Journal of Applied Mathematics

Volume 2014 (2014), Article ID 394105, 6 pages

http://dx.doi.org/10.1155/2014/394105

Research Article

Global Exponential Stability of Positive Almost Periodic Solutions for a Delay Logarithmic Population Model

Wei Chen¹ and Wentao Wang²

¹School of Mathematics and Information, Shanghai Lixin University of Commerce, Shanghai 201620, China

²College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

Received 4 October 2013; Accepted 17 April 2014; Published 13 May 2014

Academic Editor: Vu Ngoc Phat

Copyright © 2014 Wei Chen and Wentao Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with a delay logarithmic population model. Under proper conditions, we employ a novel proof to establish a criterion on guaranteeing the existence and global exponential stability of positive almost periodic solutions for the model. Moreover, an example and its numerical simulation are given to illustrate the main results.

1. Introduction

In the classic study of population dynamics, the dynamical analysis of logarithmic population model has attracted a great attention of many mathematicians and biologists in recent years. It is well known that the environment is varying periodically with time in many realistic systems and the parameters of the system usually change along with time periodically. So it is reasonable to study periodic solution for logarithmic population model and its modified model with periodic coefficients. There exist some results on the existence of periodic solution for the model; see, for example, [1–8]. Compared with periodic effects, almost periodic effects are more frequent (see [9–12]). Hence, it is of great importance to consider the dynamical behaviors of logarithmic population model with almost periodically varying coefficient. Recently, by utilizing the continuation theorem and contraction mapping principle, some criteria have been established to prove the existence and local exponential stability of positive almost periodic solutions for delay logarithmic population model and its generalized modification in the literature; see [13–17]. However, to the best of our knowledge, there is no literature considering the existence and global exponential stability of positive almost periodic solutions problem for delay logarithmic population model.

Inspired by the above discussions, in this paper, we consider the following delay logarithmic population equation:

which was proposed by Gopalsamy [18] to describe a model of single species population. Here is the size of population, is the growth rate while there are plenty of resources and there is no intraspecific competition for these resources, is the measure of the competition among the individuals, is added to generalize the model with the same interpretation of competitive effects, and is a maturation delay in the sense that competition involves adults who have matured by an age of units.

A primary purpose of this paper is to consider the existence and global exponential stability of positive almost periodic solutions of (1). A new approach will be developed to obtain a delay-independent condition for the global exponential stability of the positive almost periodic solutions of (1).

For convenience, we introduce some notations. Given a bounded continuous function defined on , let and be defined as

It will be assumed that , , are continuous almost periodic functions. Let denote the nonnegative real number space, let be the continuous functions space equipped with the usual supremun norm , and let .

Due to the biological interpretation of model (1), only positive solutions are meaningful and therefore admissible. Thus we just consider admissible initial conditions

Let ; then (1) and admissible initial conditions (3) can be rewritten in the form

Obviously, model (1) has a unique positive almost periodic solution which is globally exponentially stable if and only if system (4) has a unique almost periodic solution which is globally exponentially stable.

2. Preliminary Results

In this section, some lemmas and definitions will be presented, which are of great significance in proving our main results in Section 3.

Definition 1 (see [9, 10]). Let be continuous in . is said to be almost periodic on if, for any , the set for all is relatively dense; that is, for any , it is possible to find a real number , and for any interval with length , there exists a number in this interval such that , for all .

From the theory of almost periodic functions in [9, 10], it follows that, for any , it is possible to find a real number , and for any interval with length , there exists a number in this interval such that

for all .

Lemma 2. Suppose that and is a solution of (4) with initial conditions satisfying (5) and the following condition:

Then, for in the interval of existence

Proof. Assume, by way of contradiction, that (8) does not hold. Then suppose that there exists such that

Calculating the upper left-hand derivative of , it follows from (4) and (9) that

which is contradictory and shows that (8) holds.

Remark 3. By virtue of the boundedness of this solution, from the theory of functional differential equations in [19], it follows that the solution of system (4) with initial conditions (7) can be defined on .

Lemma 4. Let hold. Assume that is a solution of (4) with initial condition such that (7) is satisfied. Then for any , there exists , such that every interval contains at least one number for which there exists which satisfies

Proof. Define a continuous function by setting

Then, we have

which implies that there exist two constants and such that

For , we add the definition of with . Set

By Lemma 2, the solution is bounded and

which implies that the right side of (4) is also bounded, and is a bounded function on . Thus, in view of the fact that for , we obtain that is uniformly continuous on . From (6), for any , there exists , such that every interval , , contains a for which

Recall that is the same as the one mentioned in (14).

Let . For , denote

Then, for all , we get

Calculating the left upper derivative of yields

Let

It is obvious that and that is nondecreasing.

Now, we distinguish two cases to finish the proof since and are both possible.

Case  1. Consider the following:

We claim that

Assume, by way of contradiction, that (23) does not hold. Then, there exists such that . Since

there must exist such that

which contradicts (22). This contradiction implies that (23) holds. It follows that there exists such that

Case  2. There is a that . Then, in view of (14) and (20), we get

which yields that

For any , with the same approach as that in deriving of (28), we can show

if .

On the other hand, if and , we can choose such that

which, together with (29), yields

With a similar argument as that in the proof of Case  1, we can show that

which implies that

In summary, there must exist such that holds for all . The proof of Lemma 4 is now complete.

3. Main Results

In this section, we establish sufficient conditions on the existence and global exponential stability of almost periodic solutions of (4).

Theorem 5. Under the assumptions of Lemma 4, (4) has at least one almost periodic solution .

Proof. Let be a solution of (4) with initial conditions satisfying the assumptions in Lemma 4. We also add the definition of with for all . Set

where is any sequence of real numbers. By Lemma 4, the solution is bounded and

which implies that the right side of (4) is also bounded, and is a bounded function on . Thus, in view of the fact that for , we obtain that is uniformly continuous on . Then, from the almost periodicity of , and , we can select a sequence such that

Since is uniformly bounded and equiuniformly continuous, by Arzala-Ascoli lemma and diagonal selection principle, we can choose a subsequence of , such that (for convenience, we still denote by ) uniformly converges to a continuous function on any compact set of , and

In the sequel, we prove that is a solution of (4). In fact, for any and , from (36), we have

where . Consequently, (38) implies that

Therefore, is a solution of (4).

Then, we prove that is an almost periodic solution of (4). From Lemma 4, for any , there exists , such that every interval contains at least one number for which there exists which satisfies

Then, for any fixed , we can find a sufficient large positive integer such that for any ,

Let ; we obtain

which implies that is an almost periodic solution of (4). This completes the proof.

The following result implies that (4) has a unique almost periodic solution.

Theorem 6. Suppose that all conditions in Theorem 5 are satisfied. Then (4) has an almost periodic solution which is globally exponentially stable.

Proof. Theorem 5 tells us that (4) has at least one almost periodic solution. Let be one of them. We show that is globally exponentially stable. Let be an arbitrary solution of (4) and define , where . Then

We consider the Lyapunov functional

Calculating the upper left derivative of along the solution of (43), we have

We claim that

Contrarily, there must exist such that

Together with (45) and (47), we obtain

Thus,

which contradicts with (14). Hence, (46) holds. It follows that

This completes the proof of Theorem 5.

Corollary 7. Suppose that all conditions in Theorem 5 are satisfied. Then (1) has a unique positive almost periodic solution which is globally exponentially stable.

Proof. From Theorems 5 and 6, we know that (4) has a unique almost periodic solution , which is globally exponentially stable. Consider that and are the solutions of (1) and (4) with the relation and the relevant initial conditions and satisfy ; it is natural to get that (1) has a unique positive almost periodic solution , which is also globally exponentially stable.

4. An Example

In this section, we present an example to check the validity of our results obtained in the previous sections.

Example 1. Consider the following delay logarithmic population equation:

Obviously, , . It is easy to get

which implies that (51) satisfies the assumptions of Corollary 7. Therefore, (51) has a unique positive almost periodic solution , which is globally exponentially stable with the exponential convergent rate . The numerical simulation in Figure 1 strongly supports the conclusion.

Figure 1: Numerical solution of system (51) for initial value .

Remark 8. Since [14, 15] only obtain the existence and local exponential stability of positive almost periodic solutions for delay logarithmic population model (1), one can observe that all the results in this literature and the references therein can not be applicable to prove the existence and global exponential stability of positive almost periodic solution for (51). This implies that the results of this paper are essentially new.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

The authors have made this paper independently. The authors read and approved the final version.

Acknowledgments

The authors are grateful to the referees for their suggestions that improved the writing of the paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 11301341 and 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436). Innovation Program of Shanghai Municipal Education Commission (Grant no. 13YZ127), and Shanghai Fiscal Special Fund (Grant no. A4- 4902-13-12).

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