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
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:
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
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).
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 .
Lemma 2.1 (see )
Lemma 2.2 (see )
Lemma 2.3 (see )
Lemma 2.4 (see )
Lemma 2.5 (see )
Definition 2.1 (see )
A time scale is called an almost periodic time scale if
Definition 2.2 (see )
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 .
3 Main results
Assume that the coefficients of (1.1) satisfy
By Lemma 2.3, we can get
By Lemma 2.2, we can get
By Lemma 2.3, we can get
By Lemma 2.2, we can get
This completes the proof. □
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 .
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
Using the Lebesgue dominated convergence theorem, we have
This completes the proof. □
Lemma 3.3In addition to condition (H1), assume further that the coefficients of system (1.1) satisfy the following conditions:
Then system (1.1) is globally attractive.
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
By the comparison theorem and (3.12), we have
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.
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
On the other hand,
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:
By a direct calculation, we can get
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.
The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.
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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