Existence of Positive Periodic Solutions for a Class of -Species Competition Systems with Impulses
By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class of -species competition systems with impulses. Meanwhile, we point out that the conclusion of (Yan, 2009) is incorrect.
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: Owing to its theoretical and practical significance, the systems have been studied extensively by many researchers. And many excellent results which concerned with persistence, extinction, global attractivity of periodic solutions, or almost periodic solutions have been obtained.
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: where is the population density of the th species; is the intrinsic exponential growth rate of the th species; is the environmental carrying capacity of species in the absence of competition; provides a nonlinear measure of interspecific interference, provides a nonlinear measure of interspecific interference.
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 [1–7] and the references therein).
For example, by using the method of coincidence degree, Wang  considered the existence of periodic solutions for the following -species Gilpin-Ayala impulsive competition system: where the constant satisfied . What is more,  also obtained several results for the persistence and global attractivity of the periodic solution of the model.
In , Yan applied the Krasnoselskii fixed point theorem to investigate the following n-species competition system: where the constants . He obtained a necessary and sufficient condition for the existence of periodic solutions of system (1.4). Unfortunately, its last conclusion is wrong. Please see the remark in Section 3 of this paper.
Motivated by [1, 2], in this paper, we investigate the following impulsive -species competition system: where is the population density of the th species at time ; is the intrinsic exponential growth rate of the th species at time ; is the time delay; is a positive constant; measure the amount of competition between the species and ; provide a nonlinear measure of interspecific interference; () is the left (right) limits of at .
The main features of the present paper are as follows. The Gilpin-Ayala species competition system (1.5) has impulsive effects. As is known to us, there were few papers to study such system. Finally, we point out that the conclusion of  is incorrect.
For an -periodic function , let . Throughout this paper, assume the following conditions hold.(H1) are continuous -periodic functions, and , and there exists such that .(H2), is a positive constant, and .(H3), and for , there exists an positive integer such that , where .(H4) are constants, .
In order to prove our main result, now we state the fixed point theorem of cone expansion and compression.
Lemma 1.1 (see ). Let be a Banach space, and let be a cone in . Assume that are open subsets of with . Let be a completely continuous operator such that one of the following two conditions is satisfied:(i) for ; for ,(ii) for ; for ,Then, has a fixed point in .
The organization of this paper is as follows. In the next section, we introduce some lemmas and notations. In Section 3, the main result will be stated and proved on the existence of periodic solutions of system (1.5).
Let be continuous at , left continuous at , and the right limit exists for . Evidently, is a Banach space with the norm , where
Define an operator by , where where , and , where is a positive integer, .
It is obvious that the functions have the following properties.(i) for , and .(ii) for , where ,
Now, we choose a set defined by where . Clearly, is a cone in .
For the sake of convenience, we define an operator by , where
Lemma 2.1. The operator is completely continuous.
Proof. First, it is easy to see . Next, since
we have .
Observe that , for all . Hence, we obtain that, for , Thus, , that is, .
Obviously, the operator is continuous. Next, we show that is compact. Let be a bounded subset; that is, there exists such that for all . From the continuity of , we have, for all , where .
Therefore, , which implies that is uniformly bounded.
On the other hand, noticing that This guarantees that, for each , we have where ,, , , .
Consequently, is equicontinuous on , where . By the Ascoli-Arzela theorem, the function is completely continuous from to .
Lemma 2.2. The system (1.5) has a positive -periodic solution in if and only if has a fixed point in .
Proof. For satisfying , that is, , it follows from (2.2) and (2.4) that
And for ,
which implies that is a positive -periodic solution of (1.5).
Conversely, assume that is an -periodic solution of system (1.5). Then, the system (1.5) can be transformed into that is,
So, integrating the above equality from to and noticing that , we have where , and , where is a positive integer, , that is, .
Therefore, is a fixed point of the operator . The proof of the Lemma is complete.
3. Main Results
Theorem 3.1. Suppose (H1)–(H4) hold, and , where Then system (1.5) has at least one positive -periodic solution.
Proof. Let . Choose and . Then, there exists such that, for and , we have
Choose . Let .
Now, we prove that
Suppose (3.2) does not hold. Then, there exists some such that . Since , we have for , . From (2.2) and (2.4), it follows that which implies , a contradiction. Hence, for .
On the other hand, let , on the account of (H1), we know . Choose and . Then, there exists such that, for , we know that
Choose . Let . Then, there exist some integers such that ; for .
Next we show that
In fact, if there exists some such that and since , we have for , and there exists some such that . Therefore, this together with (H1) guarantees that, for , which is a contradiction. Thus, (3.5) is satisfied.
From all the above, the condition (i) of Lemma 1.1 is satisfied. So the operator has a fixed point in . That is, system (1.5) has at least one positive periodic solution.
Remark 3.2. For any , if we let , then . However, in the proof of Theorem 1.1 of , it is regarded mistakenly as . Therefore, the proof of its sufficiency is not correct. So the result of  is incorrect.
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).
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