Stability and Hopf Bifurcation in a Prey-Predator System with Disease in the Prey and Two Delays

Abstract

This paper is concerned with a prey-predator system with disease in the prey and two delays. Local stability of the positive equilibrium of the system and existence of local Hopf bifurcation are investigated by choosing different combinations of the two delays as bifurcation parameters. For further investigation, the direction and the stability of the Hopf bifurcation are determined by using the normal form method and center manifold theorem. Finally, some numerical simulations are given to support the theoretical analysis.

1. Introduction

The effect of disease in ecological system is an important issue from mathematical as well as ecological point of view. Therefore, the dynamics of epidemiological models have been investigated by many authors in recent years [1–6]. In [6], Jana and Kar proposed and investigated the following predator-prey system with disease in the prey: where , denote the population densities of the susceptible prey and the infected prey at time , respectively. denotes the population of the predator at time . The susceptible prey grows logistically with the intrinsic growth rate and the carrying capacity . The conversion from the susceptible prey to the infected prey is governed by the response function . The consumption of the susceptible prey by the predator is governed by the response function . is the removal rate of the infected prey biomass. is the intraspecific competition coefficient of the predator. is the removal rate of the predator due to natural death or harvesting. And the constant () is the time delay due to susceptible prey which becomes the infected prey. The predator-prey system with single delay has been investigated by many researchers [7–11]. Jana and Kar [6] studied the boundedness of the solutions and stability of the positive equilibrium of system (1). Existence and properties of the Hopf bifurcation were also investigated.

In recent years, there are also some papers on the bifurcations of a prey-predator system with two or multiple delays [12–16]. As is known to all, the consumption of the susceptible prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, it is reasonable to incorporate time delay due to the gestation of the predator into system (1). Based on this consideration, we consider the following system with two delays in this paper: where is the time delay due to susceptible prey which becomes the infected prey and is the time delay due to the gestation of the predator.

This paper is organized as follows. In Section 2, we investigate local stability of the positive equilibrium and existence of local Hopf bifurcation of system (2) with respect to both delays. In Section 3, by using the normal form method and center manifold theorem, the properties of the Hopf bifurcation such as direction and stability are determined. Some numerical simulations are given for the support of the analytical findings in Section 4.

2. Local Stability and Hopf Bifurcation

According to the analysis in [6], system (2) has a unique positive equilibrium if , and , where , , .

Let , , . Dropping the bars for convenience, system (2) becomes the following form: where with The linearized system of (3) is Thus, we can get that the characteristic equation of system (7) is where

Case 1 (). Equation (8) becomes where

It follows from the Routh-Hurwitz criteria that all roots of (10) have negative real parts if the following condition holds: : and . Then, the positive equilibrium of system (2) without delay is locally asymptotically stable.

Case 2 (). On substituting , (8) becomes where Let be a root of (12). Then, we have which implies that where Denote ; then (15) becomes Let
In [17], Song et al. obtained the following results on the distribution of roots of (17).

Lemma 1. For (17),(1)if , then (17) has at least one positive root;(2)if and , then (17) has no positive roots;(3)if and , then (17) has positive root if and only if and .
Suppose that (17) has at least one positive root.
Without loss of generality, we assume that (17) has three positive roots, which are denoted by , , and . Then (15) has three positive roots , . For every fixed , one can get Thus, with ; .
Let Differentiating both sides of (12) with respect to we can get Thus, we have where . Obviously, if the condition holds, then . By the Hopf bifurcation theorem in [18], we have the following results.

Theorem 2. Suppose that conditions - hold. The positive equilibrium of system (2) is asymptotically stable for and system (2) undergoes a Hopf bifurcation at when .

Case 3 (). Substitute into (8), then (8) becomes where Let () be a root of (24). Then, we get which follows that where Denote ; then (27) becomes Let
Similarly as in Case 2, we suppose that , (29) has at least one positive root. Without loss of generality, we assume that it has three positive roots and we denote them by , , and , respectively. Then (27) has three positive roots , . For every fixed , Thus, with ; .
Let Differentiating (24) regarding , we get Then, we can get where . Therefore, if the condition : holds, then . Thus, by the Hopf bifurcation theorem in [18], we have the following results.

Theorem 3. Suppose that conditions - hold. The positive equilibrium of system (2) is asymptotically stable for and system (2) undergoes a Hopf bifurcation at when .

Case 4 (). Let ; then (8) becomes where Multiplying (37) by , then (37) becomes Let be the root of (39); then we can get which follows that where Then, we can obtain where Let ; then (43) becomes

If we know all the coefficients of system (2), then we can get all the coefficients of (45) and then all the roots of (45) can be obtained by Matlab. Therefore, we give the following assumption.

Suppose that : (45) has at least one positive root.

Without loss of generality, we assume that (45) has six positive roots, which are denoted by , respectively. Then, (43) has six positive roots , . For every , with ; .

Let Next, taking the derivative of with respect to in (39), we have Then we have where

Obviously, if condition : holds, then . Thus, by the discussion above and the Hopf bifurcation theorem in [18], we have the following results.

Theorem 4. Suppose that conditions - hold. The positive equilibrium of system (2) is asymptotically stable for and system (2) undergoes a Hopf bifurcation at when .

Case 5 ( and ). Let be the root of (8). Then, we get where Then, we can have where

We suppose that , (55) has at least finite positive roots. And we denote the positive roots of (55) by . Then, for every fixed (), Thus, with ; .

Let . When , (8) has a pair of purely imaginary roots for . Next, in order to give the main results with respect to , , we give the following assumption: : .

Through the analysis above and the Hopf bifurcation theorem in [18], we have the following results.

Theorem 5. If conditions - hold and , then the positive equilibrium of system (2) is asymptotically stable for and system (2) undergoes a Hopf bifurcation at when .

3. Stability of Bifurcating Periodic Solutions

In the previous section, it is shown that system (2) undergoes a Hopf bifurcation for different combinations of and under certain conditions. In this section, the properties of Hopf bifurcation such as direction and stability are investigated with respect to for by using the normal form method and center manifold theorem in [18]. Throughout this section, we assume that where .

For convenience, let , so that is the Hopf bifurcation value of system (2). Let , , and rescale the time delay ; then system (2) can be rewritten as where with

Therefore, according to the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation such that In fact, we choose For , we define Then system (59) can be transformed into the following operator equation: where for .

For , where is the 3-dimensional space of row vectors, we define the adjoint operator of : and a bilinear inner product where .

Then and are adjoint operators. From the discussion above, we know that are eigenvalues of and they are also eigenvalues of .

Let be the eigenvectors of corresponding to the eigenvalue and the eigenvectors of corresponding to the eigenvalue .

It is not difficult to verify that From (67), we can get such that , .

In the remainder of this section, we obtain the coefficients used to determine the properties of the periodic solution by the algorithms given in [18] and using a computation process similar to that in [11]: with where and can be computed as the following equations, respectively: with Therefore, we can calculate the following values:

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

Theorem 6. For system (2),(i) determines the direction of the Hopf bifurcation. If (), then the Hopf bifurcation is supercritical (subcritical).(ii) determines the stability of the bifurcating periodic solutions. If (),   then the bifurcating periodic solutions are stable (unstable).(iii) determines the period of the bifurcating periodic solutions. If (), then the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Example

In this section, we give a numerical example to support the theoretical results in Sections 2 and 3. We use the same coefficients which are used by Jana and Kar in [6]. They are as follows: , , , , , , , , , and . Thus, we get the following particular case of system (2): which has a positive equilibrium .

For . We get ,