By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class of

In recent years, the problem of periodic solutions of the ecological species competition systems has always been one of the active areas of research and has attracted much attention. For instance, the traditional Lotka-Volterra competition system is a rudimentary model on mathematical ecology which can be expressed as follows:

However, the Lotka-Volterra competition systems ignore many important factors, such as the age structure of a population or the effect of toxic substances. So, more complicated competition systems are needed. In 1973, Ayala and Gilpin proposed several competition systems. One of the systems is the following competition system:

On the other hand, in the study of species competition systems, the effect of some impulsive factors has been neglected, which exists widely in the real world. For example, the harvesting or stocking occur at fixed time, natural disaster such as fire or flood happen unexpectedly, and some species usually migrate seasonally. Consequently, such processes experience short-time rapid change which can be described by impulses. Therefore, it is important to study the existence of the periodic solutions of competitive systems with impulse perturbation (see [

For example, by using the method of coincidence degree, Wang [

In [

Motivated by [

The main features of the present paper are as follows. The Gilpin-Ayala species competition system (

For an

In order to prove our main result, now we state the fixed point theorem of cone expansion and compression.

Let

The organization of this paper is as follows. In the next section, we introduce some lemmas and notations. In Section

Let

Define an operator

It is obvious that the functions

Now, we choose a set defined by

For the sake of convenience, we define an operator

The operator

First, it is easy to see

Observe that

Obviously, the operator

Therefore,

On the other hand, noticing that

Consequently,

The system (

For

Conversely, assume that

So, integrating the above equality from

Therefore,

Suppose (H1)–(H4) hold, and

Let

Choose

Now, we prove that

Suppose (

On the other hand, let

Choose

Next we show that

In fact, if there exists some

From all the above, the condition (i) of Lemma

For any

This research was supported by the Natural Science Foundation of Shandong Province (ZR2009AM006), the Key Project of Chinese Ministry of Education (no. 209072), and the Science and Technology Development Funds of Shandong Education Committee (J08LI10).