Abstract
This paper is concerned with a nonautonomous fishing model with a time-varying delay. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of positive almost periodic solutions of the model with almost periodic coefficients and delays. Moreover, an example and its numerical simulation are given to illustrate the main results.
1. Introduction
In the classic study of population dynamics, the differential equation
is widely used in fisheries [1–4], where denotes the population biomass, denotes the per-capita fecundity rate, denotes the per-capita mortality rate, and is the harvesting rate per-capita.
Let be a Hills type function [1, 3]
Denote . Taking account of the delay and the varying environments, Berezansky and Idels [5] proposed the following time-lag model based on (1): where , , , are almost periodic functions and parameter . Consequently, some theorems on the stability and existence of periodic solutions for (3) were established in Berezansky and Idels [5] and Wang [6]. For more details, we refer to the article of Wang [6].
In the real-world phenomena, the two foundations for the theory of nature selection are periodically and almost periodically varying environment. And the almost periodic effects are more frequent than the periodic effects (see [7, 8]). Therefore, the effects of the almost periodic environment on the evolutionary theory have been extensively studied by a large number of researchers and some of these results can be found in [9–12]. For the reason of seasonal variation, it is not necessary to let (3) be exactly periodic but almost periodic instead. The problem of finding the global stability conditions for the positive almost periodic solution of (3) with almost periodic coefficients and delays becomes important. As pointed out in [7, 8], significant differences often appear in almost periodic problem by comparison with the periodic case. For example, contrary to periodic functions, there exists an almost periodic function such that for all and . Hence, it is difficult to establish the existence, uniqueness, and global exponential stability of positive almost periodic solutions of (3). Moreover, to the best of our knowledge, there is no literature considering the global exponential stability of positive almost periodic solutions problem for (3) and its generalized equations.
Motivated by the above discussions, in this paper, a new approach will be developed to obtain a condition for the global exponential stability of the positive almost periodic solutions of (3), and the exponential convergent rate can be unveiled.
For convenience, we introduce some notations. Given a bounded continuous function defined on , let and be defined as It will be assumed that Then, Let denote nonnegative real number space, let be the continuous functions space equipped with the usual supremum norm , and let . If is defined on with , then we define where for all .
Due to the biological interpretation of model (3), only positive solutions are meaningful and therefore admissible. Thus, we just consider admissible initial conditions We denote for an admissible solution of the admissible initial value problems (3) and (9). Also, let be the maximal right-interval of existence of .
2. Preliminary Results
In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.
Definition 1 (see [7, 8]). Let be continuous in . is said to be almost periodic on if, for any , the set is relatively dense; that is, for any , it is possible to find a real number with the property that, for any interval with length , there exists a number in this interval such that for all .
From the theory of almost periodic functions in [7, 8], 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 .
From (5) and Lemma 1.1 in Berezansky and Idels [5], we obtain the following lemma.
Lemma 2. Every solution of (3) and (9) is persistent on , and ; that is, there exist two positive constants and such that
Lemma 3. For every solution of (3) and (9), there exists such that
Proof. Let . It follows from Lemma 2 that
We now give two cases to prove that there exists such that
Case i. Suppose that
Then,
By the fluctuation lemma [13, Lemma A.1], there exists a sequence such that
Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that
From (16), we get
By the boundedness of the coefficients and delays, there is a subsequence of , still denoted by , such that , , , and are convergent to , , , and , respectively. In view of the facts and , it follows from
that (taking limits)
which yields
This implies that (14) holds.
Case ii. If there exists such that , then (3) yields
and we have
Integrating (24) from to , in view of (23), we obtain
For any , with the same approach as that in derivation of (25), we can show
On the other hand, if and , we can choose such that
which, together with (26), yields
Thus, there must exist such that (14) holds.
Again from the fluctuation lemma [13, Lemma A.1], there exists a sequence such that
Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that
We next divide our proof in two steps to show that there exists such that
First, assume that
Then,
According to (30) and (33), we get
Without loss of generality, we assume that all , , , and are convergent to , , , and , respectively. This can be achieved because of almost periodicity. In view of (3) and (30), it follows from
that (taking limits)
which, together with (8), yields
This implies that (31) holds.
Second, there exists such that
This, together with (3), yields
and we have
for all . Integrating (40) from to , in view of (6), we obtain
For any , with the same approach as that in derivation of (41), we can show
On the other hand, if and , we can choose such that
which, together with (42), yields
Thus, there must exist such that (31) holds.
In summary, (14) and (31) imply that there exists such that (12) holds. This ends the proof of Lemma 3.
Lemma 4. Let where , , and . Moreover, assume that is a solution of (3) with initial condition (9) and is bounded continuous on . Then, for any , there exists , such that every interval contains at least one number for which there exists satisfying
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 3, the solution is bounded and
which implies that the right side of (3) 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 (10), for any , there exists , such that every interval , , contains a for which
Let . For , denote . Then, for all , we get
where lies between and .
Calculating the upper right derivative of yields
Let
It is obvious that and is nondecreasing.
Now, we distinguish two cases to finish the proof.
Case One. Consider
We claim that
Assume, by way of contradiction, that (57) does not hold. Then, there exists such that . Since
There must exist such that
which contradicts (56). This contradiction implies that (57) holds. It follows that there exists such that
Case Two. There is a such that . Then, in view of (49), (52), and (54), we get
which yields that
For any , with the same approach as that in derivation of (62), we can show
if .
On the other hand, if and , we can choose such that
which, together with (63), yields
With a similar argument as that in the proof of Case One, 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 (3).
Theorem 5. Under the assumptions of Lemma 4, (3) has at least one positive almost periodic solution . Moreover, is globally exponentially stable; that is, there exist constants and such that
Proof. Let be a solution of (3) 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 3, the solution is bounded and
which implies that the right side of (3) 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 it by ) uniformly converges to a continuous function on any compact set of , and
Now, we prove that is a solution of (2). In fact, for any and , from (71), we have
where . Consequently, (73) implies that
Therefore, is a solution of (3).
Secondly, we prove that is an almost periodic solution of (3). With the help of Lemma 4, for any , there exists , such that every interval contains at least one number for which there exists satisfying
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 (3).
Finally, we prove that is globally exponentially stable.
Let and , where . Then,
where and lies between and .
It follows from Lemma 3 that there exists such that
We consider the Lyapunov functional
Calculating the upper left derivative of along the solution of (78), we have
We claim that
Contrarily, there must exist such that
Together with (55), (81), and (83), we obtain
Thus,
which contradicts (49). Hence, (82) holds. It follows that
This completes the proof of Theorem 5.
4. An Example
In this section, we present an example to check the validity of the results we obtained in the previous sections.
Example 1. Consider the following fishing model with time-varying delay: Obviously, , , , , , . By calculating, we obtain which imply that (87) satisfies the assumptions of Theorem 5. Therefore, (87) 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.
Remark 6. Most recently, by using Mawhin continuation theorem, criteria ensuring the local existence of almost periodic solutions for the fishing model (3) are established in Li et al. [14], where the global exponential convergence for almost periodic solution is not touched. Unfortunately, as pointed out by Wang and Zhang [15] and Ortega [16], for the essential reason that the compact condition is not suitable for the almost periodic function family, the coincidence degree cannot be used to solve almost periodic problem. Hence, the mapping of Lemma 3.3 in Li et al. [14] is not -compact and the existence of almost periodic solutions for (3) cannot hold. Moreover, to the best of our knowledge, there is no research on the global exponential stability of positive almost periodic solutions to the fishing model (3). We also mention that all results in [5, 6, 9, 10, 12] cannot be applied to imply that all solutions of (87) with initial values (9) converge exponentially to the positive almost periodic solution. Here, we employ a novel proof to establish some criteria to guarantee the existence and global exponential stability of positive almost periodic solutions for fishing model with time-varying coefficients and delays. This implies that the results of this paper are contributed to complement the previous references in this topic.
Conflict of Interests
The authors declare no conflict of interests. They also declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this present paper.
Acknowledgments
The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. In particular, the authors express the sincere gratitude to Professor Bingwen Liu for the helpful discussion when this revision work is carried out. This work was supported by the National Natural Science Foundation of China (Grant no. 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).