The stability and bifurcation analysis of a discrete Holling-Tanner model
© Cao et al.; licensee Springer. 2013
Received: 12 June 2013
Accepted: 16 October 2013
Published: 19 November 2013
A discrete predator-prey model with Holling-Tanner functional response is formulated and studied. The existence of the positive equilibrium and its stability are investigated. More attention is paid to the existence of a flip bifurcation and a Neimark-Sacker bifurcation. Sufficient conditions for those bifurcations have been obtained. Numerical simulations are conducted to demonstrate our theoretical results and the complexity of the model.
Differential equations and difference equations are two typical mathematical approaches to modeling population dynamical systems. There have been an increasing interest and research results on discrete population dynamical systems in spite of their complexity [1–4].
Predator-prey models describe one of the most important relationships between two interacting species and have received much attention of applied mathematicians and ecologists. The stability and existence of equilibrium state, the permanence of a system, the Hopf bifurcation and the chaos of different continuous predator-prey models have been extensively investigated. However, there are less results on dynamical behaviors of discrete predator-prey models. The flip bifurcation and the Neimark-Sacker bifurcation are two important phenomena of discrete population model dynamics. Liu and Xiao  used the center manifold theorem to study the flip bifurcation and the Neimark-Sacker bifurcation. Agiza et al.  and Celik et al.  used the numerical simulations to discuss the flip bifurcation and the Neimark-Sacker bifurcation. Hu et al.  also used the center manifold theorem to study the flip bifurcation and the Neimark-Sacker bifurcation.
where and are the numbers of the prey and the predator species at time t, respectively. and are the intrinsic growth rates or biotic potential of the prey and predator, respectively. K is the prey environment carrying capacity. γ is a measure of the food quality that the prey provides for conversion into predator births. q is the maximal predator per capita consumption. a is the number of prey necessary to achieve one-half of the maximum rate q. The variables and parameters satisfy and .
where u, v, c, b, and θ are defined as in model (2). It is assumed that the initial value of solutions of system (3) satisfies , and all the parameters are positive. It is easy to prove that if the initial values are positive, then the corresponding solution is positive too.
In this paper, we study the dynamical behaviors of model (3). The existence and stability of the positive equilibrium are investigated in Section 2. The criteria for the existence of a flip bifurcation and a Neimark-Sacker bifurcation are given in Section 3. Numerical simulations are conducted to demonstrate our theoretical results and show the complexity of the model dynamics in Section 3, too. Concluding remarks and discussions are given in Section 4.
2 The existence and stability of the equilibrium
we know that is a decreasing function of c with , , and .
Then we have the following stability theorem.
Theorem 2.1 The unique positive equilibrium point of model (3) is asymptotically stable if and only if condition (7) or condition (8) holds.
We can have inequality in condition (8) by combining inequalities (13) and (17). □
Remark 1 Inequality (7) or (8) gives stability conditions for the equilibrium of model (3). Inequality (7) is directly obtained from (6), but it is not easy to verify since is dependent on c. As stated in the proof of Theorem 2.1, inequality (8) is easy to verify though it is difficult to obtain.
Remark 2 When , equivalent to , the inequality holds true automatically. The stability condition becomes , which is easy to verify.
Bifurcation may lead to different dynamical behaviors of a model when parameters pass through a critical values. Bifurcation usually occurs when the stability of an equilibrium changes. In this section, we discuss the flip bifurcation and the Neimark-Sacker bifurcation of model (3).
3.1 Flip bifurcation
We define . The stability analysis in Section 3 shows that the positive equilibrium has an eigenvalue −1 when , which means is non-hyperbolic. The flip bifurcation may occur in the neighborhood of the endemic equilibrium when θ passes through the critical point .
The eigenvalues of matrix A are and with . The following theorem confirms the flip bifurcation of model (3).
Theorem 3.1 If , then model (3) will undergo a flip bifurcation at when . That is, there exists a stable period two cycle if , where ε is a small positive number, and β is defined in the end of the proof.
Therefore, model (3) will undergo a flip bifurcation at , and the bifurcation solution of period two is stable (unstable) when () . □
From Theorem 3.1 we know that there exists a flip bifurcation of model (3) when , and the period two cycle is stable. The numerical simulation shows that the period two cycle of model (3) may be globally asymptotically stable when and is small.
3.2 Neimark-Sacker bifurcation
The Neimark-Sacker bifurcation for the discrete models is similar to the Hopf bifurcation of continuous models. In this subsection we discuss the existence of the Neimark-Sacker bifurcation of model (3).
Theorem 3.2 If , then model (3) will undergo a Neimark-Sacker bifurcation at when with , where α is defined in the proof.
Using the Neimark-Sacker bifurcation theorem in , we obtain that there exists a Neimark-Sacker bifurcation when and θ passes through . □
4 Conclusion and discussion
The horizontal axis in Figure 5 is the parameter θ, and the vertical axis is the limiting points of . When , there is only one limiting point of , which is the value of the positive equilibrium. When , the positive equilibrium loses its stability and a stable period two cycle appears. When , the period two cycle loses its stability and a stable period four cycle appears. The period doubling process continues to chaos as θ increases. The top-left subplot shows a complete bifurcation. Three different domains, , , and , in the bifurcation figure are enlarged and displayed in the other three subplots. Especially, from the bottom-left subplot we can see that there is a stable period three cycle of model (3).
The dynamics of the discrete predator-prey model with Holling-Tanner functional response is much more complicated. We have investigated the local stability of the positive equilibrium and the bifurcation of the model analytically or numerically. There are still many challenging problems on the dynamics of the model. Does the local stability of the positive equilibrium imply its global stability? Are there two invariant closed curves in the neighborhood of the positive equilibrium? The numerical simulations demonstrate that the positive equilibrium may be globally stable if it is locally stable. The numerical simulations do not give any information on the existence of two invariant closed curves. We expect that some analytical results can be obtained on those problems in the future.
This study was supported by NSFC grant 11301314, by Shaanxi Provincial Education Department grant 2013JK0599, and by Doctoral Research Foundation of Shaanxi University of Science & Technology grant BJ12-20.
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