y n Y n +Ḡ X n X n Y n +Ḡ Y n Y n Y n +ι Ḡ X n X n X n +Ḡ X n Y n Y n −F X n X n Y n −F Y n Y n Y n | (0,0) .

Conclusion
This work is related to stability and bifurcation analysis of a discrete predator-pray model. We proved that system (1) have two equilibria namely (0, 0) and A(β, α(1 − β) − 1). Moreover, simple algebra shows that if α > 1 1−β , β < 1 then system (1) has unique positive equilibrium A(β, α(1 − β) − 1). The method of linearization is used to prove the local asymptotic stability of equilibria. Linear stability analysis shows that O(0, 0) is a sink if α < 1, saddle if α > 1, and non-hyperbolic if α = 1. For the unique positive equilibrium A(β, α(1 − β) − 1), we have different topological types for possible parameters and proved that it is locally asymptotically stable and under the condition α = 1 1−2β the eigenvalues of the Jacobian matrix are a pair of complex conjugate with modulus one. This means that there exist a Neimark-Saker bifurcation when the parameters vary in the neighborhood of H A . Then we present the Neimark-Saker bifurcation for the unique positive equilibrium point A(β, α(1 − β) − 1) of system (1) by choosing α as a bifurcation parameter. We analysis the Neimark-Sacker bifurcation both by theoretical point of view and by numerical simulations. These numerical examples are experimental verifications of theoretical discussions.  (1)