Feedback Control in a General Almost Periodic Discrete System of Plankton Allelopathy

Abstract

We study the properties of almost periodic solutions for a general discrete system of plankton allelopathy with feedback controls and establish a theorem on the uniformly asymptotic stability of almost periodic solutions.

1. Introduction

Allelopathy is first used by Hans (see [1]) in 1937, which is a biological phenomenon that is characteristic of any process involving secondary metabolites produced by some plants, algae, bacteria, and fungi which influences the growth and development of biological systems. There are many investigations which have been conducted to study the toxic effect on plankton allelopathy (see [2–9]). In this paper, we investigate the following system with feedback controls: Here represent the densities of species at the th generation, are the intrinsic growth rates of species at the th generation, measure the intraspecific effects of the generation of species on their own population, and stand for the interspecific effects of the th generation of species on species ; are the rates of toxic inhibition of species by species and vice versa at the th generation, is the first-order forward difference operator , and are the feedback control variables ; . Meanwhile, , , , , , and are bounded nonnegative sequences such that for . Here, we use the notations and for any bounded sequence . For the simplicity and convenience of exposition, throughout this paper we let , , and denote the sets of all integers, and nonnegative integers, nonnegative real numbers, respectively.

From the point of view of biology, we only consider are positive; that is,

For convenience, we give the following definition of persistence of system (1).

Definition 1. System (1) is said to be persistent if there exist positive constants , , , and which are independent of the solutions of system (1), such that any positive solution of system (1) satisfies for .

The rest of this paper is organized as follows. We first give some preliminaries in Section 2. Next, we study the persistent property of (1) in Section 3 and the almost periodic property in Section 4. To the best of our knowledge, no work has been done for the discrete system (1) with feedback controls.

2. Preliminaries

In this section, we first introduce some definitions. For our purpose, we introduce the following notions with their properties. For convenience, , , , , , , , , , , and in the sequel always denote integers, and the relevant intervals and inequalities are discrete ones. This section, devoted to investigate the persistent property of system (1). To do so, we need to make some preparations.

We first introduce some definitions.

Definition 2 (see [10]). 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 for all . is called the -translation number of .

Definition 3 (see [10]). 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 for which for all and all . is called the -translation number of .

Next, we will introduce the following Lemmas.

Lemma 4 (see [10]). is an almost periodic sequence if and only if for any sequence such that converges uniformly on as . Furthermore, the limit sequence is also an almost periodic sequence.

Lemma 5 (see [11]). Assume that satisfies and for , where and are nonnegative sequences bounded above and below by positive constants. Then

Lemma 6 (see [11]). Assume that satisfies and , where and are nonnegative sequences bounded above and below by positive constants and . Then

3. Persistence

In this section, we will establish sufficient conditions for the persistence of system (1).

Proposition 7. Assume that (2) and (3) hold; any solution of system (1) satisfies where

Proof. Let be any positive solution of system (1) with initial condition (3). By assumption (2), it follows from the first and second equations of system (1) that Applying Lemma 5, one obtains Thus, for any small enough, it follows from (15) that there exists a large enough integer such that Then the third and fourth equations of system (1) lead to where , . Since , we can find a such that ; by Stolz’s theorem, we have Thus, Letting and substituting (16) in the above inequality lead to

Proposition 8. Assume that (2) and (3); furthermore for , , are satisfied, where and are the same as those in Proposition 7. Then where

Proof. For and , according to Proposition 7, there exists a such that By Lemma 6, associating (21) with (1), one has Letting in the above inequality leads to Also from (26), for each , there exists a large enough integer such that By the third and fourth equations of system (1), we can get that where , . As , similar to the analysis in the proof of Proposition 7, we have

Now, we are in a position to state Theorem 9 whose proof is a direct consequence of Propositions 7 and 8.

Theorem 9. If the inequalities in (2), (3), and (21) hold, then system (1) is persistent.

4. Almost Periodic Solution

Based on Theorem 9, in Section 3, we discuss the almost periodic property for system (1).

Consider the following almost periodic difference system: where , , and is almost periodic in uniformly for and is continuous in . The product system of system (30) is as follows:

Lemma 10 (see [2]). Suppose that there exists a Lyapunov function defined for , , satisfying that(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 (30) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of system (30) which is bounded by . In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of (30) of period .

According to Lemma 10, we first prove that there exists a bounded solution of system (1) and then construct an adaptive Lyapunov functional for system (1). We denote by the set of all solutions of system (1) satisfying , .

Lemma 11. Assume that (2), (3), and the conditions of Theorem 9 hold; then .

Proof. It is now possible to show by an inductive argument that the system (1) leads to for , . From Theorem 9, for any solution of system (1) with initial condition (2) satisfies Definition 1. Hence, for any , there exists a ; if is sufficiently large, we have Let be any integer valued sequence such that as ; we claim that there exists a subsequence of , and we still denote it by , such that uniformly in on any finite subset of as , where and is a finite number.
In fact, for any finite subset , when is large enough, , . So That is, are uniformly bounded for large enough .
Now, for , we can choose a subsequence of such that uniformly converges on for large enough.
Similarly, for , we can choose a subsequence of such that uniformly converges on for large enough.
Repeating this procedure, for , we can choose a subsequence of such that uniformly converges on for large enough.
Now pick the sequence which is a subsequence of ; we still denote it by ; then for all , we have uniformly in as .
By the arbitrary of , the conclusion is valid.

Since , , , , , and are almost periodic sequence, for the above sequence as , there exists a subsequence which we still denote it by (if necessary, we take subsequence), such that as uniformly on .

For any , we can assume that for large enough. Letting   and , by an inductive argument of (1) from to , leads to Then, for , , we have Let ; for any ,

By the arbitrariness of is a solution of system (1) on . It is clear that ; , .

So . Lemma 10 is valid.

Before stating Theorem 12, for the sake of convenience, we set

Theorem 12. Suppose that the conditions of Lemma 10 are satisfied; moreover, , where Then there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by for all .

Proof. Let ; from system (1), we have where , . From Lemma 10, it shows that system (43) has a bounded solution satisfying Thus, For , we define a norm For the product form of system (43) Let and be any two solutions of system (30) defined on ; then and , where , and For convenience, we denote Let be a function on .
Then are equivalent. It is easy to see that Let , , , and ; thus condition (i) in Lemma 10 is satisfied.
In addition, where , . And Lemma 10 (ii) is held.
Finally, calculating the along the solutions of (47) leads to By the mean value theorem we get , where lies between and .
From (49), (54), one has We get for that where Hence, where , , . That is, there exists a positive constant such that From , condition (iii) of Lemma 10 is satisfied. So, from Lemma 10, there exists a uniqueness uniformly asymptotically stable almost periodic solution of system (43) which is bounded by for all , which means that there exists a uniqueness uniformly asymptotically stable almost periodic solution of system (1) which is bounded by for all . This completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the editor and the referee for their careful reading of the original paper and many valuable comments and suggestions that greatly improved the presentation of this paper.