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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.

Let

As was pointed out by Berryman [

Considering the exploited predator-prey system (harvesting or stocking) is very valuable, for it involves the human activities. It can be referred to [

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 (

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:

Obviously, system (

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 (

The organization of this paper is as follows. In Section

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.

By

For

Since the solution of system (

The set of sequences

The function

The set of sequences

For any

For any

Let

Theoretically, one can investigate the existence, uniqueness, and stability of almost periodic solution for functional differential equations by using Lyapunov functional as follows [_{109}].

Consider the system of impulsive differential equations as follows:

Introduce the following conditions.

Function

Sequence

Suppose that there exists a Lyapunov functional

For

In this section, we establish a permanence result for system (

Assume that

If inequalities (

Assume that

Let

Suppose that

Let

Every solution

From the first equation of system (

Consider the following auxiliary system:

Then, for any constant

Let

According to Proposition

Consider the following auxiliary system:

By the second equation of system (

In view of the third equation of system (

In view of

By Propositions

Assume that

When

From the proof of Propositions

The main result of this paper is concerned with the existence of a unique uniformly asymptotically stable positive almost periodic solution for system (

Let

Assume that

there exist positive constants

Suppose that

Set

Construct a Lyapunov functional

It is obvious that

Since

For

By Lemma

When

Consider the following food chain system with impulsive perturbations:

Corresponding to system (

State variable

State variable

State variable

Stability of state variable

Stability of state variable

Stability of state variable

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

The authors declare that there is no conflict of interests regarding the publication of this paper.

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.