Abstract
A delayed predator-prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of persistence theory on infinite dimensional systems, it is proved that the system is permanent. By using Lyapunov functions and the LaSalle invariant principle, the global stability of each of the feasible equilibria of the model is discussed. Numerical simulations are carried out to illustrate the main theoretical results.
1. Introduction
The predator-prey system is very important in population modelling and has been studied by many authors (see, e.g., [1–6]). A predator-prey model generally takes the form where and are the densities of prey and predator populations at time , respectively. The function represents the growth rate of the prey; represents the death rate and intraspecific competition rate of the predator; denotes the predator response function. In 1965, Holling [7] used the function as one of the predator response functions. It is now referred to as a Holling type II functional response. We note that in the models mentioned above, it is assumed that both the immature and the mature predators have the same ability to attack prey individuals. However, in the real world, almost all animals have stage structure of immature and mature, and only mature predators can attack prey and have reproductive ability. Stage-structured models have received great attention in recent years (see, e.g., [2–6]). In [2], Wang proposed a predator-prey system with Holling type II functional response and stage structure under the assumptions that the predator is divided into two groups, one is immature and the other is mature, and that only mature predators can attack prey and have reproductive ability, while immature predators do not attack prey and have no reproductive ability.
It is generally recognized that some kinds of time delays are inevitable in population interactions and tend to be destabilizing in the sense that longer delays may destroy the stability of positive equilibria (see [8]). Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Recently, great attention has been received and a large body of work has been carried out on the existence of Hopf bifurcations in delayed population models (see, e.g., [5, 6, 8, 9] and references cited therein).
Motivated by the work of [2, 6], in the present paper, we are concerned with the combined effects of stage structure for both the predator and the prey and time delay due to the gestation of the predator on the global dynamics of a predator-prey model with Holling type II functional response. To this end, we consider the following differential system: where and represent the densities of the immature and the mature prey at time , respectively; and represent the densities of the immature and the mature predators at time , respectively. The parameters , , , , , , , , , and are positive constants, in which is the birth rate of the prey; is the intraspecific competition rate of the mature prey; , , , and are the death rates of the immature prey, mature prey, immature predators, and mature predators, respectively; and are the transformation rates from the immature individuals to mature individuals for the prey and the predators, respectively; is the capturing rate of the predators; is the conversion rate of nutrients into the reproduction of the predators; is a constant delay due to the gestation of the predators. It is assumed in (2) that the mature individual predators feed on immature prey and have the ability to reproduce.
The initial conditions for system (2) take the form where , .
It is well known by the fundamental theory of functional differential equations [10] that system (2) has a unique solution satisfying initial conditions (3). It is easy to show that all solutions of system (2) corresponding to initial conditions (3) are defined on and remain positive for all .
The organization of this paper is as follows. In the next section, we investigate the local stability of each of the feasible equilibria of system (2). The existence of a Hopf bifurcation at the coexistence equilibrium is studied. In Section 3, by means of persistence theory on infinite dimensional systems, we prove that system (2) is permanent when the coexistence equilibrium exists. In Section 4, by using Lyapunov functionals and the LaSalle invariant principle, we show that both the prey and the predators go to extinction, if both the predator-extinction equilibrium and the coexistence equilibrium are not feasible, and that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium does not exist, and sufficient conditions are obtained for the global asymptotic stability of the coexistence equilibrium of system (2). A brief discussion is given in Section 5 to conclude this work.
2. Local Stability
In this section, we discuss the local stability of each equilibrium of system (2) and the existence of a Hopf bifurcation. It is easy to show that system (2) always has a trivial equilibrium and a predator-extinction equilibrium when , where Furthermore, if the following holds: , then system (2) has a unique coexistence equilibrium , where
The characteristic equation of system (2) at the equilibrium is of the form It is readily seen from (6) that if , then is locally asymptotically stable; if , then is unstable.
The characteristic equation of system (2) at the equilibrium takes the form It is easy to show that roots of have only negative real parts if . If holds, we have ; thus (7) has at least one positive real root. Therefore, is unstable. If , we have ; then the equilibrium is locally asymptotically stable when . It is easy to show that , . Therefore, if , by Lemma B in Kuang and So [1], we see that the equilibrium is locally asymptotically stable for all .
The characteristic equation of system (2) at the equilibrium is of the form where
When , (8) becomes By calculation we derive that Hence, by the Routh-Hurwitz criterion, we see that if the following hold:, ,then the equilibrium is locally asymptotically stable when .
If is a solution of (8), separating real and imaginary parts, we have Squaring and adding the two equations of (12), it follows that It is easy to show that If , that is,, then (13) has no positive real roots. It is easy to check that holds when holds. Accordingly, by Theorem in Kuang [8], we see that if and hold, then is locally asymptotically stable.
If the inequality in is reversed, then (13) has a unique positive root ; that is, (8) has a pair of purely imaginary roots of the form . Denote By Theorem in Kuang [8], we see that remains stable for .
We now claim that This will show that there exists at least one eigenvalue with a positive real part for . Moreover, the conditions for the existence of a Hopf bifurcation [10] are then satisfied yielding a periodic solution. To this end, differentiating (8) with respect to , it follows that Hence, a direct calculation shows that We derive from (12) that Hence it follows that Therefore, the transversal condition holds and a Hopf bifurcation occurs at .
In conclusion, we have the following results.
Theorem 1. For system (2), one has the following. (i)If , then the trivial equilibrium is locally asymptotically stable; if , then is unstable.(ii)If , then the predator-extinction equilibrium is locally asymptotically stable; if , then is unstable.(iii)Let hold. If holds, then the coexistence equilibrium is locally asymptotically stable for all ; if holds and the inequality in is reversed, then there exists a positive number , such that is locally asymptotically stable if and is unstable if . Further, system (2) undergoes a Hopf bifurcation at when .
We now give an example to illustrate the main results in Theorem 1.
Example 2. In (2), let , , , , , , , , , , and . It is easy to show that and ; that is, the condition holds. Hence, system (2) has a unique coexistence equilibrium . By calculation, we have , , and . By Theorem 1, is locally asymptotically stable if and is unstable if , and system (2) undergoes a Hopf bifurcation at when . Numerical simulation illustrates this fact (see Figure 1).
(a)
(b)
3. Permanence
In this section, we are concerned with the permanence of system (2).
Definition 3. System (2) is said to be permanent if there are positive constants , , , and , such that each positive solution of system (2) satisfies
Lemma 4. There are positive constants and , such that, for any positive solution of system (2),
Proof. Let be any positive solution of system (2) with initial conditions (3). Define Calculating the derivative of along positive solutions of system (2), it follows that where . This inequality yields . If we choose and , then (22) follows. This completes the proof.
In order to study the permanence of system (2), we refer to persistence theory on infinite dimensional systems developed by Hale and Waltman in [11].
Let be a complete metric space with metric . Suppose that is a continuous map with the following properties: where denotes the mapping from to given by . The distance of a point from a subset of is defined by . Recall that the positive orbit , and its -limit set is . Define , the strong stable set of a compact invariant set , to be .
Assume that is open and dense in and , . Moreover, the semigroup on satisfies Let and be the global attractor for . Define .
Lemma 5 (Hale and Waltman [11]). Suppose that satisfies and the following conditions: (i)there is a such that is compact for ;(ii)T(t) is point dissipative in ;(iii) is isolated and has an acyclic covering , where ;(iv) for .
Then is a uniform repeller with respect to ; that is, there is an such that, for any , .
We are now able to state and prove the result on the permanence of system (2).
Theorem 6. If holds, then system (2) is permanent.
Proof. We need only to show that the boundaries of repel positive solutions of system (2) uniformly. Let denote the space of continuous functions mapping into . Define
Denote and .
In the following, we show that the conditions in Lemma 5 are satisfied. By the definition of and , it is easy to see that and are positively invariant and the condition (ii) in Lemma 5 is clearly satisfied. Using the smoothing property of solutions of delay differential equations introduced in Kuang [8] (Theorem ), it follows that condition in Lemma 5 is satisfied. Thus, we need only to show that the conditions and hold. Clearly, corresponding to and , , , respectively, there are two constant solutions in and satisfying
We now verify the condition (iii) in Lemma 5. If is a solution of system (2) initiating from , then and , which yields . If is a solution of system (2) initiating from with , then it follows from the first and second equations of system (2) that and . If holds, then , as . Noting that , we see that the invariant sets and are isolated. Hence, is isolated and is an acyclic covering satisfying the condition in Lemma 5.
We now verify that and . Here, we only prove the second equation since the proof of the first equation is simple. Assume . Then there is a positive solution satisfying
Hence, for sufficiently small, there is a such that, if , .
Since holds, we can choose sufficiently small, such that
For sufficiently small satisfying (30), it follows from the third and the fourth equations of system (2) that, for ,
Define
Since has positive off-diagonal elements, by the Perron-Frobenius theorem, there is a positive eigenvector for the maximum eigenvalue of . Noting that (30) holds, a direct calculation shows that . Using a similar argument as that in the proof of Theorem 2.1 in [2], one can show that . This contradicts Lemma 4. Hence, we have . By Lemma 5, we conclude that repels positive solutions of system (2) uniformly. Therefore, system (2) is permanent. The proof is complete.
4. Global Stability
In this section, we are concerned with the global stability of each of the feasible equilibria of system (2). The strategy of proofs is to use Lyapunov functionals and the LaSalle invariant principle.
Theorem 7. If , then the trivial equilibrium of system (2) is globally asymptotically stable.
Proof. Let be any positive solution of system (2) with initial conditions (3). By Theorem 1, we see that if , then is locally asymptotically stable. Define Calculating the derivative of along positive solutions of system (2), it follows that If , it then follows from (34) that . By Theorem in [10], solutions approach , the largest invariant subset of . Clearly, we see from (34) that if and only if , . Noting that is invariant, for each element in , we have , . It therefore follows from the second and fourth equations of system (2) that which yields , . Hence, if and only if . Accordingly, the global asymptotic stability of follows from LaSalle’s invariant principle. This completes the proof.
Theorem 8. The predator-extinction equilibrium of system (2) is globally asymptotically stable provided that.
Proof. Assume that is any positive solution of system (2) with initial conditions (3). By Theorem 1, we see that if holds, then is locally asymptotically stable. System (2) can be rewritten as
Define
where , , and . Calculating the derivative of along positive solutions of system (2), it follows that
Define
We derive from (38) and (39) that
If holds, it then follows from (40) that . By Theorem in [10], solutions approach , the largest invariant subset of . Clearly, we see from (40) that with equality if only if , , and . It follows from the fourth equation of system (2) that , which yields . Hence, if only if , , , and . Using the LaSalle invariant principle, the global asymptotic stability of follows. This completes the proof.
Theorem 9. The coexistence equilibrium of system (2) is globally asymptotically stable provided that. Here, is the uniform persistency constant for satisfying .
Proof. Let be any positive solution of system (2) with initial conditions (3). Since holds, there is a , such that
for all . Accordingly, we have
In this case, it is easy to show that and hold. By Theorem 1, is locally asymptotically stable for all .
System (2) can be rewritten as
Define
where , , . Calculating the derivative of along positive solutions of system (2), it follows that
Define
We derive from (45) and (46) that
If holds, for sufficiently enough, we have . This, together with (47), implies that , with equality if and only if
We now look for the invariant subset within the set
Since and on and consequently , which yields , it follows from the fourth equation of system (2) that , which leads to . Hence, the only invariant set in is . Using the LaSalle invariant principle, the global asymptotic stability of follows. This completes the proof.
We give an example to illustrate the result in Theorem 9.
Example 10. In (2), let , , , , , , , , , , and . It is easy to show that and ; that is, condition holds. Hence, system (2) has a unique coexistence equilibrium . Hence, by Theorem 6, system (2) is permanent. From the proof of Lemma 4, we have . Hence, for sufficiently small, there is a such that, if , . It follows from system (2) that, for , which yields By calculation, we derive that and . By Theorem 9, is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 2).
5. Discussion
In this paper, we have incorporated stage structure for both the predators and the prey into a predator-prey model with time delay due to the gestation of the predator and Holling type II functional response. By using Lyapunov functionals and the LaSalle invariant principle, we have established sufficient conditions for the globally stability of each of the feasible equilibria of the system. As a result, we have shown the threshold for the permanence and extinction of the system. By Theorems 7–9, we see that (i) if , then both the prey and the predator population go to extinction; (ii) the prey species is permanent but the predator becomes extinct if and only if ; (iii) if holds, then both the prey and predator species of system (2) are permanent.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (no. 11101117).