Research Article | Open Access

# Existence and Stability of Positive Periodic Solutions for a Neutral Multispecies Logarithmic Population Model with Feedback Control and Impulse

**Academic Editor:**Yong Ren

#### Abstract

We investigate a neutral multispecies logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence, uniqueness, and global attractivity of positive periodic solution are established. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases. We also give an example to illustrate the applicability of our results.

#### 1. Introduction

As is known to all, ecosystem in the real world is continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In recent years, the qualitative behaviors of the population dynamics with feedback control has attracted the attention of many mathematicians and biologists [1â€“5]. On the other hand, there are some other perturbations in the real world such as fires and floods, which 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 [6â€“10], 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, and optimal control; for details, see [11â€“13]. However, to the best of the authorâ€™s knowledge, to this day, no scholar considered the neutral multispecies logarithmic population model with feedback control and impulse.

The aim of this paper is to investigate the existence, uniqueness, and global attractivity of the positive periodic solution for the following neutral multispecies logarithmic population system with feedback control and impulse: where denote indirect feedback control variables. For the ecological justification of (1) and the similar types, refer to [14â€“20].

For the sake of generality and convenience, we always make the following fundamental assumptions:(), , , , , , , , , , , , , and are continuous nonnegative -periodic functions with , , , , and , , ;() are fixed impulsive points with ;() is a real sequence, , and is an -periodic function.

In the following section, some definitions and some useful lemmas are listed. In the third section, by applying the contraction mapping principle, some sufficient conditions which ensure the existence and uniqueness of positive periodic solution of system (1) are established, and then we get a few sufficient conditions ensuring the global attractivity of the positive periodic solution by employing some inequality techniques. Finally, we give an example to show our results.

#### 2. Preliminaries

In order to obtain the existence and uniqueness of a periodic solution for system (1), we first give some definitions and lemmas.

*Definition 1. *A function () is said to be a positive solution of (1), if the following conditions are satisfied:(a) is absolutely continuous on each ;(b)for each , and exist, and ;(c) satisfies the first equation of (1) for almost everywhere (for short a.e.) in and satisfies for , .

*Definition 2. *System (1) is said to be globally attractive, if there exists a positive solution of (1) such that , , for any other positive solution of the system (1).

We can easily get the following lemma.

Lemma 3. * is the positive invariable region of the system (1).*

*Proof. *In view of biological population, we obtain , . By the system (1), we have
where
Then the solution of the system (1) is positive.

Under the above hypotheses , we consider the neutral nonimpulsive system:
where
By a solution of (4), it means an absolutely continuous function defined on that satisfies (4) a.e., for , and , on .

The following lemma will be used in the proofs of our results, and the proof of the lemma is similar to that of Theorem 1 in [6].

Lemma 4. *Suppose that hold. Then*(i)*if is a solution of (4) on , then is a solution of (1) on ,*(ii)*if is a solution of (1) on , then is a solution of (4) on .*

*Proof. *(i) It is easy to see that is absolutely continuous on every interval , , ,
On the other hand, for any , ,
Thus
It follows from (6)â€“(8) that is a solution of (1).

(ii) Since is absolutely continuous on every interval , , , and in view of (8), it follows that for any ,
which implies that is continuous on . It is easy to prove that is absolutely continuous on . Similar to the proof of (i), we can check that are solutions of (4) on . The proof of Lemma 4 is completed.

Lemma 5. * is a -periodic solution of (4) if and only if is a -periodic solution of the following system:
**
where
**
and is defined by (3).*

*Proof. *The proof of Lemma 5 is similar to that of Lemma 2.2 in [2], and we omit the details here.

Obviously, the existence, uniqueness, and global attractivity of positive periodic solution of system (1) is equivalent to the existence, uniqueness, and global attractivity of periodic solution of system (10).

Lemma 6. *Assume that , are all continuously differentiable -periodic functions and is a nonnegative continuous -periodic function such that ; then
**
where .*

*Proof. *As
Denote ; then from , , it follows that . Also, when without loss of generality, we may assume that ; thus
Therefore,
and so from (13)â€“(15) it follows that
The proof Lemma 6 is complete.

#### 3. Main Theorem

In this section, by using contraction principle and some inequality techniques, several conditions on the existence, uniqueness, and global attractivity of periodic solution for system (1) are presented.

Let ; the system (10) can be reduced to where â€‰â€‰() are positive real numbers.

Obviously, the existence, uniqueness, and global attractivity of positive periodic solution of system (10) is equivalent to the existence, uniqueness, and global attractivity of periodic solution of system (17).

For the rest of this paper, we will devote ourselves to study the existence, uniqueness, and global attractivity of periodic solution of (17). We denote Our first result on the global existence of a periodic solution of system (1) is stated in the following theorem.

Theorem 7. *In addition to , assume further that there exist positive constants â€‰â€‰() and a positive constant such that*()*.**Then, system (1) has a unique -periodic solution with strictly positive components, where is defined by (18).*

*Proof. *From the above analysis, to finish the proof of Theorem 7, it is enough to prove under the conditions of Theorem 7 that system (17) has a unique -periodic solution. Let
under the norm , is a Banach space. For any , we consider the periodic solution of periodic differential equation
Since , we know that the linear system of system (20)
admits exponential dichotomies on , and so system (20) has a unique periodic solution , which can be expressed as
where
its proof is similar to that of Theorem 1 in [18]; here we omit it.

Now, by using Lemma 6, can also be expressed as
where
Now we define mapping , . Following this we will prove that is a contraction mapping; that is, there exists a constant , such that , for all . In fact, for any and , we have
Hence,
It follows from that for all . That is, is a contraction mapping. Hence, there exists a unique fixed point ; that is, . Therefore, is the unique periodic solution of system (17). It follows from (1), (4), (10), and (17) that is the unique positive periodic solution of system (1). The proof of Theorem 7 is completed.

Our next theorem is concerned with the global stability of periodic solution for system (1).

Theorem 8. *In addition to , suppose further that the following condition holds:*()*, as , .**Then system (1) has a unique periodic solution which is globally attractive.*

*Proof. *Let be the unique positive periodic solution of system (1), whose existence and uniqueness are guaranteed by Theorem 7, and let be any other solution of system (1). Let , ; then, similar to (17), we have
Let ; then
Multiply both sides of (29) with , and then integrate from to to obtain
then
Let ; we see that
Substituting (32) into (31), we get
therefore, we have
where is defined by (18). From , we have
From , we have
thus, , as , . Hence, the positive -periodic solution of (17) is globally attractive; accordingly, , as , , and by Definition 2, the positive -periodic solution of (1) is globally attractive. The proof of Theorem 8 is completed.

*Remark 9. *If , , , , then system (1) is studied by [3]. Hence, Theorems 7 and 8 generalize the corresponding results in [3].

*Remark 10. *If , , , then system (1) is studied by [4]. Hence Theorems 7 and 8 also generalize the corresponding results in [4].

#### 4. Example

Consider the following impulsive model: