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 , . Further, we have . Thus, conditions and hold. From Theorem 2, the positive equilibrium is asymptotically stable when as illustrated by Figure 1. When passes through the critical value , the positive equilibrium loses its stability and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the positive equilibrium , which can be shown as in Figure 2. Similarly, we have , for , . The corresponding waveforms and the phase plots are shown in Figures 3 and 4.
For , we can obtain and then we get . From Theorem 4, we know that when increases from zero to the critical value the positive equilibrium is asymptotically stable; then it will lose its stability and a Hopf bifurcation occurs once . These properties can be shown as in Figures 5 and 6.
For and , we can obtain , . By Theorem 5, the positive equilibrium is asymptotically stable when and is unstable when and a Hopf bifurcation occurs, which can be illustrated by Figures 7 and 8. In addition, by complex computations, we obtain , and further we have , , . By Theorem 6, we know that the Hopf bifurcation with respect to with is supercritical; the bifurcating periodic solutions are stable and decrease. From the viewpoint of ecology, if the periodic solutions bifurcating from the Hopf bifurcation are stable, the species in a prey-predator system may coexist in an oscillatory mode. Therefore, we can conclude that the three species in system (75) can coexist in an oscillatory mode, since the bifurcating periodic solutions are stable.
5. Conclusion
In this present paper a prey-predator system with disease in the prey and two delays is considered. Based on the system proposed in [6], we further incorporate the time delay due to the gestation of the predator. The main purpose of this paper is to investigate the effects of the two delays on the system. We have shown that the two delays play a complicated role in the system. By choosing the possible combinations of the two delays as bifurcation parameters, sufficient conditions for local stability and existence of local Hopf bifurcation are obtained. When the time delay is below the corresponding critical value, we get that the system is local stable. Otherwise, a local Hopf bifurcation occurs at the positive equilibrium. We also find that the delay due to the susceptible prey becoming the infected prey is more marked compared with the delay due to the gestation of the predator, because the critical value of is much smaller than that of when we only consider one of the two delays, which can be seen from the numerical simulations. Further, the properties of the bifurcated periodic solutions such as the direction and the stability are determined. And a numerical example is also given to support the theoretical results. From the numerical simulations we can see that the species in the system considered in this paper can coexist under some certain conditions.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is grateful to the referees and the editor for their valuable comments and suggestions on the paper. This work was supported by the Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2013A003, KJ2013B137).