Existence and Uniqueness of Positive (almost) Periodic Solutions for a Neutral Multi-species Logarithmic Population Model with Multiple Delays and Impulses

In this paper, by using the contraction mapping principle and constructing a suitable Lyapunov functional, we established a set of easily applicable criteria for the existence, uniqueness and global attractivity of positive periodic solution and positive almost periodic solution of a neutral multi-species Logarithmic population model with multiple delays and impulses. The results improve and generalize the known ones in [1], as an application, we also give an example to illustrate the feasibility of our main results.


Introduction
Recently, there are more works on the periodic solution of neutral type Logistic models or Lotka-Volterra models (see [2][3][4][5][6][7] for details).Only a little scholars considered the neutral Logarithmic model (see [1,[8][9][10]).In [8], Li had studied the following single species neutral Logarithmic model: He had established a set of easily applicable criteria for the existence of positive periodic solution of system (1.1) by applying the continuation theorem of the coincidence degree theory which proposed in [11] by Mawhin.In [9], Lu and Ge employed an abstract continuous theorem of k-set contractive operator to investigate the following equation: They established some criteria to guarantee the existence of positive periodic solutions of system (1.2).In [10], Chen studied the following neutral multi-species Logarithmic population model: By using the method of fixed point theory and constructing a suitable Lyapunov functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic solution (positive almost periodic solution) for system (1.3).
In [1], Wang et al. had investigated the existence, uniqueness of the positive periodic solution of the following neutral multi-species Logarithmic population model: By using an abstract continuous theorem of k-set contractive operator, the criteria is established for the existence, global attractivity of positive periodic solutions for model (1.4).
On the other hand, there are some other perturbations in the real world such as fires and floods that are not suitable to be considered continually.These perturbations bring sudden changes to the system.Systems with such sudden perturbations involving impulsive differential equations have attracted the interest of many researchers in the past twenty years [12][13][14][15][16][17][18][19][20], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process.Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, optimal control, etc.For details, see [21,22].Recently, the corresponding theory for impulsive functional differential equations has been studied by many authors [23][24][25].However there are few published papers discussing the impulsive neutral multispecies Logarithmic population model.Our method is different from that in [1,9].
In this paper, we investigate the existence, uniqueness of the positive periodic solution of the following neutral multi-species Logarithmic population system with multiple delays and impulses where 1, 2, , , , .
Throughout this paper, we make the following notations: Let 0   be a constant and with the norm defined by with the norm defined by For the sake of generality and convenience, we always make the following fundamental assumptions: are Banach spaces.
, are all positive periodic continuous functions with period , and exists; , are all almost periodic continuous functions with period  on R, and , are positive continuously differentiable almost periodic functions such that , and exists, are fixed impulsive points with ; is a real sequence such that ik 1 0 The outline of the paper is as follows.In the following section, some definitions and some useful lemmas are listed.In the third section, we first introduce a transformation, where some adjustable real parameters i   0 are introduced.After that, by using contraction mapping principle, we derive some sufficient conditions which ensure the existence and uniqueness of positive periodic solution (positive almost periodic solution) of system (1.5) and (1.6).In the fourth section, we derive a set of easily verifiable criteria for the global attractivity of the positive periodic solution (almost periodic solution) of (1.5) and (1.6) by constructing a suitable Lyapunov functional.Finally, we give an example to show our results.Here, We must point out, the idea of introducing parameters is stimulated by the recent works of [1,26,27].However, to the best of the authors knowledge, this is the first time such a technique is applied to the impulsive neutral delays ecosystem.

Preliminaries
In order to obtain the existence and uniqueness of a periodic solution for system (1.1) for almost everywhere (for short a.e.) in     0, . Definition 2.2 Let 1 be a strictly positive periodic solution (almost periodic solution) of (1.5) and (1.6).We say is globally attractive if any other solution of (1.5) and (1.6) has the property: We can easily get the following Lemma 2.1.Lemma 2.1 The region is the positive invariable region of the system (1.5).
Proof.In view of biological population,we obtain By the system (1.5), we have Then the solution of (1.5) is positive. Under the above hypotheses (H 1 )-(H 3 ), I consider the neutral non-impulsive system with initial conditions: , , where that satisfies (2.1) a.e., for t ≥ 0, and The following lemmas will be used in the proofs of our results, The proof of the first lemma is similar to that of Theorem 1 in [20] Proof.Its proof is similar to that of Theorem 1 in [20], here we omit it.
. Then the function has a unique inverse satisfying with Proof.Its proof is similar to that of Lemma 2.4 in [29], here we omit it. Lemma 2.4 (Barbalat's Lemma [30]) Let   f t be a nonnegative function defined on such that

Denote
, then from , it follows m < 1.Also, when t ≥ s without loss of generality, we may assume and so, from (2.9) it follows: The proof is complete.
Proof.Similar to the proof of Lemma 2.5, we omit it here.
Obviously, the existence, uniqueness and global attractivity of positive periodic solution (almost periodic solution) of system (1.5) is equivalent to the existence, uniqueness and global attractivity of periodic solution (almost periodic solution) of system (3.1).
follows that the linear system of system (3.2) admits exponential dichotomies on R, and so, system (3.3) has a unique continuous periodic solution   iu x t , which can be expressed as 1, 2, , .
Now, by using Lemma 2.5,

  iu
x t can also be expressed as 1, 2, , , Our main result on the global existence of a positive periodic solution of (1.5) and (1.6) is stated as follows.Theorem 3.1 In addition to (H 1 )-(H 3 ), assume further that there exist positive constants , such that .
Then (1.5) has a unique positive  -periodic solution with strictly positive components, say where where   iu g  are defined by (3.7), is a continuous periodic function, and so as follows: Following we will prove the mapping  is a contraction mapping.In fact, for any from (3.8), (3.9) and the conditions of Theorem 3.1 it follows: where This shows that  is a contraction mapping.Hence, there exists a unique fixed point   1, 2, , .
From the right-hand sides of (3.13), we know that And so, from (3.13) it follows that Copyright © 2013 SciRes.
here using the equality (3.13) again.That is is the unique continuously differentiable  -periodic solution of system (3.1), and so, is the unique positive  -periodic solution of system (1.5).The proof is complete. As a direct corollary of Theorem 3.1, one has Corollary 3.1 In addition to (H 1 )-(H 3 ), assume further that there exist positive constants , such that Then (1.5) has a unique positive  -periodic solution with strictly positive components.Our next theorem concerned with the existence of unique positive almost periodic solution of systems (1.5) and (1.6).
Let be any continuously differentiable almost periodic function, and conider equation, Since , it follows that the linear system of system (3.15) where Now, by using Lemma 2.5,

  iu
x t can also be expressed as where Then, we have Theorem 3.2 In addition to (H 4 )-(H 6 ), assume further that there exist positive constants , such that Then similarly to the prove of Theorem 3.1, we could prove that under the assumptions of Theorem 3.2, the mapping  is a contract mapping, and so system (3.19) is the unique positive almost periodic solution of system (1.5).The proof is complete.
 As a direct corollary of Theorem 3.2, one has Corollary 3.2 In addition to (H 4 )-(H 6 ), assume further that there exist positive constants , such that Then (1.5) has a unique positive almost periodic solution with strictly positive components.Consider the following equation: which is a special case of system (1.5) and (1.6) without impulse.Similarly, we can get the following results.
Theorem 3.3 In addition to (H 1 ), assume further that there exist positive constants Then (1.5) has a unique positive  -periodic solution with strictly positive components, say where Proof.Similar to the proof of Theorem 3.1, we omit it here. As a direct corollary of Theorem 3.3, one has Corollary 3.3 In addition to (H 1 ), assume further that there exist positive constants Then (1.5) has a unique positive  -periodic solution with strictly positive components.
Theorem 3.4 In addition to (H 4 ), assume further that there exist positive constants Then (1.5) has a unique positive almost periodic solution with strictly positive components, say where .
As a direct corollary of Theorem 3.4, one has Corollary 3.4 In addition to (H 4 ), assume further that there exist positive constants .
) has a unique positive almost periodic solution with strictly positive components.

Global Asymptotic Stability
In this section, we devote ourselves to the study of the global attractivity of periodic solutions (almost periodic solutions) of system (1.5), (1.6) and (3.22) (which is a special case of system (1.5) and (1.6) without impulse).Now, we state our main results of this section as follows: Theorem 4.1.Assume that the conditions in Theorem 3.1 hold.Suppose further the following conditions hold: (H 11 ) There is a positive constant M such that , .
Then system (1.5) and (1.6) has a unique periodic solution which is globally attractive.Proof.Let be the unique positive periodic solution of system (1.5) and (1.6), whose existence and uniqueness are guarantee by Theorem 2.1, and be any other then, similar to Equation (3.1), we have Then, from (4.1) and (4.2), we have Multiply both sides of (4.4) with and then integrate from 0 to to obtain t then where From (H 3 ), we have From (H 4 ), we have Proof.Similar to the proof of Theorem 4.1, we omit it here.

An Example
Now, we give an example to demonstrate our result.Let us consider the following equation: 5) has a unique positive almost periodic solution with strictly positive components, say


Then system(3.22)has a unique periodic solution (almost periodic solution) which is globally attractive, where .
Assume that the conditions in Theorem 3.2 hold.Suppose further the following conditions hold: (H 13 ) There is a positive constant m such that Then system (1.5) and (1.6) has a unique almost periodic solution which is globally attractive.Proof.Similar to the proof of Theorem 4.1, we omit it here.Theorem 4.3.Assume that the conditions in Theorem 3.3 (or Theorem 3.4) hold.Suppose further the following conditions hold: (H 15 ) There is a positive constant  such that      that is the positive  -periodic solution of (3.1) is globally attractive,  by Definition 2.2, the positive  -periodic solution of (1.5) is globally attractive.The proof is completed.Theorem 4.2.