Abstract and Applied Analysis

Volume 2014 (2014), Article ID 930762, 8 pages

http://dx.doi.org/10.1155/2014/930762

Research Article

Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

Nai-Wei Liu and Ting-Ting Kong

School of Mathematics and Information Science, Yantai University, Yantai, Shandong 264005, China

Received 25 January 2014; Accepted 14 May 2014; Published 26 May 2014

Academic Editor: Xinguang Zhang

Copyright © 2014 Nai-Wei Liu and Ting-Ting Kong. 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.

Abstract.

We consider a predator-prey system with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered. First, we give a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov function, we prove that the positive equilibrium is locally asymptotically stable under a sufficient condition. Finally, by using comparison theorem and the -limit set theory, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the global asymptotic stability.

1. Introduction

The purpose of this paper is to consider the following predator-prey system with Beddington-DeAngelis functional response and delays:

where denotes the immature prey population density, denotes the mature prey population density, and denotes the mature predator population density, and all the parameters , , , , , , , , , , and are positive. More precisely, the parameters and represent the time that the prey juveniles and the predator juveniles take to be mature, respectively. is the intrinsic growth rate of the prey and is the death rate of immature preys. The term denotes the surviving rate of the immature prey during the delay period to be mature. The constant denotes the intensity of intraspecific action on the prey and is the death rate of the predator. The function represents the Beddington-DeAngelis functional response and the function represents the predator’s growth rate that comes from the Beddington-DeAngelis functional response during time period .

Actually, the predator-prey system with Beddington-DeAngelis functional response has been extensively studied in the literature. In the original work, Beddington [1] and DeAngelis et al. [2] proposed the predator-prey system as

Shaikhet [3] considered the stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations. Indeed, the author in [3] not only obtained sufficient conditions for asymptotic mean square stability of the considered systems, but also showed that the trivial solution is asymptotically mean square stable and the positive equilibrium is stable in probability by constructing appropriate Lyapunov functions. In particular, Shaikhet expounded a general method of construction of Lyapunov functions for stochastic functional differential equations and obtained the stability conditions for functional differential equations in [4]. See also [5–10] for the corresponding results about the predator’s functional response. Particularly, Huo [8] studied the global attractiveness of delay diffusive prey-predator systems with the Michaelis-Menten functional response.

In the real world, many species usually have a life history that takes them through two stages, immature stage and mature stage. Recently, the system with the stage structure on prey has been extensively studied. In particular, She and Li [11] analyzed the following Beddington-DeAngelis model with the stage structure on prey:

where the stage structure was introduced by a constant time delay. In fact, it is the case of in system (1). See also [12–15] for predator-prey models with Beddington-DeAngelis functional response and stage structure.

More recently, Xiang et al. [16] established a virus dynamics model with Beddington-DeAngelis functional response and delays. In [16], they not only incorporated the delay , which described the time between viral entry into a target cell and the production of new virus particles, but also incorporated the delay , which described the maturation time of the newly produced viruses. Additionally, it is well known that the growth of the predator is affected by the time delay due to digestion. This motivates us to resolve the issue.

Motivated by [16], in the present paper, we study the predator-prey system (1) with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered.

More precisely, let and assume that

holds. Then, based on similar conclusions for Lemma 3.1 in [10], we obtain

which means that depends on completely. Thus, we can obtain the following system, which is separated from (1):

It is obvious that, for studying the dynamics of the predator-prey system (1) with Beddington-DeAngelis functional response and delays, it is sufficient to study the dynamics of (6).

The rest of the organization of this paper is as follows. In Section 2, we first establish the local stability of the boundary equilibria of (6) by analyzing the characteristic equations, and then, we study the local asymptotic stability of the positive equilibria of (6) by constructing an appropriate Lyapunov function. In Section 3, we consider the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively.

2. Local Stability of Equilibria

In this section, we study the local stability of the equilibria of (6). By direct calculations, we have that the system (6) has the origin equilibrium and the boundary equilibrium .

In order to investigate the existence of the positive equilibrium, we consider the equations

and obtain the following result.

Theorem 1. The system (6) has a unique positive equilibrium if and only if the condition

holds, which is equivalent to the condition

Proof. We prove the existence of the positive equilibrium by analyzing (7). For the first equation of (7), it is easy to see that its corresponding curves go through the points , , and when . Furthermore, in case of , the first equation is a hyperbolic equation and has two asymptotic lines and . If , the first equation can be simplified as , which are two lines.

For the second equation of (7), it implies a line which goes through the points and when . Their corresponding figures are as shown in Figures 1, 2, and 3.

Consequently, by analyzing Figures 1, 2, and 3 and discussions, we obtain that (6) has a unique positive equilibrium.

Figure 1: .

Figure 2: .

Figure 3: .

Condition (9) indicates that the positive equilibrium exists for all in the interval , where

Furthermore, will coincide with when and there exists no positive equilibrium for .

Now, we study the stability of equilibria and , respectively.

At the beginning, we rewrite system (6) as , where , , .

Then, for an arbitrary fixed point , by letting , , and , we have

where

Therefore, the characteristic equation of the system (6) at the point is as follows:

Thus, according to the characteristic equation at the point , we obtain the following result.

Theorem 2. The equilibrium is unstable.

Proof. The characteristic equation of the system (6) at the point is

Obviously, is a negative eigenvalue and the other eigenvalue depends on the solution of .

Noticing that the line and the curve must intersect at a unique point , where is a positive value, we can obtain that is a saddle. Thus, is unstable.

According to the characteristic equation of the point , we have the following theorem.

Theorem 3. The equilibrium is unstable if and is locally asymptotically stable if .

Proof. The characteristic equation of the system (6) at the point is

For the equation , it is clear that the line and the curve must intersect at a unique point , where is a negative value. And for the equation , noticing that the line and the curve must intersect at a unique point , we have that is positive when and is negative when . Consequently, is unstable if and locally asymptotically stable if .

In the rest of this section, by constructing an appropriate Lyapunov function, we study the stability of the positive equilibrium under condition (8). Let

then, the linearization of the system (6) is

where

Obviously, all the constants defined above are positive, and the system (17) can be recast as

By constructing an appropriate Lyapunov function, we have the following theorem.

Theorem 4. Let with defined in (10), and assume that

also holds, where

Then, the equilibrium of the system (17) is locally asymptotically stable, which implies that the positive equilibrium of the system (6) is locally asymptotically stable.

Proof. To illustrate the equilibrium of the system (17) is locally asymptotically stable, it is sufficient to study the existence of a strict Lyapunov function. By letting and differentiating with respect to , we have

Letting and differentiating with respect to , we have

Furthermore, by letting and , we have

Next, by letting and differentiating with respect to , we obtain

Letting and differentiating with respect to , we have

By letting , , and , we have

By letting , we have

Finally, we let . Obviously, and if and only if . Besides, when , . Furthermore,

Based on the conditions in Theorem 4, we have if . Thus, is a strict Lyapunov function. According to the Lyapunov stability theorem, the equilibrium of the system (17) is locally asymptotically stable, which implies that the positive equilibrium of the system (6) is locally asymptotically stable.

3. Global Stability of Equilibria

In this section, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium , respectively. We first give two lemmas which will be used later.

Lemma 5 (see [10, Lemma 3.2]). For a given system

where , , , , and for all , we have

(i) if and only if ;(ii) if and only if .

Lemma 6 (see [11, Lemma 3.2]). Let and . Then, the sets and are both invariant sets.

Based on Lemmas 5 and 6, we prove that the boundary equilibrium is globally attractive.

Theorem 7. Let be the solution of (6) with positive initial condition in .

(i)If , then is globally attractive.(ii)If , then is unstable.

Proof. If , we first prove that .

Due to the first equation (6), there is

In view of Lemma 6 and by considering the following comparison equation

we have for all . In addition, by Lemma 5, we obtain that . Thus, there exists a sufficiently small positive constant with such that, for this , there exists a such that for all . Substituting it into the second equation of (6), we have

for all . Considering the following comparison equation:

and letting , we have

Then the line and the curve must intersect at a unique point . Since , we can obtain that . Thus, we deduce that . Therefore, .

Obviously, based on the comparison theorem, we get that for all . Thus, .

Next, we prove that as .

Since , for any given , there exists a such that for all . Thus, we have

for all . That is,

Then, we consider the following comparison equation:

Since , based on Lemma 5, we have that . Additionally, we have that for all .

Since , for the above , there exists a such that for all . Similarly, since for the above , we also get that there exists a such that for all . Therefore, if we let , there is for all . Thus, we have .

At last, we prove the second part of this theorem. We can assure that and try to derive a contradiction. If , then (6) has a unique positive equilibrium by Theorem 1, which is also a solution of (6); that is, satisfies (6) and satisfies by the above arguments, which contradicts the fact that is a positive equilibrium. Thus, conclusion (ii) holds. This completes the proofs.

Remark 8. In the above theorem, we derive the global attractiveness of the boundary equilibrium . Unfortunately, we just obtain a sufficient condition that to ensure that is globally asymptotically stable. If , since the second equation of (6) is a delay diffusion equation, we cannot derive that by using comparison theorem. We leave it for a further study.

Furthermore, for the first equation of (1), we can solve explicitly as

Thus, by Theorem 7, we can obtain the following corollary directly.

Corollary 9. Let be the solution of (1) with the positive initial condition. If , then

In the rest of this section, we will study the global attractiveness of the positive equilibrium and obtain the following result.

Theorem 10. Assume that (8) and the following conditions

hold, and let be the solution of (6) with the positive initial condition in such that . Then, the positive equilibrium of (6) is globally attractive.

Proof. Provided that conditions (8) and (41) hold, based on comparison theorem and by similar arguments in [11], we can obtain that, for any solution of (6) with the positive initial condition in , there exists a point such that . Indeed, just by modifying the coefficients of (6), the discussion in [11] also holds. We omit the proof here and refer to [11] for details.

Then, in view of the property of the -limit set in [17] and by the uniqueness result of the positive equilibrium in Theorem 1, we have . Thus, we derive that is globally attractive.

4. Conclusion

In the present paper, we considered the stability of the equilibria of a predator-prey system with Beddington-DeAngelis functional response and delays. More precisely, we derived a sufficient and necessary condition for the existence of a unique positive equilibrium and proved that the positive equilibrium is locally asymptotically stable under a sufficient condition. Also, we established the local stability of the boundary equilibria by analyzing the characteristic equations. Finally, by using comparison theorem and the -limit set theory, we studied the global asymptotic stability for both the boundary equilibrium and the positive equilibrium.

In particular, in Theorem 7, we derived the global attractiveness of the boundary equilibrium . But unfortunately, we just obtained a sufficient condition that to ensure that was globally asymptotically stable. If , since the second equation of (6) was a delay diffusion equation, we could not derive that by using comparison theorem.

Moreover, as pointed out in [11], there are two interesting problems about system (1), which are related to the phenomena of bifurcations and the existence of periodic orbits of (1). We leave them for a further study.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was partially supported by NSFC (11201402, 11371179) and NSF of Shandong Province of China (ZR2010AQ006). The second author was partially supported by Graduate Innovation Foundation of Yantai University (GIFYTU).

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