# Permanence in a Discrete Mutualism Model with Infinite Deviating Arguments and Feedback Controls

Authors: Changjin Xu; Yusen Wu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We propose and deal with a discrete mutualism model with infinite deviating arguments and feedback controls. Sufficient conditions which guarantee the permanence of the system are obtained by using the difference inequality theory. The paper ends with brief conclusions.


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## Body

## 1. Introduction

It is well known that the long-term coexistence of species in mathematical ecology is an important and ubiquitous problem. Several mathematical concepts of coexistence of species are developed to deal with this aspect. Permanence is one important topic in these concepts. In recent years, permanence has received great attention and has been investigated in a number of notable studies. For example, Fan and Li [1] analyzed permanence of a delayed ratio-dependent predator-prey model with Holling type functional response. Mukherjee [2] addressed the permanence and global attractivity for facultative mutualism system with delay. Zhao and Jiang [3] focused on the permanence and extinction for nonautonomous Lotka-Volterra system. Chen [4] made a theoretical discussion on the permanence and global attractivity of Lotka-Volterra competition system with feedback control. Teng et al. [5] established the permanence criteria for a delayed discrete nonautonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [6–19].In 2007, Chen and You [20] investigated the permanence of the following two species integrodifferential model of mutualism:
(1)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))K1(t)+α1(t)∫0∞‍J2(s)N2(t-s)ds1+∫0∞‍J2(s)N2(t-s)ds],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))K2(t)+α2(t)∫0∞‍J1(s)N1(t-s)ds1+∫0∞‍J1(s)N1(t-s)ds],
where ri, Ki, αi, and σi(i=1,2) are continuous functions bounded above and below by positive constants. αi>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). For more background and biological adjustments of system (1), one could refer to [20–24] and the references cited therein.Many authors [24–33] have argued that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations. Motivated by the above viewpoint, Li and Yang [34] considered the permanence of the following discrete model of mutualism with infinite deviating arguments:
(2)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))K1(n)+α1(n)∑s=0∞J2(s)x2(n-s)1+∑s=0∞J2(s)x2(n-s)]},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))K2(n)+α2(n)∑s=0∞J1(s)x1(n-s)1+∑s=0∞J1(s)x1(n-s)]},
where xi(n)(i=1,2) is the density of mutualism species i at the generation, {ri(n)},{Ki(n)},{αi(n)},{σi(n)}(i=1,2),J1(n), and J2(n) are bounded nonnegative sequences. Applying the comparison theorem of difference equation and some lemmas, they derived some sufficient conditions which guarantee the permanence of system (2).It is well known that ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates [35]. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. To the authors’ knowledge, it is the first time to deal with system (2) with feedback control.The main objective of this paper is to investigate the following discrete mutualism model with infinite deviating arguments and feedback controls:(3)x1(n+1)=x1(n)exp{K1(n)+α1(n)∑s=0∞J2(s)x2(n-s)1+∑s=0∞J2(s)x2(n-s)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))-β1(n)u1(n)K1(n)+α1(n)∑s=0∞J2(s)x2(n-s)1+∑s=0∞J2(s)x2(n-s)]},x2(n+1)=x2(n)exp{K2(n)+α2(n)∑s=0∞J1(s)x1(n-s)1+∑s=0∞J1(s)x1(n-s)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))-β2(n)u2(n)[K2(n)+α2(n)∑s=0∞J1(s)x1(n-s)1+∑s=0∞J1(s)x1(n-s)]},Δu1(n)=-γ1(n)u1(n)+η1(n)x1(n),Δu2(n)=-γ2(n)u2(n)+η2(n)x2(n),
where xi(n)(i=1,2) is the density of mutualism species i at the generation and ui(n)(i=1,2) is the control variable. r1(n),r2(n),K1(n),K2(n),α1(n),α2(n),J1(n),J2(n),σ1(n), and σ2(n) are bounded nonnegative sequences.Throughout this paper, we assume that(H)
0<ril≤riu,0<Kil≤Kiu,0<αil≤αiu,0<βil≤βiu,0<σil≤σiu,∑n=0∞Ji(n)=1,(i=1,2).Here, for any bounded sequence {f(n)}, fu=supn∈N{f(n)} and fl=infn∈N{f(n)}.Letσ=supn∈Z{σi(n)},σ~=infn∈Z{σi(n)},i=1,2. We consider (3) together with the following initial conditions:
(4)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 (3) and (4) are well defined for all n≥0 and satisfy
(5)xi(n)>0,forn∈Z,i=1,2.
The remainder of the paper is organized as follows. In Section 2, basic definitions and lemmas are given, some sufficient conditions for the permanence of system (3) are established. Brief conclusions are presented in Section 3.


## 2. Permanence

In order to obtain the main result of this paper, we will first state the definition of permanence and several lemmas which will be useful in the proving of the main result.Definition 1.
We say that system (3) is permanence if there are positive constants M and m such that for each positive solution (x1(n),x2(n),u1(n),u2(n)) of system (3) satisfies
(6)m≤limn→+∞infxi(n)≤limn→+∞supxi(n)≤M,m≤limn→+∞infxi(n)≤limn→+∞1sup(i=1,2),m≤limn→+∞infui(n)≤limn→+∞supui(n)≤M,m≤limn→+∞infui(n)≤limn→+∞1sup(i=1,2).Let us consider the following single species discrete model:(7)N(n+1)=N(n)exp(a(n)-b(n)N(n)),
where {a(n)} and {b(n)} are strictly positive sequences of real numbers defined for n∈N={0,1,2,…} and 0<al≤au,0<bl≤bu. Similarly to the proofs Propositions 1 and  3 in [36], we can obtain the following Lemma 2.Lemma 2.
Any solution of system (7) with initial condition N(0)>0 satisfies
(8)m≤limn→+∞infN(n)≤limn→+∞supN(n)≤M,
where
(9)M=1blexp(au-1),m=albuexp(al-buM).Let us consider the first order difference equation:(10)y(n+1)=Ay(n)+B,n=1,2,…,
where A and B are positive constants. Following Theorem 6.2 of L. Wang and M. Q. Wang [37, page 125], we have the following Lemma 3.Lemma 3 (see [37]).
Assume that|A|<1, for any initial value y(0), there exists a unique solution y(n) of (10) which can be expressed as follows:
(11)y(n)=An(y(0)-y*)+y*,
where y*=B/(1-A). Thus, for any solution {y(n)} of system (10),  limn→+∞y(n)=y*.Lemma 4 (see [37]).
Letn∈Nn0+={n0,n0+1,…,n0+l,…},r≥0. For any fixed n, g(n,r) is a nondecreasing function with respect to r, and for n≥n0, the following inequalities hold:
(12)y(n+1)≤g(n,y(n)),u(n+1)≥g(n,u(n)).
If y(n0)≤u(n0), then y(n)≤u(n) for all n≥n0.Proposition 5.
Assume that the condition (H) holds, then(13)limn→+∞supxi(n)≤Mi,limn→+∞supui(n)≤Ui,i=1,2,
where
(14)Mi=1rilexp{riu(Kiu+αiu)(σ+1)-1},Ui=ηiuMiγil,(i=1,2).Proof.
Let(x1(n),x2(n),u1(n),u2(n)) be any positive solution of system (3) with the initial condition (x1(0),x2(0),u1(0),u2(0)). It follows from the first equation of system (3) that
(15)x1(n+1)≤x1(n)exp{r1(n)K1(n)+r1(n)α1(n)}.
Let x1(n)=exp{y1(n)} then (15) is equivalent to
(16)y1(n+1)-y1(n)≤r1(n)K1(n)+r1(n)α1(n).
Summing both sides of (16) from n-σ1(n) to n-1, we have
(17)∑j=n-σ1(n)n-1‍(y1(j+1)-y1(j))≤∑j=n-σ1(n)n-1‍(r1(j)K1(j)+r1(j)α1(j))≤r1u(K1u+α1u)σ,
which leads to
(18)y1(n-σ1(n))≥y1(n)-r1u(K1u+α1u)σ.
Then
(19)x1(n-σ1(n))≥x1(n)exp{-r1u(K1u+α1u)σ}.
Substituting (19) into the first equation of system (3), it follows that
(20)x1(n+1)≤x1(n)exp{r1(n)(k1(n)+α1(n))-r1(n)exp{-r1u(K1u+α1u)σ}x1(n){r1(n)(k1(n)+α1(n))}.
It follows from (20) and Lemma 2 that
(21)limn→+∞supx1(n)≤1r1lexp{r1u(K1u+α1u)(σ+1)-1}∶=M1.
For any positive constant ε>0, it follows (21) that there exists a N1>0 such that for all n>N1+σ(22)x1(n)≤M1+ε.
From the second equation of system (3), we get
(23)x2(n+1)≤x2(n)exp{r2(n)K2(n)+r2(n)α2(n)}.
Let x2(n)=exp{y2(n)} then (23) is equivalent to
(24)y2(n+1)-y2(n)≤r2(n)K2(n)+r2(n)α2(n).
Summing both sides of (24) from n-σ2(n) to n-1, we have
(25)∑j=n-σ2(n)n-1‍(y2(j+1)-y2(j))≤∑j=n-σ2(n)n-1‍(r2(j)K2(j)+r2(j)α2(j))≤r2u(K2u+α2u)σ,
which leads to
(26)y2(n-σ2(n))≥y2(n)-r2u(K2u+α2u)σ.
Then
(27)x2(n-σ2(n))≥x2(n)exp{-r2u(K2u+α2u)σ}.
Substituting (27) into the second equation of system (3), it follows that
(28)x2(n+1)≤x2(n)exp{r2(n)(k2(n)+α2(n))-r2(n)exp{-r2u(K2u+α2u)σ}x2(n)r2(n)(k2(n)+α2(n))}.
It follows from (28) and Lemma 2 that
(29)limn→+∞supx2(n)≤1r2lexp{r1u(K2u+α2u)(σ+1)-1}:=M2.
For any positive constant ε>0, it follows (29) that there exists a N2>N1+σ such that for all n>N2+σ(30)x2(n)≤M2+ε.
In view of the third and fourth equations of system (3), we can obtain
(31)Δu1(n)≤-γ1(n)u1(n)+η1(n)(M1+ε),Δu2(n)≤-γ2(n)u2(n)+η2(n)(M2+ε).
Then
(32)u1(n+1)≤(1-γ1l)u1(n)+η1u(M1+ε),u2(n+1)≤(1-γ2l)u2(n)+η2u(M2+ε).
Applying Lemmas 3 and 4, it immediately follows that
(33)limn→+∞supu1(n)≤η1u(M1+ε)γ1l,limn→+∞supu2(n)≤η2u(M2+ε)γ2l.
Setting ε→0, it follows that
(34)limn→+∞supu1(n)≤η1uM1γ1l:=U1,limn→+∞supu2(n)≤η2uM2γ2l:=U2.
This completes the proof of Proposition 5.Theorem 6.
Assume that (H) holds, then system (3) is permanent.Proof.
By applying Proposition5, we can easily see that to end the proof of Theorem 6, it is enough to show that under the conditions of Theorem 6,
(35)limn→+∞infx2(n)≥m1,limn→+∞infx2(n)≥m2,limn→+∞infu1(n)≥v1,limn→+∞infu2(n)≥v2.
In view of Proposition 5, for all ε>0, there exists a N3>0,N3∈N, for all n>N3:
(36)x1(n)≤M1+ε,x2(n)≤M2+ε,u1(n)≤U1+ε,u2(n)≤U2+ε.
It follows from the first equation of systems (3) and (36) that
(37)x1(n+1)≥x1(n)exp{r1(n)[K1(n)1+∑s=0∞J2(s)x2(n-s)-x1(n-σ1(n))K1(n)1+∑s=0∞J2(s)x2(n-s)]}≥x1(n)exp{r1lK1l1+M2+ε-(M1+ε)}
for all n>N3+σ.
Letx1(n)=exp{y1(n)}; then (37) is equivalent to
(38)y1(n+1)-y1(n)≥r1lK1l1+M2+ε-(M1+ε).
Summing both sides of (38) from n-δ1(n) to n-1 leads to
(39)∑j=n-δ1(n)n-1‍(y1(j+1)-y1(j))≥∑j=n-δ1(n)n-1‍[r1lK1l1+M2+ε-(M1+ε)]≥[r1lK1l1+M2+ε-(M1+ε)]σ~.
Then
(40)y1(n-δ1(n))≤y1(n)-[r1lK1l1+M2+ε-(M1+ε)]σ~.
Thus
(41)x1(n-δ1(n))≤x1(n)exp{-[r1lK1l1+M2+ε-(M1+ε)]σ~}.
Substituting (36) and (41) into the first equation of (3), we have
(42)x1(n+1)≥x1(n)exp{r1(n)[K1(n)1+∑s=0∞J2(s)x2(n-s)-x1(n-σ1(n))K1(n)1+∑s=0∞J2(s)x2(n-s)]}≥x1(n)exp{-[r1lK1l1+M2+ε-(M1+ε)]σ~}}r1lK1l1+M2+ε-x1(n)≥x1(n)exp≥×exp{-[r1lK1l1+M2+ε-(M1+ε)]σ~}}
for all n>N3+σ.
By applying Lemmas2 and 4, it immediately follows that
(43)limn→+∞infx1(n)≥m1ε,
where
(44)m1ε=r1lK1lr1u(1+M1+ε)×exp{r1lK1l1+M1+ε-r1u×exp[-(r1lK1l1+M2+ε-(M1+ε))σ~]M1}.
Setting ε→0 in (43), then
(45)limn→+∞infx1(n)≥m1,
where
(46)m1=r1lK1lr1u(1+M1)×exp{r1lK1l1+M1-r1u×exp[-(r1lK1l1+M2-M1)σ~]M1}.
By the second equation of systems (3) and (36), we can obtain
(47)x2(n+1)≥x2(n)×exp{r2(n)[K2(n)1+∑s=0∞J2(s)x2(n-s)-x2(n-σ2(n))K2(n)1+∑s=0∞J2(s)x2(n-s)]}≥x2(n)exp{r2lK2l1+M1+ε-(M2+ε)}
for all n>N3+σ.
Letx2(n)=exp{y2(n)}; then (47) is equivalent to
(48)y2(n+1)-y2(n)≥r2lK2l1+M1+ε-(M2+ε).
Summing both sides of (48) from n-δ2(n) to n-1 leads to
(49)∑j=n-δ2(n)n-1‍(y2(n+1)-y2(n))≥∑j=n-δ2(n)n-1‍[r2lK2l1+M1+ε-(M2+ε)]≥[r2lK2l1+M1+ε-(M2+ε)]σ~.
Then
(50)y2(n-δ2(n))≤y2(n)-[r2lK2l1+M1+ε-(M2+ε)]σ~.
Thus
(51)x2(n-δ2(n))≤x2(n)exp{-[r2lK2l1+M1+ε-(M2+ε)]σ~}.
Substituting (36) and (51) into the second equation of (3), we have
(52)x2(n+1)≥x2(n)exp{r2(n)[K2(n)1+∑s=0∞J1(s)x1(n-s)-x2(n-σ2(n))K2(n)1+∑s=0∞J1(s)x1(n-s)]}≥x2(n)exp{{-[r2lK2l1+M1+ε-(M2+ε)]σ~}}r2lK2l1+M1+ε-x2(n)×exp{-[r2lK2l1+M1+ε-(M2+ε)]σ~}}
for all n>N3+σ.
By applying Lemmas2 and 4, it immediately follows that
(53)limn→+∞infx2(n)≥m2ε,
where
(54)m2ε=r2lK2lr2u(1+M2+ε)×exp{[-(r2lK2l1+M1+ε-(M2+ε))σ~]r2lK2l1+M2+ε-r2u×exp[-(r2lK2l1+M1+ε-(M2+ε))σ~]M2[-(r2lK2l1+M1+ε-(M2+ε))σ~]}.
Setting ε→0 in (53), then
(55)limn→+∞infx2(n)≥m2,
where
(56)m2=r2lK2lr2u(1+M2)×exp{[-(r2lK2l1+M1-M2)σ~]M2}r2lK2l1+M2-r2u×exp[-(r2lK2l1+M1-M2)σ~]M2}.
Without loss of generality, we assume that ε<(1/2)min{m1,m2}. For any positive constant ε small enough, it follows from (45) and (55) that there exists enough large N4>N3+σ such that
(57)x1(n)≥m1-ε,x2(n)≥m2-ε
for any n≥N4.
From the third and fourth equations of systems (3) and (57), we can derive that
(58)Δu1(n)≥-γ1(n)u1(n)+η1(n)(m1-ε),Δu2(n)≥-γ2(n)u2(n)+η2(n)(m2-ε).
Hence
(59)u1(n+1)≥(1-γ1u)u1(n)+η1l(m1-ε),u2(n+1)≥(1-γ2u)u2(n)+η2l(m2-ε).
By applying Lemmas 2 and 3, it immediately follows that
(60)limn→+∞infu1(n)≥η1l(m1-ε)γ1u,(61)limn→+∞infu2(n)≥η2l(m2-ε)γ2u.
Setting ε→0 in the previous inequality leads to
(62)limn→+∞infu1(n)≥η1lm1γ1u,limn→+∞infu2(n)≥η2lm2γ2u.
This completes the proof of Theorem 6.


## 3. Conclusions

In the present paper, we have investigated the permanence of a discrete mutualism model with infinite deviating arguments and feedback controls. Sufficient conditions which ensure the permanence of the system are established. We have shown the effect of delay to the permanence of system and concluded that delay is an important factor to decide the permanence of the system.



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