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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 724325, 10 pages
Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response
Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang, Hebei 050003, China
Received 21 May 2013; Accepted 27 October 2013
Academic Editor: Qi-Ru Wang
Copyright © 2013 Lili Wang and Rui Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A Holling type II predator-prey model with time delay and stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. The existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and the sufficient conditions are obtained for the global stability of the coexistence equilibrium.
In population dynamics, the functional response of predator to prey density refers to the change in the density of prey attacked per unit time per predator as the prey density changes . Based on experiments, Holling  suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. The most popular functional response used in the modelling of predator-prey systems is Holling type II with which takes into account the time a predator uses in handing the prey being captured. There has been a large body of work about predator-prey systems with Holling type II functional responses, and many good results have been obtained (see, e.g., [1, 3, 4]).
Time delays of one type or another have been incorporated into biological models by many researchers. We refer to the monographs of Gopalsamy , Kuang , and Wangersky and Cunningham  on delayed predator-prey systems. In these research works, it is shown that a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. Hence, delay differential equations exhibit more complex dynamics than ordinary differential equations. Time delay due to gestation is a common example, since generally the consumption of prey by the predator throughout its past history decides the present birth rate of the predator. In , Wangersky and Cunningham proposed and studied the following non-Kolmogorov-type predator-prey model: In this model, it is assumed that a duration of time units elapses when an individual prey is killed and the moment when the corresponding addition is made to the predator population.
In natural world, there are many species whose individuals have a history that can be divided into two stages immature and mature. Usually the dynamics-eating habits of predator are often quite different in different stages. Generally speaking, the immature predators are raised by their parents and do not have the ability to attack prey, so the rate at which they attack prey and the reproductive rate can be ignored. Hence, it is of ecological importance to investigate predator-prey models with stage structure. In recent years, the predator-prey population models with stage structure have received much attraction (see, e.g., [8–10]). In , it was assumed that feeding on prey can only make contribution to the increasing of the physique of the predator and does not make contribution to the reproductive ability, and the following strengthen type predator-prey model with stage structure was studied: where represents the density of the prey at time and and represent the densities of the immature and the mature predator at time , respectively. describes the Holling type II functional response; and represent the effects of capturing rate and handling time, respectively. denotes the intraspecific competition rate of the prey, denotes the birth rate of immature predator, denotes the intrinsic growth rate of the prey, denotes the rate of conversing prey into new mature predator, denotes the death rate of the immature predator, denotes the death rate of the mature predator, and denotes the rate of immature predator becoming mature predator. All parameters are positive constants. In , sufficient conditions were derived in for the global asymptotic stability of nonnegative equilibria of the model by constructing suitable Lyapunov functions.
Motivated by the work of Wangersky and Cunningham  and Tian and Xu , we are concerned with the combined effects of the stage structure for the predator and time delay due to the gestation of mature predator on the global dynamics of a predator-prey model with Holling type II functional response. To this end, we consider the following delay differential system: where the meanings of the variables , , , and the parameters , , , , , , , , are the same as those of system (2) and the constant denotes the time delay due to the gestation of the mature predator. This is based on the assumption that the change rate of predators depends on the number of prey and of mature predators present at some previous time.
The initial conditions for system (3) take the form where
By the fundamental theory of functional differential equations , it is well known that system (3) has a unique solution satisfying initial conditions (4). Further, it is easy to show that all solutions of system (3) are defined on and remain positive for all .
The paper is organized as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (3) is discussed and the existence of Hopf bifurcations at the coexistence equilibrium is established. In Section 3, permanence of the system (3) is proved by means of the persistence theory on infinite dimensional systems. In Section 4, by using Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are received for the global asymptotic stability of the predator-extinction equilibrium and the coexistence equilibrium.
2. Local Stability and Hopf Bifurcation
In this section, we discuss the local stability of each of feasible equilibria of system (3) and the existence of Hopf bifurcations at the coexistence equilibrium.
It is easy to show that system (3) always has a trivial equilibrium and a predator-extinction equilibrium . Furthermore, if the following holds.
(H1) , then system (3) has a coexistence equilibrium , where
We now study the local stability of the trivial equilibrium and the predator-extinction equilibrium .
The characteristic equation of system (3) at is where , .
Denote If , then Hence, has at least one positive root; the predator-extinction equilibrium is unstable.
If , it is readily seen from (9) that is locally asymptotically stable when . Denote where Hence, has no positive root. By Theorem 3.4.1 in , we know that if , the predator-extinction equilibrium is locally asymptotically stable for all .
Concluding the above discussions, we obtain the following results.
Theorem 1. For system (3), one has the following. (i) The trivial equilibrium is always unstable. (ii) If , the predator-extinction equilibrium is unstable; if , is locally asymptotically stable for all .
In the following, we discuss the local stability of the coexistence equilibrium and the existence of Hopf bifurcations at .
The characteristic equation of system (3) at is where
When , (14) becomes By calculations, we obtain that If (H1) holds, . Hence, by the Routh-Hurwitz criterion, we know that the coexistence equilibrium is locally asymptotically stable provided the following.
(H2) , , and is unstable if one of the inequality in (H2) is reversed.
Letting , (20) can be rewritten as
Denote If , then has two unequal roots:
By Lemma 2.1 in , we can obtain the following conclusion.
Lemma 2. Suppose that (H1) is satisfied and is defined by (24). (i) If , then (22) has at least one positive root. (ii) If and , then (22) has no positive roots. (iii) If , then (22) has positive roots if and only if , .
Denote then is a pair of purely imaginary roots of (14) with , ; .
Define , . By Lemma 2.2 in , the following result can be obtained.
Let be the root of (14) satisfying , . The conditions in the following lemma ensure that the transversality condition holds.
Lemma 4. If and the conditions (ii) in Lemma 3 are satisfied, then
Proof. Differentiating (14) with respect , it follows that
On substituting into (28), then
Since , If , there exists a root satisfying for and close to , which contradicts (ii) in Lemma 3. The proof is complete.
Theorem 5. For system (3), suppose that (H1) and (H2) are satisfied. (i) If and , then the coexistence equilibrium is locally asymptotically stable for all . (ii) If , or , , , then is locally asymptotically stable for and is unstable for . (iii) If the conditions in (ii) are satisfied, and , then system (3) exhibits the Hopf bifurcation at when .
Example 6. In (2), let , , , , , , , , and . System (3) with above coefficients has a unique coexistence equilibrium . A direct calculation shows that , , , . By Theorem 5, we see that the positive is locally asymptotically stable if and is unstable if . Numerical simulation illustrates our result (see Figure 1).
In the following, we show that system (3) is permanent.
Definition 7. System (3) is said to be permanent if there are positive constants , , , and such that each positive solution of system (3) satisfies
Firstly, we prove that system (3) is ultimately bounded.
Lemma 8. If , the arbitrary solution of system (3) is ultimately bounded.
Proof. Let be any positive solution of system (3). Define
where is a constant satisfying .
Calculating the derivative of along positive solutions of system (3), we get where .
It is easy to know that for . According to (33), we get If we choose then, for sufficiently small, there exists a such that if , That is, the arbitrary solution of system (3) is ultimately bounded if . This completes the proof.
Let be a complete metric space. Suppose that , , and . Assume that is a semigroup on satisfying
Let and be the global attractor for . The following is a small variant of Theorem 4.1 in .
Lemma 9 (see ). Suppose that satisfies (37) and one has the following. (i) There is a such that is compact for ; (ii) 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 , , where is the distance of from .
Now we state and prove the permanence of system (3).
Theorem 10. If (H1) holds, then system (3) is permanent.
Proof. We need only to show that the boundaries of repel positive solutions of system (3) uniformly.
Let denote the space of continuous functions mapping into . Define Denote , , .
Now, we verify that the conditions in Lemma 9 are satisfied. According to the definition of and , it is easy to know that and are positively invariant, so condition (ii) in Lemma 9 is satisfied. The solution of system (3) is ultimately bounded if (H1) holds by Lemma 8. Thus, by the smoothing property of solutions of delay differential equations introduced in [6, Theorem 2.2.8], condition (i) is satisfied.
Next, we verify condition (iii) in Lemma 9. There are two constant solutions in , corresponding to and , respectively, which satisfy If is a solution of system (3) initiating from , then Obviously, if (H1) holds, and as . If is a solution of system (3) initiating from , then , which yields , as . Noting that , we know the invariant and are isolated. So, is isolated and is an acyclic covering satisfying conditions (iii) in Lemma 9.
We now show that and .
Assume that . Then there is a positive solution of system (3) satisfying Then for sufficiently small such that , there exists a such that if , Together with (43) and the first equation of system (3), we derive that Thus, , which contradicts . Then .
Assume that . Then, there is a positive solution of system (3) satisfying Since (H1) holds, we can choose small enough such that Since , for satisfying (46), there exists a such that if ,
From the last two equations of system (3), it is easy to know that if , Let us consider the following auxiliary system: with initial conditions (4).
Consider the following matrix defined by Since admits positive off-diagonal elements, the Perron-Frobenius theorem implies that there is a positive eigenvector for the maximum root of . Since (46) holds, it is shown that the maximum root by a simple computation.
Let be a solution of system (49) through for , where satisfies , for . Since the semiflow of (49) is monotone and , it follows from  that is strictly increasing and as , . By comparison, , as , contradicting Lemma 8. Hence, .
By Lemma 9, we conclude that repels positive solutions of system (3) uniformly. Hence, system (3) is permanent. This proof is complete.
4. Global Stability
In this section, we study the global stability of the predator-extinction equilibrium and the coexistence equilibrium , respectively, by means of Lyapunov functionals and LaSalle's invariance principle.
First, we discuss the global stability of the predator-extinction equilibrium .
Theorem 11. If , then the predator-extinction equilibrium is globally asymptotically stable.
Proof. Let be any positive solution of system (3) with initial conditions (4). By Theorem 1, we know that is locally asymptotically stable if .
Define where , .
Calculating the derivative of along positive solutions of system (3), we obtain that On substituting into (52), we derive that Define By (53) and (54), it follows that If , then . By Theorem 5.3.1 in , solutions limit to , the largest invariant sunset of . We can see from (55) that if and only if , . Since is invariant, for each element in , we have , . Therefore, it follows from the last equation of system (3) that which yields . Consequently, if and only if , , . Hence, by Lasalle’s invariance principle, is globally asymptotically stable. The proof is complete.
Next, we prove the global stability of the coexistence equilibrium .
Theorem 12. If (H1) holds, then the coexistence equilibrium is globally asymptotically stable provided the following.
Here, is the persistency constant for satisfying .
Proof. Let be any positive solution of system (3) with initial conditions (4). Since , we know that there exists a such that for and also that . By Theorem 5 in , it is shown that , . If (H1) holds, according to (21), , , for . Then, (20) has no positive roots. Hence, is locally asymptotically stable for all .
Define where , .
Calculating the derivative of along positive solutions of system (3), we obtain that On substituting into (58), we derive that Noticing that , (59) can be rewritten as
Define According to (60) and (61), it follows that
Noting that we derive from (62) that
If for , then and the equality holds only for . Since the arithmetic mean is greater than or equal to the geometric mean, it is shown that with equality if and only if . Together with (64), it follows that if for , , with equality if and only if , . Hence, we now look for the invariant subset within the set Since on , then , which yields . It follows from the last equation of system (3) that , which yields . Thus, the only invariant set in is . Hence, by Lasalle's invariance principle, is globally asymptotically stable. The proof is complete.
Remark 13. By Theorem 12, it is shown that if (H1) holds and , then the coexistence equilibrium is globally asymptotically stable. We now give a sufficient condition for this inequality. By (35), has an ultimately upper bound . Hence, we derive from the first equation of system (3) that , which yields . A brief calculation shows that we need only to choose ; then .
This work was supported by the National Natural Science Foundation of China (11071254) and the Nature Science Foundation for Yong Scientists of Hebei Province, China (A2013506012).
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