# Periodic Solutions of Multispecies Mutualism System with Infinite Delays

Authors: Wenbo Zhao; Caocuan Ma; Taotao Zheng; Xiao-Ke Sun
Journal: Abstract and Applied Analysis
 (2014)

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## 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 forn-species mutualism system with infinite delays.


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## Body

## 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:
(1)y˙i(t)=yi(t)[ri(t)-aii(t)yi(t)αij(t)+∑j=1,j≠in‍aij(t)yjαij(t)mmmm+∑j=1,j≠in‍bij(t)yjβij(t-τij)mmmm+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)yjγij(t+s)ds],mmmmmmmmmmmmmm1111mmlli=1,2,…n,
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:
(2)yi(t0)=yi0,yi0>0,i=1,2,…n.


## 2. Existence of Periodic Solutions

For convenience, we introduce some notations, definitions, and lemmas. Ifg(t) is a continuous ω-periodic function defined on R, denote
(3)g_=min⁡t∈[0,ω]|g(t)|,g¯=max⁡t∈[0,ω]|g(t)|,m(g)=1ω∫0ω‍g(t)dt.
We also denote the spectral radius of the matrix A by ρ(A). Denote
(4)X={x(t)=(x1(t),x2(t),…,xn(t))T∈C1(R,Rn)∣x(t+ω)=x(t)∀t∈R},Z={x(t)=(x1(t),x2(t),…,xn(t))T∈C(R,Rn)∣x(t+ω)=x(t)∀t∈R(x1(t),x2(t),…,xn(t))T}.Lemma 1 (see [8]).
LetΩ⊂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¯→X is compact). Assume(i)
for eachλ∈(0,1), x∈∂Ω∩Dom⁡L, Lx≠λNx;(ii)
for eachx∈∂Ω∩Ker⁡L, QNx≠0 and deg⁡{JQN,Ω∩Ker⁡L,0}≠0.Then Lx=Nx has at least one solution in Ω¯∩Dom⁡L.Definition 2 (see [9, 10]).
A realn×n matrix A=(aij) is said to be an M-matrix if aij≤0, i,j=1,2,…,n, i≠j, and A-1≥0.Lemma 3 (see [9, 10]).
LetA≥0 be an n×n matrix and ρ(A)<1; then (En-A)-1≥0, where En denotes the identity matrix of size n.Theorem 4.
Assume the following.(H1)
The algebraic equation(5)f(u)∶=(m(ri)-m(aii)uiαii+∑j=1n‍m(aij)ujαij+∑j=1n‍m(bij)ujβij+∑j=1,j≠in‍m(cij)ujγij)n×1=0has finite solutions (u1*,u2*,…,un*)T∈R+n with ui*>0 and ∑u*‍sgn⁡Jf(u*)≠0. (H2)
αji≤αii, βji≤αii, γji≤γii(j≠i),i,j=1,2,…,n.(H3)
ρ(K)<1, where K=(Γij)n×n and
(6)Γij={0,i=j,a¯ij+b¯ij+c¯ija_jj,i≠j.Then system (1) has at least one positive ω-periodic solution.Proof.
Note that every solutiony(t)=(y1(t),y2(t),...yn(t))T of system (1) with the initial value condition is positive. Make the change of variables
(7)yi(t)=exi(t),i=1,2,…,n.
Then system (1) is the same as
(8)xi(t)=ri(t)-aii(t)eαij(t)+∑j=1,j≠in‍aij(t)eαij(t)+∑j=1,j≠in‍bij(t)eβij(t-τij)+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγij(t+s)ds,mmmmmmmmmmmmmmli=1,2,…n.
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
(9)Δi(x,t)=ri(t)-aii(t)eαij(t)+∑j=1,j≠in‍aij(t)eαij(t)+∑j=1,j≠in‍bij(t)eβij(t-τij)+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγij(t+s)ds.
And define L:Dom⁡L⊂X→Z and N:X→Z as follows:
(10)X∋x(t)⟶(Lx)(t)=dx(t)dt∈Z,X∋x(t)⟶(Nx)(t)=((Nx)1(t),(Nx)2(t),…,(Nx)n(t))T∈Z,
where
(11)(Nx)i(t)=Δi(x,t),i=1,2,…,n.
The projectors are defined by P:X→X and Q:Z→Z by
(12)Px(t)=1ω∫0ω‍x(t)dt,Qz(t)=1ω∫0ω‍z(t)dt,mmmmmmmmmmmmmmmmnx∈X,z∈Z.
It is easy to follow that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) KP:Im⁡L→Dom⁡L∩Ker⁡P exists, which is given by
(13)KP(y)=∫0t‍y(s)ds-1ω∫0ω‍∫0t‍y(s)dsdt.
Then QN:X→Z and KP(I-Q)N:X→X are defined by
(14)QNx=(1ω∫0ω‍Δ1(x,t)dt,1ω∫0ω‍Δ2(x,t)dt,mmmmm…,1ω∫0ω‍Δn(x,t)dt)T,KP(I-Q)Nx=(Ψ1(x,t),Ψ2(x,t),…,Ψn(x,t))T,
where
(15)Ψk(x,t)=∫0t‍Δk(x,s)ds-1ω∫0ω‍∫0t‍Δk(x,s)dsdt-(tω-12)∫0ω‍Δk(x,s)ds,k=1,2,…,n.
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,
(16)x˙i(t)=λ[ri(t)-aii(t)eαij(t)+∑j=1,j≠in‍aij(t)eαij(t)mmll+∑j=1,j≠in‍bij(t)eβij(t-τij)mmll+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγij(t+s)ds],mmmmmmmmmmmmmmmmi=1,2,…n.
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
(17)ri(ti)-aii(ti)eαiixi(ti)+∑j=1,j≠in‍aij(ti)eαijxj(ti)+∑j=1,j≠in‍bij(ti)eβijxj(ti-τij)+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγijxj(ti+s)ds=0,mmmmmmmmmmmmmmi=1,2,…,n.
That is,
(18)aii(ti)eαiixi(ti)=ri(ti)+∑j=1,j≠in‍aij(ti)exj(ti)+∑j=1,j≠in‍bij(ti)eβijxj(ti-τij)+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγijxj(ti+s)ds,i=1,2,…n.
Noticing that xj(tj)=max⁡t∈[0,ω]|xj(t)| implies
(19)xj(ti-τij)≤xj(tj).
It follows from (H2) that
(20)a_iieαiixi(ti)≤|aii(ti)eαiixi(ti)|=|ri(ti)+∑j=1,j≠in‍aij(ti)eαijxj(ti)+∑j=1,j≠in‍bij(ti)eβijxj(ti-τij)+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγijxj(ti+s)ds|≤r¯i+∑j=1,j≠in‍a¯ijeαijxj(ti)+∑j=1,j≠in‍b¯ijeβijxj(ti-τij)+∑j=1,j≠in‍c¯ij(t)eγijxj(ti+s)∫0∞‍Hij(s)ds=r¯i+∑j=1,j≠in‍a¯ijeαijxj(ti)+∑j=1,j≠in‍b¯ijeβijxj(ti-τij)+∑j=1,j≠in‍c¯ij(t)eγijxj(ti+s)≤r¯i+∑j=1,j≠in‍a¯ijeαjjxj(tj)+∑j=1,j≠in‍b¯ijeαjjxj(tj)+∑j=1,j≠in‍c¯ij(t)eαjjxj(tj)=r¯i+∑j=1,j≠in‍(a¯ij+b¯ij+c¯ij)eαjjxj(tj).
Here we used (H2). Letting (a_ii+b_ii)eαiixi(ti)=zi(ti), it follows from (20) that
(21)zi(ti)≤r¯i+∑j=1,j≠in(a¯ij+b¯ij+c¯ij)a_jj-1zj(tj),
or
(22)zi(ti)-∑j=1,j≠in‍a¯ij+b¯ij+c¯ija_jjzj(tj)≤r¯i,
which implies
(23)(1-a¯12+b¯12+c¯12a_22⋯-a¯1n+b¯1n+c¯1na_nn-a¯21+b¯21+c¯21a_111⋯-a¯2n+b¯2n+c¯2na_nn⋯⋯⋯⋯-a¯n1+b¯n1+c¯n1a_11-a¯n2+b¯n2+c¯n2a_22⋯1)×(z1(t1)z2(t2)⋯zn(tn))≤(r¯1r¯2⋯r¯n)
Set D=(D1,D2,…,Dn)T=(r¯1,r¯2,…,r¯n)T. It follows from (23) that
(24)(E-K)(z1(t1),z2(t2),…,zn(tn))T≤D.
In view of ρ(K)<1 and Lemma 3, (En-K)-1≥0. Let
(25)H=(h~1,h~2,…,h~n)T∶=(E-K)-1D≥0.
Then it follows from (24) and (25) that
(26)(z1(t1),z2(t2),…,zn(tn))T≤H,orzi(ti)≤h~i,i=1,2,…,n,
which implies
(27)|xi(t)|0=max⁡t∈[0,ω]|xi(t)|=xi(ti)≤1αiiln⁡h~ia_ii+b_ii,mmmmmmmmmmmmlmmmmmi=1,2,…,n.
On the other hand, it follows from (25) that
(28)(E-K)H=D,orH=KH+D,thatis,h~i=∑j=1n‍Γijh~j+Di,i=1,2,…,n.
Estimating (16), by using (26) and (28), we have
(29)|x˙i(t)|0=λ|ri(t)-aii(t)eαiixi(t)+∑j=1,j≠in‍aij(t)eαijxj(t)mmll+∑j=1,j≠in‍bij(t)eβijxj(t-τij)mmll+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγijxj(ti+s)ds|0≤r¯i+|aii(t)eαiixi(ti)+∑j=1,j≠in‍aij(t)eαijxj(t)mmmm+∑j=1,j≠in‍bij(t)eβijxj(tj)mmmm+∑j=1,j≠in‍cij(t)∫0∞‍Hij(s)eγijxj(ti+s)ds|0=r¯i+∑j=1,j≠in‍a¯ij+b¯ij+c¯ija_jjzj(tj)+a¯iia_iizi(ti)≤Di+∑j=1n‍Γijh~j+a¯iia_iizi(ti)≤h~i+a¯iia_iih~i=[1+a¯iia_ii]h~i.
We can choose a large enough real number (d>1) such that
(30)1αiiln⁡dh~ia_ii>1αiiln⁡h~ia_ii+[1+a¯iia_ii]h~i.
Let hi=(1/αii)ln⁡(dh~i/a_ii). Then, for any solution of Lx=λNx, we have |xi(t)|1=|xi(t)|0+|x˙i(t)|0≤ln⁡(h~i/a_ii)+2h~i<hi for all i=1,2,…,n. Obviously, hi are independent of λ and the choice of x(t). Thus, taking hi=ln⁡(dh~i/a_ii), the open subset Ω satisfies that Lx≠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 and deg⁡{JQN,Ω∩Ker⁡L,0}≠0.
Hence, by Lemma1, 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.



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*Source: 127876-2014-08-27.xml*