Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting
A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delay passes through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.
The classical predator-prey systems have been extensively investigated in recent years, and they will continue to be one of the dominant themes in the future due to their universal existence and importance. Many biological phenomena are always described by differential equations, difference equations, and other type equations. In general, delay differential equations exhibit more complicated dynamical behaviors than ordinary ones; for example, the delay can induce the loss of stability, various oscillations, and periodic solutions. The dynamical behaviors of delay differential equations, stability, bifurcation and chaos, and so forth have been paid much attention by many researchers. Especially, the direction and stability of Hopf bifurcation to delay differential equations have been investigated extensively in recent work (see [1–7] and references therein).
After the classical predator-prey model was first proposed and discussed by May in , there were some similar topics, regarding persistence, local and global stabilities of equilibria, and other dynamical behaviors (see [5, 9, 10] and references therein). Recently, Song and Wei in  had considered a delayed predator-prey system as follows: where and were the densities of prey species and predator species at time , respectively. The local Hopf bifurcation and the existence of the periodic solution bifurcating of system (1) was investigated in . When selective harvesting was put into the predator-prey model similar to (1), Kar  studied two predator-prey models with selective harvesting; that is, in the first model, selective harvesting of predator species: and, in the second model, selective harvesting of prey species: had been considered by incorporating time delay on the harvesting term. They found that the delay for selective harvesting could induce the switching of stability and Hopf bifurcation occurred at .
Recently, Kar and Ghorai  had investigated a predator-prey model with harvesting: They obtained the local stability, global stability, influence of the harvesting, direction of Hopf bifurcation and the stability to system (4). Motivated by models (1)–(4), we will consider a predator-prey system with delay incorporating harvests to predator and prey: where and represent the population densities of prey species and predator species, respectively, at time ; , , , , , and are model parameters assuming only positive values; measures the scale whose environment provides protection to prey ; denotes the scale whose environment provides protection to predator ; means the period of pregnancy; represents the number of prey species which was born at time and still survived at time ; and represent the coefficients of prey species and predator species, respectively. We always assume that in this paper.
The organization of the paper is as follows. The stability of the positive equilibrium and the existence of the Hopf bifurcation are discussed in Section 2. The effect of harvesting to prey species and predator species is investigated in Section 3. The direction of Hopf bifurcation and stability of the corresponding periodic solution are obtained in Section 4. Numerical simulations are carried out to illustrate our results in Section 5.
2. Stability of Positive Equilibrium and Hopf Bifurcation
By simple computation, if holds, system (5) admits a unique positive equilibrium : Let , , and then we get the linear system of (5): where , , , . From linear system (5) the characteristic equation is as follows: Roots of system (8) imply the stability of the equilibrium and Hopf bifurcation of system (5). Obviously, is not a root of system (8). For , system (8) becomes It is obvious that the root of system (9) has negative real part. Now, for , if is a root of (8), then we have Furthermore, which lead to polynomial equation It is easy to see that (12) has one positive root where . By (11), one gets that Let be a pair of purely imaginary roots of (8), such that Next, we will prove meets the transversality conditions; taking the derivative of system (8) with respect to , one derives that which, together with (11), leads to So, we have Thus, we can obtain the following lemma.
Lemma 1. If holds, then the following results are true:(i)when , the positive equilibrium of of system (5) is locally asymptotically stable;(ii)when , the positive equilibrium of of system (5) is locally asymptotically stable, and is unstable when , where , () can be defined in (13), (14).
3. The Influence of Harvesting
Next, we will discuss the influence of the harvesting on system (5).
Case 1 (only predator species is harvested). For , and the positive equilibrium of system (5) changes to , where it is obvious that and if and only if . Obviously, and are the continuous differentiable functions with respect to ; then, we have
Theorem 2. If holds, then is the monotonic increasing function of , is the monotonic decreasing function of ; that is, when increases, the density of prey species will increase, the density of predator species will decrease.
Case 2 (only prey species is harvested). For , and the positive equilibrium of system (5) changes to , where it is obvious that and if and only if . Obviously, and are the continuous differentiable functions with respect to ; then, one get that
Theorem 3. If holds, then and are the monotonic decreasing functions of ; that is, if increases, then the density of prey species and predator species will decrease; on the contrary, if decreases, the density of prey species and predator species will increase.
Case 3 (predator species and prey species are harvested simultaneously). For , the mixed derivative of and are given by
Theorem 4. If is valid, then the densities of prey species and predator species will both decrease when harvesting rate increases; on the contrary, the density of prey species will increase and predator species will decrease when harvesting rate increases.
4. Direction and Stability of Hopf Bifurcation
Motivated by the ideas of Hassard et al. , by applying the normal form theory and the center manifold theorem, the properties of the Hopf bifurcation at the critical value are derived in this section.
By Riesz representation theorem, there exists a function of bounded variation for , such that We choose where is the Dirac delta function. For , we define Then, system (25) can be transformed into an operator differential equation of the form where , for . For , we define and a bilinear inner product where ; then, and are adjoint operators. Noting that are eigenvalues of , thus, they are also eigenvalues of . In order to calculate the eigenvector of corresponding to the eigenvalue and of corresponding to the eigenvalue , let be the eigenvector of corresponding to ; then, .
In view of standardization of and ; that is, , we have Thus, choose . Next, we will quote the same notation (see ), we first compute the coordinates to describe the center manifold at . Define On the center manifold , we have and are local coordinates for center manifold in the direction and ; noting that is real if is real, we only consider real solution of (25). Since , then we have We rewrite this equation as where Noting and , we have From (27), (42), we obtain that Because contains and , from (32) and (38), we have where Substituting the corresponding series into (45) and comparing the coefficients, we have From (45), we know that for , we have Comparing the coefficient with (46) yields that for From (47), (49) and the definition of , it follows that taking notice of ; hence, where is a constant vector. By the similar way, we have where is a constant vector.
Next, computing and , from the definition of and (47), one then obtains where . Furthermore, we haveSubstituting (52) and (56) into (54) and noting that it implies that Namely, Then it yields thatwhere Similarly, we getwhere Through simple computation, we determine , from (52) and (53); further, we can determine . Therefore, in (44) can be expressed by the parameter and delay; hence, which determine the qualities of bifurcation periodic solution of the critical value .
Theorem 5. (i) determines the direction of Hopf bifurcation: if (), then Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist for ().
(ii) determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if (). determines the period of the bifurcating periodic solution: the period increases (decrease) if ().
5. Numerical Simulations
In this section, we consider a delayed predator-prey system with harvesting as follows: Because holds, from (14), we obtain that The unique positive equilibrium is .
If , when decreases, then prey species decreases and predator species increases (see Figure 1); when increases, prey species increases and predator species decreases (see Figure 2); If , when the values of harvesting decreases, then both predator species and prey species will increase (see Figure 3); on the other hand, when increases, then both predator species and prey species will decrease (see Figure 4).
When parameter is little bigger than the critical value , system (5) will become unstable and predator species and prey species can coexist; when increases much more, prey species will go to extinct (see Figure 5). Moreover, from Figure 6, we can see that system (5) is unstable when passes through the critical value . By controlling the harvesting rates and , respectively, the stability of positive equilibrium to system (5) can been changed. Similarly, when , system (5) is stable; if we decrease the harvesting rate , then the stable system becomes unstable one (see Figure 7).
Since , , Hopf bifurcation is subcritical and the positive equilibrium is asymptotically stable for (see Figure 8); when , loses its stability and Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from (see Figure 9).
In our model, the harvesting term has been introduced into the model (5); by applying the normal form theorem and the center manifold theorem, we investigate the dynamical behaviors of the delayed predator-prey model with harvesting term and obtain the influence of harvesting term on the prey species and predator species. Further, we prove that the influence of the harvesting rates and to the stability of system (5), by controlling harvesting rates and of prey species and predator species, which makes the unstable (stable) system become stable (unstable).
Our results show that Hopf bifurcations occur as the delay passes through critical values . When , the positive equilibrium of system (5) is asymptotically stable; when , the positive equilibrium of system (5) loses its stability and Hopf bifurcations occur.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by National Natural Science Foundation of China (no. 11201075) and Natural Science Foundation of Fujian Province of China (no. 2010J01005).
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