Abstract
This paper is concerned with a singlespecies 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 scale1 Introduction
In the past few years, different types of ecosystems with periodic coefficients have been studied extensively; see, for example, [15] 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 [610] 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 singlespecies system with feedback control on time scales as follows:
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
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 leftdense if and , leftscattered if , rightdense if and , and rightscattered if . If has a leftscattered maximum m, then ; otherwise . If has a rightscattered minimum m, then ; otherwise .
A function is rightdense continuous provided it is continuous at a rightdense point in and its leftside limits exist at leftdense points in . If f is continuous at each rightdense point and each leftdense 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 rdcontinuous 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
Lemma 2.2 (see [14])
implies
Lemma 2.3 (see [14])
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 εtranslation 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 εtranslation number of f, and is called the inclusion length of .
For relevant definitions and the properties of almost periodic functions, see [1618]. 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
Lemma 3.1Letbe any positive solution of system (1.1) with initial condition (1.2). If (H_{1}) holds, then system (1.1) is permanent, that is, any positive solutionof system (1.1) satisfies
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
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 ,
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 ,
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 ,
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 .
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 equicontinuous 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
Inequalities (3.7) and (3.8) show that and are equicontinuous on . By the arbitrariness of , the conclusion is valid.
By the AscoliArzela 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 (H_{1}), assume further that the coefficients of system (1.1) satisfy the following conditions:
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
and
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
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), (H_{2}) and (H_{3}) that for ,
By the comparison theorem and (3.12), we have
that is,
then
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), (H_{2}) and (H_{3}) that for ,
where . By the comparison theorem and (3.14), we have
that is,
then
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 (H_{1})(H_{3}) 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 equicontinuous. By the AscoliArzela 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,
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:
By a direct calculation, we can get
then
that is, conditions (H_{1})(H_{3}) 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.
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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