Abstract
In order to obtain a more accurate description of the ecological system perturbed by human exploitation activities such as planting and harvesting, we need to consider the impulsive differential equations. Therefore, by applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a food chain system with almost periodic impulsive perturbations. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.
1. Introduction
Let and denote the sets of real numbers and integers, respectively. Related to a continuous function , we use the following notations:
As was pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Food chain predator-prey system, as one of the most important predator-prey systems, has been extensively studied by many scholars; many excellent results were concerned with the persistent property and positive periodic solution of the system; see [2–8] and the references cited therein. Recently, Shen considered the following three species food chain predator-prey system with Holling type IV functional response: where , , denotes the density of species at time , is the predator of the first species , and is the predator of the second species . By applying the comparison theorem of the differential equation and constructing the suitable Lyapunov function, sufficient conditions which guarantee the permanence and the global attractivity of the system are obtained.
Considering the exploited predator-prey system (harvesting or stocking) is very valuable, for it involves the human activities. It can be referred to [9], in which the human activities always happen in a short time or instantaneously. The continuous action of human is then removed from the model and replaced with an impulsive perturbation. These models are subject to short-term perturbations which are often assumed to be in the form of impulsive in the modelling process. Consequently, impulsive differential equations provide a natural description of such systems [10–13]. Then, in [14], Zhang and Tan studied the following Holling II functional responses food chain system with periodic constant impulsive perturbation of predator: where is the release amount of top predator at and is the period of the impulsive effect. By using the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we consider the local stability of prey and top predator eradication periodic solution.
In real world phenomenon, the environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting. So, it is usual to assume the periodicity of parameters in system (2). 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 the assumption of almost periodicity is more realistic, more important, and more general when we consider the effects of the environmental factors. In recent years, there have been many mathematical studies for the existence, uniqueness, and stability of positive almost periodic solution of biological models governed by differential equations in the literature (see [11, 15–25] and the references cited therein). Therefore, Bai and wang in [15] studied the following nonautonomous food chains system with Holling's type II functional response: By applying the comparison theorem and the Lyapunov method of ordinary differential equations, some sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of system (4) are obtained.
Stimulated by the above reason, this paper is concerned with the following almost periodic food chain system with almost periodic impulsive perturbations and general functional responses: where , , and , , , are all continuous almost periodic functions which are bounded above and below by positive constants; are positive constants; are constants; are impulse points with ; and the set of sequences , is uniformly almost periodic (see Definition 1 in Section 2).
Obviously, system (2)–(4) is special case of system (5).
The main purpose of this paper is to establish some new sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of system (5) by using the comparison theorem and the Lyapunov method of the impulsive differential equations [10, 11] (see Theorems 11 and 14 in Sections 3 and 4).
The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Section 3, by using the comparison theorem of the impulsive differential equations [10], we give the permanence of system (5). In Section 4, we study the existence of a unique uniformly asymptotically stable positive almost periodic solution of system (5) by applying the Lyapunov method of the impulsive differential equations [11].
2. Preliminaries
Now, let us state the following definitions and lemmas, which will be useful in proving our main result.
By , , , , we denote the set of all sequences that are unbounded and strictly increasing. Introduce the following notations.
For , is the space of all piecewise continuous functions from to with points of discontinuity of the first kind , at which it is left continuous. By the basic theories of impulsive differential equations in [10, 11], system (5) has a unique solution .
Since the solution of system (5) is a piecewise continuous function with points of discontinuity of the first kind , , we adopt the following definitions for almost periodicity.
Definition 1 (see [11]). The set of sequences , is said to be uniformly almost periodic if for arbitrary there exists a relatively dense set of -almost periods common for any sequences.
Definition 2 (see [11]). The function is said to be almost periodic, if the following hold. (1)The set of sequences , is uniformly almost periodic.(2)For any , there exists a real number such that if the points and belong to one and the same interval of continuity of and satisfy the inequality , then .(3)For any , there exists a relatively dense set such that if , then for all satisfying condition , . The elements of are called -almost periods.
Lemma 3 (see [11]). Let . Then, there exists a positive integer such that, on each interval of length , one has no more than elements of the sequence ; that is, where is the number of the points in the interval .
Theoretically, one can investigate the existence, uniqueness, and stability of almost periodic solution for functional differential equations by using Lyapunov functional as follows [11, P109].
Consider the system of impulsive differential equations as follows: where , , , , , and is an open set in .
Introduce the following conditions.Function is almost periodic in uniformly with respect to .Sequence , , is almost periodic uniformly with respect to .
Lemma 4 (see [11, P109]). Suppose that there exists a Lyapunov functional defined on satisfying the following conditions. (1), where with is continuous increasing function and as .(2), where is a constant.(3)For , ; for , , , where is a constant.Moreover, one assumes that system (7) has a solution that remains in a compact set . Then, system (7) has a unique almost periodic solution which is uniformly asymptotically stable.
3. Permanence
In this section, we establish a permanence result for system (5).
Lemma 5 (see [10]). Assume that with points of discontinuity at and is left continuous at for and where , , and is nondecreasing in for . Let be the maximal solution of the scalar impulsive differential equation as existing on . Then, implies for .
Remark 6. If inequalities (8) in Lemma 5 are reversed and is the minimal solution of system (9) existing on , then implies for .
Lemma 7. Assume that ; then, the following impulsive logistic equation has a unique globally asymptotically stable positive almost periodic solution which can be expressed as follows: where is defined as that in Lemma 3, , , , , and
Proof. Let ; then, system (10) changes to
We can easily obtain that system (13) has a unique almost periodic solution which can be expressed as follows:
Then, system (10) has a unique almost periodic solution which can be expressed by (11). By Lemma 3, we have
On the other hand,
Suppose that is another positive solution of system (10). Define a Lyapunov function as
For , , calculating the upper right derivative of along the solution of system (10), we have
For , , we have
Therefore, is nonincreasing. Integrating (18) from to leads to
that is,
which implies that
Thus, the almost periodic solution of system (10) is globally asymptotically stable. This completes the proof.
Let
Proposition 8. Every solution of system (5) satisfies where , , and are defined as those in (27), (32), and (35), respectively.
Proof. From the first equation of system (5), we have
Consider the following auxiliary system:
By Lemma 5, , where is the solution of system (26) with . By Lemma 7, system (26) has a unique globally asymptotically stable positive almost periodic solution which can be expressed as follows:
where
Then, for any constant , there exists such that for . So,
For any , there exists such that
From the second equation of system (5), we have
Similar to the above argument as that in (29), one has
Then, there exists such that
By the third equation of system (5), we have
Similar to the above argument as that in (32), we have
This completes the proof.
Proposition 9. Let , , and be defined as those in (42)–(48), respectively. Then, every solution of system (5) satisfies if the following condition holds: (), , and .
Proof. According to Proposition 8, there exist and such that
From the first equation of system (5), we have
Consider the following auxiliary system:
By Remark 15, for , where is the solution of system (39) with . By Lemma 7, system (39) has a unique globally asymptotically stable positive almost periodic solution which can be expressed as follows:
Similar to the above argument as that in (29), we have . By the arbitrariness of , it leads to
Then, there exist and such that
By the second equation of system (5), we have
Similar to the above argument as that in (42), one has
Then, there exist and such that
In view of the third equation of system (5), we have
Similar to the above argument as that in (45), one has
This completes the proof.
Remark 10. In view of in Proposition 9, the values of impulse coefficients and the number of the impulse points in each interval of length have negative effect on the permanence of system (5).
By Propositions 8 and 9, we have the following theorem.
Theorem 11. Assume that holds; then, system (5) is permanent.
Remark 12. When in system (5), then Theorem 11 changes to the corresponding permanence result in Bai and Wang [15]. So, Theorem 11 extends the corresponding result in Bai and Wang [15]. Further, Theorem 11 gives the sufficient conditions for the permanence of system (5) with almost periodic impulsive perturbations. Therefore, Theorem 11 provides a possible method to study the permanence of the models with impulsive perturbations in biological populations.
Remark 13. From the proof of Propositions 8 and 9, we know that, under the conditions of Theorem 11, set is an invariant set of system (5).
4. Almost Periodic Solution
The main result of this paper is concerned with the existence of a unique uniformly asymptotically stable positive almost periodic solution for system (5).
Let
Theorem 14. Assume that holds; suppose further that ()there exist positive constants , , , and such that then, system (5) admits a unique positive almost periodic solution, which is uniformly asymptotically stable.
Proof. Suppose that and are any two solutions of system (5). Consider the product system of system (5) as
Set , which is an invariant set of system (51) directly from Remark 13.
Construct a Lyapunov functional defined on as follows:
It is obvious that
where . Further, we have
where ; thus, (1) in Lemma 4 is satisfied.
Since
(2) in Lemma 4 holds.
For , , calculating the upper right derivative of along the solution of system (51), we have
For , , we have
In view of (56)-(57), (3) in Lemma 4 is satisfied.
By Lemma 4, system (5) admits a unique uniformly asymptotically stable positive almost periodic solution . This completes the proof.
Remark 15. When in system (5), then Theorem 14 changes to the corresponding permanence result in Bai and Wang [15]. So, Theorem 14 extends the corresponding result in Bai and Wang [15]. Further, Theorem 14 gives the sufficient conditions for the uniform asymptotical stability of a unique positive almost periodic solution of system (5), in which and are allowed to be any real-valued positive number. Therefore, Theorem 14 provides a possible method to study the existence, uniqueness, and stability of positive almost periodic solution of the models with impulsive perturbations in biological populations.
5. An Example and Numerical Simulations
Example 1. Consider the following food chain system with impulsive perturbations: then, system (58) is permanent and admits a unique uniformly asymptotically stable positive almost periodic solution.
Proof. Corresponding to system (2), , , , , , , , , , , , , . Taking , the result is easy to obtain from Theorems 11 and 14; we would omit it (see Figures 1, 2, 3, 4, 5, and 6). This completes the proof.
6. Conclusion
By using the comparison theorem and the Lyapunov method of the impulsive differential equations, sufficient conditions are obtained which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of a food chain system with almost periodic impulsive perturbations. Proposition 9 and Theorem 14 imply that the values of impulse coefficients and the number of the impulse points in each interval of length are harm for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the model. The main results obtained in this paper are completely new and the method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the reviewers for their valuable comments and constructive suggestions, which considerably improve the presentation of this paper. This work was supported by the Scientific Research Fund of Yunnan Provincial Education Department.