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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 236484, 19 pages
Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays
1Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education, Jiangnan University, Wuxi 214122, China
2School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
3Department of Science, Bengbu College, Bengbu 233030, China
Received 9 August 2012; Revised 17 September 2012; Accepted 4 October 2012
Academic Editor: Kunquan Lan
Copyright © 2012 Zizhen Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A modified Holling-Tanner predator-prey system with multiple delays is investigated. By analyzing the associated characteristic equation, the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are established. Direction and stability of the periodic solutions are obtained by using normal form and center manifold theory. Finally, numerical simulations are carried out to substantiate the analytical results.
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 [1, 2]. Many population models investigating the dynamic relationship between predators and their preys have been proposed and studied. For example, Lotka-Volterra model [3–5], Leslie-Gower model [6–10], and Holling-Tanner model [11–16]. Among these widely used models, Holling-Tanner model plays a special role in view of the interesting dynamics it possesses. Holling-Tanner model for predator-prey interaction is governed by the following nonlinear coupled ordinary differential equations: where and denote the population densities of prey species and predator species at time , respectively. The first equation in system (1.1) shows that the prey grows logistically with the carrying capacity and the intrinsic growth rate in the absence of the predator. And the growth of the prey is hampered by the predator at a rate proportional to the functional response in the presence of the predator. The second equation shows that the predator consumes the prey according to the functional response and grows logistically with the intrinsic growth rate and carrying capacity proportional to the number of the prey. The parameter denotes the maximal predator per capita consumption rate. is a saturation value; it corresponds to the number of prey necessary to achieve one half the maximum rate . The parameter denotes the number of prey required to support one predator at equilibrium when equals .
Recently, there has been considerable interest in predator-prey systems with the Beddington-DeAngelis functional response. And it has been shown that the predator-prey systems with the Beddington-DeAngelis functional response have rich but biologically reasonable dynamics. For more details about this functional response one can refer to [17–21]. Zhang , Lu and Liu  considered the following modified Holling-Tanner delayed predator-prey system: where is incorporated in the negative feedback of the predator density. is the Beddington-DeAngelis functional response. The parameters , , , and are assumed to be positive. is the maximum value at which per capita reduction rate of the prey can attain. measures the extent to which environment provides protection to the prey. describes the effect of handling time on the feeding rate. describes the magnitude of interference among predators. Zhang  investigated the local Hopf bifurcation of system (1.2). Lu and Liu  proved that system (1.2) is permanent under some conditions and obtained the sufficient conditions of local and global stability of system (1.2). Since both the species are growing logistically, it is reasonable to assume delay in prey species as well. Based on this consideration, we incorporate the negative feedback of the prey density into system (1.2) and obtain the following system: where and are the feedback time delays of the prey density and the predator density respectively. Let , , , , , system (1.3) can be transformed into the following nondimensional form: where are the non-dimensional parameters and they are positive.
The main purpose of this paper is to consider the effect of multiple delays on system (1.4). The local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. By employing normal form and center manifold theory, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined. Finally, some numerical simulations are also included to illustrate the theoretical analysis.
2. Local Stability and the Existence of Hopf Bifurcation
In this section, we study the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the positive equilibrium. Obviously, system (1.4) has a unique boundary equilibrium and a unique positive equilibrium , where The Jacobian matrix of system (1.4) at takes the form The characteristic equation of system (1.4) at is of the form Clearly, the boundary equilibrium is unable.
Next, we discuss the existence of Hopf bifurcation at the positive equilibrium . Let , , and still denote and by and , respectively, then system (1.4) becomes where Then we can obtain the linearized system of system (1.4) The characteristic equation of system (2.6) is
Case 1. .
The characteristic equation of system (1.4) becomes where It is easy to verify that Therefore, if , the roots of (2.8) must have negative real parts. Then, we know that the positive equilibrium of system (1.4) is locally stable in the absence of delay, if holds.
Case 2. .
The associated characteristic equation of the system is where Multiplying on both sides of (2.11), we can obtain Now, for , if be a root of (2.13). Then, we have It follows from (2.14) that Then we have where Let , then (2.16) becomes Next, we give the following assumption. : (2.18) has at least one positive real root.
Suppose that holds. Without loss of generality, we assume that (2.18) has four real positive roots, which are defined by , , , and , respectively. Then (2.16) has four positive roots . Therefore, Then we can know that ± are a pair of purely imaginary roots of (2.11) with . Define Let be the root of (2.11) near which satisfies . Taking the derivative of with respect to in (2.13), we obtain it follows that Thus Let Substitute into (2.23), we can get Noting that Therefore, we make the following assumption in order to give the main results: . Then, by Corollary 2.4 in [23, 24], we have the following theorem.
Theorem 2.1. For system (1.4), if the conditions hold, then the equilibrium of system (1.4) is asymptotically stable for and unstable when . And system (1.4) has a branch of periodic solution bifurcation from the zero solution near .
Case 3. , and .
The associated characteristic equation of the system is where We consider (2.27) with in its stable interval, regarding as a parameter. Without loss of generality, we consider system (1.4) under the case considered in , and . is defined as in  and can be obtained by with Let be a root of (2.27). Then we obtain It follows from (2.31) that With Then we have where Suppose that : (2.34) has at least finite positive roots. If holds, we define the roots of (2.34) as . Then, for every fixed , there exists a sequence which satisfies (2.34). Let When , (2.27) has a pair of purely imaginary roots for .
To verify the transversality condition of Hopf bifurcation, we take the derivative of with respect to in (2.27), we can obtain Thus Substitute into (2.38), we can get where Next, we make the following assumption: : .
Thus, by the discussion above and by the general Hopf bifurcation theorem for FDEs in Hale , we have the following results.
Theorem 2.2. For , is defined by (2.29). If the conditions hold, then the equilibrium of system (1.4) is asymptotically stable for and unstable when . System (1.4) has a branch of periodic solution bifurcation from the zero solution near .
3. Direction and Stability of Bifurcated Periodic Solutions
In this section, we shall investigate the direction of the Hopf bifurcation and the stability of bifurcating periodic solution of system (1.4) w.r. to for , and is defined by (2.29). The idea employed here is the normal form and center manifold theory described in Hassard et al. . Throughout this section, it is considered that system (1.4) undergoes the Hopf bifurcation at , at . Let so that the Hopf bifurcation occurs at . Without loss of generality, we assume that , where .
Let , , and rescaling the time delay , Then system (1.4) can be transformed into an FDE in as: where and , are given, respectively, by with where Using Riesz representation theorem, there exists a matrix function whose elements are of bounded variation, such that In fact, choosing For , we define Then system (3.1) can be transformed into the following operator equation where .
For , where is the 2-dimensional space of row vectors, we further define the adjoint operator of A(0): and a bilinear inner product: where .
Since ± are eigenvalues of , they are also eigenvalues of . Let be the eigenvectors of corresponding to and be the eigenvectors of corresponding to . By a simple computation, we can get Then , .
In the remainder of this section, Following the algorithms given in  and using similar computation process to that in , we can get that the coefficients which will be used to determine the important qualities of the bifurcating periodic solutions, 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 , 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). The period of the bifurcating periodic solutions is determined by the sign of : if , the period of the bifurcating periodic solutions increases (decreases).
4. Numerical Example
In order to support the analytic results obtained above, we give some numerical simulations in this section. We only study the most important steady state, namely, the positive steady state. We consider the following system by taking the same coefficients as in : where we get the positive equilibrium . For system (4.1), we can get that , namely, the condition holds.
For . By a simple computation, we obtain that (2.18) has two positive roots: , . Thus, we know that the condition holds. Further, we get , . In addition, we have . Therefore, the condition is satisfied. Hence, from Theorem 2.1, we conclude that the positive equilibrium is asymptotically stable when . The corresponding waveform and the phase plot are illustrated by Figure 1. When the delay passes through the critical value the positive equilibrium will lose its stability and a Hopf bifurcation occurs, and a family of periodic solution bifurcates from the positive equilibrium . This property is illustrated by the numerical simulation in Figure 2.
For , and . Regard as a parameter and let , and from (2.29) we can obtian . Then by a simple computation, we can obtain (2.34) has a positive root . Thus, the condition holds. Further, we we can get and . Namely, the condition holds. Thus, by Theorem 2.2, the positive equilibrium is asymptotically stable when and unstable when , and a family of periodic solution bifurcates from the positive equilibrium . The corresponding waveform and the phase plot are illustrated by Figures 3 and 4, respectively. In addition, from (3.16), we can get , then , , . From Theorem 3.1, we know that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable.
In this paper, a delayed predator-prey system with Beddington-DeAngelis functional response has been investigated. The bifurcation of a predator-prey system with single delay has been studied by many researchers [16, 27–30]. However, there are few papers considering the bifurcation of a predator-prey system with multiple delays (see [31–33]). Compared with the system considered in , the system in this paper accounts for not only the feedback delay of the prey density but also the feedback delay of the predator. The sufficient conditions for the stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation at the positive equilibrium are obtained when and with . Special attention is paid to the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. By computation, we find that the feedback delay of the predator is marked because the critical value of is much smaller when we only consider it. The feedback delay of the prey is unremarkable because the critical value of is much bigger when we consider it with in its stable interval. Furthermore, Zhang  has obtained that the two species in system (1.4) with only the feedback delay of the predator could coexist. However, we get that the two species could also coexist with some available feedback delays of the prey and the predator. This is valuable from the view of ecology.
Unfortunately, the existence of the periodic solutions remain valid only in a small neighborhood of the critical value. It is definitely an interesting work to investigate whether these nonconstant periodic solutions which are obtained through local Hopf bifurcation can still exist for large values of the corresponding parameter time delay. The global continuation of the local Hopf bifurcation is left as the future work.
The authors are grateful to the two anonymous reviewers for their helpful comments and valuable suggestions on improving the paper. This work was supported by Doctor Candidate Foundation of Jiangnan University (JUDCF12030) and Anhui Provincial Natural Science Foundation under Grant no. 1208085QA11.
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