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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 835310, 16 pages

http://dx.doi.org/10.1155/2014/835310
Research Article

Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

1Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

2Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received 27 November 2013; Accepted 31 March 2014; Published 11 May 2014

Academic Editor: Yuming Chen

Copyright © 2014 Yunxian Dai 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.

Abstract

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delays .

1..Introduction

In [1], Xu and Chaplain studied the following delayed predator-prey model with Michaelis-Menten type functional response: with initial conditions where , and denote the densities of the prey, predator, and top predator population, respectively. are positive constants. , and are nonnegative constants. denote the delay in the negative feedback of the prey, predator, and top predator crowding, respectively. , are constant delays due to gestation; that is, mature adult predators can only contribute to the production of predator biomass. . are continuous bounded functions in the interval . The authors proved that the system is uniformly persistent under some appropriate conditions. By means of constructing suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of the system.

Time delays of one type or another have been incorporated into systems by many researchers since a time delay could cause a stable equilibrium to become unstable and fluctuation. In [212], authors showed effects of two delays on dynamical behaviors of system.

It is well known that periodic solutions can arise through the Hopf bifurcation in delay differential equations. However, these periodic solutions bifurcating from Hopf bifurcations are generally local. Under some circumstances, periodic solutions exist when the parameter is far away from the critical value. Therefore, global existence of Hopf bifurcation is a more interesting and difficult topic. A great deal of research has been devoted to the topics [1321]. In this paper, let in (1); we consider Hopf bifurcation and global periodic solutions of the following system with two unequal and nonzero delays: with initial conditions Our goal is to investigate the possible stability switches of the positive equilibrium and stability of periodic orbits arising due to a Hopf bifurcation when one of the delays is treated as a bifurcation parameter. Special attention is paid to the global continuation of local Hopf bifurcation when the delays .

This paper is organized as follows. In Section 2, by analyzing the characteristic equation of the linearized system of system (3) at positive equilibrium, the sufficient conditions ensuring the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained [22]. Some explicit formulas determining the direction and stability of periodic solutions bifurcating from Hopf bifurcations are demonstrated by applying the normal form method and center manifold theory due to Hassard et al. [23] in Section 3. In Section 4, we consider the global existence of these bifurcating periodic solutions [24] with two different delays. Some numerical simulation results are included in Section 5.

2. Stability of the Positive Equilibrium and Local Hopf Bifurcations

In this section, we first study the existence and local stability of the positive equilibrium and then investigate the effect of delay and the conditions for existence of Hopf bifurcations.

There are at most four nonnegative equilibria for system (3): where and satisfy where is a nonnegative equilibrium point if there is a positive solution of (6), and is a nonnegative equilibrium point if there is a positive solution of (7).

Let(H1) ; (H2) ; (H3) ; (H4) . From [1, 25], we know that if , and hold, and always exist as nonnegative equilibria.

Let be the arbitrary equilibrium point, and let ; still denote by , respectively; then the linearized system of the corresponding equations at is as follows: where all the others of , and are .

The characteristic equation for system (8) is where

We consider the following cases.

(1) . The characteristic equation reduces to There are always a positive root and two negative roots of (12); hence is a saddle point.

(2) . Equation (10) takes the form There is a positive root if ; hence, is a saddle point. If , is locally asymptotically stable.

(3) . The characteristic equation is We will analyse the distribution of the characteristic root of (14) from Ruan and Wei [26], which is stated as follows.

Lemma 1. Consider the exponential polynomial where and are constants. As vary, the sum of the order of the zeros of on the open right half plane can change only if a zero appears on or crosses the imaginary axis.

By using Lemma 1, we can easily obtain the following results.

Lemma 2. If is a nonnegative equilibrium point, then(1) is unstable if ;(2) is locally asymptotically stable if , and .

Proof. (1) is a root of (14); if , then is unstable.

(2) Clearly, is not a root of (14); we should discuss the following equation instead of (14): Assume that with is a solution of (16). Substituting into (16) and separating the real and imaginary parts yield which implies If , that is , there is no real root of (16). Hence there is no purely imaginary root of (18). When , (16) reduces to If and , both roots of (19) have negative real parts. Thus, by using Lemma 1, when , and , is locally asymptotically stable.

(4) . The characteristic equation about is (10). In the following, we will analyse the distribution of roots of (10). We consider four cases.

Case a. Consider

.

The associated characteristic equation of system (3) is Let(H5) . By Routh-Hurwitz criterion, we have the following.

Theorem 3. For , assume that hold. Then when , the positive equilibrium of system (3) is locally asymptotically stable.

Case b. Consider

, .

The associated characteristic equation of system (3) is

We want to determine if the real part of some root increases to reach zero and eventually becomes positive as varies. Let be a root of (21); then we have Separating the real and imaginary parts, we have It follows that where .

Denoting , (24) becomes

Let we have

If , then . We can know that (25) has at least one positive root.

If , we obtain that when , (25) has no positive roots for . On the other hand, when , the following equation has two real roots: . Because of and are the local minimum and the local maximum of , respectively. By the above analysis, we immediately obtain the following.

Lemma 4. (1) If and , (25) has no positive root for .

(2) If and , (25) has at least one positive root if and only if and .

(3) If   , (25) has at least one positive root.

Without loss of generality, we assume that (25) has three positive roots, defined by , respectively. Then (24) has three positive roots: From (23) we have Thus, if we denote where ; then is a pair of purely imaginary roots of (21) corresponding to . Define

Let be the root of (21) near satisfying Substituting into (21) and taking the derivative with respect to , we have Therefore, When ,         +   .

According to (35), we have where . Notice that , then we have the following lemma.

Lemma 5. Suppose that and , where is defined by (26); then has the same sign with .

From Lemmas 1, 4, and 5 and Theorem 3, we can easily obtain the following theorem.

Theorem 6. For , suppose that hold. (i)If and , then all roots of (10) have negative real parts for all , and the positive equilibrium is locally asymptotically stable for all .(ii)If either or , and , then has at least one positive roots, and all roots of (23) have negative real parts for , and the positive equilibrium is locally asymptotically stable for .(iii)If (ii) holds and , then system (3)undergoes Hopf bifurcations at the positive equilibrium for .

Case c. Consider

, .

The associated characteristic equation of system (3) is

Similar to the analysis of Case , we get the following theorem.

Theorem 7. For , suppose that hold. (i)If and , then all roots of (38) have negative real parts for all , and the positive equilibrium is locally asymptotically stable for all .(ii)If either or , , and , then has at least one positive root , and all roots of (38) have negative real parts for , and the positive equilibrium is locally asymptotically stable for .(iii)If (ii) holds and , then system (3) undergoes Hopf bifurcations at the positive equilibrium for ,

where where ; ; then is a pair of purely imaginary roots of (38) corresponding to . Define

Case d. Consider

, .

The associated characteristic equation of system (3) is We consider (41) with   in its stable interval . Regard as a parameter.

Let be a root of (41); then we have Separating the real and imaginary parts, we have It follows that where

Denote . If , then We can obtain that (44) has at most six positive roots . For every fixed ,   , there exists a sequence , such that (43) holds.

Let When , (41) has a pair of purely imaginary roots for .

In the following, we assume that(H6) .

Thus, by the general Hopf bifurcation theorem for FDEs in Hale [22], we have the following result on the stability and Hopf bifurcation in system (3).

Theorem 8. For , , , suppose that is satisfied. If and , then the positive equilibrium is locally asymptotically stable for . System (3) undergoes Hopf bifurcations at the positive equilibrium for .

3. Direction and Stability of the Hopf Bifurcation

In Section 2, we obtain the conditions under which system (3) undergoes the Hopf bifurcation at the positive equilibrium . In this section, we consider with and regard as a parameter. We will derive the explicit formulas determining the direction, stability, and period of these periodic solutions bifurcating from equilibrium at the critical values by using the normal form and the center manifold theory developed by Hassard et al. [23]. Without loss of generality, denote any one of these critical values by , at which (43) has a pair of purely imaginary roots and system (3) undergoes Hopf bifurcation from .

Throughout this section, we always assume that . Let and . Then is the Hopf bifurcation value of system (3). System (3) may be written as a functional differential equation in where , and where , and Obviously, is a continuous linear function mapping into . By the Riesz representation theorem, there exists a matrix function    , whose elements are of bounded variation such that In fact, we can choose where is Dirac-delta function. For , define Then when , the system is equivalent to where ,   . For , define and a bilinear inner product where . Let ; then and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of . Thus, they are also eigenvalues of . We first need to compute the eigenvector of and corresponding to and , respectively. Suppose that is the eigenvector of corresponding to . Then . From the definition of , and we can easily obtain , where and . Similarly, let be the eigenvector of corresponding to . By the definition of , we can compute From (57), we have Thus, we can choose such that .

In the remainder of this section, we follow the ideas in Hassard et al. [23] and use the same notations as there to compute the coordinates describing the center manifold at . Let be the solution of (48) when . Define On the center manifold , we have where and are local coordinates for center manifold in the direction of and . Note that is real if is real. We consider only real solutions. For the solution of (48), since , we have where By (62), we have , and then It follows together with (50) that <