Permanence of a Discrete Nonlinear Prey-Competition System with Delays

Abstract

A discrete nonlinear prey-competition system with m-preys and (n-m)-predators and delays is considered. Two sets of sufficient conditions on the permanence of the system are obtained. One set is delay independent, while the other set is delay dependent.

1. Introduction

In this paper, we investigate the following discrete nonlinear prey-competition system with delays:

𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥^𝛼𝑖𝑗_𝑗(𝑘)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑘)𝑥^𝛽𝑖𝑗_𝑗(𝑘−𝜏_𝑖𝑗𝑥(𝑘))],𝑖=1,2,…,𝑚,_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp[−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥^𝛼𝑖𝑗_𝑗(𝑘)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)𝑥^𝛽𝑖𝑗_𝑗(𝑘−𝜏_𝑖𝑗−(𝑘))_{𝑛𝑗=𝑚+1}𝑎_𝑖𝑗(𝑘)𝑥^𝛼𝑖𝑗_𝑗(𝑘)−^𝑛_𝑗=𝑚+1𝑏_𝑖𝑗(𝑘)𝑥^𝛽𝑖𝑗_𝑗(𝑘−𝜏_𝑖𝑗(𝑘))],𝑖=𝑚+1,2,…,𝑛, (1.1) where 𝑥_𝑖(𝑘)(𝑖=1,2,…,𝑚) is the density of prey species 𝑖 at 𝑘th generation, 𝑥_𝑖(𝑘)(𝑖=𝑚+1,…,𝑛) is the density of predator species 𝑖 at 𝑘th generation. In this system, the competition among predator species and among prey species is simultaneously considered. For more background and biological adjustments of system (1.1), we can see [1–5] and the references cited therein.

Throughout this paper, we always assume that for all 𝑖,𝑗=1,2,…,𝑛,

(𝐻₁)𝑟_𝑖(𝑘),𝑎_𝑖𝑗(𝑘),𝑏_𝑖𝑗(𝑘) are all bounded nonnegative sequences and 𝑎^𝑙_𝑖𝑖≥0,𝑏^𝑙_𝑖𝑖≥0,𝑎^𝑙_𝑖𝑖+𝑏^𝑙_𝑖𝑖>0. Here, for any bounded sequence 𝑓^𝑢=sup_𝑘∈𝑁𝑓(𝑘),𝑓^𝑙=inf_𝑘∈𝑁𝑓(𝑘);

(𝐻₂)𝜏_𝑖𝑗(𝑘) are bounded nonnegative integer sequences, and 𝛼_𝑖𝑗,𝛽_𝑖𝑗 are all positive constants.

By a solution of system (1.1), we mean a sequence {𝑥₁(𝑘),…,𝑥_𝑛(𝑘)} which defined for 𝑁={0,1,…} and which satisfies system (1.1) for𝑁={0,1,…}. Motivated by application of system (1.1) in population dynamics, we assume that solutions of system (1.1) satisfy the following initial conditions:

𝑥_i(𝜃)=𝜙_𝑖(𝜃),𝜃∈𝑁[−𝜏,0]=[−𝜏,−𝜏+1,…,0],𝜙_𝑖(0)>0, (1.2) where 𝜏=max{𝜏_𝑖𝑗(𝑘),𝑖,𝑗=1,2,…,𝑛}. The exponential forms of system (1.1) assure that the solution of system (1.1) with initial conditions (1.2) remains positive.

Recently, Chen et al. in [1] proposed the following nonlinear prey-competition system with delays:

𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)[𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗(𝑡−𝜏_𝑖𝑗𝑥̇(𝑡))],𝑖=1,2,…,𝑚,_𝑖(𝑡)=𝑥_𝑖(𝑡)[−𝑟_𝑖(𝑡)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗(𝑡−𝜏_𝑖𝑗−(𝑡))_{𝑛𝑗=𝑚+1}𝑎_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=𝑚+1𝑏_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))],𝑖=𝑚+1,2,…,𝑛. (1.3)

By using Gaines and Mawhins continuation theorem of coincidence degree theory and by constructing an appropriate Lyapunov functional, they obtained a set of sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions of the system (1.3). In addition, sufficient conditions are obtained for the permanence of the system (1.3) in [2].

On the other hand, though most population dynamics are based on continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations [3–15]. Therefore, it is reasonable to study discrete time prey-competition models governed by difference equations.

As we know, a more important theme that interested mathematicians as well as biologists is whether all species in a multispecies community would survive in the long run, that is, whether the ecosystems are permanent. In fact, no such work has been done for system (1.1).

The main purpose of this paper is, by developing the analytical technique of [4, 8, 16], to obtain two sets of sufficient conditions which guarantee the permanence of system (1.1).

2. Main Results

Firstly, we introduce a definition and some lemmas which will be useful in the proof of the main results of this section.

Definition 2.1. System (1.1) is said to be permanent, if there are positive constants 𝑚 and 𝑀, such that each positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) of system (1.1) satisfies 𝑚≤liminf_𝑘→+∞𝑥_𝑖(𝑘)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀,𝑖=1,2,…,𝑛. (2.1)

Lemma 2.2 (see [8]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘)>0 and 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑟(𝑘)(1−𝑎𝑥(𝑘))} (2.2) for 𝑘∈[𝑘₁,+∞), where𝑎is a positive constant. Then limsup_𝑘→+∞1𝑥(𝑘)≤𝑎𝑟^𝑢exp(𝑟^𝑢−1). (2.3)

Lemma 2.3 (see [8]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘+1)≥𝑥(𝑘)exp{𝑟(𝑘)(1−𝑎𝑥(𝑘))},𝑘≥𝐾₀, (2.4)limsup_𝑘→+∞𝑥(𝑘)≤𝑥^∗ and 𝑥(𝐾₀)>0, where 𝑎 is a constant such that 𝑎𝑥^∗>1and𝐾₀∈𝑁. Then liminf_𝑘→+∞1𝑥(𝑘)≥𝑎exp(𝑟^𝑢(1−𝑎𝑥^∗)). (2.5)

For system (1.1), we will consider two cases, 𝑎^𝑙_𝑖𝑖>0,𝑏^𝑙_𝑖𝑖≥0 and 𝑎^𝑙_𝑖𝑖≥0,𝑏^𝑙_𝑖𝑖>0 respectively, and then we obtain Lemmas 2.4–2.6.

Lemma 2.4. Assume that𝑎^𝑙_𝑖𝑖>0. Then for every positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) of system (1.1) with initial condition (1.2), one has limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2,…,𝑛, (2.6) where 𝑀_𝑖1=(𝛼_𝑖𝑖𝑎^𝑙_𝑖𝑖)^{1/𝛼𝑖𝑖}exp[𝑟^𝑢_𝑖−1𝛼_𝑖𝑖𝑀],𝑖=1,2,…,𝑚,_𝑖1=(𝛼_𝑖𝑖𝑎^𝑙_𝑖𝑖)^{1/𝛼𝑖𝑖}exp[−𝑟^𝑙_𝑖+^𝑛_𝑗=1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗+^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗−1𝛼_𝑖𝑖],𝑖=𝑚+1,…,𝑛. (2.7)

Proof. Let 𝑥(𝑘)=(𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) be any positive solution of system (1.1) with initial condition (1.2), for 𝑖=1,2,…,𝑚, it follows from system (1.1) that 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘)], (2.8) thus 𝑥^𝛼𝑖𝑖_𝑖(𝑘+1)≤𝑥^𝛼𝑖𝑖_𝑖(𝑘)exp[𝛼_𝑖𝑖(𝑟_𝑖(𝑘)−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘))]. (2.9) Let𝑢_𝑖(𝑘)=𝑥^𝛼𝑖𝑖_𝑖(𝑘), we can have 𝑢_𝑖(𝑘+1)≤𝑢_𝑖(𝑘)exp[𝛼_𝑖𝑖(𝑟_𝑖(𝑘)−𝑎_𝑖𝑖(𝑘)𝑢_𝑖(𝑘))]≤𝑢_𝑖(𝑘)exp[𝛼_𝑖𝑖𝑟_𝑖𝑎(𝑘)(1−^𝑙𝑖𝑖𝑟^𝑢_𝑖𝑢_𝑖(𝑘))]. (2.10) By applying Lemma 2.2 to (2.10), we obtain limsup_𝑘→+∞𝑢_𝑖1(𝑘)≤(𝛼_𝑖𝑖𝑎^𝑙_𝑖𝑖)exp[𝛼_𝑖𝑖𝑟^𝑢_𝑖−1]=̇𝐿_𝑖; (2.11) so, we immediately get limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2,…,𝑚. (2.12) For any𝜀>0 small enough, it follows from (2.12) that there exists enough large𝐾₁ such that for all𝑖=1,2,…,𝑚 and 𝑘≥𝐾₁𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀. (2.13) For 𝑖=𝑚+1,…,𝑛 and 𝑘≥𝐾₁+𝜏, (2.13) combining with the 𝑖-th equation of system (1.1) leads to 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘)], (2.14) thus 𝑥^𝛼𝑖𝑖_𝑖(𝑘+1)≤𝑥^𝛼𝑖𝑖_𝑖(𝑘)exp[𝛼_𝑖𝑖(−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘))]. (2.15) Similarly, let 𝑢_𝑖(𝑘)=𝑥^𝛼𝑖𝑖_𝑖(𝑘), we get 𝑢_𝑖(𝑘+1)≤𝑢_𝑖(𝑘)exp[𝛼_𝑖𝑖(−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑢_𝑖(𝑘))]≤𝑢_𝑖(𝑘)exp[𝛼_𝑖𝑖(−𝑟_i(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗)𝑎×(1−^𝑙𝑖𝑖−𝑟^𝑙_𝑖+∑^𝑚_𝑗=1𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+∑^𝑚_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗𝑢_𝑖(𝑘))]. (2.16) By using (2.16), for 𝑖=𝑚+1,…,𝑛, according to Lemma 2.2, it follows that limsup_𝑘→+∞𝑢_𝑖1(𝑘)≤(𝛼_𝑖𝑖𝑎^𝑙_𝑖𝑖)exp[𝛼_𝑖𝑖(−𝑟^𝑙_𝑖+^𝑛_𝑗=1𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗)−1]; (2.17) setting𝜀→0 in above inequality, we have limsup_𝑘→+∞𝑢_𝑖1(𝑘)≤(𝛼_𝑖𝑖𝑎^𝑙_𝑖𝑖)exp[𝛼_i𝑖(−𝑟^𝑙_𝑖+^𝑛_𝑗=1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗+^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗)−1]=̇𝐿_𝑖, (2.18) then limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=𝑚+1,…,𝑛. (2.19)
This completes the proof.

For convenience, we introduce the following notation.

For 𝑖=1,2,…,𝑚

𝐴_𝑖=𝑎^𝑢_𝑖𝑖𝑟^𝑙_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−∑^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗,𝑅^𝑢_𝑖=𝑟^𝑢_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎^𝑙_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−^𝑛_𝑗=1𝑏^𝑙_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗. (2.20) For 𝑖=𝑚+1,…,𝑛

𝐴_𝑖=𝑎^𝑢_𝑖𝑖−𝑟^𝑢_𝑖+∑^𝑚_𝑗=1𝑎^𝑙_𝑖𝑗𝑚^𝛼𝑖𝑗_𝑗+∑^𝑛_𝑗=1𝑏^𝑙_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗−∑^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−∑^𝑛_𝑗=𝑚+1𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗,𝑅^𝑢_𝑖=−𝑟^𝑙_𝑖+^𝑚_𝑗=1𝑎^𝑢_𝑖𝑗𝑚^𝛼𝑖𝑗_𝑗+^𝑚_𝑗=1𝑏^𝑢_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗−^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑎^𝑙_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−^𝑛_𝑗=𝑚+1𝑏^𝑙_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗. (2.21)

Lemma 2.5. Assume that𝑎^𝑙_𝑖𝑖>0 and min_{1≤𝑖≤𝑛}𝐿_𝑖𝐴_𝑖>1, (2.22) hold. Then for any positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) of system (1.1) with initial condition (1.2), one has liminf_𝑘→+∞𝑥_𝑖(𝑘)≥𝑚_𝑖, (2.23) where 𝑚_𝑖1=(𝐴_𝑖)^{1/𝛼𝑖𝑖}exp[𝑅^𝑢_𝑖(1−𝐴_𝑖𝐿_𝑖)],𝑖=1,2,…,𝑛. (2.24)

Proof. Let 𝑥(𝑘)=(𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) be any positive solution of system (1.1) with initial condition (1.2). From Lemma 2.4, we know that there exists 𝐾₂>𝐾₁+𝜏, such that for 𝑖=1,2,…,𝑛 and 𝑘≥𝐾₂𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀. (2.25) For 𝑖=1,…,𝑚 and 𝑘≥𝐾₂+𝜏, (2.25) combining with the 𝑖-th equation of system (1.1) lead to 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘)], (2.26) thus 𝑥^𝛼𝑖𝑖_𝑖(𝑘+1)≥𝑥^𝛼𝑖𝑖_𝑖(𝑘)exp[𝛼_𝑖𝑖(𝑟_𝑖(𝑘)−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘))]; (2.27) let 𝑢_𝑖(𝑘)=𝑥^𝛼𝑖𝑖_𝑖(𝑘), we can have 𝑢_𝑖(𝑘+1)≥𝑢_𝑖(𝑘)exp[𝛼_𝑖𝑖(𝑟_𝑖(𝑘)−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑢_𝑖(𝑘))]≥𝑢_𝑖(𝑘)exp[𝛼_𝑖𝑖𝑅_𝑖𝜀(𝑘)(1−𝐴_𝑖𝜀𝑢_𝑖(𝑘))], (2.28) where 𝑅_𝑖𝜀(𝑘)=𝑟_𝑖(𝑘)−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗,𝐴_𝑖𝜀=𝑎^𝑢_𝑖𝑖𝑟^𝑙_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−∑^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗. (2.29) According to Lemma 2.3, we obtain liminf_𝑘→+∞𝑢_𝑖1(𝑘)≥𝐴_𝑖𝜀exp[𝛼_𝑖𝑖𝑅^𝑢_𝑖𝜀(1−𝐴_𝑖𝜀𝐿_𝑖)], (2.30) where 𝑅^𝑢_𝑖𝜀=𝑟^𝑢_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎^𝑙_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=1𝑏^𝑙_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗. (2.31) Setting𝜀→0 in (2.30) leads to liminf_𝑘→+∞𝑢_𝑖1(𝑘)≥𝐴_𝑖exp[𝛼_𝑖𝑖𝑅^𝑢_𝑖(1−𝐴_𝑖𝐿_𝑖)], (2.32) therefore liminf_𝑘→+∞𝑥_𝑖(𝑘)≥𝑚_𝑖,𝑖=1,2,…,𝑚. (2.33) For any 𝜀>0 small enough, it follows from (2.33) that there exists enough large 𝐾₃>𝐾₂+𝜏 such that for all𝑖=1,…,𝑚 and𝑘≥𝐾₃𝑥_𝑖(𝑘)≥𝑚_𝑖−𝜀, (2.34) and so, for𝑖=𝑚+1,…,𝑛and 𝑘≥𝐾₃+𝜏, it follows from system (1.1) that 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp[−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑚_𝑗−𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑚_𝑗−𝜀)^𝛽𝑖𝑗−^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=𝑚+1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑎_𝑖𝑖(𝑘)𝑥^𝛼𝑖𝑖_𝑖(𝑘)]≥𝑥_𝑖(𝑘)exp[𝑅_𝑖𝜀(𝑘)(1−𝐴_𝑖𝜀𝑥^𝛼𝑖𝑖_𝑖(𝑘))], (2.35) where 𝑅_𝑖𝜀(𝑘)=−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑚_𝑗−𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑚_𝑗−𝜀)^𝛽𝑖𝑗−^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=𝑚+1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗,𝐴_𝑖𝜀=𝑎^𝑢_𝑖𝑖/{−𝑟^𝑢_𝑖+^𝑚_𝑗=1𝑎^𝑙_𝑖𝑗(𝑚_𝑗−𝜀)^𝛼𝑖𝑗+^𝑛_𝑗=1𝑏^𝑙_𝑖𝑗(𝑚_𝑗−𝜀)^𝛽𝑖𝑗−^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=𝑚+1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗}, (2.36) by using (2.35), similarly to the analysis of (2.33), for𝑖=𝑚+1,…,𝑛liminf_𝑘→+∞𝑢_𝑖1(𝑘)≥𝐴_𝑖exp[𝛼_𝑖𝑖𝑅^𝑢_𝑖(1−𝐴_𝑖𝐿_𝑖)], (2.37) and therefore, we easily get liminf_𝑘→+∞𝑥_𝑖(𝑘)≥𝑚_𝑖,𝑖=𝑚+1,…,𝑛. (2.38) This ends the proof of Lemma 2.5.

Denote for 𝑖=1,2,…,𝑚

𝐿_𝑖=1𝛽_𝑖𝑖𝑏^𝑙_𝑖𝑖exp[𝛽_𝑖𝑖𝑟^𝑢_𝑖Γ(𝜏+1)−1];_𝑖=𝑟^𝑙_𝑖−^𝑛_𝑗=1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗;𝐴_𝑖=𝑏^𝑢_𝑖𝑖exp[−𝛽_𝑖𝑖Γ_𝑖𝜏]𝑟^𝑙_𝑖−∑^𝑛_𝑗=1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗;𝑅^𝑢_𝑖=𝑟^𝑢_𝑖−^𝑛_𝑗=1𝑎^𝑙_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏^𝑙_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗. (2.39) For 𝑖=𝑚+1,…,𝑛

Υ_𝑖=−𝑟^𝑙_𝑖+^𝑚_𝑗=1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗+^𝑚_𝑗=1𝑏^𝑢_𝑖𝑗𝑀^𝛽𝑖𝑗_𝑗;𝐿_𝑖=1𝛽_𝑖𝑖𝑏^𝑙_𝑖𝑖exp[𝛽_𝑖𝑖Υ_𝑖Γ(𝜏+1)−1];_𝑖=−𝑟^𝑢_𝑖+^𝑚_𝑗=1𝑎^𝑙_𝑖𝑗𝑚^𝛼𝑖𝑗_𝑗+^𝑚_𝑗=1𝑏^𝑙_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗−^𝑛_𝑗=𝑚+1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−^𝑛_𝑗=𝑚+1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗;𝐴_𝑖=𝑏^𝑢_𝑖𝑖exp[−𝛽_𝑖𝑖Γ_𝑖𝜏]−𝑟^𝑢_𝑖+∑^𝑚_𝑗=1𝑎^𝑙_𝑖𝑗𝑚^𝛼𝑖𝑗_𝑗+∑^𝑛_𝑗=1𝑏^𝑙_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗−∑^𝑛_𝑗=𝑚+1𝑎^𝑢_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−∑^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑏^𝑢_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗;𝑅^𝑢_𝑖=−𝑟^𝑙_𝑖+^𝑚_𝑗=1𝑎^𝑢_𝑖𝑗𝑚^𝛼𝑖𝑗_𝑗+^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗−^𝑛_𝑗=𝑚+1𝑎^𝑙_𝑖𝑗𝑀^𝛼𝑖𝑗_𝑗−^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑏^𝑙_𝑖𝑗𝑚^𝛽𝑖𝑗_𝑗. (2.40)

Lemma 2.6. Assume that 𝑏^𝑙_𝑖𝑖>0 and (𝐻₃)min_{1≤𝑖≤𝑛}𝐿_𝑖𝐴_𝑖>1, (2.41) hold. Then for any positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) of system (1.1) with initial condition (1.2), one has 𝑚_𝑖≤liminf_𝑘→+∞𝑥_𝑖(𝑘)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2,…,𝑛. (2.42) where 𝑀_𝑖=𝐿^{1/𝛽𝑖𝑖}_𝑖,𝑚_𝑖1=(𝐴_𝑖)^{1/𝛽𝑖𝑖}exp[𝑅^𝑢_𝑖(1−𝐴_𝑖𝐿_𝑖)]. (2.43)

Proof. Let 𝑥(𝑘)=(𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) be any positive solution of system (1.1) with initial condition (1.2), for 𝑖=1,2,…,𝑚, it follows from system (1.1) that 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)]≤𝑥_𝑖(𝑘)exp[𝑟^𝑢_𝑖𝑥],_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−𝑏_𝑖𝑖(𝑘)𝑥^𝛽𝑖𝑖_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))], (2.44)(2.45) It follows from (2.44)that ^𝑘−1_{𝑗=𝑘−𝜏𝑖𝑖(𝑘)}𝑥_𝑖(𝑗+1)𝑥_𝑖≤(𝑗)^𝑘−1_{𝑗=𝑘−𝜏𝑖𝑖(𝑘)}exp[𝑟^𝑢_𝑖]≤exp[𝑟^𝑢_𝑖𝜏], (2.46) and hence 𝑥_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))≥𝑥_𝑖(𝑘)exp[−𝑟^𝑢_𝑖𝜏], (2.47) which, together with (2.45), produces 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−𝑏_𝑖𝑖(𝑘)exp[−𝛽_𝑖𝑖𝑟^𝑢_𝑖𝜏]𝑥^𝛽𝑖𝑖_𝑖(𝑘)]≤𝑥_𝑖(𝑘)exp[𝑟_𝑖𝑏(𝑘)(1−^𝑙𝑖𝑖exp[−𝛽_𝑖𝑖𝑟^𝑢_𝑖𝜏]𝑟^𝑢_𝑖𝑥^𝛽𝑖𝑖_𝑖(𝑘))], (2.48) similar to the analysis of (2.11) and (2.12), for 𝑖=1,2,…,𝑚limsup_𝑘→+∞𝑢_𝑖(𝑘)≤𝐿_𝑖, (2.49) and thus, we immediately get limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2,…,𝑚. (2.50) For any 𝜀>0 small enough, it follows from (2.50) that there exists enough large 𝐾₁ such that for all 𝑖=1,2,…,𝑚 and 𝑘≥𝐾₁𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀. (2.51) For 𝑖=𝑚+1,…,𝑛 and 𝑘≥𝐾₁+𝜏, (2.51) combining with the 𝑖-th equation of system (1.1) lead to 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[−𝑟^𝑙_𝑖+^𝑚_𝑗=1𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗]=𝑥_𝑖(𝑘)exp[Υ_𝑖𝜀𝑥],_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑏_𝑖𝑖(𝑘)𝑥^𝛽𝑖𝑖_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))], (2.52)(2.53) from (2.53), similar to the argument of (2.44) and (2.47), for 𝑘≥𝐾₁+𝜏, we have 𝑥_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))≥𝑥_𝑖(𝑘)exp[−Υ_𝑖𝜀𝜏], (2.54) substituting (2.54) into (2.53), we get 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp[−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑏_𝑖𝑖(𝑘)exp[−𝛽_𝑖𝑖Υ_𝑖𝜀𝜏]𝑥^𝛽𝑖𝑖_𝑖(𝑘)], (2.55) similar to the analysis of (2.18) and (2.19), for𝑖=𝑚+1,…,𝑛limsup_𝑘→+∞𝑢_𝑖(𝑘)≤𝐿_𝑖, (2.56) then, limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=𝑚+1,…,𝑛. (2.57) For any 𝜀>0 small enough, it follows from (2.51) and (2.57) that there exists enough large𝐾₂>𝐾₁+𝜏 such that for all 𝑖=1,2,…,𝑛 and 𝑘≥𝐾₂𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀. (2.58) Hence, for𝑖=1,2,…,𝑚, and 𝑘≥𝐾₂+𝜏, it follows from system (1.1) that 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp[𝑟^𝑙_𝑖−^𝑛_𝑗=1𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗]=𝑥_𝑖(𝑘)exp[Γ_𝑖𝜀𝑥],_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑏_𝑖𝑖(𝑘)𝑥^𝛽𝑖𝑖_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))], (2.59)(2.60) from(2.59), similar to the argument of (2.44) and (2.47), for 𝑘≥𝐾₂+𝜏, we have 𝑥_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))≤𝑥_𝑖(𝑘)exp[−Γ_𝑖𝜀𝜏], (2.61) and this combined with (2.60) gives 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑏_𝑖𝑖(𝑘)exp[−𝛽_𝑖𝑖Γ_𝑖𝜀𝜏]𝑥^𝛽𝑖𝑖_𝑖(𝑘)]. (2.62) Similar to the argument of (2.32) and (2.33), for 𝑘≥𝐾₂+𝜏, we obtain liminf_𝑘→+∞𝑢_𝑖1(𝑘)≥𝐴_𝑖exp[𝛽_𝑖𝑖𝑅^𝑢_𝑖(1−𝐴_𝑖𝐿_𝑖)], (2.63) then liminf_𝑘→+∞𝑥_𝑖(𝑘)≥𝑚_𝑖,𝑖=1,2,…,𝑚. (2.64) For any𝜀>0 small enough, it follows from (2.63) that there exists enough large 𝐾₃>𝐾₂+𝜏 such that for all𝑖=1,…,𝑚 and 𝑘≥𝐾₃𝑥_𝑖(𝑘)≥𝑚_𝑖−𝜀, (2.65) and so, for𝑖=𝑚+1,…,𝑛 and 𝑘≥𝐾₃+𝜏, it follows from system (1.1) that 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp[−𝑟^𝑢_𝑖+^𝑚_𝑗=1𝑎^𝑙_𝑖𝑗(𝑚_𝑗−𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏^𝑙_𝑖𝑗(𝑚_𝑗−𝜀)^𝛽𝑖𝑗−^𝑛_𝑗=𝑚+1𝑎^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_𝑗=𝑚+1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)^𝛽𝑖𝑗]=𝑥_𝑖(𝑘)exp[Γ_𝑖𝜀𝑥],_i(𝑘+1)≥𝑥_𝑖(𝑘)exp[−𝑟_𝑖(𝑘)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑘)(𝑚_𝑗−𝜀)^𝛼𝑖𝑗+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑘)(𝑚_𝑗−𝜀)^𝛽𝑖𝑗−^𝑛_𝑗=𝑚+1𝑎_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛼𝑖𝑗−^𝑛_{𝑗=𝑚+1,𝑗≠𝑖}𝑏_𝑖𝑗(𝑘)(𝑀_𝑗+𝜀)^𝛽𝑖𝑗−𝑏_𝑖𝑖(𝑘)𝑥^𝛽𝑖𝑖_𝑖(𝑘−𝜏_𝑖𝑖(𝑘))]. (2.66)
Similar to the argument of (2.61) and (2.62), for 𝑘≥𝐾₃+𝜏, we have
liminf_𝑘→+∞𝑢_𝑖1(𝑘)≥𝐴_𝑖exp[𝛽_𝑖𝑖𝑅^𝑢_𝑖(1−𝐴_𝑖𝐿_𝑖)], (2.67) then liminf_𝑘→+∞𝑥_𝑖(𝑘)≥𝑚_𝑖,𝑖=𝑚+1,…,𝑛. (2.68) This ends the proof of Lemma 2.6.

Denote (𝐻₃)

𝑎^𝑙_𝑖𝑖>0,min_{1≤𝑖≤𝑛}𝐿_𝑖𝐴_𝑖>1, (2.69) or

𝑏^𝑙_𝑖𝑖>0,min_{1≤𝑖≤𝑛}𝐿_𝑖𝐴_𝑖>1, (2.70)

Our main result in this paper is the following theorem about the permanence of system (1.1).

Theorem 2.7. Assume that (𝐻₁),(𝐻₂), and (𝐻₃) hold, then system (1.1) is permanent.

Proof. Let 𝑥(𝑘)=(𝑥₁(𝑘),…,𝑥_𝑛(𝑘)) be any positive solution of system (1.1) with initial condition (1.2). Suppose 𝑀=max_{𝑖=1,…,𝑛}{𝑀_𝑖,𝑀_𝑖},𝑚=min_{𝑖=1,…,𝑛}{𝑚_𝑖,𝑚_𝑖}. By Lemmas 2.4–2.6, if system (1.1) satisfies (𝐻₁),(𝐻₂), and (𝐻₃), then we have 𝑚≤liminf_𝑘→+∞𝑥_𝑖(𝑘)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀,𝑖=1,2,…,𝑛. (2.71) The proof is completed.

In this paper, we study a discrete nonlinear predator-prey system with 𝑚-preys and (𝑛-𝑚)-predators and delays, which can be seen as the modification of the traditional Lotka-Volterra prey-competition model. From our main results, Theorem 2.7 gives two sets of sufficient conditions on the permanence of the system (1.1). One set is delay independent, while the other set is delay dependent.

Acknowledgments

The author is grateful to the Associate Editor, Professor Xue-Zhong He, and two referees for a number of helpful suggestions that have greatly improved her original submission. This research is supported by the Nation Natural Science Foundation of China (no. 10171010), Key Project on Science and Technology of Education Ministry of China (no. 01061), and Innovation Group Program of Liaoning Educational Committee No. 2007T050.