Periodic Solutions of Multispecies Mutualism System with Infinite Delays

Abstract

We studied the delayed periodic mutualism system with Gilpin-Ayala effect. Some new and interesting sufficient conditions are obtained to guarantee the existence of periodic solution for the multispecies mutualism system with infinite delays. Our method is based on Mawhin's coincidence degree. To the best knowledge of the authors, there is no paper considering the existence of periodic solutions for n-species mutualism system with infinite delays.

1. Introduction

Recently, there are many papers considering the existence of periodic solutions for competitive Lotka-Volterra system based on Mawhin's coincidence degree theory (see [1–4]). But there are few papers considering the periodicity of mutualism system; for example, one can refer to [5–7]. However, the references mentioned above only considered two-dimensional mutualism system. To the best knowledge of the authors, there is no paper considering the existence of periodic solutions for -species mutualism system. It should be noted that the method used in [5–7] is difficult to be extended to the -dimensional system. So, we employ the method used in [2–4]. However, the problem considered in this paper is completely different from those mentioned above. On the other hand, the above-mentioned works considered the models with constant discrete delays or without delays. In practice, there will be a distribution of transmission delays. In this case, the transmission of species is no longer instantaneous and cannot be modelled with discrete delays. A more appropriate way is to incorporate distributed delays. Therefore, the studies of the model with distributed delays have more important significance than the ones of the model with discrete delays. Thus, in this paper, we considered the following mutualism system with distributed delays: where , and , , are -periodic functions, that is, , , , and , and , and are constants, . From biological view, , and , , are nonnegative, , and is positive. System (1) is associated with the IVP as follows:

2. Existence of Periodic Solutions

For convenience, we introduce some notations, definitions, and lemmas. If is a continuous -periodic function defined on , denote We also denote the spectral radius of the matrix by . Denote

Lemma 1 (see [8]). Let be an open and bounded set. Let be a Fredholm mapping of index zero and be -compact on (i.e., is bounded and is compact). Assume(i)for each , , ;(ii)for each , and .Then has at least one solution in .

Definition 2 (see [9, 10]). A real matrix is said to be an -matrix if , , , and .

Lemma 3 (see [9, 10]). Let be an matrix and ; then , where denotes the identity matrix of size .

Theorem 4. Assume the following. The algebraic equation has finite solutions with and . , , ., where and Then system (1) has at least one positive -periodic solution.

Proof. Note that every solution of system (1) with the initial value condition is positive. Make the change of variables Then system (1) is the same as
Obviously, if system (8) has at least one -periodic solution, then system (1) has at least one -periodic solution. To prove Theorem 4, we should find an appropriate open set satisfying Lemma 1. We divide the proof into three steps.
Step 1. We verify that (i) of Lemma 1 is satisfied. For any , by periodicity, it is easy to check that And define and as follows: where The projectors are defined by and by It is easy to follow that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) exists, which is given by Then and are defined by where Using similar arguments to Step 1 in [2], it is easy to show that is relatively compact in the space .
Step 2. In this step, we are in a position to search for an appropriate open bounded subset satisfying condition (i) of Lemma 1. Specifically, our aim is to search for an appropriate defined by in Step 1 such that satisfies condition (i) of Lemma 1. To this end, assume that is a solution of the equation for each ; that is, Since , each , as components of , is continuously differentiable and -periodic. In view of continuity and periodicity, there exists such that , . Accordingly, and we arrive at That is, Noticing that implies It follows from () that Here we used (). Letting , it follows from (20) that or which implies Set . It follows from (23) that In view of and Lemma 3, . Let Then it follows from (24) and (25) that which implies On the other hand, it follows from (25) that Estimating (16), by using (26) and (28), we have We can choose a large enough real number () such that Let . Then, for any solution of , we have for all . Obviously, are independent of and the choice of . Thus, taking , the open subset satisfies that for each , ; that is, the open subset satisfies the assumption (i) of Lemma 1.
Using similar arguments to Step 3 in [2], it is not difficult to show that, for each , and .
Hence, by Lemma 1, system (8) has at least one positive -periodic solution in . By (7), system (1) has at least one positive -periodic solution, denoted by . This completes the proof of Theorem 4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.