Positive Almost Periodic Solution for Impulsive Nicholson's Blowflies Model with Linear Harvesting Term on the Bounded Domain

In this paper, impulsive Nicholson's blowflies model with linear harvesting term is studied. By using the contraction mapping fixed point theorem, we obtain sufficient conditions for the existence of a unique positive almost periodic solution. In addition, the exponential convergence of positive almost periodic solution is investigated. The dynamic behaviors of biological models are very important research topics. In 1980, Gurney [1] proposed the following delay differential equation () () () () xt x t ax t bx t e         to describe the population of the Australian sheep-blowfly and to agree with the experimental data obtained by Nicholson in [2]. Since this equation explains Nicholson's data of blowfly more accurately, this model and its modifications have been now referred to as the Nicholson's blowflies model. The theory of Nicholson's blowflies model has made a remarkable progress[3-10,16-19,21-23]. The assumption that the environment is constant is rarely the case in real life. When the environmental fluctuation is taken into account, a model must be nonautonomous. Due to the various seasonal effects of the environmental factors in real life situation, it is rational and practical to study the biological system with periodic coefficients or almost periodic coefficients. Many authors [4,6,7,10,16-18] have studied nonautonomous differential equations with periodic coefficients of the above Nicholson's blowflies model and its generalized models. Recently, L. Berezansky [9] pointed out an open problem: How about the dynamic behaviors of the Nicholson's blowflies model with linear harvesting term. In the natural biological systems, there exist many impulsive phenomena. If impulsive factors are introduced into biological models, the models must be governed by impulsive differential equations. The theory of impulsive differential equation has been well developed [11-13]. In this paper, motivated by the above mentioned facts, we will study the following impulsive Nicholson's blowflies model with linear harvesting term


INTRODUCTION
The dynamic behaviors of biological models are very important research topics.In 1980, Gurney [1] proposed the following delay differential equation      to describe the population of the Australian sheep-blowfly and to agree with the experimental data obtained by Nicholson in [2].Since this equation explains Nicholson's data of blowfly more accurately, this model and its modifications have been now referred to as the Nicholson's blowflies model.The theory of Nicholson's blowflies model has made a remarkable progress [3][4][5][6][7][8][9][10][16][17][18][19][21][22][23].
The assumption that the environment is constant is rarely the case in real life.When the environmental fluctuation is taken into account, a model must be nonautonomous.Due to the various seasonal effects of the environmental factors in real life situation, it is rational and practical to study the biological system with periodic coefficients or almost periodic coefficients.Many authors [4,6,7,10,[16][17][18] have studied nonautonomous differential equations with periodic coefficients of the above Nicholson's blowflies model and its generalized models.Recently, L. Berezansky [9] pointed out an open problem: How about the dynamic behaviors of the Nicholson's blowflies model with linear harvesting term.
In the natural biological systems, there exist many impulsive phenomena.If impulsive factors are introduced into biological models, the models must be governed by impulsive differential equations.The theory of impulsive differential equation has been well developed [11][12][13].
In this paper, motivated by the above mentioned facts, we will study the following impulsive Nicholson . The admissible initial condition associated with equation (1.1) is , where   , by [15] we know that (1.1) has a unique solution    and satisfying the initial condition: In the study of biological systems, an important problem is concerned with the existence of positive periodic solutions or positive almost periodic solutions.Many authors have investigated the existence of positive periodic solution by using Krasnoselskii cone fixed point theorem and Mawhin's coincidence degree theory.
The almost periodicity is closer to the reality of biological systems.In this paper, we aim to obtain sufficient conditions that guarantee the existence of unique positive almost periodic solution of model (1.1) by using contraction mapping fixed point theorem.We also investigate the exponential convergence of positive almost periodic solution by means of Liapunov functional.For the impulsive Nicholson's blowflies model with linear harvesting term, we give answers to the open problem proposed in [9] by L. Berezansky.The results of this paper are valuable in applications, which complement the previously obtained results in [3][4][5][6][7][8][9][10][16][17][18][19][21][22][23].

PRELIMINARIES
We denote is uniformly almost periodic, and .
Thus, it follows that By the Lemma 3 in [11], we have the following Lemma 5. Lemma 5. Let the conditions is the solution of the following integral equation is the almost periodic solution of equation ( 1.

1) if and only if
x is the fixed point of operator We make assumptions: 1 () S 1 1 1 . Now we prove that every solution (2.2) Suppose the claim (2.2) is not true, then there must exist a Thus,  (2.4)By (2.3) and (2.4), we have . Thus, there must exist an interval (2.5) Suppose the claim (2.5) is not true, then there must exist a Thus, By (2.6) and (2.7) , we have

Let
Firstly, we prove that A   .For x   , we have ( ) ( ) Again, we get

Next, we show that
A is a contraction mapping.For ) .

i n s y s i i t s y s e ds H W t s x s y s ds
For the function () . Hence, by means of the mean value theorem, we get where  lies between ( ) ( Since the function Thus, from (3.5), we have i i s x s s y s i i i i s x s e s y s e s x s s y s Hence, (3.4) and (3.6) imply that   , we know that A is a contraction mapping.So the operator A exists a unique fixed point x  in  .Moreover, from the inequality (3.1), we have

which means that
x  is positive.This implies that equation (1.1) exists a unique almost periodic positive solution The proof is completed.

EXPONENTIAL CONVERGENCE OF POSITIVE ALMOST PERIODIC SOLUTION
Theorem 2. Assume that , then there exists a constant (0,1)   , such that ( ) 0 Now, we prove that Suppose that (4.3) does not hold true, then there must exist Using the mean value theorem, we get where  lies between Hence, (4.4) and (4.5) imply that

CONCLUSION
Impulsive phenomena exist extensively in natural biological systems, almost periodicity is closer to real world.This paper has studied the almost periodic impulsive Nicholson's blowflies model with linear harvesting term.By applying the contraction mapping fixed point theorem, we obtain sufficient conditions for the existence of unique positive almost periodic solution.By constructing Liapunov functional, we study the exponential convergence of positive almost periodic solution.The dynamic behaviors have close relations to the harvesting term and impulsive term.For the impulsive almost periodic Nicholson's blowflies model with linear harvesting term, we answer the open problem proposed in [9].Our results complement the previous results of some past literatures.

3 (
From Theorem 1, we know that equation (1.1) exists a unique almost periodic positive 's blowflies model with linear harvesting term is easy to check the global existence of the positive solution . . ) by Lemma 1 and Lemma 5, we can deduce that sup