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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 264870, 20 pages
http://dx.doi.org/10.1155/2012/264870
Research Article

Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters

Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, China

Received 13 January 2012; Accepted 20 August 2012

Academic Editor: Benchawan Wiwatanapataphee

Copyright © 2012 Changjin Xu. 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

A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

1. Introduction

Over the past decade, a great many predator-prey models have been developed to describe the interaction between predator and prey. Their dynamical phenomena have been extensively studied because of the wide application in the field of biomathematics. In particular, the appearance of a cycle bifurcating from the equilibrium of an ordinary or a delayed predator-prey model with a single parameter, which is known as a Hopf bifurcation, has attracted much attention due to its theoretical and practical significance [15]. But most of the research literature on these models are only connected with parameters which are independent of time delay; thus, the corresponding characteristic equations are easy to deal with. While in most applications of delay differential equations in population dynamics, the need of incorporation of a time delay is often the result of existence of some stage structure [68]. Indeed, every population goes through some distinct life stages [9, 10]. Since the through-stage survival rate is often a function of time delay, it is easy to conceive that these models will inevitably involve some delay-dependent parameters. Thus, the corresponding characteristic equations dependent on the delay become more complicated. In view of the fact that it is often difficult to analytically study models with delay-dependent parameters even if only a single discrete delay is present, we resort to the help of computer programs.

In 2008, Wang et al. [11] introduced and investigated the following predator-prey interaction model with time delay and delay-dependent parameters: where and stand for prey and predator density at time , respectively. are real positive parameters and the time delay is a positive constant. Wang et al. [11] obtained the conditions that guarantee the system asymptotically stable and permanent. For more knowledge about the model, one can see [11].

It is well known that time delays which occur in the interaction between predator-prey will affect the stability of a model by creating instability, oscillation, and chaos phenomena. Based on the discussion above, the main purpose of this paper is to investigate the stability and the properties of Hopf bifurcation of the model (1.1) which involves some delay-dependent parameters. Recently, there are few papers on the topic that involves some delay-dependent parameters, for example, Liu and Zhang [12] investigated the stability and Hopf bifurcation of the following SIS model with nonlinear birth rate: Jiang and Wei [13] studied the stability and Hopf bifurcation of the following SIR model: It worth pointing out that Liu and Zhang [12] investigated the Hopf bifurcation of system (1.2) by choosing (not delay ) as the bifurcation parameters and Jiang and Wei [13] studied the Hopf bifurcation of system (1.3) by choosing (not delay ) as the bifurcation parameters. In this paper, we will investigate the Hopf bifurcation by regarding the delay as the bifurcation parameter which is different from the papers [12, 13]. To the best of our knowledge, it is the first time to deal with the stability and Hopf bifurcation of system (1.1).

This paper is organized as follows. In Section 2, the stability of the equilibrium and the existence of Hopf bifurcation at the equilibrium are studied. In Section 3, the direction of Hopf bifurcation and the stability and periodic of bifurcating periodic solutions on the center manifold are determined. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 5.

2. Stability of the Equilibrium and Local Hopf Bifurcations

Throughout this paper, we assume that the following condition(H1) holds.

The hypothesis (H1) implies that system (1.1) has a unique positive equilibrium , where The linearized system of (1.1) around takes the form where The associated characteristic equation of (2.2) is where When , then (2.4) becomes where It is easy to obtain the following result.

Lemma 2.1. If the condition
   (H2),
holds, then the positive equilibrium of system (1.1) is asymptotically stable.

In the following, one investigates the existence of purely imaginary roots of (2.4). Equation (2.4) takes the form of a second-degree exponential polynomial in , which some of the coefficients of P and Q depend on . Beretta and Kuang [14] established a geometrical criterion which gives the existence of purely imaginary roots of a characteristic equation with delay-dependent coefficients. In order to apply the criterion due to Beretta and Kuang [14], one needs to verify the following properties for all , where is the maximum value which exists.(a); (b); (c); (d) has a finite number of zeros;(e)Each positive root of is continuous and differentiable in whenever it exists. Here, and are defined as in (2.5), respectively.

Let . It is easy to see that which implies that (a) is satisfied, and (b) From (2.4), one has Therefore, (c) follows. Let be defined as in (d). From one obtain Obviously, property (d) is satisfied, and by implicit function theorem, (e) is also satisfied.

Now let be a root of (2.4). Substituting it into (2.4) and separating the real and imaginary parts yields From (2.13), it follows that By the definitions of and as in (2.5), respectively, and applying the property (a), then (2.14) can be written as which yields . Assume that is the set where is a positive root of and for is not definite. Then for all in satisfied . The polynomial function can be written as where is a second degree polynomial, defined by It is easy to see that has only one positive real root if the following condition (H3) holds:(H3).

One denotes this positive real root by . Hence, (2.17) has only one positive real root . Since the critical value of and are impossible to solve explicitly, so one will use the procedure described in Beretta and Kuang [14]. According to this procedure, one defines such that and are given by the righthand sides of (2.14), respectively, with given by (2.19). This define in a form suitable for numerical evaluation using standard software. And the relation between the argument and in (2.18) for must be ,  .

Hence, one can define the maps: given by where a positive root of exists in I. Let us introduce the functions , which are continuous and differentiable in . Thus, one gives the following theorem which is due to Beretta and Kuang [14].

Theorem 2.2. Assume that is a positive root of (2.4) defined for , and at some for some . Then, a pair of simple conjugate pure imaginary roots exists at which crosses the imaginary axis from left to right if and crosses the imaginary axis from right to left if , where .

Applying Lemma 2.1 and the Hopf bifurcation theorem for functional differential equation [5], we can conclude the existence of a Hopf bifurcation as stated in the following theorem.

Theorem 2.3. For system (1.1), if (H1)–(H3) hold, then there exists s such that the positive equilibrium is asymptotically stable for and becomes unstable for staying in some right neighborhood of , with a Hopf bifurcation occurring when .

3. Direction and Stability of the Hopf Bifurcation

In the previous section, we obtained some conditions which guarantee that the stage-structured predator-prey model with time delay undergoes the Hopf bifurcation at some values of . In this section, we will derive the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium at these critical value of , by using techniques from normal form and center manifold theory [15]. Throughout this section, we always assume that system (1.1) undergoes Hopf bifurcation at the positive equilibrium for , and then is corresponding purely imaginary roots of the characteristic equation at the equilibrium .

For convenience, let . Then is the Hopf bifurcation value of (1.1). Thus, one will study Hopf bifurcation of small amplitude periodic solutions of (1.1) from the positive equilibrium point for close to 0.

Let , , , , then system (1.1) can be transformed into an functional differential equation (FDE) in as where and are given, respectively, by where where where where

Clearly, is a linear continuous operator from to . By the Riesz representation theorem, there exists a matrix function with bounded variation components such that In fact, we can choose where is the Dirac delta function.

For , define Then (1.1) is equivalent to the abstract differential equation where , , .

For , define

For and , define the bilinear form where . We have the following result on the relation between the operators and .

Lemma 3.1. and are adjoint operators.

The proof is easy from (3.12), so we omit it.

By the discussions in Section 2, we know that are eigenvalues of , and they are also eigenvalues of corresponding to and , respectively. We have the following result.

Lemma 3.2. The vector where is the eigenvector of corresponding to the eigenvalue , and where is the eigenvector of corresponding to the eigenvalue ; moreover, , where

Proof. Let be the eigenvector of corresponding to the eigenvalue and be the eigenvector of corresponding to the eigenvalue , namely, and . From the definitions of and , we have and . Thus, and . In addition, That is, Therefore, we can easily obtain And so and hence On the other hand, Namely, Therefore, we can easily obtain and so and hence In the sequel, one will verify that . In fact, from (3.12), we have
Next, we use the same notations as those in Hassard et al. [15], and we first compute the coordinates to describe the center manifold at . Let be the solution of (1.1) when .
Define on the center manifold , and we have where and and are local coordinates for center manifold in the direction of and . Noting that is also real if is real, we consider only real solutions. For solutions of (1.1), That is, where Hence, we have where Noticing and , we have From (3.34) and (3.35), we have and we obtain For unknown in , we still need to compute them.
From (3.10) and (3.29), we have where Comparing the coefficients, we obtain and we know that for , Comparing the coefficients of (3.42) with (3.45) gives that From (3.43) and (3.46) and the definition of , we get Noting that , we have where is a constant vector.
Similarly, from (3.44) and (3.47) and the definition of , we have where is a constant vector.
In what follows, one will seek appropriate ,  in (3.49) and (3.51), respectively. It follows from the definition of and (3.46) and (3.47) that where .
From (3.43), we have where From (3.44), we have where Noting that and substituting (3.49) and (3.54) into (3.52), we have That is, then Hence, where Similarly, substituting (3.51) and (3.56) into (3.53), we have Then, That is, Hence, where From (3.49) and (3.51), we can calculate and derive the following values: These formulae give a description of the Hopf bifurcation periodic solutions of (1.1) at on the center manifold. From the discussion above, we have the following result.

Theorem 3.3. The periodic solution is supercritical (subcritical) if . The bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if   . The periods of the bifurcating periodic solutions increase (decrease) if .

4. Numerical Examples

In this section, we present some numerical results to verify the analytical predictions obtained in the previous section. As an example, we consider the following special case of system (1.1) with the parameters . Then system (1.1) becomes which has a positive equilibrium . By some complicated computation by means of Matlab 7.0, we get only one critical values of the delay . Thus, we derive . We obtain that the conditions indicated in Theorem 2.3 are satisfied. Furthermore, it follows that and . Thus, the positive equilibrium is stable when which is illustrated by the computer simulations (see Figures 1(a)1(d)). When passes through the critical value , the positive equilibrium loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the positive equilibrium . Since and , the direction of the Hopf bifurcation is , and these bifurcating periodic solutions from at are stable, which are depicted in Figures 2(a)2(d).

fig1
Figure 1: The time histories and phase portrait of system (4.1) with the following parameters: , and . The positive equilibrium is asymptotically stable. The initial value is (0.2, 2).
fig2
Figure 2: The time histories and phase portrait of system (4.1) with the following parameters: , and . Hopf bifurcation occurs from the positive equilibrium . The initial value is (0.2, 2).

5. Conclusions

In this paper, the main object is to investigate the local stability and Hopf bifurcation and also to study the stability of bifurcating periodic solutions and some formulae for determining the direction of Hopf bifurcation for a stage-structured predator-prey model with time delay and delay. By choosing the delay as a bifurcation parameter, It is shown that under certain condition, the positive equilibrium of system (1.1) is asymptotically stable for all and unstable for and under another condition; when the delay increases, the equilibrium loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium , that is, a family of periodic orbits bifurcate from the positive equilibrium . At the same time, using the normal form theory and the center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed. Finally, numerical simulations are carried out to validate the theorems obtained.

Acknowledgments

This work is supported by The National Natural Science Foundation of China (no. 11261010), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J (2012) 2100), Governor Foundation of Guizhou Province ((2012)53), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).

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