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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 358594, 31 pages
Permanence and Almost Periodic Solutions of a Discrete Ratio-Dependent Leslie System with Time Delays and Feedback Controls
School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China
Received 10 July 2012; Accepted 2 September 2012
Academic Editor: Wolfgang Ruess
Copyright © 2012 Gang Yu and Hongying Lu. 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 discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls. Sufficient conditions are obtained for the permanence and global attractivity of the system. Furthermore, by using an almost periodic functional Hull theory, we show that the almost periodic system has a unique globally attractive positive almost periodic solution.
Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The predator-prey models have been extensively studied by many scholars [1–4]. Under the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food, Leslie  proposed the famous Leslie-Gower predator-prey model where the carry capacity of the predator’s environment is proportional to the number of prey: where and represent prey and predator densities at time , respectively. is the predator’s rate of feeding upon prey, that is, the so-called predator’s functional response. If we assume that the predator consumes the prey according to the functional response , then system (1.1) formulates as the following: where is prey-dependent functional responses. Owing to its theoretical and practical significance, system (1.2) and its various generalized forms have been studied extensively and seen great progress (see, for example, [6–9]).
However, in the study of the dynamic behaviors of predator-prey system, many scholars argued that the ratio-dependent predator-prey systems are more realistic [10, 11]. A ratio-dependent predator-prey system with Leslie-Gower term takes the form of
Though much progress has been seen in the asymptotic convergence of solutions of population systems, such systems are not well studied in the sense that most results are continuous-time cases related. Already, many scholars have paid attention to the nonautonomous discrete population models, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world (see [12–23]). Furthermore, the asymptotic convergence of solutions of difference equations with delay is one of the most important topics in the study of population dynamics, and many excellent results have already been obtained and seen great progress (see [24–26] and the references cited therein).
Since time delays occur so often in nature, a number of ecological systems can be described as systems with time delays (see [14, 15, 19, 21–27]). One of the most important problems for this type of system is to analyze the effect of time delays on the stability of the system. Furthermore, as we know, ecological systems in the real world are often distributed by unpredictable forces which can result in changes in biological parameters such as survival rates, so it is reasonable to study models with control variables which are so-called disturbance functions [22, 23].
So it is very interesting to study dynamics of the following discrete ratio-dependent Leslie system with time delays and feedback controls: where , stand for the density of the prey and the predator at time , respectively. , are the control variables at time . is a positive constant, denoting the constant of capturing half-saturation.
In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global dynamics of the discrete ratio-dependent Leslie systems with feedback controls. To do so, for system (1.4) we always assume that for , , , , , , , , , are all bounded nonnegative almost periodic sequences such that Here, we let , denote the sets of all integers, nonnegative integers, respectively, and use the notations: , , for any bounded sequence defined on .
Let , we consider system (1.4) with the following initial conditions:
The principle aim of this paper is to study the dynamic behaviors of system (1.4), such as permanence, global attractivity, and existence of a unique globally attractive positive almost periodic solution of the system. To the best of our knowledge, no work has been done for the nonautonomous difference system (1.4).
The organization of this paper is as follows. In the next section, we introduce some definitions and several useful lemmas. In Section 3, we explore the permanent property of system (1.4). We study globally attractive property of system (1.4) in Section 4 and the almost periodic property of system (1.4) in Section 5. Finally, the conclusion ends with brief remarks.
In this section, we will introduce some basic definitions and several useful lemmas.
Definition 2.2. Suppose that is any solution of system (1.4). is said to be a strictly positive solution in if for and such that
Definition 2.3 (see ). A sequence is called an almost periodic sequence if the -translation set of is a relatively dense set in for all ; that is, for any given , there exists an integer such that each interval of length contains an integer with is called an -translation number of .
Definition 2.4 (see ). The hull of , denoted by , is defined by for some sequence , where is any compact set in .
Lemma 2.5 (see ). Assume that satisfies and for , where is a positive constant and . Then
Lemma 2.6 (see ). Assume that satisfies and for , , where is a constant such that and . Then
Lemma 2.7 (see ). Assume that and . Suppose that Then for any integer , Especially, if and is bounded above with respect to , then
Lemma 2.8 (see ). Assume that and . Suppose that Then for any integer , Especially, if and is bounded below with respect to , then
In this section, we establish a permanent result for system (1.4).
Theorem 3.1. Assume that holds; assume further thatholds. Then system (1.4) is permanent.
Proof. Let be any positive solution of system (1.4), from the first equation of system (1.4), it follows that
It follows from (3.1) that
which implies that
Substituting (3.3) into the first equation of (1.4), it immediately follows that
By applying Lemma 2.5 to (3.4), we have
For any small enough, it follows from (3.5) that there exists enough large such that for ,
From the second equation of system (1.4) it follows that
From (3.7), similar to the argument of (3.1), one has
By applying Lemma 2.5 to (3.8) again, we have
Setting in the above inequality, we have
For any small enough, it follows from (3.5) and (3.10) that there exists enough large such that for and For , (3.11) combining with the third and fourth equations of system (1.4) leads to that is, By applying Lemma 2.7, it follows from (3.13) that Letting in the above inequality, we have
For any small enough, it follows from (3.11) and (3.15) that there exists enough large such that for and Thus, from (3.16) and the first equation of system (1.4), it follows that where For any integer , it follows from (3.17) that and hence From the third equation of system (1.4), we have where and . Therefore, for any , by Lemma 2.7, (3.20), and (3.21), one has Since , we have . So
By the conditions , for any solution of system (1.4), there exists a positive integer such that for . In fact, we can choose . Then we get where .
Substituting (3.20) and (3.24) into the first equation of system (1.4), one has where In particular, we have where we use the inequality for . By applying Lemma 2.6 to (3.25), it follows that Setting in the above inequality, we have From (3.29) we know that there exists enough large such that for , From (3.30) and the second equation of system (1.4), it follows that where .
This implies for any integer From (3.32) and the fourth equation of system (1.4), by a procedure similar to the discussion of (3.21)–(3.24), we can verify that where Substituting (3.32) and (3.33) into the second equation of system (1.4), one has where In particular, we have By applying Lemma 2.6 to (3.35) again, it follows that Setting in the above inequality, we have From (3.29) and (3.39) we know that there exists enough large such that for , For , (3.40) combining with the third and fourth equations of system (1.4) produces By applying Lemma 2.8, it follows from (3.41) that Setting in the above inequality, we have
Consequently, combining (3.5), (3.10), (3.15), (3.29), (3.39) with (3.43), system (1.4) is permanent. This completes the proof of Theorem 3.1.
4. Global Attractivity
Firstly, we prove two lemmas which will be useful to our main result.
Lemma 4.1. For any two positive solutions and of system (1.4), one has where
Proof. It follows from the first equation of system (1.4) that where Hence Since By the mean value theorem, one has Combining (4.6) with (4.8), we can easily obtain (4.1). The proof of Lemma 4.1 is completed.
Lemma 4.2. For any two positive solutions and of system (1.4), one has where
Now we are in the position of stating the main result on the global attractivity of system (1.4).
Theorem 4.3. In addition to , assume further that there exist positive constants , , such that holds, where , , , , , are defined by (4.37). Then for any two positive solutions and of system (1.4), one has
Proof. For two arbitrary nontrivial solutions and of system (1.4), we have from Theorem 3.1 that there exist positive constants and , , , such that for all and Firstly, we define From (4.1), we have where By the mean value theorem, we have that is, where lies between and . So, we have For , it follows that Secondly, we define Then Thirdly, we define By a simple calculation, it follows that We now define Then for all , it follows from (4.14)–(4.24) that Similarly, we define where