Almost periodic solutions of a single-species system with feedback control on time scales
© Hu and Lv; licensee Springer 2013
Received: 30 January 2013
Accepted: 19 June 2013
Published: 3 July 2013
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.
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.
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 .
A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . Define the set .
Lemma 2.1 (see )
Lemma 2.2 (see )
Lemma 2.3 (see )
Lemma 2.4 (see )
Lemma 2.5 (see )
Definition 2.1 (see )
Definition 2.2 (see )
τ is called the ε-translation number of f, and is called the inclusion length of .
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
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 .
Inequalities (3.7) and (3.8) show that and are equi-continuous on . By the arbitrariness of , the conclusion is valid.
This completes the proof. □
Lemma 3.3 In addition to condition (H1), assume further that the coefficients of system (1.1) satisfy the following conditions:
Then system (1.1) is globally attractive.
Let . We divide the proof into two cases.
From the above discussion, we can see that system (1.1) is globally attractive. This completes the proof. □
Theorem 3.1 Assume that conditions (H1)-(H3) hold, then system (1.1) has a unique globally attractive positive almost periodic solution.
is an almost periodic function. It means that , .
So, the limit , , exists.
Next, we shall prove that is an almost solution of system (1.1).
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
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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