Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

and Applied Analysis 3 which, together with (11), leads to Re [dλ dτ ] −1


Introduction
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][2][3][4][5][6][7] and references therein).
After the classical predator-prey model was first proposed and discussed by May in [8], 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 [7] 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 [7].When selective harvesting was put into the predator-prey model similar to (1), Kar [11] 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  =  0 .
ẋ () =  1  () −  1  2 () −  1  ()  ()  () +  1 −  1  () , 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 ; , , ℎ 1 , ℎ 2 ,  1 , and  2 are model parameters assuming only positive values;  1 measures the scale whose environment provides protection to prey ;  2 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 ; ℎ 1 and ℎ 2 represent the coefficients of prey species and predator species, respectively.We always assume that 0 ≤ ℎ 1 ≤ ℎ 2 < 1 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.

The Influence of Harvesting
Next, we will discuss the influence of the harvesting on system (5).
Case 2 (only prey species is harvested).For ℎ 2 = 0, and the positive equilibrium of system (5) , where it is obvious that  * 2 > 0 and  * 2 > 0 if and only if  1 −  2  > 0. Obviously,  * 2 and  * 2 are the continuous differentiable functions with respect to ℎ 1 ; then, one get that and  * 2 are the monotonic decreasing functions of ℎ 1 ; that is, if ℎ 1 increases, then the density of prey species and predator species will decrease; on the contrary, if ℎ 1 decreases, the density of prey species and predator species will increase.Case 3 (predator species and prey species are harvested simultaneously).For ℎ 1 ℎ 2 ̸ = 0, the mixed derivative of  * and  * are given by Theorem 4. If  1 +  2 (ℎ 2 − 1) > 0 is valid, then the densities of prey species and predator species will both decrease when harvesting rate ℎ 1 increases; on the contrary, the density of prey species will increase and predator species will decrease when harvesting rate ℎ 2 increases.

Direction and Stability of Hopf Bifurcation
Motivated by the ideas of Hassard et al. [12], 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.

Numerical Simulations
In this section, we consider a delayed predator-prey system with harvesting as follows: Because ( 1 ) holds, from (14), we obtain that The unique positive equilibrium is  * = (2.907,3.436).If ℎ 1 = 0.4, when ℎ 2 decreases, then prey species decreases and predator species increases (see Figure 1); when ℎ 2 increases, prey species increases and predator species decreases (see Figure 2); If ℎ 2 = 0.5, when the values of harvesting ℎ 1 decreases, then both predator species and prey species will increase (see Figure 3); on the other hand, when ℎ 1 increases, then both predator species and prey species will decrease (see Figure 4).
When parameter  is little bigger than the critical value  0 , 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  0 .By controlling the harvesting rates ℎ 1 and ℎ 2 , respectively, the stability of positive equilibrium to system (5) can been changed.Similarly, when  <  0 , system (5) is stable; if we decrease the harvesting rate ℎ 2 , then the stable system becomes unstable one (see Figure 7).
Since  2 < 0,  < 0, Hopf bifurcation is subcritical and the positive equilibrium  * is asymptotically stable for 0 <  <  0 (see Figure 8); when  >  0 ,  * loses its stability and Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from  * (see Figure 9).As discussed, our results show that the delay  affects the stability of system (5) and harvesting rates ℎ 1 and ℎ 2 also affect the stability of system (5).

Conclusion
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 ℎ 1 and ℎ 2 to the stability of system (5), by controlling harvesting rates ℎ 1 and ℎ 2 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  0 ≈ 2.8015.When  <  0 , the positive equilibrium  * of system (5) is asymptotically stable; when  >  0 , the positive equilibrium  * of system (5) loses its stability and Hopf bifurcations occur.