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.

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Wenbo Zhao 1 and Caocuan Ma 2 and Taotao Zheng 3 and Xiao-Ke Sun 2

Academic Editor:Yonghui Xia

1, School of Physics and Information, Tianshui Normal University, Tianshui, Gansu 741000, China
2, School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741000, China
3, Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310000, China

Received 27 June 2014; Accepted 9 August 2014; 27 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 n -species mutualism system. It should be noted that the method used in [5-7] is difficult to be extended to the n -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: [figure omitted; refer to PDF] where _ri ,_aij ,_bij , and _cij , i,j=1,2,...,n , are ω -periodic functions, that is, _ri (t+ω)=_ri (t) , _aij (t+ω)=_aij (t) , _bij (t+ω)=_bij (t) , and _cij (t+ω)=_cij (t) , and _αij ,_βij , and _γij are constants, i,j=1,2,...,n . From biological view, _ri ,_aij ,_bij , and _cij , i,j=1,2,...,n , are nonnegative, ^∫0∞ ..._Hij (s)ds=1 , and _aii is positive. System (1) is associated with the IVP as follows: [figure omitted; refer to PDF]

2. Existence of Periodic Solutions

For convenience, we introduce some notations, definitions, and lemmas. If g(t) is a continuous ω -periodic function defined on R , denote [figure omitted; refer to PDF] We also denote the spectral radius of the matrix A by ρ(A) . Denote [figure omitted; refer to PDF]

Lemma 1 (see [8]).

Let Ω⊂X be an open and bounded set. Let L be a Fredholm mapping of index zero and N be L -compact on Ω¯ (i.e., QN(Ω¯) is bounded and _KP (I-Q)N:Ω¯[arrow right]X is compact). Assume

(i) for each λ∈(0,1) , x∈∂Ω∩Dom...L , Lx...0;λNx ;

(ii) for each x∈∂Ω∩Ker...L , QNx...0;0 and deg...{JQN, Ω∩Ker...L,0}...0;0 .

Then Lx=Nx has at least one solution in Ω¯∩Dom...L .

Definition 2 (see [9, 10]).

A real n×n matrix A=(_aij ) is said to be an M -matrix if _aij ...4;0 , i,j=1,2,...,n , i...0;j , and ^A-1 ...5;0 .

Lemma 3 (see [9, 10]).

Let A...5;0 be an n×n matrix and ρ(A)<1 ; then (_En -A^)-1 ...5;0 , where _En denotes the identity matrix of size n .

Theorem 4.

Assume the following.

(_H1 ) : The algebraic equation [figure omitted; refer to PDF]

has finite solutions (^u1* ,^u2* ,...,^un*)T ∈^R+n with ^ui* >0 and _∑^u* ...sgn..._Jf (^u* )...0;0 .

(_H2 ) : _αji ...4;_αii , _βji ...4;_αii , _γji ...4;_γii (j...0;i),i,j=1,2,...,n .

(_H3 ) : ρ(K)<1 , where K=(_Γij)n×n and [figure omitted; refer to PDF]

Then system (1) has at least one positive ω -periodic solution.

Proof.

Note that every solution y(t)=(_y1 (t),_y2 (t),..._yn (t)^)T of system (1) with the initial value condition is positive. Make the change of variables [figure omitted; refer to PDF] Then system (1) is the same as [figure omitted; refer to PDF]

Obviously, if system (8) has at least one ω -periodic solution, then system (1) has at least one ω -periodic solution. To prove Theorem 4, we should find an appropriate open set Ω satisfying Lemma 1. We divide the proof into three steps.

Step 1 . We verify that (i) of Lemma 1 is satisfied. For any x(t)∈X , by periodicity, it is easy to check that [figure omitted; refer to PDF] And define L:Dom...L⊂X[arrow right]Z and N:X[arrow right]Z as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The projectors are defined by P:X[arrow right]X and Q:Z[arrow right]Z by [figure omitted; refer to PDF] It is easy to follow that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L ) _KP :Im...L[arrow right]Dom...L∩Ker...P exists, which is given by [figure omitted; refer to PDF] Then QN:X[arrow right]Z and _KP (I-Q)N:X[arrow right]X are defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Using similar arguments to Step 1 in [2], it is easy to show that (_KP (I-Q)Nx)(Ω¯) is relatively compact in the space (X,||·_||1 ) .

Step 2 . In this step, we are in a position to search for an appropriate open bounded subset Ω satisfying condition (i) of Lemma 1. Specifically, our aim is to search for an appropriate _hi defined by Ω in Step 1 such that Ω satisfies condition (i) of Lemma 1. To this end, assume that x(t)∈X is a solution of the equation Lx=λNx for each λ∈(0,1) ; that is, [figure omitted; refer to PDF] Since x(t)∈X , each _xi (t),i=1,2,...,n , as components of x(t) , is continuously differentiable and ω -periodic. In view of continuity and periodicity, there exists _ti ∈[0,ω] such that _xi (_ti )=_{max...t∈[0,ω]} |_xi (t)| , i=1,2,...,n . Accordingly, _{x i} (_ti )=0 and we arrive at [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] Noticing that _xj (_tj )=_{max...t∈[0,ω]} |_xj (t)| implies [figure omitted; refer to PDF] It follows from (_H2 ) that [figure omitted; refer to PDF] Here we used (_H2 ). Letting (_{a_ii} +_{b_ii} )^{e_αiixi (_ti )} =_zi (_ti ) , it follows from (20) that [figure omitted; refer to PDF] or [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] Set D=^{(_D1 ,_D2 ,...,_Dn )T} =^{(_r¯1 ,_r¯2 ,...,_r¯n )T} . It follows from (23) that [figure omitted; refer to PDF] In view of ρ(K)<1 and Lemma 3, (_En -K^)-1 ...5;0 . Let [figure omitted; refer to PDF] Then it follows from (24) and (25) that [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] On the other hand, it follows from (25) that [figure omitted; refer to PDF] Estimating (16), by using (26) and (28), we have [figure omitted; refer to PDF] We can choose a large enough real number (d>1 ) such that [figure omitted; refer to PDF] Let _hi =(1/_αii )ln...(d_h~i /_{a_ii} ) . Then, for any solution of Lx=λNx , we have |_xi (t)_|1 =|_xi (t)_|0 +|_{x i} (t)_|0 ...4;ln...(_h~i /_{a_ii} )+2_h~i <_hi for all i=1,2,...,n . Obviously, _hi are independent of λ and the choice of x(t) . Thus, taking _hi =ln...(d_h~i /_{a_ii} ) , the open subset Ω satisfies that Lx...0;λNx for each λ∈(0,1) , x∈∂Ω∩Dom...L ; that is, the open subset Ω satisfies the assumption (i) of Lemma 1.

Using similar arguments to Step 3 in [2], it is not difficult to show that, for each x∈∂Ω∩Ker...L , QNx...0;0 and deg...{JQN,Ω∩Ker...L,0}...0;0 .

Hence, by Lemma 1, system (8) has at least one positive ω -periodic solution in Dom...L∩Ω¯ . By (7), system (1) has at least one positive ω -periodic solution, denoted by y~(t) . 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.

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