^{1, 2}

^{1}

^{3}

^{1}

^{2}

^{3}

We acquire some sufficient and realistic conditions for the existence of
positive periodic solution of a general neutral impulsive

In this paper, we consider the existence of the positive periodic solution of the following impulsive

In 1991, in [

It is of course very interesting to study the neutral delay population model for higher dimensional systems. In fact, in [

Moreover, in some situations, people may wish to change the position of the existing periodic solution but keep its stability. This is of significance in the control of ecology balance. One of the methods for its realization is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control schemes or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics with feedback control have been studied extensively; see [

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, see [

In [

In [

Recently, in [

However, to this day, no scholars had done works on the existence of positive periodic solution of the system (

For the sake of generality and convenience, we make the following notations and assumptions: let

The organization of this paper is as follows. In the following section, we introduce some lemmas and an important existence theorem developed in [

In this section, in order to obtain the existence of a periodic solution for system (

For a fixed

there exists

there exists a constant

By using the continuation theorem for composite coincidence degree, in [

Assume that there exists a constant

for any

One has that

The following remark is introduced by Fang (see Remark 1 in [

Lemma

(ii*) there exists a constant

We will also need the following lemmas.

Suppose that

Suppose that

Consider that

In view of biological population, we obtain

A function

for each

Consider the following nonimpulsive delay differential equation:

Suppose that (A)–(C) hold, then

if

if

(i) It is easy to see that

(ii) Since

Consider that

The proof of Lemma

From Lemmas

Since

From Lemma

Similarly,

Here, we have the following notations:

Suppose that the following conditions hold:

the system of algebraic equations

has a unique positive solution

To prove the previous theorem, we make the change of variables

We define the following maps:

If the assumptions of Theorem

For

If the assumptions of Theorem

Let

Based on the previous results, we can now apply Lemma

Obviously, for

Consider the following:

Suppose that the following conditions hold;

the system of algebraic equations

has a unique positive solution

Its proof is similar to the proof of Theorem

Similarly, we can get the following results.

Assume that conditions of Theorem

Its proof is similar to the proof of Theorem

Assume that conditions of Corollary

Its proof is similar to the proof of Theorem

When

In order to illustrate some features of our main result, in the following, we will apply Theorem

We consider an

Assume that the following conditions are satisfied:

the system of algebraic equations

has a unique positive solution

Then, (

When

We consider the single specie neutral delay logistic equation with impulse:

and

Assume that the following conditions are satisfied:

Then, system (

When

This research is supported by NSF of China (nos. 10971229, 11161015, and 11371367), PSF of China (no. 2012M512162), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, and 13C084), and the construct program of the key discipline in Hunan province.