Permanence of a Discrete Model of Mutualism with Infinite Deviating Arguments

Abstract

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=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠))/(1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠))−𝑥₁(𝑛−𝜎₁(𝑛))]},𝑥₂(𝑛+1)=𝑥₂(𝑛)exp{𝑟₂(𝑛)[(𝐾₂(𝑛)+𝛼₂(𝑛)∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠))/(1+∑^∞_𝑠=0𝐽₁(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)𝑁₁𝐾(𝑡)[¹(𝑡)+𝛼₁(𝑡)∫^∞₀𝐽₂(𝑠)𝑁₂(𝑡−𝑠)𝑑𝑠1+∫^∞₀𝐽₂(𝑠)𝑁₂(𝑡−𝑠)𝑑𝑠−𝑁₁(𝑡−𝜎₁(𝑡))],𝑑𝑁₂(𝑡)𝑑𝑡=𝑟₂(𝑡)𝑁₂𝐾(𝑡)[²(𝑡)+𝛼₂(𝑡)∫^∞₀𝐽₁(𝑠)𝑁₁(𝑡−𝑠)𝑑𝑠1+∫^∞₀𝐽₁(𝑠)𝑁₁(𝑡−𝑠)𝑑𝑠−𝑁₂(𝑡−𝜎₂(𝑡))], (1.1) where 𝑟_𝑖,𝐾_𝑖,𝛼_𝑖, and 𝜎_𝑖,𝑖=1,2 are continuous functions bounded above and below by positive constants: 𝑎_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 (1.1). 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{𝑟₁𝐾(𝑛)[¹(𝑛)+𝛼₁(𝑛)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)−𝑥₁(𝑛−𝜎₁𝑥(𝑛))]},₂(𝑛+1)=𝑥₂(𝑛)exp{𝑟₂𝐾(𝑛)[²(𝑛)+𝛼₂(𝑛)∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₁(𝑠)𝑥₁(𝑛−𝑠)−𝑥₂(𝑛−𝜎₂(𝑛))]}, (1.2) where 𝑥_𝑖(𝑛),𝑖=1,2 is the density of mutualism species 𝑖 at the 𝑛th generation. For {𝑟_𝑖(𝑛)},{𝐾_𝑖(𝑛)},{𝛼_𝑖(𝑛)},{𝐽_𝑖(𝑛)}, and {𝜎_𝑖(𝑛)},𝑖=1,2 are bounded nonnegative sequences such that

0<𝑟^𝑙_𝑖≤𝑟^𝑢_𝑖,0<𝛼^𝑙_𝑖≤𝛼^𝑢_𝑖,0<𝐾^𝑙_𝑖≤𝐾^𝑢_𝑖,0<𝜎^𝑙_𝑖≤𝜎^𝑢_𝑖,^∞_𝑛=0𝐽_𝑖(𝑛)=1. (1.3) Here, for any bounded sequence {𝑎(𝑛)}, 𝑎^𝑢=sup_𝑛∈𝑁𝑎(𝑛),𝑎^𝑙=inf_𝑛∈𝑁𝑎(𝑛).

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

𝑥_𝑖(𝜃)=𝜑_𝑖(𝜃)≥0,𝜃∈𝑁[−𝜏,0]={−𝜏,−𝜏+1,…,0},𝜑_𝑖(0)>0. (1.4)

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

𝑥_𝑖(𝑛)>0,for𝑛∈𝑍,𝑖=1,2. (1.5)

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∈𝑁⁺_𝑘0={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)}, (2.1) where {𝑎(𝑘)} and {𝑏(𝑘)} are strictly positive sequences of real numbers defined for 𝑘∈𝑁={0,1,2,…} and 0<𝑎^𝑙≤𝑎^𝑢,0<𝑏^𝑙≤𝑏^𝑢. 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≤lim_𝑘→+∞inf𝑁(𝑘)≤lim_𝑘→+∞sup𝑁(𝑘)≤𝑀, (2.2) where 1𝑀=𝑏^𝑙exp{𝑎^𝑢𝑎−1},𝑚=^𝑙𝑏^𝑢exp{𝑎^𝑙−𝑏^𝑢𝑀}. (2.3)

Lemma 2.3 (see [14]). Let 𝑥(𝑛) and 𝑏(𝑛) be nonnegative sequences defined on 𝑁, and 𝑐≥0 is a constant. If 𝑥(𝑛)≤𝑐+^𝑛−1_𝑠=0𝑏(𝑠)𝑥(𝑠),for𝑛∈𝑁, (2.4) then 𝑥(𝑛)≤𝑐^𝑛−1_𝑠=0[1+𝑏(𝑠)],for𝑛∈𝑁. (2.5)

Lemma 2.4 (see [2]). Let 𝑥∶𝑍→𝑅 be a nonnegative bounded sequences, and let 𝐻∶𝑁→𝑅 be a nonnegative sequence such that ∑^∞_𝑛=0𝐽_𝑖(𝑛)=1. Then lim_𝑛→+∞inf𝑥(𝑛)≤lim_𝑛→+∞inf^𝑛_𝑠=−∞𝐻(𝑛−𝑠)𝑥(𝑠)≤lim_𝑛→+∞sup^𝑛_𝑠=−∞𝐻(𝑛−𝑠)𝑥(𝑠)≤lim_𝑛→+∞sup𝑥(𝑛). (2.6)

Proposition. Let (𝑥₁(𝑛),𝑥₂(𝑛)) be any positive solution of system (1.2), then lim_𝑛→+∞sup𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2, (2.7) where 𝑀_𝑖=exp{2𝑟^𝑢_𝑖[𝐾^𝑢_𝑖+𝛼^𝑢_𝑖]},𝑖=1,2. (2.8)

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{𝑟₁𝐾(𝑛)[¹(𝑛)+𝛼₁(𝑛)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)]}=𝑥₁(𝑛)exp{𝑟₁𝐾(𝑛)[¹(𝑛)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂+𝛼(𝑛−𝑠)¹(𝑛)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)1+∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)]}≤𝑥₁(𝑛)exp{𝑟₁𝐾(𝑛)[¹(𝑛)1+𝛼₁(𝑛)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)∑^∞_𝑠=0𝐽₂(𝑠)𝑥₂(𝑛−𝑠)]}=𝑥₁(𝑛)exp{𝑟₁(𝑛)[𝐾₁(𝑛)+𝛼₁(𝑛)]}≤𝑥₁(𝑛)exp{𝑟^𝑢₁[𝐾^𝑢₁+𝛼^𝑢₁]}. (2.9) Let 𝑥₁(𝑛)=exp{𝑢₁(𝑛)}, then 𝑢₁(𝑛+1)≤𝑢₁(𝑛)+𝑟^𝑢₁[𝐾^𝑢₁+𝛼^𝑢₁]=𝑟^𝑢₁[𝐾^𝑢₁+𝛼^𝑢₁]+^𝑛_𝑠=0𝑏(𝑠)𝑥(𝑠), (2.10) where 𝑏(𝑠)=0,0≤𝑠≤𝑛−1,1,𝑠=𝑛. (2.11) When 𝑢₁(𝑛) is nonnegative sequence, by applying Lemma 2.3, it immediately follows that 𝑢₁(𝑛+1)≤2𝑟^𝑢₁[𝐾^𝑢₁+𝛼^𝑢₁]. (2.12) When 𝑢₁(𝑛) is negative sequence, (2.12) also holds. From (2.12), we have lim_𝑛→+∞sup𝑥₁(𝑛)≤exp{2𝑟^𝑢₁[𝐾^𝑢₁+𝛼^𝑢₁]}∶=𝑀₁. (2.13) By using the second equation of system (1.2), similar to the above analysis, we can obtain lim_𝑛→+∞sup𝑥₂(𝑛)≤exp{2𝑟^𝑢₂[𝐾^𝑢₂+𝛼^𝑢₂]}∶=𝑀₂. (2.14) 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 𝑚_𝑖,𝑀_𝑖,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 𝑚_𝑖≤lim_𝑛→+∞inf𝑥_𝑖(𝑛)≤lim_𝑛→+∞sup𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2. (2.15)

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 2.6lim_𝑛→+∞inf𝑥_𝑖(𝑛)≥𝑚_𝑖. (2.16) From Proposition 2.5, For all 𝜀>0, there exists a 𝑁₁>0,𝑁₁∈𝑁,For all 𝑛>𝑁₁, 𝑥_𝑖(𝑛)≤𝑀_𝑖+𝜀. (2.17) According to Lemma 2.4, from (2.13) and (2.14) we have lim_𝑛→+∞sup^∞_𝑠=0𝐽_𝑖(𝑠)𝑥_𝑖(𝑛−𝑠)=lim_𝑛→+∞sup^𝑛_𝑘=−∞𝐽_𝑖(𝑛−𝑘)𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2. (2.18) For above 𝜀>0, according to (2.18), there exists a positive integer 𝑁₂, such that, for all 𝑛>𝑁₂, ^∞_𝑠=0𝐽_𝑖(𝑠)𝑥_𝑖(𝑛−𝑠)≤𝑀_𝑖+𝜀,𝑖=1,2. (2.19) 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{𝑟₁𝐾(𝑛)[^𝑙11+(𝑀₂+𝜀)−(𝑀₁+𝜀)]}≥𝑥₁𝑟(𝑛)exp{^𝑙1𝐾^𝑙₁1+(𝑀₂+𝜀)−𝑟^𝑢₁(𝑀₁+𝜀)}. (2.20) It follows that, for 𝑛≥𝜎₁(𝑛), ^𝑛−1_{𝑖=𝑛−𝜎1(𝑛)}𝑥₁(𝑖+1)≥^𝑛−1_{𝑖=𝑛−𝜎1(𝑛)}𝑥₁𝑟(𝑖)exp{^𝑙1𝐾^𝑙₁1+(𝑀₂+𝜀)−𝑟^𝑢₁(𝑀₁+𝜀)}. (2.21) Hence 𝑥₁(𝑛)≥𝑥₁(𝑛−𝜎₁𝑟(𝑛))exp{^𝑙1𝐾^𝑙₁1+(𝑀₂𝜎+𝜀)^𝑙1−𝑟^𝑢₁(𝑀₁+𝜀)𝜎^𝑢₁}. (2.22) In other words, 𝑥₁(𝑛−𝜎₁(𝑛))≤𝑥₁𝑟(𝑛)exp{−^𝑙1𝐾^𝑙₁1+(𝑀₂𝜎+𝜀)^𝑙1+𝑟^𝑢₁(𝑀₁+𝜀)𝜎^𝑢₁}. (2.23) From the first equation of system (1.2) and (2.23), for all 𝑛>𝑚𝑎𝑥{𝑁₁,𝑁₂}+𝜎, it follows that 𝑥₁(𝑛+1)≥𝑥₁𝑟(𝑛)exp{−^𝑙1𝐾^𝑙₁1+(𝑀₂+𝜀)−𝑟^𝑢₁𝑟exp{−^𝑙1𝐾^𝑙₁1+(𝑀₂𝜎+𝜀)^𝑙1+𝑟^𝑢₁(𝑀₁+𝜀)𝜎^𝑢₁}𝑥₁(𝑛)}. (2.24) By applying Lemmas 2.1 and 2.2 to (2.24), it immediately follows thatlim_𝑛→+∞inf𝑥₁𝑟(𝑛)≥^𝑙1𝐾^𝑙₁𝑟^𝑢₁(1+(𝑀₂𝑟+𝜀))exp{^𝑙1𝐾^𝑙₁1+(𝑀₂𝜎+𝜀)^𝑙1−𝑟^𝑢₁(𝑀₁+𝜀)𝜎^𝑢₁}𝑟×exp{^𝑙1𝐾^𝑙₁1+(𝑀₂+𝜀)−𝑟^𝑢₁𝑟exp{−^𝑙1𝐾^𝑙₁1+(𝑀₂𝜎+𝜀)^𝑙1+𝑟^𝑢₁(𝑀₁+𝜀)𝜎^𝑢₁}𝑀₁}. (2.25) Setting 𝜀→0, it follows that lim_𝑛→+∞inf𝑥₁𝑟(𝑛)≥^𝑙1𝐾^𝑙₁𝑟^𝑢₁(1+𝑀₂)𝑟exp{^𝑙1𝐾^𝑙₁1+𝑀₂𝜎^𝑙₁−𝑟^𝑢₁𝑀₁𝜎^𝑢₁}𝑟×exp{^𝑙1𝐾^𝑙₁1+𝑀₂−𝑟^𝑢₁𝑟exp{−^𝑙1𝐾^𝑙₁1+𝑀₂𝜎^𝑙₁+𝑟^𝑢₁𝑀₁𝜎^𝑢₁}𝑀₁}. (2.26) Similar to the above analysis, from the second equation of system (1.2), we have that lim_𝑛→+∞inf𝑥₂𝑟(𝑛)≥^𝑙2𝐾^𝑙₂𝑟^𝑢₂(1+𝑀₁)𝑟exp{^𝑙2𝐾^𝑙₂1+𝑀₁𝜎^𝑙₂−𝑟^𝑢₂𝑀₂𝜎^𝑢₂}𝑟×exp{^𝑙2𝐾^𝑙₂1+𝑀₁−𝑟^𝑢₂𝑟exp{−^𝑙2𝐾^𝑙₂1+𝑀₁𝜎^𝑙₂+𝑟^𝑢₂𝑀₂𝜎^𝑢₂}𝑀₂}. (2.27) This completes the proof of Theorem 2.6.