Qualitative Theory of Functional Differential and Integral EquationsView this Special Issue
Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response and Feedback Controls
We consider a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis functional response and feedback controls as follows: , , , and . Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained.
In recent years, the modified predator-prey systems with periodic or almost periodic coefficients have been studied extensively.
Leslie  proposed the famous Leslie predator-prey system as follows: where and stand for the population of the prey and the predator at time , respectively, and is the so-called predator functional response to the prey. The term is the Leslie-Gower term which measures the loss in the predator population due to rarity of its favorite food.
Global stability of the positive locally asymptotically stable equilibrium in a class of predator-prey systems has been introduced by Hsu and Huang , and the system is as follows: When the functional response equals , then (2) turns into a Leslie-Gower system .
On the other hand, the periodic solution (almost periodic solution) and some other properties of Leslie-Gower predator-prey models were studied (see [4–9]). In particular, Zhang  discussed the almost periodic solution of a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis function response as follows: where is the size of prey population and is the size of predator population.
Stimulated by the above reasons, in this paper, we incorporate the feedback control into model (3) and consider the following model: where and all the coefficients , , , , , , , , , , and are all continuous, almost periodic functions on .
Associated with (4), we consider a group of initial conditions with the following form (we assume, without loss of generality, that the initial time ):
Let be a continuous bounded function on and we set Throughout this paper, we assume that the coefficients of the almost periodic system (4) satisfy By constructing a suitable Lyapunov functional, we obtain some sufficient conditions for the existence of a globally attractive positive almost periodic solution of system (4) with initial conditions (5).
In this section, we give some definitions and results that we will use in the rest of the paper.
Lemma 1 (see ). If , , and , when and , one has
Lemma 2 (see ). If , , and , when and , one has
Set the following:
Theorem 3. Suppose that system (4) with initial condition (5) satisfies the following condition: Then system (4) is permanent; that is, any positive solution of the system (4) satisfies
Proof. From the first equation of (4), we have the following:
Applying Lemma 1 to (13) leads to
From (14), we know that there exists an enough large such that
so there exists an enough large such that
It follows from (16) and the second equation of system (4) that, for ,
Applying Lemma 2 to (17) leads to
By using a similar argument as that in the proof of (14) and (18), we can get the following:
From (18) and the first equation of system (4) we know
Applying Lemma 1 and (11) to the above leads to
Therefore, we know that there exists an enough large such that
From the second equation of system (4) we have the following:
Applying Lemma 2 to the above, we obtain the following:
By using a similar method as that in the proof of (21) and (24), it follows that
This completes the proof.
We denote by the set of all solutions of system (4) satisfying , , , and for all .
Theorem 4. Consider the following: .
Proof. From the properties of almost periodic function there exists a sequence with as such that as uniformly on . Let be a solution of system (4) satisfying , , , and for . Clearly, the sequence is uniformly bounded and equicontinuous on each bounded subset of . Therefore, by the Arzelà-Ascoli theorem, there exists a subsequence which converges to a continuous function as uniformly on each bounded subset of . Let be given. We may assume that for all . For , we have the following: Applying Lebesgue’s dominated convergence theorem and letting in (27), we obtain the following: Since is arbitrarily given, is a solution of system (4) on . It is clear that , , , for . Thus . This completes the proof.
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 5 (see ). 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 a such that the inequality is satisfied for all , . The number is called an number of .
Definition 6 (see ). A function is said to be asymptotically almost periodic function, if there exists an almost periodic function and a continuous function such that , and as .
Lemma 7 (see ). Let be a nonnegative, integral, and uniformly continuous function defined on ; then .
Theorem 8. Suppose that all conditions of Theorem 3 hold; furthermore assume that(H), where , Then system (4) with initial conditions (5) is globally attractive.
Proof. Let , , and then system (4) is transformed into
Suppose that and are any two positive solutions of (31).
Let , where Calculating the right derivative of along the solution of (31), we have the following: Further, it follows that Therefore, we have the following: Integrating the above inequality on internal , it follows that, for , Then, for , we obtain that By Lemma 7, we obtain Then the solution of systems (4) and (5) is globally attractive.
Theorem 9. Suppose that all conditions of Theorem 8 hold; then there exists a unique almost periodic solution of systems (4) and (5).
Proof. According to Theorem 4, there exists a bounded positive solution of (4) and (5). Then there exists a sequence , as , such that is a solution of the following system:
According to Theorem 3, we get that not only but also 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
This is to say, are asymptotically almost periodic functions; hence there exist four almost periodic functions and four continuous functions such that
are an almost periodic function.
Therefore, On the other hand, So exist. Now we will prove that is an almost periodic solution of system (4).
From properties of almost periodic function, there exits a sequence , as , such that as uniformly on .
It is easy to know that as , and then we have the following: By using a similar argument as that in the above, we have the following:
This proves that is a nonnegative almost periodic solution of systems (4) and (5); by Theorem 8, it follows that there exists a globally asymptotically stable nonnegative almost periodic solution of system (4). The proof is complete.
4. An Example
Consider the following system:
By a simple calculation, we check that all conditions in Theorems 8 and 9 are fulfilled. Therefore, by Theorems 8 and 9, system (48) has a unique globally asymptotically stable nonnegative almost periodic solution (see Figure 1).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the NNSF of China (no. 11171085) and Hunan Provincial Natural Science Foundation of China (no. 10JJ6002).
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