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

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.

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: 𝑑𝑁₁(𝑡)𝑑𝑡=𝑟₁(𝑡)𝑁₁(𝑡)×𝐾₁(𝑡)+𝛼₁(𝑡)∫^∞₀𝐽₂(𝑠)𝑁₂(𝑡−𝑠)𝑑𝑠1+∫^∞₀𝐽₂(𝑠)𝑁₂(𝑡−𝑠)𝑑𝑠−𝑁₁𝑡−𝜎₁(𝑡),𝑑𝑁₂(𝑡)𝑑𝑡=𝑟₂(𝑡)𝑁₂(𝑡)×𝐾₂(𝑡)+𝛼₂(𝑡)∫^∞₀𝐽₁(𝑠)𝑁₁(𝑡−𝑠)𝑑𝑠1+∫^∞₀𝐽₁(𝑠)𝑁₁(𝑡−𝑠)𝑑𝑠−𝑁₂𝑡−𝜎₂(𝑡), (1) where 𝑟_𝑖, 𝐾_𝑖, 𝛼_𝑖, and 𝜎_𝑖(𝑖=1,2) are continuous functions bounded above and below by positive constants. 𝛼_𝑖>𝐾_𝑖,𝑖=1,2; 𝐽_𝑖∈𝐶([0,+∞),[0,+∞)) and ∫^∞₀𝐽_𝑖(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: 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp𝑟₁(𝑛𝐾)¹(𝑛)+𝛼₁(𝑛)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)−𝑥₁𝑛−𝜎₁(𝑛),𝑥₂(𝑛+1)=𝑥₂(𝑛)exp𝑟₂(𝑛𝐾)²(𝑛)+𝛼₂(𝑛)∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)−𝑥₂𝑛−𝜎₂(𝑛), (2) where 𝑥_𝑖(𝑛)(𝑖=1,2) is the density of mutualism species 𝑖 at the generation, {𝑟_𝑖(𝑛)},{𝐾_𝑖(𝑛)},{𝛼_𝑖(𝑛)},{𝜎_𝑖(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: 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp𝑟₁(𝑛)×𝐾₁(𝑛)+𝛼₁(𝑛)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)−𝑥₁𝑛−𝜎₁(𝑛)−𝛽₁(𝑛)𝑢₁(𝑛,𝑥₂)(𝑛+1)=𝑥₂(𝑛)exp𝑟₂(𝑛)×𝐾₂(𝑛)+𝛼₂(𝑛)∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)−𝑥₂𝑛−𝜎₂(𝑛)−𝛽₂(𝑛)𝑢₂(𝑛),Δ𝑢₁(𝑛)=−𝛾₁(𝑛)𝑢₁(𝑛)+𝜂₁(𝑛)𝑥₁(𝑛),Δ𝑢₂(𝑛)=−𝛾₂(𝑛)𝑢₂(𝑛)+𝜂₂(𝑛)𝑥₂(𝑛), (3) where 𝑥_𝑖(𝑛)(𝑖=1,2) is the density of mutualism species 𝑖 at the generation and 𝑢_𝑖(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<𝑟^𝑙_𝑖≤𝑟^𝑢_𝑖,0<𝐾^𝑙_𝑖≤𝐾^𝑢_𝑖,0<𝛼^𝑙_𝑖≤𝛼^𝑢_𝑖,0<𝛽^𝑙_𝑖≤𝛽^𝑢_𝑖,0<𝜎^𝑙_𝑖≤𝜎^𝑢_𝑖,∑^∞_𝑛=0𝐽_𝑖(𝑛)=1,(𝑖=1,2). Here, for any bounded sequence {𝑓(𝑛)}, 𝑓^𝑢=sup_𝑛∈𝑁{𝑓(𝑛)} and 𝑓^𝑙=inf_𝑛∈𝑁{𝑓(𝑛)}.

Let 𝜎=sup_𝑛∈𝑍{𝜎_𝑖(𝑛)},𝜎=inf_𝑛∈𝑍{𝜎_𝑖(𝑛)},𝑖=1,2. We consider (3) together with the following initial conditions: 𝑥_𝑖(𝜃)=𝜑_𝑖(𝜃)[]={}≥0,𝜃∈𝑁−𝜏,0−𝜏,−𝜏+1,…,0,𝜑_𝑖(0)>0. (4) It is not difficult to see that solutions of (3) and (4) are well defined for all 𝑛≥0 and satisfy 𝑥_𝑖(𝑛)>0,for𝑛∈𝑍,𝑖=1,2. (5) 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 𝑚≤lim_𝑛→+∞inf𝑥_𝑖(𝑛)≤lim_𝑛→+∞sup𝑥_𝑖(𝑛)(),≤𝑀,𝑖=1,2𝑚≤lim_𝑛→+∞inf𝑢_𝑖(𝑛)≤lim_𝑛→+∞sup𝑢_𝑖(𝑛)().≤𝑀,𝑖=1,2 (6)

Let us consider the following single species discrete model: 𝑁(𝑛+1)=𝑁(𝑛)exp(𝑎(𝑛)−𝑏(𝑛)𝑁(𝑛)), (7) where {𝑎(𝑛)} and {𝑏(𝑛)} are strictly positive sequences of real numbers defined for 𝑛∈𝑁={0,1,2,…} and 0<𝑎^𝑙≤𝑎^𝑢,0<𝑏^𝑙≤𝑏^𝑢. 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 𝑚≤lim_𝑛→+∞inf𝑁(𝑛)≤lim_𝑛→+∞sup𝑁(𝑛)≤𝑀, (8) where 1𝑀=𝑏^𝑙exp𝑎^𝑢𝑎−1,𝑚=^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑀. (9)

Let us consider the first order difference equation: 𝑦(𝑛+1)=𝐴𝑦(𝑛)+𝐵,𝑛=1,2,…, (10) 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: 𝑦(𝑛)=𝐴^𝑛𝑦(0)−𝑦^∗+𝑦^∗, (11) where 𝑦^∗=𝐵/(1−𝐴). Thus, for any solution {𝑦(𝑛)} of system (10),  lim_𝑛→+∞𝑦(𝑛)=𝑦^∗.

Lemma 4 (see [37]). Let 𝑛∈𝑁⁺_𝑛0={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: 𝑦(𝑛+1)≤𝑔𝑛,𝑦(𝑛),𝑢(𝑛+1)≥𝑔(𝑛,𝑢(𝑛)). (12) If 𝑦(𝑛₀)≤𝑢(𝑛₀), then 𝑦(𝑛)≤𝑢(𝑛) for all 𝑛≥𝑛₀.

Proposition 5. Assume that the condition (H) holds, then lim_𝑛→+∞sup𝑥_𝑖(𝑛)≤𝑀_𝑖,lim_𝑛→+∞sup𝑢_𝑖(𝑛)≤𝑈_𝑖,𝑖=1,2, (13) where 𝑀_𝑖=1𝑟^𝑙_𝑖exp𝑟^𝑢_𝑖𝐾^𝑢_𝑖+𝛼^𝑢_𝑖(),𝑈𝜎+1−1_𝑖=𝜂^𝑢_𝑖𝑀_𝑖𝛾^𝑙_𝑖,(𝑖=1,2). (14)

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 𝑥₁(𝑛+1)≤𝑥₁(𝑛)exp𝑟₁(𝑛)𝐾₁(𝑛)+𝑟₁(𝑛)𝛼₁(𝑛). (15) Let 𝑥₁(𝑛)=exp{𝑦₁(𝑛)} then (15) is equivalent to 𝑦₁(𝑛+1)−𝑦₁(𝑛)≤𝑟₁(𝑛)𝐾₁(𝑛)+𝑟₁(𝑛)𝛼₁(𝑛). (16) Summing both sides of (16) from 𝑛−𝜎₁(𝑛) to 𝑛−1, we have ^𝑛−1_{𝑗=𝑛−𝜎1(𝑛)}𝑦₁𝑗+1−𝑦₁𝑗≤^𝑛−1_{𝑗=𝑛−𝜎1(𝑛)}𝑟₁𝑗𝐾₁𝑗+𝑟₁𝑗𝛼₁𝑗≤𝑟^𝑢₁𝐾^𝑢₁+𝛼^𝑢₁𝜎, (17) which leads to 𝑦₁𝑛−𝜎₁(𝑛)≥𝑦₁(𝑛)−𝑟^𝑢₁𝐾^𝑢₁+𝛼^𝑢₁𝜎. (18) Then 𝑥₁𝑛−𝜎₁(𝑛)≥𝑥₁(𝑛)exp−𝑟^𝑢₁𝐾^𝑢₁+𝛼^𝑢₁𝜎. (19) Substituting (19) into the first equation of system (3), it follows that 𝑥₁(𝑛+1)≤𝑥₁(𝑛)exp𝑟₁(𝑛)𝑘₁(𝑛)+𝛼₁(𝑛)−𝑟₁(𝑛)exp−𝑟^𝑢₁𝐾^𝑢₁+𝛼^𝑢₁𝜎𝑥₁(𝑛). (20) It follows from (20) and Lemma 2 that lim_𝑛→+∞sup𝑥₁(𝑛)≤1𝑟^𝑙₁exp𝑟^𝑢₁𝐾^𝑢₁+𝛼^𝑢₁(𝜎+1)−1∶=𝑀₁. (21) For any positive constant 𝜀>0, it follows (21) that there exists a 𝑁₁>0 such that for all 𝑛>𝑁₁+𝜎𝑥₁(𝑛)≤𝑀₁+𝜀. (22) From the second equation of system (3), we get 𝑥₂(𝑛+1)≤𝑥₂(𝑛)exp𝑟₂(𝑛)𝐾₂(𝑛)+𝑟₂(𝑛)𝛼₂(𝑛). (23) Let 𝑥₂(𝑛)=exp{𝑦₂(𝑛)} then (23) is equivalent to 𝑦₂(𝑛+1)−𝑦₂(𝑛)≤𝑟₂(𝑛)𝐾₂(𝑛)+𝑟₂(𝑛)𝛼₂(𝑛). (24) Summing both sides of (24) from 𝑛−𝜎₂(𝑛) to 𝑛−1, we have ^𝑛−1_{𝑗=𝑛−𝜎2(𝑛)}𝑦₂𝑗+1−𝑦₂𝑗≤^𝑛−1_{𝑗=𝑛−𝜎2(𝑛)}𝑟₂𝑗𝐾₂𝑗+𝑟₂𝑗𝛼₂𝑗≤𝑟^𝑢₂𝐾^𝑢₂+𝛼^𝑢₂𝜎, (25) which leads to 𝑦₂𝑛−𝜎₂(𝑛)≥𝑦₂(𝑛)−𝑟^𝑢₂𝐾^𝑢₂+𝛼^𝑢₂𝜎. (26) Then 𝑥₂𝑛−𝜎₂(𝑛)≥𝑥₂(𝑛)exp−𝑟^𝑢₂𝐾^𝑢₂+𝛼^𝑢₂𝜎. (27) Substituting (27) into the second equation of system (3), it follows that 𝑥₂(𝑛+1)≤𝑥₂(𝑛)exp𝑟₂(𝑛)𝑘₂(𝑛)+𝛼₂(𝑛)−𝑟₂(𝑛)exp−𝑟^𝑢₂𝐾^𝑢₂+𝛼^𝑢₂𝜎𝑥₂(𝑛). (28) It follows from (28) and Lemma 2 that lim_𝑛→+∞sup𝑥₂(𝑛)≤1𝑟^𝑙₂exp𝑟^𝑢₁𝐾^𝑢₂+𝛼^𝑢₂(𝜎+1)−1∶=𝑀₂. (29) For any positive constant 𝜀>0, it follows (29) that there exists a 𝑁₂>𝑁₁+𝜎 such that for all 𝑛>𝑁₂+𝜎𝑥₂(𝑛)≤𝑀₂+𝜀. (30) In view of the third and fourth equations of system (3), we can obtain Δ𝑢₁(𝑛)≤−𝛾₁(𝑛)𝑢₁(𝑛)+𝜂₁(𝑛)𝑀₁+𝜀,Δ𝑢₂(𝑛)≤−𝛾₂(𝑛)𝑢₂(𝑛)+𝜂₂(𝑛)𝑀₂+𝜀. (31) Then 𝑢₁(𝑛+1)≤1−𝛾^𝑙₁𝑢₁(𝑛)+𝜂^𝑢₁𝑀₁,𝑢+𝜀₂(𝑛+1)≤1−𝛾^𝑙₂𝑢₂(𝑛)+𝜂^𝑢₂𝑀₂+𝜀. (32) Applying Lemmas 3 and 4, it immediately follows that lim_𝑛→+∞sup𝑢₁(𝑛)≤𝜂^𝑢₁𝑀₁+𝜀𝛾^𝑙₁,lim_𝑛→+∞sup𝑢₂(𝑛)≤𝜂^𝑢₂𝑀₂+𝜀𝛾^𝑙₂. (33) Setting 𝜀→0, it follows that lim_𝑛→+∞sup𝑢₁(𝑛)≤𝜂^𝑢₁𝑀₁𝛾^𝑙₁∶=𝑈₁,lim_𝑛→+∞sup𝑢₂(𝑛)≤𝜂^𝑢₂𝑀₂𝛾^𝑙₂∶=𝑈₂. (34) This completes the proof of Proposition 5.

Theorem 6. Assume that (H) holds, then system (3) is permanent.

Proof. By applying Proposition 5, we can easily see that to end the proof of Theorem 6, it is enough to show that under the conditions of Theorem 6, lim_𝑛→+∞inf𝑥₂(𝑛)≥𝑚₁,lim_𝑛→+∞inf𝑥₂(𝑛)≥𝑚₂,lim_𝑛→+∞inf𝑢₁(𝑛)≥𝑣₁,lim_𝑛→+∞inf𝑢₂(𝑛)≥𝑣₂. (35) In view of Proposition 5, for all 𝜀>0, there exists a 𝑁₃>0,𝑁₃∈𝑁, for all 𝑛>𝑁₃: 𝑥₁(𝑛)≤𝑀₁+𝜀,𝑥₂(𝑛)≤𝑀₂𝑢+𝜀,₁(𝑛)≤𝑈₁+𝜀,𝑢₂(𝑛)≤𝑈₂+𝜀. (36) It follows from the first equation of systems (3) and (36) that 𝑥₁(𝑛+1)≥𝑥₁(𝑛)exp𝑟₁(𝑛𝐾)¹(𝑛)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)−𝑥₁𝑛−𝜎₁(𝑛)≥𝑥₁(𝑛)𝑟exp^𝑙1𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀 (37) for all 𝑛>𝑁₃+𝜎.
Let 𝑥₁(𝑛)=exp{𝑦₁(𝑛)}; then (37) is equivalent to 𝑦₁(𝑛+1)−𝑦₁(𝑛)≥𝑟^𝑙₁𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀. (38) Summing both sides of (38) from 𝑛−𝛿₁(𝑛) to 𝑛−1 leads to ^𝑛−1_{𝑗=𝑛−𝛿1(𝑛)}𝑦₁𝑗+1−𝑦₁𝑗≥^𝑛−1_{𝑗=𝑛−𝛿1(𝑛)}𝑟^𝑙₁𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹≥𝑟+𝜀_𝑙1𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀𝜎. (39) Then 𝑦₁𝑛−𝛿₁(𝑛)≤𝑦₁(𝑛)−𝑟^𝑙₁𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀𝜎. (40) Thus 𝑥₁𝑛−𝛿₁(𝑛)≤𝑥₁(𝑛)−𝑟exp^𝑙1𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀𝜎. (41) Substituting (36) and (41) into the first equation of (3), we have 𝑥₁(𝑛+1)≥𝑥₁(𝑛)exp𝑟₁(𝑛𝐾)¹(𝑛)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)−𝑥₁𝑛−𝜎₁(𝑛)≥𝑥₁(𝑛)𝑟exp^𝑙1𝐾^𝑙₁1+𝑀₂+𝜀−𝑥₁(𝑛)−𝑟×exp^𝑙1𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀𝜎 (42) for all 𝑛>𝑁₃+𝜎.
By applying Lemmas 2 and 4, it immediately follows that lim_𝑛→+∞inf𝑥₁(𝑛)≥𝑚^𝜀₁, (43) where 𝑚^𝜀₁=𝑟^𝑙₁𝐾^𝑙₁𝑟^𝑢₁1+𝑀₁𝑟+𝜀×exp^𝑙1𝐾^𝑙₁1+𝑀₁+𝜀−𝑟^𝑢₁−𝑟×exp^𝑙1𝐾^𝑙₁1+𝑀₂−𝑀+𝜀¹+𝜀𝜎𝑀₁. (44) Setting 𝜀→0 in (43), then lim_𝑛→+∞inf𝑥₁(𝑛)≥𝑚₁, (45) where 𝑚₁=𝑟^𝑙₁𝐾^𝑙₁𝑟^𝑢₁1+𝑀₁𝑟×exp^𝑙1𝐾^𝑙₁1+𝑀₁−𝑟^𝑢₁−𝑟×exp^𝑙1𝐾^𝑙₁1+𝑀₂−𝑀₁𝜎𝑀₁. (46) By the second equation of systems (3) and (36), we can obtain 𝑥₂(𝑛+1)≥𝑥₂(𝑛)×exp𝑟₂(𝑛𝐾)²(𝑛)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)−𝑥₂𝑛−𝜎₂(𝑛)≥𝑥₂(𝑛)𝑟exp^𝑙2𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀 (47) for all 𝑛>𝑁₃+𝜎.
Let 𝑥₂(𝑛)=exp{𝑦₂(𝑛)}; then (47) is equivalent to 𝑦₂(𝑛+1)−𝑦₂(𝑛)≥𝑟^𝑙₂𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀. (48) Summing both sides of (48) from 𝑛−𝛿₂(𝑛) to 𝑛−1 leads to ^𝑛−1_{𝑗=𝑛−𝛿2(𝑛)}𝑦₂(𝑛+1)−𝑦₂(𝑛≥_{𝑛−1𝑗=𝑛−𝛿2(𝑛)})𝑟^𝑙₂𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²≥𝑟+𝜀_𝑙2𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀𝜎. (49) Then 𝑦₂𝑛−𝛿₂(𝑛)≤𝑦₂(𝑛)−𝑟^𝑙₂𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀𝜎. (50) Thus 𝑥₂𝑛−𝛿₂(𝑛)≤𝑥₂(𝑛)−𝑟exp^𝑙2𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀𝜎. (51) Substituting (36) and (51) into the second equation of (3), we have 𝑥₂(𝑛+1)≥𝑥₂(𝑛)exp𝑟₂(𝑛𝐾)²(𝑛)1+∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)−𝑥₂𝑛−𝜎₂(𝑛)≥𝑥₂(𝑛)𝑟exp^𝑙2𝐾^𝑙₂1+𝑀₁+𝜀−𝑥₂(𝑛)−𝑟×exp^𝑙2𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀𝜎 (52) for all 𝑛>𝑁₃+𝜎.
By applying Lemmas 2 and 4, it immediately follows that lim_𝑛→+∞inf𝑥₂(𝑛)≥𝑚^𝜀₂, (53) where 𝑚^𝜀₂=𝑟^𝑙₂𝐾^𝑙₂𝑟^𝑢₂1+𝑀₂𝑟+𝜀×exp^𝑙2𝐾^𝑙₂1+𝑀₂+𝜀−𝑟^𝑢₂−𝑟×exp^𝑙2𝐾^𝑙₂1+𝑀₁−𝑀+𝜀²+𝜀𝜎𝑀₂. (54) Setting 𝜀→0 in (53), then lim_𝑛→+∞inf𝑥₂(𝑛)≥𝑚₂, (55) where 𝑚₂=𝑟^𝑙₂𝐾^𝑙₂𝑟^𝑢₂1+𝑀₂𝑟×exp^𝑙2𝐾^𝑙₂1+𝑀₂−𝑟^𝑢₂−𝑟×exp^𝑙2𝐾^𝑙₂1+𝑀₁−𝑀₂𝜎𝑀₂. (56) 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 𝑥₁(𝑛)≥𝑚₁−𝜀,𝑥₂(𝑛)≥𝑚₂−𝜀 (57) for any 𝑛≥𝑁₄.
From the third and fourth equations of systems (3) and (57), we can derive that Δ𝑢₁(𝑛)≥−𝛾₁(𝑛)𝑢₁(𝑛)+𝜂₁(𝑛)𝑚₁−𝜀,Δ𝑢₂(𝑛)≥−𝛾₂(𝑛)𝑢₂(𝑛)+𝜂₂(𝑛)𝑚₂−𝜀. (58) Hence 𝑢₁(𝑛+1)≥1−𝛾^𝑢₁𝑢₁(𝑛)+𝜂^𝑙₁𝑚₁,𝑢−𝜀₂(𝑛+1)≥1−𝛾^𝑢₂𝑢₂(𝑛)+𝜂^𝑙₂𝑚₂−𝜀. (59) By applying Lemmas 2 and 3, it immediately follows that lim_𝑛→+∞inf𝑢₁(𝑛)≥𝜂^𝑙₁𝑚₁−𝜀𝛾^𝑢₁,lim_𝑛→+∞inf𝑢₂(𝑛)≥𝜂^𝑙₂𝑚₂−𝜀𝛾^𝑢₂. (60)(61) Setting 𝜀→0 in the previous inequality leads to lim_𝑛→+∞inf𝑢₁(𝑛)≥𝜂^𝑙₁𝑚₁𝛾^𝑢₁,lim_𝑛→+∞inf𝑢₂(𝑛)≥𝜂^𝑙₂𝑚₂𝛾^𝑢₂. (62) 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.

Acknowledgments

This work is supported by thr National Natural Science Foundation of China (no. 11261010 and no. 11101126), the Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), the Governor Foundation of Guizhou Province ([2012]53), and the Doctoral Foundation of Guizhou University of Finance and Economics (2010).