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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 834047, 9 pages
Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response
Sunshine College, Fuzhou University, Fuzhou, Fujian 350015, China
Received 5 April 2013; Revised 8 July 2013; Accepted 11 July 2013
Academic Editor: Mingxin Wang
Copyright © 2013 Zhimin Zhang. 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.
We consider a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov function, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. Our results not only supplement but also improve some existing ones. One example is presented to verify our main results.
Let be a continuous bounded function on , and we set Leslie  introduced the famous Leslie predator-prey system where , stand for the population (the density) of the prey and the predator at time , respectively, and is the so-called predator functional response to prey. The term is the Leslie-Gower term which measures the loss in the predator population due to rarity (per capita ) of its favorite food. Leslie and Gower , Pielou  obtain some excellent results on the system (2) with the functional response which is called Holling-type I. By applying the Dulac's criterion and constructing Lyapunov functions, Hsu and Huang  establish the global stability of system (2).
Recently, Aziz-Alaoui and Daher Okiye  pointed out that in the case of severe scarcity, can switch over to other populations, but its growth will be limited by the fact that its most favorite food is not available in abundance. To solve such problem, they suggested to add a positive constant to the denominator and proposed the following predator-prey model with modified Leslie-Gower and Holling-type II schemes: with initial conditions and , where and represent the population densities at time . , , , , , , and are model parameters assuming only positive values. is the growth rate of prey , measures the strength of competition among individuals of species , is the maximum value of the per capita reduction rate of due to , (resp., ) measures the extent to which the environment provides protection to prey (resp., to the predator ), describes the growth rate of , and has a similar meaning to . The authors studied the boundedness and global stability of positive equilibrium of the system (3). Since then, system (2) and its nonautonomous versions have been studied by incorporating delay, impulses, harvesting, and so on (see, e.g., [6–13]). In particular, Zhu and Wang  consider the following nonautonomous model: Under the assumption that the coefficients of the system (4) are all continuous -periodic functions, by utilizing the coincidence degree theorem and constructing a suitable Lyapunov function, they obtained sufficient conditions for the existence and global attractivity of positive periodic solutions of the system (4). More precisely, Zhu and Wang  obtained the following theorem (see [13, Theorems 3.1 and 3.2]).
Theorem 1. Suppose that holds, and further suppose that one of the following conditions: holds; then system (4) has at least one positive -periodic solution.
Corollary 6 given in Section 2 of this paper shows that when or does not satisfy, the conclusion of Theorem 1 also holds. Moreover, as we know, permanence is one of the most important topics in the study of population dynamics; however, Zhu and Wang  did not investigate this property of the system (4). One aim of this work is to obtain a set of sufficient conditions which guarantee the permanence of the system (4).
On the other hand, as Fan and Kuang  have mentioned, the Holling II type functional responses fail to model the interference among predators and have been facing challenges from the biology and physiology communities (see [15–17]). Some biologists have argued that in many situations, especially when predators have to search for food (and therefore have to share or compete for food), the functional response in a prey-predator model should be predator-dependent.
Stimulated by the previous reasons, in this paper we will incorporate the Beddington-DeAngelis functional response into model (4) and consider the following model which is the generalization of the model (4): where is the size of prey population and is the size of predator population.
Li and Zhang  pointed out that in real world phenomenon, the environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting. So it is usual to assume the periodicity of parameters in the system (8). However, if the various constituent components of the temporally nonuniform environment are with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. For this reason, the assumption of almost periodicity is more realistic, more important, and more general when we consider the effects of the environmental factors. So it is assumed that the coefficients , , , , , , , are all continuous, almost periodic functions and satisfy
We consider system (8) with the following initial conditions:
The aim of this paper is to obtain sufficient conditions for the existence of a unique globally attractive almost periodic solution of systems (8) and (10), by utilizing the comparison theorem of the differential equation and applying the analysis technique of papers [19–21].
The organization of this paper is as follows. In Section 2, by applying the theory of differential inequality, we present the permanence results for system (8). In Section 3, by constructing a suitable Lyapunov function, a set of sufficient conditions which ensure the existence and uniqueness of almost periodic solutions of system (8) are obtained. Then, in Section 4, a suitable example together with its numeric simulations is given to illustrate the feasibility of the main results. We end this paper by a brief discussion.
Now let us state several definitions and lemmas which will be useful in the proving of the main result of this section.
Lemma 2 (see ). If , and , when and , we have If , and , when and , we have
Proof. From condition , we can choose a small enough such that The first equation of (8) yields Applying Lemma 2 to (16) leads to Equation (17) shows that there exists a large enough such that for all , It follows from (18) and the second equation of system (8) that, for , According to Lemma 2 and (19), one has Thus, for previous , there exists a , such that Equation (21) together with the first equation of (8) leads to From (22), according to (15) and Lemma 2, we can obtain Hence, for previous , there exists a , such that From (24) and the second equation of system (8), we know that for , Applying Lemma 2 to (25) leads to Setting , it follows from the previous discussion that Obviously, and are independent of the solution of system (8); (17) and (27) show that the conclusion of Theorem 3 holds. The proof is completed.
Theorem 4. Suppose that system (8) with initial condition (10) satisfies the following conditions:Then system (8) is permanent, that is, any positive solution of the system (8) satisfies where , are defined in Theorem 3 and , .
Proof. From (17) and (20), one can derive
where and are defined in Theorem 3.
It follows from the first equation of system (8) that According to , by applying Lemma 2 to (31), we obtain Hence, for a small enough , there exists a , such that From (33) and the second equation of system (8), we know that for , Applying Lemma 2 to (34) leads to Setting , it follows from the previous discussion that Obviously, and are independent of the solution of system (8); (30), (32) and (36) show that the conclusion of Theorem 4 holds. The proof is completed.
Corollary 5. Suppose that or holds, then system (8) admits at least one positive -periodic solution if , , , , , , , are all continuous positive -periodic functions.
Corollary 6. Suppose thatholds; then system (4) is permanent and admits at least one positive -periodic solution if , , are all continuous positive -periodic functions.
3. Existence of a Unique Almost Periodic Solution
Now let us state several definitions and lemmas which will be useful in the proving of the main result of this section.
Definition 8 (see [24, 25]). A function , where is an -vector, is a real scalar, and is an -vector, is said to be almost periodic in uniformly with respect to , if is continuous in and , and if for any , there is a constant , such that in any interval of length there exists such that the inequality is satisfied for all , . The number is called an -translation number of .
We denote by the set of all solutions of system (8) satisfying , for all .
Lemma 10. .
Proof. Since , , , , , , , are almost periodic functions, there exists a sequence , as such that as uniformly on . Let be solution of systems (8) and (10) satisfying , for . Obviously, the sequence is uniformly bounded and equicontinuous on each bounded subset of . Therefore, by Ascoli-Arzela theorem, there exists a subsequence of , and we still denote it as , such that , as uniformly on each bounded subset of . For any , we may assume that for all . For , we haveApplying Lebesgue's dominated convergence theorem and letting in the previous equations, we obtain for all . Since is arbitrarily given, is a solution of system (8) on . It is clear that , for . That is to say . This completes the proof.
Lemma 11 (see ). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then .
Proof. It follows from conditions and that there exists a small enough such that
Let , be any two positive solutions of system (8) with initial conditions (10). For previous , according to Lemma 10 and Theorem 3 (or Theorem 4), there exists a , when ,
Let , where Calculating the upper right derivatives of along the solution of (8) leads to
Calculating the upper right derivatives of along the solution of (8), one has
It follows from (49)-(50) that for , It follows from (46) and (51) that for , which implies is nonincreasing on . Integrating the previous inequality from to leads to Then for , we obtain that Hence, , . By system (8) and Theorem 3 (or Theorem 4), we get , , , , and their derivatives are bounded on , which implies that both and are uniformly continuous on . By Lemma 11, we obtain Then the solution of systems (8) and (10) is globally attractive.
Proof. According to Lemma 10, there exists a bounded positive solution of (8) with initial condition (10). Then there exists a sequence , as , such that is a solution of the following system:
According to Theorem 3 (or Theorem 4) and the fact that , , , , , , , are all continuous, positive almost periodic functions, we know that both and their derivative function are uniformly bounded; thus are uniformly bounded and equicontinuous. By Ascoli's theorem, there exists a uniformly convergent subsequence such that for any , there exists a with the property that if , then
That is to say are asymptotically almost periodic functions; hence there exists two almost periodic functions and two continuous functions such that
are also almost periodic functions.
Therefore, On the other hand, So exist. Now we will prove that is an almost solution of system (8).
From properties of almost periodic function, there exists an sequence , as , such that as uniformly on .
It is easy to know that as ; then we have
These show that satisfies system (8). Hence, is a positive almost periodic solution of (8). Then, it follows from Theorem 12 that system (8) has a unique positive almost periodic solution. The proof is completed.
4. Examples and Numeric Simulations
Considering and , we choose , . Hence, Equation (66) means that all conditions of Theorem 13 are satisfied in system (64). Thus, it admits a unique, globally attractive, positive, almost periodic solution. Figure 1 shows the dynamic behaviors of the solution with the four group initial values , , , and . From the figure, we could easily see that the solution is asymptotic to the unique, almost periodic solution of the system (64).
In this paper, we consider a predator-prey with modified Leslie-Gower model and Beddington-DeAngelis functional response. When , , , , , , (8) we discussed reduces to (4) which was studied by Zhu and Wang . By utilizing the coincidence degree theorem and constructing a suitable Lyapunov function, the authors in  investigated the existence and global attractivity of positive periodic solutions of (4) and obtained Theorem 1. More precisely, comparing Theorem 1 with Corollary 6, we find that conditions or of Theorem 1 are redundant, which implies that our results improve those of . Example together with numeric simulation shows the feasibility of our main results.
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