Research

# Almost periodic solutions of a single-species system with feedback control on time scales

Meng Hu* and Haiyan Lv

Author Affiliations

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan, 455000, People’s Republic of China

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Advances in Difference Equations 2013, 2013:196  doi:10.1186/1687-1847-2013-196

 Received: 30 January 2013 Accepted: 19 June 2013 Published: 3 July 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper is concerned with a single-species system with feedback control on time scales. Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are obtained. Finally, an example and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

##### Keywords:
permanence; almost periodic solution; global attractivity; time scale

### 1 Introduction

In the past few years, different types of ecosystems with periodic coefficients have been studied extensively; see, for example, [1-5] and the references therein. 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. Therefore, if we consider the effects of the environmental factors (e.g., seasonal effects of weather, food supplies, mating habits and harvesting), the assumption of almost periodicity is more realistic, more important and more general. Almost periodicity of different types of ecosystems has received more recently researchers’ special attention; see [6-10] and the references therein.

However, in the natural world, there are many species whose developing processes are both continuous and discrete. Hence, using the only differential equation or difference equation cannot accurately describe the law of their development; see, for example, [11,12]. Therefore, there is a need to establish correspondent dynamic models on new time scales.

To the best of the authors’ knowledge, there are few papers published on the existence of an almost periodic solution of ecosystems on time scales.

Motivated by the above, in the present paper, we shall study an almost periodic single-species system with feedback control on time scales as follows:

(1.1)

where , is an almost time scale. All the coefficients , , , , , , are continuous, almost periodic functions.

For convenience, we introduce the notation

where f is a positive and bounded function. Throughout this paper, we assume that the coefficients of almost periodic system (1.1) satisfy

The initial condition of system (1.1) is in the form

(1.2)

The aim of this paper is, by using the properties of almost periodic functions and constructing a suitable Lyapunov functional, to obtain sufficient conditions for the existence of a unique globally attractive positive almost periodic solution of system (1.1).

In this paper, the time scale considered is unbounded above, and for each interval of , we denote .

### 2 Preliminaries

Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators and the graininess are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum m, then ; otherwise . If has a right-scattered minimum m, then ; otherwise .

A function is right-dense continuous provided it is continuous at a right-dense point in and its left-side limits exist at left-dense points in . If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on .

For the basic theories of calculus on time scales, one can see [13].

A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . Define the set .

If r is a regressive function, then the generalized exponential function is defined by

for all , with the cylinder transformation

Let be two regressive functions, define

Lemma 2.1 (see [13])

Ifare two regressive functions, then

(i) and;

(ii) ;

(iii) ;

(iv) ;

(v) ;

(vi) .

Lemma 2.2 (see [14])

Assume that, and. Then

implies

Lemma 2.3 (see [14])

Assume that, . Then

implies

Let be a time scale with at least two positive points, one of them being always one: . There exists at least one point such that . Define the natural logarithm function on the time scale by

Lemma 2.4 (see [15])

Assume thatis strictly increasing andis a time scale. Ifexists for, then

Lemma 2.5 (see [13])

Assume thatare differentiable at, thenis differentiable attwith

Definition 2.1 (see [16])

A time scale is called an almost periodic time scale if

Definition 2.2 (see [16])

Let be an almost periodic time scale. A function is called an almost periodic function if the ε-translation set of f

is a relatively dense set in for all ; that is, for any given , there exists a constant such that in any interval of length , there exists at least a such that

τ is called the ε-translation number of f, and is called the inclusion length of .

For relevant definitions and the properties of almost periodic functions, see [16-18]. Similar to the proof of Corollary 1.2 in [18], we can get the following lemma.

Lemma 2.6Letbe an almost periodic time scale. If, are almost periodic functions, then, for any, is a nonempty relatively dense set in; that is, for any given, there exists a constantsuch that in any interval of length, there exists at least asuch that

Remark 2.1 Lemma 2.6 is a special case of Theorem 3.22 in [16].

### 3 Main results

Assume that the coefficients of (1.1) satisfy

(H1) .

Lemma 3.1Letbe any positive solution of system (1.1) with initial condition (1.2). If (H1) holds, then system (1.1) is permanent, that is, any positive solutionof system (1.1) satisfies

(3.1)

(3.2)

especially if, , then

where

Proof Assume that is any positive solution of system (1.1) with initial condition (1.2). From the first equation of system (1.1), we have

(3.3)

By Lemma 2.3, we can get

Then, for an arbitrarily small positive constant , there exists a such that

From the second equation of system (1.1), when ,

Let , then

(3.4)

By Lemma 2.2, we can get

Then, for an arbitrarily small positive constant , there exists a such that

On the other hand, from the first equation of system (1.1), when ,

Let , then

(3.5)

By Lemma 2.3, we can get

Then, for an arbitrarily small positive constant , there exists a such that

From the second equation of system (1.1), when ,

Let , then

(3.6)

By Lemma 2.2, we can get

Then, for arbitrarily small positive constant , there exists a such that

In special case, if , , by Lemma 2.2 and Lemma 2.3, it follows from (3.3)-(3.6) that

This completes the proof. □

Let be a set of all solutions of system (1.1) satisfying , for all .

Lemma 3.2.

Proof By Lemma 3.1, we see that for any with , , system (1.1) has a solution satisfying , , . Since , , , , , , , are almost periodic, it follows from Lemma 2.6 that there exists a sequence , as such that , , , , , , , as uniformly on .

We claim that and are uniformly bounded and equi-continuous on any bounded interval in .

In fact, for any bounded interval , when n is large enough, , then , . So, , for any , that is, and are uniformly bounded. On the other hand, , from the mean value theorem of differential calculus on time scales, we have

(3.7)

(3.8)

Inequalities (3.7) and (3.8) show that and are equi-continuous on . By the arbitrariness of , the conclusion is valid.

By the Ascoli-Arzela theorem, there exists a subsequence of , we still denote it as , such that

as uniformly in t on any bounded interval in . For any , we can assume that for all n. Let , integrating both equations of system (1.1) from to , we have

and

Using the Lebesgue dominated convergence theorem, we have

This means that is a solution of system (1.1), and by the arbitrariness of θ, is a solution of system (1.1) on . It is clear that

This completes the proof. □

Lemma 3.3In addition to condition (H1), assume further that the coefficients of system (1.1) satisfy the following conditions:

(H2) ;

(H3) .

Then system (1.1) is globally attractive.

Proof Let and be any two positive solutions of system (1.1). It follows from (3.1)-(3.2) that for a sufficiently small positive constant , there exists a such that

(3.9)

and

(3.10)

Since , , are positive, bounded and differentiable functions on , then there exists a positive, bounded and differentiable function , , such that , , are strictly increasing on . By Lemma 2.4 and Lemma 2.5, we have

Here, we can choose a function such that is bounded on , that is, there exist two positive constants and such that , .

Set

where is a constant (if , then ; if , then ). It follows from the mean value theorem of differential calculus on time scales for that

(3.11)

Let . We divide the proof into two cases.

Case I. If , set and . Calculating the upper right derivatives of along the solution of system (1.1), it follows from (3.9)-(3.11), (H2) and (H3) that for ,

(3.12)

By the comparison theorem and (3.12), we have

that is,

then

(3.13)

Since and , then . It follows from (3.13) that

Case II. If , set , then and . Calculating the upper right derivatives of along the solution of system (1.1), it follows from (3.9)-(3.11), (H2) and (H3) that for ,

(3.14)

where . By the comparison theorem and (3.14), we have

that is,

then

(3.15)

It follows from (3.15) that

From the above discussion, we can see that system (1.1) is globally attractive. This completes the proof. □

Theorem 3.1Assume that conditions (H1)-(H3) hold, then system (1.1) has a unique globally attractive positive almost periodic solution.

Proof By Lemma 3.2, there exists a bounded positive solution , then there exists a sequence , as , such that is a solution of the following system:

From the above discussion and Lemma 2.1, we have that not only , , but also , , are uniformly bounded, thus , , are uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of such that for any , there exists a with the property that if then

It shows that , , are asymptotically almost periodic functions, then , , are the sum of an almost periodic function , , and a continuous function , , defined on , that is,

where

is an almost periodic function. It means that , .

On the other hand,

So, the limit , , exists.

Next, we shall prove that is an almost solution of system (1.1).

From the properties of an almost periodic function, there exists a sequence , as , such that , , , , , , , as uniformly on .

It is easy to know that , as , then we have

This proves that is a positive almost periodic solution of system (1.1). Together with Lemma 3.3, system (1.1) has a unique globally attractive positive almost periodic solution. This completes the proof. □

### 4 Example and simulations

Consider the following system on time scales:

(4.1)

By a direct calculation, we can get

then

that is, conditions (H1)-(H3) hold. According to Theorem 3.1, system (4.1) has a unique globally attractive positive almost periodic solution. For dynamic simulations of system (4.1) with and , see Figures 1 and 2, respectively.

Figure 1. . Dynamics behavior of system (4.1) with initial condition.

Figure 2. . Dynamics behavior of system (4.1) with initial condition.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.

### Acknowledgements

This work was supported by the National Natural Sciences Foundation of China (Grant No. 11071143) and the Natural Sciences Foundation of Henan Educational Committee (Grant No. 2011A110001).

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