Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response

Abstract

A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.

1. Introduction

The dynamic relationship between prey and predators has long been and will continue to be one of the dominant subjects in mathematical ecology due to its universal existence and importance [1–12]. In [13, 14], the author proposed the following predator-prey model based on the model in May [15]: where and denote the population of prey and predator, respectively, and and are the intrinsic growth rates of prey and predator, respectively. The parameter represents the carrying capacity of the prey and the ratio represents the carrying capacity of the predator. It has been assumed that both prey and predator populations grow logistically and that the predator consumes the prey according to a functional .

In recent years, models with time delay have been extensively studied by many authors [5, 16–26]. The authors of [1] discussed model (1) with a discrete delay: and obtained the stability of equilibria, the existence of Hopf bifurcation, and the direction of bifurcating periodic solutions. This paper focuses mainly on the effects of both spatial diffusion and time delay on system (1). It is assumed that the delay affects predation and consumption and that the system has homogeneous Neumann boundary conditions: where and are the diffusion coefficients of prey and predator, respectively, denotes the Laplacian operator, and is the outward unit normal vector on . For convenience, it is assumed that , and that all parameters are positive.

The rest of this paper is structured as follows. In Section 2, the local stability of equilibria is analyzed using the associated characteristic equations, and the occurrence of the Hopf bifurcation with time delays is presented. In Section 3, the global asymptotical stability of the interior equilibrium for any is proved by means of the upper-lower solution method. In Section 4, using normal form theory and the center manifold theorem, the stability and direction of bifurcating periodic orbits are investigated. Finally, numerical simulations and a brief discussion are presented.

2. Local Stability and Hopf Bifurcation Analysis

In this section, the local stability of the equilibria of system (3) is analyzed. Denote for , ; then is a Hilbert space. In the abstract space , system (3) can be regarded as an abstract functional differential equation.

System (3) has two nonnegative equilibria and , where and is the positive root of the equation Let ; then , , which guarantee the existence of .

From analysis of the characteristic equation of , it can easily be determined that it always has a saddle point. To analyze the stability of the positive equilibrium , the first step is to linearize system (3) at : where , and is defined as for .

The characteristic equation of (7) is

Recall that under the Neumann boundary condition has eigenvalues and , with the corresponding eigenfunctions . Substituting into (10), the following expression results: Hence, it can be concluded that the characteristic equation (10) is equivalent to the equation where

The stability of the positive equilibrium can be determined by the distribution of the roots of (13). It is locally asymptotically stable if all the roots of (13) have negative real parts for all . Obviously, 0 is not a root of (13) for all . When as well as , (13) can be simplified as It can be verified that that is, which requires that or Therefore, the following theorem can be stated.

Theorem 1. If holds, the interior equilibrium of system (3) with is asymptotically stable.
When , assume that is a root of (13). Substituting it into (13) yields: that is, where

This leads to the following theorem.

Theorem 2. If and hold for , then the interior equilibrium of system (3) is asymptotically stable for all .

Proof. If , then there exists such that, for , (20) has a unique positive real root: From (20), it follows that Then (13) has a pair of pure imaginary roots when where Substituting into (13) and taking the derivatives with respect to lead to For , considering , it follows that In other words, .

From the above discussion, the following theorem can be stated.

Theorem 3. If holds, then the following statements are true.(1)If  , then the interior equilibrium of system (3) is asymptotically stable.(2)If , then the interior equilibrium of system (3) is unstable.(3)System (3) undergoes a Hopf bifurcation at the interior equilibrium for .

3. Global Stability

This section mainly proves that the interior equilibrium is globally asymptotically stable with the upper-lower solution method in [27, 28]. For simplicity, let denote and .

Lemma 4 (see [29]). Assume that is defined by Then .

Theorem 5. If , then for system (3), the positive equilibrium is globally asymptotically stable.

Proof. From the maximum principle of parabolic equations, it is known that for any initial value , the corresponding nonnegative solution is strictly positive for . Because , it is possible to choose satisfying Because according to Lemma 4 and the comparison principle of parabolic equations, there exists such that, for any , . This in turn implies that for .
Hence there exists such that, for any , Consequently, for .
Because , it can be easily verified that Hence, there exists such that, for any , This implies that for . Again it can be verified that and hence there exists such that, for any , Therefore, for , it is possible to obtain and satisfy This implies that and are a pair of coupled upper and lower solutions of system (3), as in the definition in [29], for the reason that (3) is a mixed quasimonotonic system.
It is clear that there exists such that, for any , Define two iteration sequences and as follows: for , where , .
Then, for , and there exist and such that , , , , and Because is the unique positive constant equilibrium of system (3), it must hold that .
According to the results of [27, 28], the solution of system (3) satisfies Hence, the constant equilibrium is globally asymptotically stable.

4. Direction and Stability of Hopf Bifurcation

Part Two has already shown that system (3) undergoes Hopf bifurcation at the interior equilibrium for . In this section, the direction, stability, and period of the periodic solutions from the steady state will be studied by applying the method introduced by Hassard et al. [30] and the center manifold theorem due to [31–35].

For convenience, let , , . Then system (3) is equal to a single-delay system:

Let , , . Then is the Hopf bifurcation value of system (48), which can be written as follows: where with for .

From the previous discussion, it is known that is a pair of simple purely imaginary eigenvalues of the linear system (7) and the following linear functional differential equation: By the Riesz representation theorem, there exists a 2 × 2 matrix function , whose elements are bounded variables, such that In fact, it is possible to choose where is a Dirac delta function satisfying For , define as For , define Then and are adjoint operators under the bilinear form: Therefore, are eigenvalues of as well as . Next, the eigenvectors of and corresponding to the eigenvalues and can be calculated. Let Under the condition , that is, it follows that Similarly, let , and with , that is, it is possible to obtain According to the conditions and , Therefore, Let , , ; then Therefore, the center subspace of system (52) is , and the adjoint subspace is . Let , where Using the notion from [30], it is also possible to define for .

Define for and for , .

Hence, for . Then the center subspace of linear system (7) is given by , where and , where is a stable subspace.

According to [30], it is known that the infinitesimal generator of linear system (7) satisfies Moreover, if and only if Setting in (49), the center manifold can be obtained in . The flow of system (49) can be written as where From Taylor’s formula, where Let , where From (77),

In the last formula, , , , and are still unknown. Therefore, it is necessary to compute and , as described below.

From (74), it is possible to obtain In addition, satisfies Therefore, For , and therefore From (84) and (86), Solving for , Similarly, where and are both two-dimensional vectors and can be determined as follows.

For , according to the definition of and the first two equations of (84), Then Note that It follows that Therefore, it is possible to obtain where Similarly, Then Therefore, it is possible to obtain where Then is expressed by the parameters. Therefore, the following values can be computed:

Theorem 6. If , , and are defined as above, then the following are true:(i)if (), then the Hopf bifurcation is supercritical (subcritical);(ii)if (), then the periodic solutions are stable (unstable);(iii)if (), then the period of the bifurcating periodic solutions of system (3) increases (decreases).

Remark 7. The direction and stability of the Hopf bifurcation have already been computed when . The other Hopf bifurcation value can be analyzed using the same procedure.

5. Simulations and Discussion

This paper has considered a delayed predator-prey model with modified Holling-Tanner functional response. The effect of time delays on the positive equilibrium of system (3) has been investigated. The results obtained here show that the time delay plays an important role in determining system stability. Theorem 3 shows that the time delay can cause a stable equilibrium to become unstable or, in other words, that there is a critical value , such that, for , equilibrium is stable; when approaches , equilibrium bifurcates into periodic solutions; and when , the equilibrium becomes unstable.

Numerical simulations are performed to illustrate the dynamic behavior of system (3). Assume that , , , , , , , , and . In this case, . As shown in Figure 1, the positive equilibrium is stable when , and the predators and prey tend to a steady state. However, if , the positive equilibrium loses its stability and Hopf bifurcation occurs, leading to oscillatory behavior. These results are identical to the numerical results illustrated in Figure 2.

(a)

(b)

Figure 1 

Convergence to positive equilibrium. Here the parameters are assumed to be , , and the initial values are , , and , .

(a)

(b)

Figure 2 

Convergence to positive equilibrium. Here the parameters are assumed to be , , and the initial values , , and , .

In both figures, the upper diagram represents the prey and the lower diagram represents the predator.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001), by the National Natural Science Foundation of China (Grant no. 31370381), by the National Key Basic Research Program of China (973 Program, Grant no. 2012CB426510), and by the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010010).