Abstract and Applied Analysis

Volume 2013 (2013), Article ID 454097, 12 pages

http://dx.doi.org/10.1155/2013/454097

Research Article

Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

Ruiqing Shi,1,2 Junmei Qi,² and Sanyi Tang¹

¹College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China

²School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 3 March 2013; Accepted 27 May 2013

Academic Editor: Peixuan Weng

Copyright © 2013 Ruiqing Shi 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 propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delay as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.

1. Introduction

Since the pioneering theoretical works by Lotka [1] and Volterra [2], there were a lot of authors who studied all kinds of predator-prey models modeled by ordinary differential equations (ODEs). To reflect that the dynamical behavior of the models depends on the past history of the system, it is often necessary to incorporate time delays into the models. Therefore, a more realistic predator-prey model should be described by delayed differential equations (DDEs) [3–11]. In general, delay differential equations exhibit more complicated dynamics on stability, periodic structure, bifurcation, and so on [12–26]. In [27, 28], the authors investigated the effect of the discrete delay on the stability of the model. In [29], the effect of the distributed delay on the stability of the model was investigated. In [11], the authors proposed a Logistic model with discrete and distributed delays:

where the parameters , , , are positive constants. The function in (1) is called the delayed kernel, which is the weight given to the population time units ago. And it was assumed that for all , together with the normalization condition

which ensures that the steady states of the model (1) are unaffected by the delay. They studied the stability of the positive equilibrium and existence of Hopf bifurcations, and direction and stability of the Hopf bifurcation were also analyzed. In [7], the authors proposed and investigated the following predator-prey model with time delay:

where and can be interpreted as the population densities of the prey and the predator at time , respectively. denotes the intrinsic growth rate of the prey, and denotes the death rate of the predator. For the convenience of computation, they chose the same as delays; the delay represents the feedback time delay of the prey species to the growth of itself in term , represents the feedback time delay of the predator species to the growth of itself in term , represents the hunting delay in term , and represents the time of the predator maturation in term . The parameters are all positive constants. They studied the stability of the positive equilibrium and existence of Hopf bifurcations.

Motivated by [7, 11, 27–29] and the references cited therein, in the present paper, we will consider the following predator-prey model with discrete and distributed delay:

where and can be interpreted as the population densities of the prey and the predator at time , respectively. denotes the intrinsic growth rate of the prey and denotes the death rate of the predator; represents the feedback time delay of the prey species to the growth of itself in term , represents the feedback time delay of the predator species to the growth of itself in term , and represents the hunting delay in term ; . The function is the same as the function in system (1). Following the ideas of Cushing [30], we define as the following weak kernel function:

Next, we define a new variable:

then using the linear chain trick technique, system (4) can be transformed into the following equivalent system:

The organization of this paper is as follows. In Section 2, we will consider the existence and stability of equilibria of system (7). The existence of Hopf bifurcation is also discussed. In Section 3, by use of normal form theory and central manifold argument, we illustrate the direction and stability of Hopf bifurcation. In Section 4, we provide an example with some numerical simulations to verify the theoretical results, and we also give some brief discussion.

2. Local Stability of Equilibria and the Existence of Hopf Bifurcations

In this section, we will finish two tasks: (a) investigating the existence and stability of equilibriums of system (7) and (b) studying the effect of time delay on the system (7); that is, we will choose as bifurcating parameter to analyze Hopf bifurcation.

Let the right equations of system (7) equal zero; we get the following algebraic equations:

By simple computation, we know that the trivial equilibrium and boundary equilibrium of system (7) always exist with values and , respectively. In addition, we have the following results.

(i)The eigenvalues of characteristic equations at the trivial equilibrium are , , and , which means that this equilibrium is always unstable.(ii)The eigenvalues of characteristic equations at the boundary equilibrium are and , and other eigenvalues are determined by . When , it is easy to see that , which means that this equilibrium is locally stable; whereas when , the sign of the real part of the eigenvalues can not be determined, which means that this equilibrium may be locally stable or unstable.

In fact, there exists a unique positive equilibrium for system (7) provided that , holds. Here

Next, we always assume that

(H1) holds.

The characteristic equation for system (7) at the equilibrium takes the form

where

Multiplying on both sides of (10), we obtain equivalent characteristic equation as

When , characteristic equation (10) or (12) becomes

It is easy to confirm that , and . By the Routh-Hurwitz criterion we know that all the roots of (13) have negative real parts. Thus, the positive equilibrium is locally asymptotically stable for .

Next, we will consider the eigenvalues of (12) for . Suppose that there is a pure imaginary root , , then we get

Separating the real and imaginary parts, we have

By simple calculation, we can obtain the following equations:

Let

then , can be written as

By adding the square of (18) and (19), we obtain

where

Denote , then (20) becomes

Let

then the following assumption holds true.

(H2) Equation (22) has at least one positive real root.

In fact, if all the parameters of system (7) are given, it is easy to calculate the root of (22) by using a computer. Since , we conclude that if , then (22) has at least one positive real root. Without loss of generality, we assume that (22) has six positive roots, defined by , , , , , and , respectively. Then (20) has six positive roots:

By (19), we get

If we denote

where and , then is a pair of purely imaginary roots of (12). Define

Here, we use the method by [25–27], which is different from [31].

In order to obtain the main result, it is necessary to make the following assumption.

(H3).

Taking the derivative of with respect to in (12), it is easy to obtain

which is equivalent to

By (12), we have

Take into the above equation, we get

Let

Then we get

Note that

Now, we can use the following lemma to get our result.

Lemma 1 (see [32]). 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.

Theorem 2. Suppose that (H1), (H2), and (H3) hold; then the following results hold.

(i)The positive equilibrium of system (7) (or the positive equilibrium of system (4)) is asymptotically stable for .(ii)The positive equilibrium of system (7) or system (4) undergoes a Hopf bifurcation when . That is, system (7) has a branch of periodic of solutions bifurcating from the positive equilibrium near .

3. Direction and Stability of the Hopf Bifurcation

In this section, following the ideas of [33], we derive the explicit formulae for determining the properties of the Hopf bifurcation at critical value of by using the normal form and the center manifold theory. Throughout this section, we always assume that system (7) undergoes Hopf bifurcation at the positive equilibrium for , and then is the corresponding purely imaginary roots of the characteristic equation at the positive equilibrium .

Let , , dropping the bars for simplification of notations; then system (7) is transformed into functional differential equations in as

where , , , and

where . By the Riesz representation theorem, there exists a function of bounded variation for , such that

In fact, we can choose

where is the Dirac delta function. For , define

Then system (37) is equivalent to

where . For , define

and a bilinear inner product

where . Then and are adjoint operators. By the discussion in Section 2, we know are eigenvalues of . Thus, they are also eigenvalues of . We need to compute the eigenvector of and corresponding to and , respectively.

Suppose that is the eigenvalues of corresponding to : then . It follows from the definition of and that

because ; then we get

where

Similarly, let be the eigenvalues of corresponding to ; according to the definition of we get

where , from which we obtain

By (45) we get

then we choose

such that , .

Next, we will use the ideas in [33] to compute the coordinates describing center manifold at . 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 (43), since , we have

where

From (53) and (54), we have

then we can obtain

From the definition of , we have

Comparing the coefficients with (56), we obtain

In order to determine we need to compute and . From (43) and (53), we have

where

Note that on the center manifold near to the origin,

thus we obtain

By (62) we know that for ,

Comparing the coefficients with (62), we get that

From (64), (66), and the definition of , we have

Noting , hence

where is a constant vector. Similarly, we have

where is a constant vector. In the following, we will find out and . From the definition of and (64), we can obtain

where . From (61) and (62) we have

That is,

By (39) and (53) we have

Thus,

Since is the eigenvalues of and is the corresponding eigenvector, we obtain

Thus, substituting (68) and (75) into (70), we have

or

From which we can get

Similarly, substituting (69) and (76) into (71), we can get

or

Thus, we can determine and from (68) and (69). Furthermore, can be expressed by the parameters and delay. Thus, we can compute the following values:

which determine the qualities of bifurcating periodic solution in the center manifold at critical value ; that is, determine the direction of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for ; determines the stability of the bifurcating periodic solution: the bifurcating periodic solution is stable (unstable) if ; and determines the period of the bifurcating periodic solution: the period increases (decreases) if .

4. Numerical Investigations and Discussion

In this paper, we propose a two-dimensional predatory-prey model with discrete and distributed delay. Then, by introducing a new variable, the original system is transformed into an equivalent three-dimensional system. In Section 2, we analyze the existence and local stability of the equilibria of the three-dimensional system. The condition for the existence of a Hopf bifurcation is also obtained. In Section 3, by the use of normal form theory and central manifold argument, we establish the formulae for the direction and the stability of the Hopf bifurcation.

In order to confirm our main results obtained in this work, we consider the following special system:

By simple calculation, it is easy to see that model (84) exists a unique positive equilibrium and . Note that the parameter set provided in model (84) satisfies the conditions of Theorem 2. When , the positive equilibrium is asymptotically stable, as shown in Figures 1(a) and 1(c). It follows from the discussion in Section 2 that , , and . Thus, is stable when , as indicated in Figures 1(a) and 1(c).

(a)

(b)

(c)

(d)

Figure 1: The stability of unique positive equilibrium . (a), (c) The equilibrium is stable for . (b), (d) The equilibrium is unstable and a stable periodic solution appears for .

When passes through the critical value , loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from , as shown in Figures 1(b) and 1(d). Since and , the Hopf bifurcation is supercritical and the direction of the bifurcation is and these bifurcating periodic solutions from are stable; please see Figures 1(b) and 1(d) and Figure 2(a) for . Note that the model (84) may have very complex dynamics if we choose the time delay as a bifurcation parameter. It follows from Figure 2 that the period of periodic solution is doubled as increase, and around (Figure 2(b)). If the time delay is increasing further, a periodic solution with 4-time period appears around (Figure 2(c)). Finally, the chaotic solution exists once the time delay reaches around (Figure 2(d)).

(a)

(b)

(c)

(d)

Figure 2: Bifurcation of model (84) with bifurcation parameter .

Both our theoretical and numerical results show that the positive equilibrium is asymptotically stable if , which indicates that the dynamical behavior is simple for the considered system. However, if , bifurcation and chaos may occur, which means that the considered system can take on very complex dynamics, and this may explain some complex phenomenon in the natural world.

Acknowledgments

The authors would like to thank the referees for their helpful suggestions, which improved the quality of this paper greatly. The first author is supported by the Postdoctoral Science Foundation of China (no. 2011M501428) and Young Science Funds of Shanxi (no. 2013021002-2). The third author is supported by the National Natural Science Foundation of China (nos. 11171199 and 81161120403).

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