Research Article | Open Access
Positive Almost Periodic Solutions for a Discrete Competitive System Subject to Feedback Controls
This paper concerns a discrete competitive system subject to feedback controls. By using Lyapunov function and some preliminary lemmas, the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system are investigated. Numerical simulations suggest the feasibility of our theoretical results.
Many real world phenomena are studied through discrete mathematical models involving difference equations which are more suitable than the continuous ones when the populations have nonoverlapping generations. On the other hand, discrete models can also provide efficient computational models of continuous models for numerical stimulations; therefore, the studies of dynamic systems governed by difference equations have attracted more attention from scholars. Many good results concerned with discrete systems are deliberated (see [1–7] in detail).
Recently, in  we consider the following discrete two-species competitive almost system Here stand for the densities of species at the th generation, represent the natural growth rates of species at the th generation, are the intraspecific effects of the th generation of species on own population, and measure the interspecific effects of the th generation of species on species . The coefficients , and are bounded positive almost periodic sequences. We established a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (1) (see ).
Note that ecosystems in the real world are often disturbed by outside continuous forces. In the language of control, we call the disturbance functions control variables and they can be regarded as feedback controls. For more discussions on this direction, we can refer to [8–14] and the references cited therein. Motivated by the above ideas we can establish the discrete two-species competitive almost system with feedback controls where and represent control variables and are the forward difference operators. and are bounded positive almost periodic sequences, where . To belong to the direction of , in this contribution, we continue to discuss the effect of feedback controls and establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (2).
The rest of this paper is organized as follows. In the next section, we introduce some notations, definitions, and lemmas which are available for our main results. Sufficient conditions for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (2) are established in Section 3. In Section 4, we carry out numerical simulations to substantiate our analytical results.
In this section, we give some notations, definitions, and lemmas which will be useful for the later sections.
, and denote the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. and denote the cone of -dimensional and -dimensional real Euclidean space, respectively. For an almost periodic sequence defined on , the notations below will be used
Definition 1 (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 discrete interval of length contains a such that is called the -translation number of .
Definition 2 (see ). Let , where is an open set in . is said to be almost periodic in uniformly for , or uniformly almost periodic for short, if for any and any compact set in , there exists a positive integer such that any interval of length contains a integer for which for all and all . is called the -translation number of .
Lemma 3 (see ). is an almost periodic sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limit sequence is also an almost periodic sequence.
Consider the following almost periodic difference system where , and is almost periodic in uniformly for and is continuous in . The product system of (7) is the following system: and Zhang  established the following result.
Lemma 4 (see ). Suppose that there exists a Lyapunov function defined for , satisfying the following conditions:(i), where with and is increasing};(ii), where is a constant;(iii), where is a constant and Moreover, if there exists a solution of system (7) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of system (7) which satisfies . In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of system (7) of periodic .
3. Main Results
We first give the following two propositions which are useful for our main results.
Proposition 5. Any positive solution of system (2) satisfies where
Proof. We first prove that ; to do so, we consider Cases (1) and (2).
Case (1). Suppose that there exists an such that ; it follows from the first equation of system (2) that which implies Hence, where we use the fact that for .
It is claimed that for all . By way of contradiction, assume that there is a such that , then . Set ; that is to say, and ; then . It is easy to see that from the above argument, which is a contradiction. Therefore, for all , then . This proves the claim.
Case (2). Suppose that for all . In particular, exists, denoted by . We will prove that by way of contradiction, if , then by taking limit in the first equation in system (2) we have Notice that , so we have which is a contradiction. This proves the claim. Hence, .
Similar to , we can prove that .
Next, we prove . For any , there exists a large enough integer such that for all . We have from the third equation of system (2) that where . Since , we can find a positive number such that , then by using Stolz’s theorem, we obtain that Thus . Since is arbitrary, is valid. Analogously, we can prove . Hence, the proof of Proposition 5 is complete.
Proposition 6. If the following inequalities hold. Then any positive solution of system (2) satisfies where
Proof. For any small enough , which satisfies , according to Proposition 5, there exists such that
We first present Cases (1) and (2) to prove that .
Case (1). Assuming that there exists a positive integer such that , we have from the first equation of system (2) that Therefore, which implies that Then Hence , where
We claim that for . By way of contradiction, assume that there exists a such that , then . Let , that is, and , then , and the above argument produces that , which is a contradiction. Thus, for all ; since can be sufficiently small, it obtains that . This proves the claim.
Case (2). We assume that for all . Then exists, denoted by . We claim that By way of contradiction, assume that . Taking limit in the first equation in system (2) yields However, which is a contradiction. It implies that . By the fact that , we obtain that , which means . From (27), we know that . Therefore, .
Since can be sufficiently small, we have By a similar argument, we can prove that
Now we prove that . For any small enough , there exists a positive integer such that for .
By the third equation of system (2), we obtain that where . Since , we can find a positive number such that , then by Stolz’s theorem, we have Thus , by the arbitrary of , is valid. The conclusion about can be obtained in a similar way. Thus the proof of Proposition 6 is complete.
Theorem 7. If the assumptions in (19) are satisfied, then .
Proof. We have from system (2) that
for . Based on Propositions 5 and 6, any solution of system (2) satisfies (10) and (20). Thus, for any , there exists large enough such that
Setting to be any positive integer valued sequence such that as , it is easy to show that there exists a subsequence of still denoted by , such that uniformly in on any finite subset of as , where and is a finite number.
In reality, for any finite subset , when is large enough. Thus, , which mean that are uniformly bounded for large enough.
Now, for , we can select a subsequence of such that uniformly converge on for large enough.
Similarly, for , we can also select a subsequence of such that uniformly converge on for large enough.
Repeating the above process, for , we choose a subsequence of such that uniformly converge on for large enough.
Then we choose the sequence which is a subsequence of still denoted by ; for all , we have uniformly in as . Therefore, the conclusion is true by the arbitrary of .
Consider the almost periodicity of , and , for the above sequence , as , there exists a subsequence still denoted by such that as uniformly on .
For any , we can presume that for large enough. Let , by an inductive argument of system (2) from to which results in for , so it derives that Letting , one has By the arbitrary of , we can easily see that is a solution of system (2) on , and Since is an arbitrarily small positive number, we obtain that So . This completes the proof.
The following theorem concerns the existence and uniformly asymptotical stability of unique positive almost periodic solution of system (2).
Theorem 8. Suppose the inequalities in (19) are satisfied; furthermore, , where and , then there exists a unique uniformly asymptotically stable positive almost periodic solution of system (2) which is bounded by for all .
Proof. We first make the change of variables
then system (2) can be reformulated as
By Theorem 7, it is easy to see that there exists a bounded solution of system (46) satisfying
Then , where , and . Let , where .
The following associate product system of system (46) can be expressed as Suppose that are any two solutions of system (46) defined on , then where , and
Let us construct the following Lyapunov function defined on : Obviously, is equivalent to ; that is, there exist two constants , such that Consequently, Denote , thus the condition (i) of Lemma 4 is satisfied.
In addition, for any , we find that where ,, and . Hence, the condition (ii) of Lemma 4 is satisfied.
At last, we calculate the of along the solutions of system (48) and obtain that By the mean-value theorem, one has where and