Research Article | Open Access
Stability and Bifurcation in a Delayed Holling-Tanner Predator-Prey System with Ratio-Dependent Functional Response
We analyze a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. The local asymptotic stability and the existence of the Hopf bifurcation are investigated. Direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by deriving the equation describing the flow on the center manifold. Finally, numerical simulations are presented for the support of our analytical findings.
Predator-prey dynamics has long been and will continue to be of interest to both applied mathematicians and ecologists due to its universal existence and importance . Although the early Lotka-Volterra model has given way to more sophisticated models from both a mathematical and biological point of view, it has been challenged by ecologists for its functional response suffers from paradox of enrichment and biological control paradox. The ratio-dependent models are discussed as a solution to these difficulties and found to be a more reasonable choice for many predator-prey interactions [2–4]. One type of the ratio-dependent models which plays a special role in view of the interesting dynamics it possesses is the ratio-dependent Holling-Tanner predator-prey system [5, 6]. A ratio-dependent Holling-Tanner predator-prey system takes the form of where and represent the population of prey species and predator species at time . It is assumed that in the absence of the predator, the prey grows logistically with carrying and intrinsic growth rate . The predator growth equation is of logistic type with a modification of the conventional one. The parameter represents the maximal predator per capita consumption rate, and is the half capturing saturation constant. The parameter is the intrinsic growth rate of the predator and is the number of prey required to support one predator at equilibrium, when equals . All the parameters are assumed to be positive.
Liang and Pan  established the sufficient conditions for the global stability of positive equilibrium of system (1.1) by constructing Lyapunov function. Considering the effect of time delays on the system, Saha and Chakrabarti  considered the following delayed system where is the negative feedback delay of the prey. Saha and Chakrabarti  proved that the system (1.2) is permanent under certain conditions and obtained the conditions for the local and global stability of the positive equilibrium. It is well known that studies on dynamical systems not only involve a discussion of stability and persistence, but also involve many dynamical behaviors such as periodic phenomenon, bifurcation, and chaos [8–10]. In particular, the Hopf bifurcation has been studied by many authors [11–13]. Based on this consideration and since both species are growing logistically, we consider the Hopf bifurcation of the following system with two delays: where and represent the negative feedbacks in prey and predator growth.
This paper is organized as follows. In the next section, we will consider the local stability of the positive equilibrium and the existence of Hopf bifurcation of system (1.4). In Section 3, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Some numerical simulations are also given to illustrate the theoretical prediction in Section 4.
2. Local Stability and Hopf Bifurcation
Considering the ecological significance of system (1.4), we are interested only in the positive equilibrium of system (1.4). It is not difficult to verify that system (1.4) has a unique positive equilibrium , where , if : holds.
Case 2 (). Equation (2.4) becomes Let be a root of (2.7). Then, we have From (2.8), we can get If the condition : holds, then (2.9) has a unique positive root The corresponding critical value of time delay is Next, differentiating (2.7) with respect to and substituting , then we get From (2.10) and (2.12), we have Therefore, if the condition : holds, then . Thus, we have the following results.
Theorem 2.1. For system (1.4), if the conditions hold, then the positive equilibrium of system (1.4) is asymptotically stable for and unstable when , system (1.4) undergoes a Hopf bifurcation at when .
Case 3 (). Equation (2.4) becomes Let be a root of (2.14). Then, we get It follows that If the condition holds, then . Thus, (2.16) has a unique positive root , The corresponding critical value of time delay is Similar as in Case 2, we know that if the condition holds, then we have In conclusion, we have the following results.
Theorem 2.2. For system (1.4), if the condition holds, then the positive equilibrium of system (1.4) is asymptotically stable for and unstable when , system (1.4) undergoes a Hopf bifurcation at when .
Case 4 (). Equation (2.4) becomes
Multiplying on both sides of (2.20), we have
Let be a root of (2.21). Then, we get
Then, we can get
Thus, we can obtain
Let , then (2.25) can be transformed into the following form
Next, we suppose that : (2.27) has at least one positive root. Without loss of generality, we suppose that it has four positive roots which are denoted as , , , and . Thus, (2.25) has four positive roots , . The corresponding critical value of time delay is
Let , , .
Differentiating (2.21) regarding and substituting , we obtain where Obviously, if the condition : holds, then . Namely, the transversality condition is satisfied if holds. From the above analysis, we have the following theorem.
Theorem 2.3. For system (1.4), if the conditions , , and hold, then the positive equilibrium of system (1.4) is asymptotically stable for and unstable when , system (1.4) undergoes a Hopf bifurcation at when .
Case 5 ( and , ). We consider (2.4) with in its stable interval and is considered as a parameter. Let be the root of (2.4). Then we have where It follows that where Suppose that : (2.33) has at least finite positive roots.
If the condition holds, we denote the roots of (2.33) as . Then, for every fixed , the corresponding critical value of time delay is Let . The corresponding purely imaginary roots of (2.33) are denoted as . Next, we give the following assumption. : . Hence, we have the following theorem.
Theorem 2.4. Suppose that the conditions , , and hold and . The positive equilibrium of system (1.4) is asymptotically stable for and unstable when , system (1.4) undergoes a Hopf bifurcation at when .
3. Direction and Stability of Bifurcated Periodic Solutions
In this section, we will employ the normal form method and center manifold theorem introduced by Hassard  to determine the direction of Hopf bifurcation and stability of bifurcating periodic solutions of system (1.4) at .
We denote as , , , Then is the Hopf bifurcation value of system (1.4). For convenience, we first rescale the time by , , and still denote , then system (1.4) can be transformed to the following form: where and are given by where , and with
Hence, by the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation such that In fact, we choose where is the Dirac delta function, then (3.3) is satisfied.
For , we define Then system (3.1) can be transformed into the following operator equation The adjoint operator of is defined by associated with a bilinear form where .
From the above discussion, we know that that are eigenvalues of and they are also eigenvalues of .
We assume that are the eigenvectors of belonging to the eigenvalue and are the eigenvectors of belonging to . Thus, Then, we can obtain
Next, we get the coefficients used in determining the important quantities of the periodic solution by using a computation process similar to that in : with where and can be computed as the following equations, respectively with Therefore, we can calculate the following values: Based on the discussion above, we can obtain the following results.
Theorem 3.1. The direction of the Hopf bifurcation is determined by the sign of : if , then the Hopf bifurcation is supercritical (subcritical). The stability of bifurcating periodic solutions is determined by the sign of : if , the bifurcating periodic solutions are stable (unstable).
4. Numerical Example
In this section, to illustrate the analytical results obtained in the previous sections, we present some numerical simulations. Let , , , then we have the following particular case of system (1.4): Obviously, . Thus, the condition holds. Then we can get the unique positive equilibrium of system (4.1). By a simple computation, , . Namely, the condition holds.
Firstly, we can obtain that . Namely, the condition is satisfied for , . Further, we have , . By Theorem 2.1, we can know that the positive equilibrium is asymptotically stable for and unstable when . Let , then the positive equilibrium is asymptotically stable, which can be seen from Figure 1. When , it can be seen from Figure 2 that the positive equilibrium is unstable and a Hopf bifurcation occurs. Similarly, we have , . For , the positive equilibrium is asymptotically stable from Theorem 2.2 and this property can be shown in Figure 3. If , the positive equilibrium is unstable and a Hopf bifurcation occurs, and the corresponding waveform and phase plots are shown in Figure 4.
Secondly, we can get , , and for . From Theorem 2.3, we know that is asymptotically stable for , which can be illustrated by Figure 5. As can be seen from Figure 5 that when the positive equilibrium is asymptotically stable. However, if , then the positive equilibrium becomes unstable and a family of bifurcated periodic solutions occur, which is illustrated by Figure 6. In addition, from (3.18), we get , . Thus, by Theorem 3.1, we know that the Hopf bifurcation is supercritical and the bifurcated periodic solutions are stable.
Lastly, regard as a parameter and let , we can obtain that . Further we have . Let , we can know that the positive equilibrium is asymptotically stable from Theorem 2.4, which can be shown by Figure 7. When then the positive equilibrium becomes unstable and a Hopf bifurcation occurs, which can be illustrated in Figure 8.
In the present paper, a Holling-Tanner predator-prey system with ratio-dependent functional response and two delays is investigated. We prove that the system is asymptotically stable under certain conditions. Compared with the system considered in , we not only consider the feedback delay of the prey but also the feedback delay of the predator. By choosing the delay as a bifurcation parameter, we show that the Hopf bifurcations can occur as the delay crosses some critical values. Furthermore, we get that the two species could also coexist with some available delays of the prey and the predator. This is valuable from the view of biology. In addition, Saha and Chakrabarti  only considered the stability of the system. It is well known that there are also some other behaviors for dynamical systems. Based on this consideration, we investigate the Hopf bifurcation and properties of the bifurcated periodic solutions of the system. The direction and the stability of the bifurcated periodic solutions are determined by applying the normal theory and the center manifold theorem. If the bifurcated periodic solutions are stable, then the two species may coexist in an oscillatory mode from the viewpoint of biology. Some numerical simulations supporting the theoretical results are also included.
The authors are grateful to the referees and the editor for their valuable comments and suggestions on the paper. This work is supported by Anhui Provincial Natural Science Foundation under Grant no. 1208085QA11.
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