We propose a discrete model of mutualism with infinite deviating arguments, that is x1(n+1)=x1(n)exp{r1(n)[(K1(n)+α1(n)∑s=0∞J2(s)x2(n−s))/(1+∑s=0∞J2(s)x2(n−s))−x1(n−σ1(n))]},x2(n+1)=x2(n)exp{r2(n)[(K2(n)+α2(n)∑s=0∞J1(s)x1(n−s))/(1+∑s=0∞J1(s)x1(n−s))−x2(n−σ2(n))]}. By some Lemmas, sufficient conditions are obtained for the permanence of the system.

1. Introduction

Chen and You [1] studied the following two species integro-differential model of mutualism:

dN1(t)dt=r1(t)N1(t)[K1(t)+α1(t)∫0∞J2(s)N2(t-s)ds1+∫0∞J2(s)N2(t-s)ds-N1(t-σ1(t))],dN2(t)dt=r2(t)N2(t)[K2(t)+α2(t)∫0∞J1(s)N1(t-s)ds1+∫0∞J1(s)N1(t-s)ds-N2(t-σ2(t))],
where ri,Ki,αi, and σi,i=1,2 are continuous functions bounded above and below by positive constants: ai>Ki,i=1,2;Ji∈C([0,+∞),[0,+∞)) and ∫0∞Ji(s)ds=1,i=1,2. Using the differential inequality theory, they obtained a set of sufficient conditions to ensure the permanence of system (1.1). For more background and biological adjustments of system(1.1), one could refer to [1–4] and the references cited therein.

However, many authors [5–12] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Also, since discrete time models can also provide efficient computational models of continuous models for numerical simulations, it is reasonable to study discrete time models governed by difference equations. Another permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. It is reasonable to ask for conditions under which the system is permanent.

Motivated by the above question, we consider the permanence of the following discrete model of mutualism with infinite deviating arguments:

x1(n+1)=x1(n)exp{r1(n)[K1(n)+α1(n)∑s=0∞J2(s)x2(n-s)1+∑s=0∞J2(s)x2(n-s)-x1(n-σ1(n))]},x2(n+1)=x2(n)exp{r2(n)[K2(n)+α2(n)∑s=0∞J1(s)x1(n-s)1+∑s=0∞J1(s)x1(n-s)-x2(n-σ2(n))]},
where xi(n),i=1,2 is the density of mutualism species i at the nth generation. For {ri(n)},{Ki(n)},{αi(n)},{Ji(n)}, and {σi(n)},i=1,2 are bounded nonnegative sequences such that

0<ril≤riu,0<αil≤αiu,0<Kil≤Kiu,0<σil≤σiu,∑n=0∞Ji(n)=1.
Here, for any bounded sequence {a(n)}, au=supn∈Na(n),al=infn∈Na(n).

Let σ=supn{σi(n),i=1,2}, we consider (1.2) together with the following initial condition:

xi(θ)=φi(θ)≥0,θ∈N[-τ,0]={-τ,-τ+1,…,0},φi(0)>0.

It is not difficult to see that solutions of (1.2) and (1.4) are well defined for all n≥0 and satisfy

xi(n)>0,forn∈Z,i=1,2.

The aim of this paper is, by applying the comparison theorem of difference equation and some lemmas, to obtain a set of sufficient conditions which guarantee the permanence of system (1.2).

2. Permanence

In this section, we establish permanence results for system (1.2).

Following Comparison Theorem of difference equation is Theorem 2.6 of [13, page 241].

Lemma.

Let k∈Nk0+={k0,k0+1,…,k0+l,…},r≥0. For any fixed k,g(k,r) is a non-decreasing function with respect to r, and for k≥k0, following inequalities hold: y(k+1)≤g(k,y(k)),u(k+1)≥g(k,u(k)). If y(k0)≤u(k0), then y(k)≤u(k) for all k≥k0.

Now let us consider the following single species discrete model:

N(k+1)=N(k)exp{a(k)-b(k)N(k)},
where {a(k)} and {b(k)} are strictly positive sequences of real numbers defined for k∈N={0,1,2,…} and 0<al≤au,0<bl≤bu. Similar to the proof of Propositions 1 and 3 in [6], we can obtain the following.

Lemma.

Any solution of system (2.1) with initial condition N(0)>0 satisfies
m≤limk→+∞infN(k)≤limk→+∞supN(k)≤M,
where
M=1blexp{au-1},m=albuexp{al-buM}.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B13">14</xref>]).

Let x(n) and b(n) be nonnegative sequences defined on N, and c≥0 is a constant. If
x(n)≤c+∑s=0n-1b(s)x(s),forn∈N,
then
x(n)≤c∏s=0n-1[1+b(s)],forn∈N.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let x:Z→R be a nonnegative bounded sequences, and let H:N→R be a nonnegative sequence such that ∑n=0∞Ji(n)=1. Then
limn→+∞infx(n)≤limn→+∞inf∑s=-∞nH(n-s)x(s)≤limn→+∞sup∑s=-∞nH(n-s)x(s)≤limn→+∞supx(n).

Proposition.

Let (x1(n),x2(n)) be any positive solution of system (1.2), then
limn→+∞supxi(n)≤Mi,i=1,2,
where
Mi=exp{2riu[Kiu+αiu]},i=1,2.

Proof.

Let (x1(n),x2(n)) be any positive solution of system (1.2), then from the first equation of system (1.2) we have
x1(n+1)≤x1(n)exp{r1(n)[K1(n)+α1(n)∑s=0∞J2(s)x2(n-s)1+∑s=0∞J2(s)x2(n-s)]}=x1(n)exp{r1(n)[K1(n)1+∑s=0∞J2(s)x2(n-s)+α1(n)∑s=0∞J2(s)x2(n-s)1+∑s=0∞J2(s)x2(n-s)]}≤x1(n)exp{r1(n)[K1(n)1+α1(n)∑s=0∞J2(s)x2(n-s)∑s=0∞J2(s)x2(n-s)]}=x1(n)exp{r1(n)[K1(n)+α1(n)]}≤x1(n)exp{r1u[K1u+α1u]}.
Let x1(n)=exp{u1(n)}, then
u1(n+1)≤u1(n)+r1u[K1u+α1u]=r1u[K1u+α1u]+∑s=0nb(s)x(s),
where
b(s)={0,0≤s≤n-1,1,s=n.
When u1(n) is nonnegative sequence, by applying Lemma 2.3, it immediately follows that
u1(n+1)≤2r1u[K1u+α1u].
When u1(n) is negative sequence, (2.12) also holds. From (2.12), we have
limn→+∞supx1(n)≤exp{2r1u[K1u+α1u]}:=M1.
By using the second equation of system (1.2), similar to the above analysis, we can obtain
limn→+∞supx2(n)≤exp{2r2u[K2u+α2u]}:=M2.
This completes the proof of Proposition 2.5.

Now we are in the position of stating the permanence of system (1.2).

Theorem.

Under the assumption(1.3), system (1.2) is permanent, that is, there exist positive constants mi,Mi,i=1,2 which are independent of the solutions of system (1.2) such that, for any positive solution (x1(n),x2(n)) of system(1.2) with initial condition (1.4), one has
mi≤limn→+∞infxi(n)≤limn→+∞supxi(n)≤Mi,i=1,2.

Proof.

By applying Proposition 2.5, we see that to end the proof of Theorem 2.6 it is enough to show that under the conditions of Theorem 2.6limn→+∞infxi(n)≥mi.
From Proposition 2.5, For all ε>0, there exists a N1>0,N1∈N,For all n>N1,
xi(n)≤Mi+ε.
According to Lemma 2.4, from (2.13) and (2.14) we have
limn→+∞sup∑s=0∞Ji(s)xi(n-s)=limn→+∞sup∑k=-∞nJi(n-k)xi(k)≤Mi,i=1,2.
For above ɛ>0, according to (2.18), there exists a positive integer N2, such that, for all n>N2,
∑s=0∞Ji(s)xi(n-s)≤Mi+ɛ,i=1,2.
Thus, for all n>max{N1,N2}+σ, from the first equation of system(1.2), it follows that
x1(n+1)≥x1(n)exp{r1(n)[K1l1+(M2+ε)-(M1+ε)]}≥x1(n)exp{r1lK1l1+(M2+ε)-r1u(M1+ε)}.
It follows that, for n≥σ1(n),
∏i=n-σ1(n)n-1x1(i+1)≥∏i=n-σ1(n)n-1x1(i)exp{r1lK1l1+(M2+ε)-r1u(M1+ε)}.
Hence
x1(n)≥x1(n-σ1(n))exp{r1lK1l1+(M2+ε)σ1l-r1u(M1+ε)σ1u}.
In other words,
x1(n-σ1(n))≤x1(n)exp{-r1lK1l1+(M2+ε)σ1l+r1u(M1+ε)σ1u}.
From the first equation of system (1.2) and (2.23), for all n>max{N1,N2}+σ, it follows that
x1(n+1)≥x1(n)exp{-r1lK1l1+(M2+ε)-r1uexp{-r1lK1l1+(M2+ε)σ1l+r1u(M1+ε)σ1u}x1(n)}.
By applying Lemmas 2.1 and 2.2 to (2.24), it immediately follows thatlimn→+∞infx1(n)≥r1lK1lr1u(1+(M2+ε))exp{r1lK1l1+(M2+ε)σ1l-r1u(M1+ε)σ1u}×exp{r1lK1l1+(M2+ε)-r1uexp{-r1lK1l1+(M2+ε)σ1l+r1u(M1+ε)σ1u}M1}.
Setting ɛ→0, it follows that
limn→+∞infx1(n)≥r1lK1lr1u(1+M2)exp{r1lK1l1+M2σ1l-r1uM1σ1u}×exp{r1lK1l1+M2-r1uexp{-r1lK1l1+M2σ1l+r1uM1σ1u}M1}.
Similar to the above analysis, from the second equation of system (1.2), we have that
limn→+∞infx2(n)≥r2lK2lr2u(1+M1)exp{r2lK2l1+M1σ2l-r2uM2σ2u}×exp{r2lK2l1+M1-r2uexp{-r2lK2l1+M1σ2l+r2uM2σ2u}M2}.
This completes the proof of Theorem 2.6.

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