Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 282908, 28 pages

doi:10.1155/2012/282908

Research Article

Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey

Zizhen Zhang1,2 and Huizhong Yang¹

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

Received 22 August 2012; Revised 3 October 2012; Accepted 5 October 2012

Academic Editor: M. De la Sen

Copyright © 2012 Zizhen Zhang and Huizhong Yang. 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.

This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included.

1. Introduction

Predator-prey dynamics continues to draw interest from both applied mathematicians and ecologists due to its universal existence and importance. Many kinds of predator-prey models have been studied extensively [1–6]. It is well known that there are many species whose individual members have a life history that takes them through immature stage and mature stage. To analyze the effect of a stage structure for the predator or the prey on the dynamics of a predator-prey system, many scholars have investigated predator-prey systems with stage structure in the last two decades [7–15]. In [7], Wang considered the following predator-prey system with stage structure for the predator and obtained the sufficient conditions for the global stability of a coexistence equilibrium of the system:

where represents the density of the prey at time . and represent the densities of the immature predator and the mature predator at time , respectively. For the meanings of all the parameters in system (1.1), one can refer to [7]. Considering the gestation time of the mature predator, Xu [8] incorporated the time delay due to the gestation of the mature predator into system (1.1) and considered the effect of the time delay on the dynamics of system (1.1).

There has also been a significant body of work on the predator-prey system with stage structure for the prey. In [12], Xu considered a delayed predator-prey system with a stage structure for the prey:

where and denote the population densities of the immature prey and the mature prey at time , respectively. denotes the population density of the predator at time . All the parameters in system (1.2) are assumed positive. is the birth rate of the immature prey. is the transformation rate from immature individual to mature individuals. is the intraspecific competition coefficient of the mature prey. and are the death rates of the immature and the mature prey, respectively. is the death rate of the predator. and are the interspecific interaction coefficients between the mature prey and the predator, respectively. is the response function of the predator. And is a constant delay due to the gestation of the predator. In [12], Xu investigated the persistence of system (1.2) by means of the persistence theory on infinite dimensional systems, and sufficient conditions are obtained for the global stability of nonnegative equilibrium of the model by constructing appropriate Lyapunov function. But studies on the predator-prey system not only involve the persistence and stability, but also involve many other behaviors such as periodic phenomenon, attractivity, and bifurcation [16–19]. In particular, the properties of periodic solutions are of great interest [20–24]. Therefore, F. Li and H. W. Li [14] considered the property of periodic solutions of the following system:

Motivated by the work of Xu [12] and F. Li and H. W. Li [14] and considering the intraspecific competition of the immature prey population, we consider the following system:

where and denote the population densities of the immature prey and the mature prey at time , respectively. denotes the population density of the predator at time . The parameters , , , , , , , , and are defined as in system (1.3). is the intraspecific competition of the immature prey, is the feedback delay of the mature prey, and is the time delay due to the gestation of the predator.

The organization of this paper is as follows. In Section 2, by analyzing the corresponding characteristic equations, the local stability of the positive equilibrium of system (1.4) is discussed, and the existence of Hopf bifurcation at the positive equilibrium is established. In Section 3, we determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem in [20]. And numerical simulations are carried out in Section 4 to illustrate the main theoretical results. Finally, main conclusions are included.

2. Local Stability and Hopf Bifurcation

From the viewpoint of biology, we are only interested in the positive equilibrium of system (1.4). It is not difficult to verify that system (1.4) has a positive equilibrium , where

if the following conditions hold: , .

Let , , , and we still denote , , and by , , and . Then system (1.4) can be transformed to the following form:

where

Then we can get the linearized system of system (2.2)

Therefore, the corresponding characteristic equation of system (2.4) is

where , , , , , , , , , .

Next, we consider the local stability of the positive equilibrium and the Hopf bifurcation of system (1.4) for the different combination of and .

Case 1. . The characteristic equation (2.5) becomes

where , , .

It is not difficult to verify that and . Thus, all the roots of (2.6) must have negative real parts, if the following condition holds: . Namely, the positive equilibrium is locally stable in the absence of time delay, if holds.

Case 2. . On substituting , (2.5) becomes

where , , , , , .

Let be a root of (2.7). Then, we have

Squaring both sides and adding them up, we get the following sixth-degree polynomial equation:

where , , .

Let , then (2.9) becomes

Define

If , it is easy to know that (2.10) has at least one positive root. On the other hand, if , according to Lemma 2.2 in [25], (2.10) has positive roots if and , hold. Therefore, we give the following assumption.

: equation (2.10) has at least one positive root.

Without loss of generality, we assume that it has three positive roots which are denoted as , , and . Thus, (2.9) has three positive roots , . The corresponding critical value of time delay is

where , , , , , .

Let , , .

To verify the transversality condition of Hopf bifurcation, differentiating the two sides of (2.7) with respect to , and noticing that is a function of , we can obtain

Thus,

Therefore,

From (2.9), we can get

Then, we have

where .

Therefore, if holds. Notice that and have the same sign. Then we have if holds. In conclusion, we have the following results.

Theorem 2.1. Suppose that the conditions and hold. The positive equilibrium of system (1.4) is asymptotically stable for and unstable when . Further, system (1.4) undergoes a Hopf bifurcation when .

Case 3. . On substituting , (2.5) becomes

where , , , , , .

Let be a root of (2.18). Then, we get

which follows that

where , , .

Let , then (2.20) becomes

Define

Similar as in case (2), we give the following assumption.

: equation (2.21) has at least one positive root.

Without loss of generality, we assume that it has three positive roots denoted by , , and . Thus, (2.20) has three positive roots .

The corresponding critical value of time delay is

where , , , , , .

Let , , .

Similar as in case (1), next, we suppose that the condition holds, where . Then we have . By the above analysis, we have the following results.

Theorem 2.2. Suppose that the conditions and hold. The positive equilibrium of system (1.4) is asymptotically stable for and unstable when . Further, system (1.4) undergoes a Hopf bifurcation when .

Case 4. .

For , (2.5) can be rewritten in the following form:

where , , , , , , , .

Multiplying on both sides of (2.24), it is obvious to get

Let be the root of (2.25). Then, we have

where , , , , , .

It follows that

where , , , , , , , , .

From (2.27), we can get

where , , , , , .

Let , then (2.28) becomes

Suppose that (2.29) has at least one positive root, and, without loss of generality, we assume that it has six positive roots which are denoted as , , , , , and . Then, (2.28) has six positive roots .

The corresponding critical value of time delay is

Let , , .

Next, we verify the transversality condition. Differentiating (2.25) regarding and substituting , we get

where

Thus, if the condition holds, the transversality condition is satisfied.

Theorem 2.3. Suppose that the conditions and hold. The positive equilibrium of system (1.4) is asymptotically stable for and unstable when . Further, system (1.4) undergoes a Hopf bifurcation when .

Case 5. .

We consider (2.5) with in its stable interval, and is considered as a parameter.

Let be the root of (2.5). Then, we have

where

From (2.33), we can get the following transcendental equation:

where , , , , , , , , .

In order to give the main results, we suppose that (2.35) has finite positive root. We denote the positive roots of (2.35) as . For every , the corresponding critical value of time delay is

Let , and is the corresponding root of (2.35) with .

In the following, we differentiate the two sides of (2.5) with respect to to verify the transversality condition.

Taking the derivative of with respect to in (2.5) and substituting , we get

where

Obviously, if the condition holds, the transversality condition is satisfied. Through the above analysis, we have the following results.

Theorem 2.4. Suppose that the conditions and hold and . The positive equilibrium of system (1.4) is asymptotically stable for and unstable when . Further, system (1.4) undergoes a Hopf bifurcation when .

Case 6. .

We consider (2.5) with in its stable interval, and is considered as a parameter.

Substitute into (2.5). Then, we get

where , , , , , , , , .

Similar as in case (5), we give the following assumption. : (2.39) has finite positive root.

The positive roots of (2.39) are denoted as , ,,. For every , the corresponding critical value of time delay is

where

Let , and is the corresponding root of (2.39) with .

Then, we suppose that holds. By the general Hopf bifurcation theorem for FDEs in Hale [26], we have the following results.

Theorem 2.5. Suppose that the conditions and hold and . The positive equilibrium of system (1.4) is asymptotically stable for and unstable when . Further, system (1.4) undergoes a Hopf bifurcation at when .

3. Direction and Stability of Bifurcated Periodic Solutions

In Section 2, we have obtained the conditions under which a family of periodic solutions bifurcate from the positive equilibrium of system (1.4) when the delay crosses through the critical value. In this section, we will determine the direction of Hopf bifurcation and stability of bifurcating periodic solutions of system (1.4) with respect to for by using the normal form method and center manifold theorem introduced by Hassard et al. [20]. It is considered that system (1.4) undergoes Hopf bifurcation at . Without loss of generality, we assume that , where .

Let , , , , . We still denote by . Then, system (1.4) can be transformed into the following system:

where and , are given, respectively, by

where

Hence, by the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation such that

In fact, we choose

For , we define

and

Then, system (3.1) can be transformed into the following operator equation:

For , where is the 3-dimensional space of row vectors, we define the adjoint operator of :

For and , we define a bilinear inner product:

where .

By the discussion in Section 2, we know that are eigenvalues of . Thus, they are also eigenvalues of .

Suppose that is the eigenvector of corresponding to and is the eigenvector of corresponding to . By direction computation, we can get

Then, from (3.10), we can get

Therefore, we can choose

such that and .

In the remainder of this section, following the algorithms given in [20] and using similar computation process in [27], we can get the coefficients that can be used to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions:

with

where , can be determined by the following equations, respectively,

with

Then, we can calculate the following values:

Based on the above discussion, we can obtain the following results.

Theorem 3.1. From (3.20) one has

(i)the direction of the Hopf bifurcation is determined by the sign of : if , then the Hopf bifurcation is supercritical (subcritical);(ii)the stability of bifurcating periodic solutions is determined by the sign of : if , the bifurcating periodic solutions are stable (unstable);(iii)the period of the bifurcating periodic solution is determined by the sign of : if , the bifurcating periodic solution increases (decreases).

4. Numerical Example

In this section, we give some numerical simulations to verify the theoretical analysis in Sections 2 and 3. Let , , , , , , , , , and . Then, we have the following particular case of system (1.4):

It is not difficult to verify that , , namely, the conditions and hold. Therefore, system (4.1) has at least a positive equilibrium. By means of Matlab, we can get that the positive equilibrium of (4.1) is .

For , , we can get , . From Theorem 2.2, we know that the positive equilibrium is asymptotically stable when . The corresponding waveform and the phase plot are illustrated by Figure 1. When the time delay passes through the critical value , the positive equilibrium will lose its stability and a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the positive equilibrium . This property is illustrated by the numerical simulation in Figure 2. Similarly, we have , , when , . The corresponding waveform and the phase plots are shown in Figures 3 and 4.

Figure 1: is locally asymptotically stable for with initial value “2.31; 1.9; 2.34.”

Figure 2: is unstable for with initial value “2.31; 1.9; 2.34.”

Figure 3: is locally asymptotically stable for with initial value “2.31; 1.9; 2.34.”

Figure 4: is unstable for with initial value “2.31; 1.9; 2.34.”

For , we can obtain , . From Theorem 2.2, we know that, when the time delay increases from zero to , the positive equilibrium is asymptotically stable. Once the time delay passes through the critical value , the positive equilibrium will lose its stability and a Hopf bifurcation occurs. This property is illustrated by the numerical simulation in Figures 5 and 6.

Figure 5: is locally asymptotically stable for with initial value “2.31; 1.9; 2.34.”

Figure 6: is unstable for with initial value “2.31; 1.9; 2.34.”

For and , we have , . According to Theorem 2.2, the positive equilibrium is asymptotically stable when and unstable when , which can be depicted by the numerical simulation in Figures 7 and 8. In addition, from (3.20), we can obtain , , , . Thus, from Theorem 2.3, we know that the Hopf bifurcation with respect to with is subcritical, the bifurcating periodic solutions are stable and increase. Similarly, we have , , for and . The corresponding waveform and the phase plots are shown in Figures 9 and 10.

Figure 7: is locally asymptotically stable for and with initial value “2.31; 1.9; 2.34.”

Figure 8: is unstable for and with initial value “2.31; 1.9; 2.34.”

Figure 9: is locally asymptotically stable for and with initial value “2.31; 1.9; 2.34.”

Figure 10: is unstable for and with initial value “2.31; 1.9; 2.34.”

5. Conclusions

In this paper, a delayed predator-prey system with Holling type III functional response and stage structure for the prey population is investigated. Compared with literature [14], we consider not only the time delay due to the gestation of the predator but also the negative feedback of the mature prey density and the intraspecific competition of the immature prey population. F. Li and H. W. Li [14] has obtained that the species in system (4.1) with only the time delay due to the gestation of the predator could coexist. However, we get that the species could also coexist with some available time delays of the mature prey and the predator. This is valuable from the view of ecology.

The sufficient conditions for the local stability of the positive equilibrium and the existence of local Hopf bifurcation for the possible combinations of two delays are obtained. The main results are given in Theorems 2.1–2.5. By computation, we find that the time delay due to the gestation of the predator is marked because the critical value of is smaller than that of when we only consider them, respectively. Furthermore, the explicit formulae which determines the direction of the bifurcation and the stability of the bifurcating periodic solutions is established when and by using the normal form theory and center manifold theorem. The main results are given in Theorem 2.3. Finally, numerical simulations are carried out to support the obtained theoretical results.

Acknowledgments

The authors are grateful to the anonymous reviewers for their helpful comments and valuable suggestions on improving the paper. This work was supported by the National Natural Science Foundation of China (61273070), Doctor Candidate Foundation of Jiangnan University (JUDCF12030) and Anhui Provincial Natural Science Foundation under Grant no. 1208085QA11.

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