Extinction in Nonautonomous Discrete Lotka-Volterra Competitive System with Pure Delays and Feedback Controls

Abstract

The paper discusses a nonautonomous discrete time Lotka-Volterra competitive system with pure delays and feedback controls. New sufficient conditions for which a part of the 𝑛-species is driven to extinction are established by using the method of multiple discrete Lyapunov functionals.

1. Introduction

The coexistence and global stability of population models are the interesting subjects in mathematical biology. Many authors have argued that the discrete time models are governed by differential equations which are more appropriate than the continuous ones to describe the dynamics of population when the population has nonoverlapping generations, a lot has been done on discrete Lotka-Volterra systems.

May in [1] firstly considered the following autonomous discrete two-species Lotka-Volterra competitive system:

𝑥(𝑛+1)=𝑥(𝑛)exp{𝑟₁−𝑎₁₁𝑥(𝑛)−𝑎₁₂𝑦(𝑛)},𝑦(𝑛+1)=𝑦(𝑛)exp{𝑟₂−𝑎₂₁𝑥(𝑛)−𝑎₂₂𝑦(𝑛)} (1.1) and studied the stable points, stable cycles, and the chaos behaivor. Further, Lu and Wang [2] studied the permanence and global attractivity of this system.

Chen and Zhou in [3] considered the following periodic discrete two-species Lotka-Volterra competitive system:

𝑥(𝑛+1)=𝑥(𝑛)exp{𝑟₁(1−𝑥(𝑛)𝐾₁(𝑛)−𝜇₂(𝑛)𝑦(𝑛))},𝑦(𝑛+1)=𝑦(𝑛)exp{𝑟₂(1−𝜇₁𝑥(𝑛)−𝑦(𝑛)𝐾₂(𝑛))} (1.2) and studied the permanence and existence of a periodic solution, and further, sufficient conditions are established on the global stability of the periodic solution.

Zhang and Zhou in [4] investigated the following nonautonomous discrete two-species Lotka-Volterra competitive system:

𝑥(𝑛+1)=𝑥(𝑛)exp{𝑟₁(𝑛)−𝑎₁₁(𝑛)𝑥(𝑛−𝑘₁)−𝑎₁₂(𝑛)𝑦(𝑛−𝑘₂)},𝑦(𝑛+1)=𝑦(𝑛)exp{𝑟₂(𝑛)−𝑎₂₁(𝑛)𝑥(𝑛−𝑙₁)−𝑎₂₂(𝑛)𝑦(𝑛−𝑙₂)}. (1.3) Some sufficient conditions were obtained for the permanence of the system.

Wang et al. in [5] studied the following general nonautonomous discrete 𝑛-species Lotka-Volterra systems:

𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−^𝑛𝑚_𝑗=1𝑙=0𝑎^𝑙_𝑖𝑗(𝑘)𝑥_𝑗(𝑘−𝑙)−𝑎₁₂(𝑛)𝑦(𝑛−𝑘₂)},𝑖=1,2,…,𝑛. (1.4) By applying the linear method and constructing the appropriate Lyapunov functionals, the author established the sufficient conditions which guarantee that any positive solution of this system is stable and attracts others, and obtained some applications of main results.

Muroya in [6, 7] considered the following general nonautonomous discrete 𝑛-species Lotka-Volterra systems:

𝑁_𝑖(𝑝+1)=𝑁_𝑖(𝑝)exp{𝑐_𝑖(𝑝)−𝑎_𝑖(𝑝)𝑁_𝑖(𝑝)−^𝑛𝑚_𝑗=1𝑙=0𝑎^𝑙_𝑖𝑗(𝑝)𝑁_𝑗(𝑝−𝑘_𝑙𝑁)},𝑝=0,1,2,…,_𝑖(𝑠)=𝑁_𝑖𝑠≥0∀𝑠≤0,𝑁_𝑖0>0,𝑖=1,2,…,𝑛, (1.5) and related pure delays models, that is, 𝑎_𝑖(𝑝)=0 for all 𝑝≥0. The author obtained the permanence and the global asymptotically stable by applying mean-value conditions and the method of constructing discrete Lyapunov functionals.

Liao et al. in [8] discussed the following general discrete nonautonomous 𝑛-species competitive system with feedback controls:

𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑏_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘)−^𝑛_{𝑗=1,𝑗≠𝑖}𝑐_𝑖𝑗(𝑘)𝑥_𝑖(𝑘)𝑥_𝑗(𝑘)−𝑑_𝑖(𝑘)𝑢_𝑖(𝑘)},𝑘=0,1,2,…,Δ𝑢_𝑖(𝑘)=𝑟_𝑖(𝑘)−𝑒_𝑖(𝑘)𝑢_𝑖(𝑘)+𝑓_𝑖(𝑘)𝑥_𝑖(𝑘),𝑖=1,2,…,𝑛. (1.6) Some sufficient conditions are established on the permanence and the global stability of the system.

Recently, we see that in [9, 10] the authors studied the following nonautonomous continuous Lotka-Volterra competitive system with pure delays and feedback controls:

𝑥̇(𝑡)=𝑥_𝑖(𝑡)[𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))−^𝑛_𝑗=1⁰_{−𝜎𝑖𝑗}𝑏_𝑖𝑗(𝑡,𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑑_𝑖(𝑡)𝑢_𝑖(𝑡−𝜏_𝑖̇(𝑡))],𝑢(𝑡)=−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑓_𝑖(𝑡)𝑥_𝑖(𝑡)+𝑔_𝑖(𝑡)𝑥_𝑖(𝑡−𝛿_𝑖(𝑡)),𝑖=1,2,…,𝑛. (1.7) The sufficient conditions for which a part of the 𝑛-species is driven to extinction and the surplus part of the 𝑛-species remains permanence are established.

However, we see that for general discrete 𝑛-species population systems the results for which a part of the 𝑛-species is driven to extinction and the surplus part of the 𝑛-species remains the permanence, up to now, are still not obtained. Therefore, motivated by the above works, in this paper we study the following discrete nonautonomous Lotka-Volterra competitive system with pure delays and feedback controls

𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑖𝑗)−^𝑛𝜎_𝑗=1^𝑖𝑗_𝑙=0𝑏_𝑖𝑗𝑙(𝑘)𝑥_𝑗(𝑘−𝑙)−𝑑_𝑖(𝑘)𝑢_𝑖(𝑘−𝜏_𝑖𝑢)},_𝑖(𝑘+1)=(1−𝑒_𝑖(𝑘))𝑢_𝑖(𝑘)+𝑔_𝑖(𝑘)𝑥_𝑖(𝑘−𝛿_𝑖),𝑖=1,2,…,𝑛. (1.8) The main purpose is to establish a criterion for which guarantee the part species 𝑥_𝑟+1,𝑥_𝑟+2,…,𝑥_𝑛 in system (1.8) is driven to extinction. The method used in this paper is to constructing the multiple discrete Lyapunov functions. On the permanence of the surplus species 𝑥₁,𝑥₂,…,𝑥_𝑟, owing to the length of this paper, we will give the discussion in another paper.

This paper is organized as follows. In next section, as preliminaries, some assumptions and useful lemmas are introduced. In Section 3, the main results of this paper on the extinction of a part of the 𝑛-species of system (1.8) are established. In Section 4, an example is presented to illustrate the feasibility of our results.

2. Preliminaries

Let 𝑍 denote the set of all nonnegative integers. For any bounded sequence 𝑦(𝑘), we denote 𝑦=sup_𝑘∈𝑍{𝑦(𝑘)}, 𝑦=inf_𝑘∈𝑍{𝑦(𝑘)}. Throughout this paper, we introduce the following assumptions.

(𝐻₁)𝑟_𝑖(𝑘) is a bounded sequence defined on 𝑍; 𝑎_𝑖𝑗(𝑘), 𝑑_𝑖(𝑘), 𝑔_𝑖(𝑘) and 𝑏_𝑖𝑗𝑙(𝑘)(𝑖,𝑗=1,2,…,𝑛,𝑙=0,1,…,𝜎_𝑖𝑗) are nonnegative bounded sequences defined on 𝑍; 𝜏_𝑖𝑗, 𝜏_𝑖, 𝛿_𝑖, and 𝜎_𝑖𝑗(𝑖,𝑗=1,2,…,𝑛) are nonnegative integers.(𝐻₂)Sequences 𝑒_𝑖(𝑘)(𝑖=1,2,…,𝑛) satisfy 0<𝑒_𝑖(𝑘)≤1 for all 𝑘∈𝑍.(𝐻₃)There exist positive integers 𝜔 and 𝜆 such that for each 𝑖=1,2,…,𝑛liminf_𝑛→∞^{𝑛+𝜔−1}_𝑠=𝑛𝑟_𝑖(𝑠)>0,liminf_𝑛→∞^{𝑛+𝜆−1}_𝑠=𝑛[𝑎_𝑖𝑖(𝑠)+^𝜎𝑖𝑖_𝑙=0𝑏_𝑖𝑖𝑙(𝑠)]>0. (2.1)(𝐻₄)There exists positive integer 𝛽 such that for each 𝑖=1,2,…,𝑛limsup_𝑛→∞^{𝑛+𝛽−1}_𝑛(1−𝑒_𝑖(𝑠))<1. (2.2)

Let 𝑅₊=[0,∞) and 𝑅^𝑛₊={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∶𝑥_𝑖∈𝑅₊,𝑖=1,2,…,𝑛}. We denote by int𝑅^𝑛₊ the interior of 𝑅^𝑛₊. For any nonnegative constants 𝑎 and 𝑏 with 𝑎<𝑏, we denote by [𝑎,b]_𝑍 the set of all integers in the interval [𝑎,𝑏]. For some integer 𝑚≥0, we denote by 𝐷₊[−𝑚,0]_𝑍 the space of all nonnegative discrete time function 𝜙∶[−𝑚,0]_𝑍→𝑅^𝑛₊ with norm ‖𝜙‖=sup{|𝜙(𝑠)|∶𝑠∈[−𝑚,0]_𝑍}.

Let 𝜏=max{𝜏_𝑖𝑗,𝜏_𝑖,𝛿_𝑖,𝜎_𝑖𝑗∶𝑖,𝑗=1,2,…,𝑛}. Motivated by the biological background of system (1.8), in this paper we only consider all solutions of system (1.8) that satisfy the following initial conditions:

𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠)≥0,𝑢_𝑖(𝑠)=𝜓_𝑖(𝑠)≥0,𝑠∈[−𝑚,0]_𝑍,𝑖=1,2,…,𝑛, (2.3) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝐷₊[−𝑚,0]_𝑍 and 𝜓=(𝜓₁,𝜓₂,…,𝜓_𝑛)∈𝐷₊[−𝑚,0]_𝑍. For any 𝜙,𝜓∈𝐷₊[−𝑚,0]_𝑍, let 𝑧=(𝜙,𝜓), by the fundamental theory of difference equations, system (1.8) has a unique solution (𝑥(𝑠,𝑧),(𝑠,𝑧)) satisfying the initial condition (2.3), where 𝑥(𝑠,𝑧)=(𝑥₁(𝑠,𝑧),𝑥₂(𝑠,𝑧),…,𝑥_𝑛(𝑠,𝑧)) and 𝑢(𝑠,𝑧)=(𝑢₁(𝑠,𝑧),𝑢₂(𝑠,𝑧),…,𝑢_𝑛(𝑠,𝑧)). It is obvious that solution (𝑥(𝑠,𝑧),(𝑠,𝑧)) is positive, that is, 𝑥_𝑖(𝑠,𝑧)>0 and 𝑢_𝑖(𝑠,𝑧)>0(𝑖=1,2,…,𝑛) for all 𝑠∈𝑍.

We first consider the following nonautonomous difference inequality system:

𝑦(𝑛+1)≤𝑦(𝑛)exp{𝛼(𝑛)−𝛽(𝑛)𝑦(𝑛)},𝑛∈𝑍, (2.4) where 𝛼(𝑛) and 𝛽(𝑛) are bounded sequences defined on 𝑍 and 𝛽(𝑛)≥0 for all 𝑛∈𝑍. We have the following result.

Lemma 2.1 (see [11]). Assume that there exists an integer 𝜆>0 such that liminf_𝑛→∞^𝑛+𝜆_𝑘=𝑛𝛽(𝑘)>0, (2.5) then there exists a constant 𝑀>0 such that for any nonnegative solution 𝑦(𝑛) of system (2.4) with initial value 𝑦(𝑛₀)=𝑦₀≥0, where 𝑛₀∈𝑍 is some integer, limsup_𝑛→∞𝑦(𝑛)<𝑀. (2.6)

Next, we consider the following nonautonomous linear difference equation:

𝑣(𝑛+1)=𝛾(𝑛)𝑣(𝑛)+𝜔(𝑛), (2.7) where 𝛾(𝑛) and 𝜔(𝑛) are nonnegative bounded sequences defined on 𝑍. We have the following results.

Lemma 2.2 (see [11]). Assume that there exists an integer 𝜆>0 such that limsup_𝑛→∞^𝑛+𝜆_𝑘=𝑛𝛾(𝑘)<1, (2.8) then there exists a constant 𝑀>0 such that for any nonnegative solution 𝑣(𝑛) of system (2.7) with initial value 𝑣(𝑛₀)=𝑣₀≥0, where 𝑛₀∈Z is some integer, limsup_𝑛→∞𝑣(𝑛)<𝑀. (2.9)

Lemma 2.3 (see [11]). Assume that the conditions of Lemma 2.2 hold, then for any constants 𝜀>0 and 𝑀₁>0 there exist positive constants 𝛿=𝛿(𝜀) and 𝑛=𝑛(𝜀,𝑀₁) such that for any 𝑛₀∈𝑍 and 0≤𝑣₀≤𝑀₁, when 𝜔(𝑛)<𝛿 for all 𝑛≥𝑛₀, one has 𝑣(𝑛,𝑛₀,𝑣₀)<𝜀∀𝑛≥𝑛₀+𝑛, (2.10) where 𝑣(𝑛,𝑛₀,𝑣₀) is the solution of (2.7) with initial value 𝑣(𝑛₀)=𝑣₀.

Lemma 2.4. Assume that assumptions (𝐻₁)-(𝐻₄) hold, then there exists a constant 𝑀₀>0 such that limsup_𝑘→∞𝑥_𝑖(𝑘)<𝑀₀,limsup_𝑘→∞𝑢_𝑖(𝑘)<𝑀₀,𝑖=1,2,…,𝑛 (2.11) for any positive solution (𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(𝑘)) of system (1.8).

Proof. Let (𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(𝑘)) be any positive solution of system (1.8). For each 𝑖∈{1,2,…,𝑛}, we have 𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑖𝑗)−^𝑛𝜎_𝑗=1^𝑖𝑗_𝑙=0𝑏_𝑖𝑗𝑙(𝑘)𝑥_𝑗(𝑘−𝑙)−𝑑_𝑖(𝑘)𝑢_𝑖(𝑘−𝜏_𝑖)}≤𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)}, (2.12) then, for any integer 𝑘≥0 and 𝜃≤0 with 𝑘+𝜃≥0, summing inequality (2.12) from 𝑘+𝜃 to 𝑘−1, we obtain 𝑥_𝑖(𝑘+𝜃)≥𝑥_𝑖(𝑘)exp{−^𝑘−1_{𝑠=𝑘+𝜃}𝑟_𝑖(𝑠)}. (2.13) Therefore, for any integer 𝑘≥𝜏, from (2.13) and the first equation of system (1.8) we obtain 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−𝑎_𝑖𝑖(𝑘)𝑥_𝑖(𝑘−𝜏_𝑖𝑖)−^𝜎𝑖𝑖_𝑙=0𝑏_𝑖𝑖𝑙(𝑘)𝑥_𝑖(𝑘−𝑙)}≤𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−𝑎_𝑖𝑖(𝑘)𝑥_𝑖(𝑘)exp{−^𝑘−1_{𝑠=𝑘+𝜃}𝑟_𝑖−(𝑠)}_{𝜎𝑖𝑖𝑙=0}𝑏_𝑖𝑖𝑙(𝑘)𝑥_𝑖(𝑘)exp{−^𝑘−1_{𝑠=𝑘−𝑙}𝑟_𝑖(𝑠)}}. (2.14) Since for any 𝑘≥𝜏 and 𝑙∈[0,𝜏]_𝑍^𝑘−1_{𝑠=𝑘+𝜃}𝑟_𝑖(𝑠)≤𝜏𝑟^𝑢_𝑖, (2.15) where 𝑟^𝑢_𝑖=sup_𝑘∈𝑍𝑟_𝑖(𝑘), we have from (2.14), 𝑥_𝑖(𝑘+𝑖)≤𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−ℎ_𝑖(𝑎_𝑖𝑖(𝑘)+^𝜎𝑖𝑖_𝑙=0𝑏_𝑖𝑖𝑙(𝑘))𝑥_𝑖(𝑘)} (2.16) for any 𝑘≥𝜏, where ℎ_𝑖=exp{−𝜏𝑟^𝑢_𝑖}.
We consider the following auxiliary equation:
𝑦_𝑖(𝑘+1)≤𝑦_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−𝛽_𝑖(𝑘)𝑦_𝑖(𝑘)},𝑘∈𝑍, (2.17) where 𝛽_𝑖(𝑘)=ℎ_𝑖(𝑎_𝑖𝑖(𝑘)+∑^𝜎𝑖𝑖_𝑙=0𝑏_𝑖𝑖𝑙(𝑘)), then by assumption (𝐻₃) and applying Lemma 2.1 there exists a constant 𝑁_𝑖>0 such that limsup_𝑘→∞𝑥_𝑖(𝑘)<𝑁_𝑖 (2.18) for any positive solution 𝑦_𝑖(𝑘) of (2.17). Therefore, from the comparison theorem of difference equation, we finally obtain limsup_𝑘→∞𝑥_𝑖(𝑘)<𝑁_𝑖. (2.19) Further form inequality (2.19), there exists a positive constant 𝑘_𝑖≥𝜏 such that 𝑥_𝑖(𝑘)<𝑁_𝑖,𝑥_𝑖(𝑘−𝛿_𝑖)<𝑁_𝑖∀𝑘≥𝑘_𝑖. (2.20) Thus, from the second equation of system (1.8), we obtain 𝑢_𝑖(𝑘+1)≤(1−𝑒_𝑖(𝑘))𝑢_𝑖(𝑘)+(𝑓_𝑖(𝑘)+𝑔_𝑖(𝑘))𝑁_𝑖 (2.21) for all 𝑘≥𝑘_𝑖. We consider the following auxiliary equation: 𝑣_𝑖(𝑘+1)=(1−𝑒_𝑖(𝑘))𝑣_𝑖(𝑘)+(𝑓_𝑖(𝑘)+𝑔_𝑖(𝑘))𝑁_𝑖, (2.22) then by assumption (𝐻₄) and applying Lemma 2.2, there exists a constant 𝑁^∗_𝑖 such that limsup_𝑘→∞𝑣_𝑖(𝑘)<𝑁^∗_𝑖 (2.23) for any positive solution 𝑣_𝑖(𝑘) of (2.22). Let 𝑣^∗_𝑖(𝑘) be the solution of (2.22) with initial value 𝑣^∗_𝑖(𝑘_𝑖)=𝑢_𝑖(𝑘_𝑖), then from the comparison theorem of difference equation, we have 𝑢_𝑖(𝑘)≤𝑣^∗_𝑖(𝑘_𝑖) for all 𝑘≥𝑘_𝑖. Thus, we finally obtain limsup_𝑘→∞𝑢_𝑖(𝑘)<𝑁^∗_𝑖. (2.24) Let 𝑁=max_{1≤𝑖≤𝑛}{𝑁_𝑖,𝑁^∗_𝑖}, then from (2.19) and (2.24) we finally see that the conclusions of Lemma 2.4 hold.

3. Main Results

In this section, we discuss the extinction of the part of species 𝑥_𝑟+1,…,𝑥_𝑛 of system (1.8). Define functions as follows:

𝐴_𝑖𝑗(𝑘)=𝑎_𝑖𝑗(𝑘+𝜏_𝑖𝑗)+^𝜎𝑖𝑗_𝑙=0𝑏_𝑖𝑗𝑙𝐷(𝑘+𝑙),_𝑖(𝑘)=𝑑_𝑖(𝑘+𝜏_𝑖),𝑅_𝑖(𝑘)=𝑔_𝑖(𝑘+𝛿_𝑖), (3.1) where 𝑖,𝑗=1,2,…,𝑛.

Theorem 3.1. Assume that assumptions (𝐻₁)-(𝐻₄) hold and there exists an integer 1≤𝑟<𝑛 such that for any ℎ>𝑟 there exists an integer 𝑖_ℎ<ℎ such that limsup_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ(𝑠)<liminf_𝑘→∞𝐴_ℎ𝑗(𝑘)𝐴_𝑖ℎ𝑗(𝑘)∀𝑗≤ℎ,liminf_𝑘→∞𝐷_ℎ(𝑘)𝑒_ℎ(𝑘)>limsup_𝑘→∞(𝐴_𝑖ℎℎ(𝑘)𝑅_ℎ(𝑘)limsup_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ−𝐴(𝑠)^ℎℎ(𝑘)𝑅_ℎ(𝑘)),limsup_𝑘→∞𝐷_𝑖ℎ(𝑘)𝑒_𝑖ℎ(𝑘)<liminf_𝑘→∞(𝐴_ℎ𝑖ℎ(𝑘)𝑅_𝑖ℎ(𝑘)liminf_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ−𝐴(𝑠)^{𝑖ℎ𝑖ℎ}(𝑘)𝑅_𝑖ℎ(𝑘)), (3.2) then for each 𝑖=𝑟+1,…,𝑛 one has lim_𝑘→∞𝑥_𝑖(𝑘)=0,lim_𝑘→∞𝑢_𝑖(𝑘)=0,^∞_𝑘=0𝑥_𝑖(𝑘)<∞ (3.3) for any positive solution (𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(k)) of system (1.8).

Proof. From assumption (𝐻₂), there exist constant 𝜂₀>0 and integer 𝐾₀>0 such that ^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖(𝑠)≥𝜂₀∀𝑘≥𝐾₀. (3.4) We first prove that lim_𝑘→∞𝑥_𝑛(𝑘)=0. Let ℎ=𝑛 and 𝑖_ℎ=𝑝. From conditions (3.2) we can find positive constants 𝛼_𝑛1,𝛼_𝑛2,𝛼_𝑛3,𝛼_𝑛4,𝜀_𝑛 and integer 𝐾_𝑛≥𝐾₀ such that ∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑛(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑝<𝛼(𝑠)𝛼^𝑛1_𝑛2−𝜀_𝑛<𝛼_𝑛1𝛼_𝑛2<𝐴_𝑛𝑗(𝑘)𝐴_𝑝𝑗,𝐷(𝑘)_𝑛(𝑘)𝑒_𝑛>𝛼(𝑘)𝛼^𝑛4_𝑛2>𝛼_𝑛1𝐴_𝑝𝑛(𝑘)−𝛼_𝑛2𝐴_𝑛𝑛(𝑘)𝛼_𝑛2𝑅_𝑛,𝐷(𝑘)_𝑝(𝑘)𝑒_𝑝<𝛼(𝑘)𝛼^𝑛3_𝑛1<𝛼_𝑛2𝐴_𝑛𝑝(𝑘)−𝛼_𝑛1𝐴_𝑝𝑝(𝑘)𝛼_𝑛1𝑅_𝑝(𝑘) (3.5) for all 𝑘≥𝐾_𝑛 and 𝑗=1,2,…,𝑛. Consequently, ^{𝑘+𝜔−1}_𝑠=𝑘(−𝛼_𝑛1𝑟_𝑝(𝑠)+𝛼_𝑛2𝑟_𝑛(𝑠))<−𝜀_𝑛𝛼_𝑛2𝜂₀,𝛼_𝑛1𝐴_𝑝𝑗(𝑘)−𝛼_𝑛2𝐴_𝑛𝑗𝛼(𝑘)<0,_𝑛4𝑒_𝑛(𝑘)−𝛼_𝑛2𝐷_𝑛𝛼(𝑘)<0,_𝑛1𝐴_𝑝𝑝(𝑘)−𝛼_𝑛2𝐴_𝑛𝑝(𝑘)+𝛼_𝑛3𝑅_𝑝(𝑘)<0,−𝛼_𝑛3𝑒_𝑝(𝑘)+𝛼_𝑛1𝐷_𝑝𝛼(𝑘)<0_𝑛1𝐴_𝑝𝑛(𝑘)−𝛼_𝑛2𝐴_𝑛𝑛(𝑘)−𝛼_𝑛4𝑅_𝑛(𝑘)<0 (3.6)(3.7)(3.8)(3.9)(3.10)(3.11) for all 𝑘≥𝐾_𝑛 and 𝑗=1,2,…,𝑛.
Let (𝑥(𝑘),𝑢(𝑘))=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(𝑘)) be any positive solution of system (1.8). Constructing the following discrete Lyapunov functional
𝑉_𝑛(𝑘)=𝑥^−𝛼𝑛1_𝑝(𝑘)𝑥^𝛼𝑛2_𝑛(𝑘)×exp{𝛼_𝑛3𝑢_𝑝(𝑘)−𝛼_𝑛4𝑢_𝑛(𝑘)+𝛼^𝑛_𝑛1𝑗=1^𝑘−1_{𝑠=𝑘−𝜏𝑝𝑗}𝑎_𝑝𝑗(𝑠+𝜏_𝑝𝑗)𝑥_𝑗(𝑠)+𝛼^𝑛_𝑛1^𝜎_𝑗=1^𝑝𝑗_𝑙=0^𝑘−1_{𝑠=𝑘−𝑙}𝑏_𝑝𝑗𝑙(𝑠+𝑙)𝑥_𝑗(𝑠)+𝛼_𝑛1^𝑘−1_{𝑠=𝑘−𝜏𝑝}𝑑_𝑝(𝑠+𝜏_𝑝)𝑢_𝑝(𝑠)−𝛼^𝑛_𝑛2𝑗=1^𝑘−1_{𝑠=𝑘−𝜏𝑛𝑗}𝑎_𝑛𝑗(𝑠+𝜏_𝑛𝑗)𝑥_𝑗(𝑠)−𝛼^𝑛_𝑛2^𝜎_𝑗=1^𝑛𝑗_𝑙=0^𝑘−1_{𝑠=𝑘−𝑙}𝑏_𝑛𝑗𝑙(𝑠+𝑙)𝑥_𝑗(𝑠)−𝛼_𝑛2^𝑘−1_{𝑠=𝑘−𝜏𝑛}𝑑_𝑛(𝑠+𝜏_𝑛)𝑢_𝑛(𝑠)+𝛼_𝑛3^𝑘−1_{𝑠=𝑘−𝛿𝑝}𝑔_𝑝(𝑠+𝛿_𝑝)𝑥_𝑝(𝑠)−𝛼_𝑛4^𝑘−1_{𝑠=𝑘−𝛿𝑛}𝑔_𝑛(𝑠+𝛿_n)𝑥_𝑛(𝑠)}. (3.12) By calculating, we obtain 𝑉_𝑛(𝑘+1)𝑉_𝑛(𝑘)=exp{−𝛼_𝑛1[𝑟_𝑝(𝑘)−^𝑛_𝑗=1𝑎_𝑝𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑝𝑗)−^𝑛𝜎_𝑗=1^𝑝𝑗_𝑙=0𝑏_𝑝𝑗𝑙(𝑘)𝑥_𝑗(𝑘+𝑙)−𝑑_𝑝(𝑘)𝑢_𝑝(𝑘−𝜏_𝑝)]+𝛼_𝑛2[𝑟_𝑛(𝑘)−^𝑛_𝑗=1𝑎_𝑛𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑛𝑗)−^𝑛𝜎_𝑗=1^𝑛𝑗_𝑙=0𝑏_𝑛𝑗𝑙(𝑘)𝑥_𝑗(𝑘+𝑙)−𝑑_𝑛(𝑘)𝑢_𝑛(𝑘−𝜏_𝑛)]+𝛼_𝑛3[−𝑒_𝑝(𝑘)𝑢_𝑝(𝑘)+𝑔_𝑝(𝑘)𝑥_𝑝(𝑘−𝛿_𝑝)]−𝛼_𝑛4[−𝑒_𝑛(𝑘)𝑢_𝑛(𝑘)+𝑔_𝑛(𝑘)𝑥_𝑛(𝑘−𝛿_𝑛)]+𝛼^𝑛_𝑛1𝑗=1[𝑎_𝑝𝑗(𝑘+𝜏_𝑝𝑗)𝑥_𝑗(𝑘)−𝑎_𝑝𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑝𝑗)]+𝛼^𝑛_𝑛1^𝜎_𝑗=1^𝑝𝑗_𝑙=0[𝑏_𝑝𝑗𝑙(𝑘+𝑙)𝑥_𝑗(𝑘)−𝑏_𝑝𝑗𝑙(𝑘)𝑥_𝑗(𝑘−𝑙)]+𝛼_𝑛1[𝑑_𝑝(𝑘+𝜏_𝑝)𝑢_𝑝(𝑘)−𝑑_𝑝(𝑘)𝑢_𝑝(𝑘−𝜏_𝑝)]−𝛼^𝑛_𝑛2𝑗=1[𝑎_𝑛𝑗(𝑘+𝜏_𝑛𝑗)𝑥_𝑗(𝑘)−𝑎_𝑛𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑛𝑗)]−𝛼^𝑛_𝑛2^𝜎_𝑗=1^𝑛𝑗_𝑙=0[𝑏_𝑛𝑗𝑙(𝑘+𝑙)𝑥_𝑗(𝑘)−𝑏_𝑛𝑗𝑙(𝑘)𝑥_𝑗(𝑘−𝑙)]−𝛼_𝑛2[𝑑_𝑛(𝑘+𝜏_𝑛)𝑢_𝑛(𝑘)−𝑑_𝑛(𝑘)𝑢_𝑛(𝑘−𝜏_𝑛)]+𝛼_𝑛3[𝑔_𝑝(𝑘+𝛿_𝑝)𝑥_𝑝(𝑘)−𝑔_𝑝(𝑘)𝑥_𝑝(𝑘−𝛿_𝑝)]−𝛼_𝑛4[𝑔_𝑛(𝑘+𝛿_𝑛)𝑥_𝑛(𝑘)−𝑔_𝑛(𝑘)𝑥_𝑛(𝑘−𝛿_𝑛)]}=exp{−𝛼_𝑛1𝑟_𝑝(𝑘)+𝛼_𝑛2𝑟_𝑛(𝑘)−(𝛼_𝑛3𝑒_𝑝(𝑘)−𝛼_𝑛1𝐷_𝑝(𝑘))𝑢_𝑝(𝑘)−(𝛼_𝑛2𝐷_𝑛(𝑘)−𝛼_𝑛4𝑒_𝑛(𝑘))𝑢_𝑛(𝑘)+^𝑛−1_𝑗≠𝑝(𝛼_𝑛1𝐴_𝑝𝑗(𝑘)−𝛼_𝑛2𝐴_𝑛𝑗(𝑘))𝑥_𝑗(𝑘)+(𝛼_𝑛1𝐴_𝑝𝑝(𝑘)−𝛼_𝑛2𝐴_𝑛𝑝(𝑘)−𝛼_𝑛3𝑅_𝑝(𝑘))𝑥_𝑝(𝑘)+(𝛼_𝑛1𝐴_𝑝𝑛(𝑘)−𝛼_𝑛2𝐴_𝑛𝑛(𝑘)−𝛼_𝑛4𝑅_𝑛(𝑘))𝑥_𝑛(𝑘)}. (3.13) From inequalities (3.7)–(3.11), we can obtain 𝑉_𝑛(𝑘+1)≤𝑉_𝑛(𝑘)exp{−𝛼_𝑛1𝑟_𝑝(𝑘)+𝛼_𝑛2𝑟_𝑛(𝑘)}∀𝑘≥𝐾_𝑛. (3.14) For any 𝑘>𝐾_𝑛, we choose an integer 𝑚≥0, such that 𝑘∈[𝐾_𝑛+𝑚𝜔,𝐾_𝑛+(𝑚+1)𝜔), then from (3.6) and (3.14) we further have 𝑉_𝑛(𝑘)≤𝑉_𝑛(𝐾_𝑛)exp{^𝑘−1_{𝑠=𝐾𝑛}(−𝛼_𝑛1𝑟_𝑝(𝑠)+𝛼_𝑛2𝑟_𝑛(𝑠))}≤𝑉_𝑛(𝐾_𝑛)exp{(^{𝐾𝑛+𝑚𝜔−1}_{𝑠=𝐾𝑛}+^𝑘−1_{𝑠=𝐾𝑛+𝑚𝜔})(−𝛼_𝑛1𝑟_𝑝(𝑠)+𝛼_𝑛2𝑟_𝑛(𝑠))}≤𝑉_𝑛(𝐾_𝑛)exp{−𝜀_𝑛𝛼_𝑛2𝜂₀𝑚+𝑀_𝑛}≤𝑉_𝑛(𝐾_𝑛)exp{−𝜆_𝑛𝑘+𝑀^∗_𝑛}, (3.15) where 𝜆_𝑛=𝜀_𝑛𝛼_𝑛2𝜂₀𝜔,𝑀^∗_𝑛=𝑀_𝑛+𝜀_𝑛𝛼_𝑛2𝜂₀𝐾(1+^𝑛𝜔𝑀),_𝑛=sup_𝑘∈𝑍(−𝛼_𝑛1𝑟_𝑝(𝑘)+𝛼_𝑛2𝑟_𝑛(𝑘))𝜔. (3.16) On the other hand, from assumptions (𝐻₁), we have 𝑉_𝑛(𝑘)≥𝑥^−𝛼𝑛1_𝑝(𝑘)𝑥^𝛼𝑛2_𝑛(𝑘)×exp{−𝛼_𝑛4𝑢_𝑛(𝑘)−^𝑛_𝑗=1𝛼_𝑛2^𝑘−1_{𝑠=𝑘−𝜏𝑛𝑗}𝑎_𝑛𝑗(𝑠+𝜏_𝑛𝑗)𝑥_𝑗−(𝑠)_𝑛𝑗=1𝛼^𝜎_𝑛2^𝑛𝑗_𝑙=0^𝑘−1_{𝑠=𝑘−𝑙}𝑏_𝑛𝑗𝑙(𝑠+𝑙)𝑥_𝑗(𝑠)−𝛼_𝑛2^𝑘−1_{𝑠=𝑘−𝜏𝑛}𝑑_𝑛(𝑠+𝜏_𝑛)𝑢_𝑛(𝑠)−𝛼_𝑛4^𝑘−1_{𝑠=𝑘−𝛿𝑛}𝑔_𝑛(𝑠+𝛿_𝑛)𝑥_𝑛(𝑠)}≥𝑥^−𝛼𝑛1_𝑝(𝑘)𝑥^𝛼𝑛2_𝑛(𝑘)×exp{−𝛼_𝑛4𝑀₀−𝛼_𝑛2𝜏^𝑛_𝑗=1𝑎_𝑛𝑗𝑀₀−𝛼^𝑛_𝑛2^𝜎_𝑗=1^𝑛𝑗_𝑙=0𝑏_𝑛𝑗𝑙𝑙𝑀₀−𝛼_𝑛2𝜏𝑑_𝑛𝑀₀−𝛼_𝑛4𝜏𝑔_𝑛𝑀₀} (3.17) for all 𝑘≥𝜏, where constant 𝑀₀ is given in Lemma 2.4. Hence, there exist a positive constant 𝑄_𝑛>0 such that 𝑉_𝑛(𝑘)≥𝑥^−𝛼𝑛1_𝑝(𝑘)𝑥^𝛼𝑛2_𝑛(𝑘)𝑄_𝑛∀𝑘≥𝜏. (3.18) From (3.15) and (3.18), we obtain 𝑥_𝑛(𝑘)≤[𝑄_𝑛⁻¹𝑥^𝛼𝑛1_𝑝(𝑘)𝑉_𝑛(𝐾_𝑛)exp{−𝜆_𝑛𝑘+𝑀^∗_𝑛}]^1/𝛼𝑛2≤𝑄^∗_𝑛exp{−𝜆^∗_𝑛𝑘} (3.19) for all 𝑘≥𝐾_𝑛+𝜏, where 𝑄^∗_𝑛=sup_𝑘∈𝑍[𝑥^𝛼𝑛1_𝑝(𝑘)𝑉_𝑛(𝐾_𝑛)𝑄_𝑛⁻¹exp𝑀^∗_𝑛]^1/𝛼𝑛2,𝜆^∗_𝑛=𝜆_𝑛𝛼_𝑛2. (3.20) From (3.19), we finally obtain lim_𝑘→∞𝑥_𝑛(𝑘)=0,^∞_𝑘=0𝑥_𝑛(𝑘)<∞. (3.21)
Next, we consider the second equation of system (1.8), applying Lemma 2.3 we can easily obtain 𝑢_𝑛(𝑘)→0 as 𝑘→∞.
Now, we suppose that for any 𝑡>𝑟, we have obtained
lim_𝑘→∞𝑥_𝑖(𝑘)=0,lim_𝑘→∞𝑢_𝑖(𝑘)=0,^∞_𝑘=0𝑥_𝑛(𝑘)<∞ (3.22) for all 𝑖>𝑡. We further will prove lim_𝑘→∞𝑥_𝑡(𝑘)=0,^∞_𝑘=0𝑥_𝑡(𝑘)<∞. (3.23) From conditions (3.2), we can choose positive constants 𝛼_𝑡1, 𝛼_𝑡2, 𝛼_𝑡3, 𝛼_𝑡4, 𝜀_𝑡 and integer 𝐾_𝑡≥𝐾_𝑛 such that ∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑡(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑞<𝛼(𝑠)𝛼^𝑡1_𝑡2−𝜀_𝑡<𝛼_𝑡1𝛼_𝑡2<𝐴_𝑡𝑗(𝑘)𝐴_𝑞𝑗,𝐷(𝑘)_𝑡(𝑘)𝑒_𝑡>𝛼(𝑘)𝛼^𝑡4_𝑡2>𝛼_𝑡1𝐴_𝑞𝑛(𝑘)−𝛼_𝑡2𝐴_𝑡𝑡(𝑘)𝛼_𝑡2𝑅_𝑡,𝐷(𝑘)_𝑞(𝑘)𝑒_𝑞<𝛼(𝑘)𝛼^𝑡3_𝑡1>𝛼_𝑡2𝐴_𝑡𝑞(𝑘)−𝛼_𝑡1𝐴_𝑞𝑞(𝑘)𝛼_𝑡1𝑅_𝑞(𝑘) (3.24) for all 𝑘≥𝐾_𝑡 and 𝑗=1,2,…,𝑡, where 𝑖_ℎ=𝑞. Consequently, ^{𝑘+𝜔−1}_𝑠=𝑘(−𝛼_𝑡1𝑟_𝑞(𝑠)+𝛼_𝑡2𝑟_𝑡(𝑠))<−𝜀_𝑡𝛼_𝑡2𝜂₀,𝛼_𝑡1𝐴_𝑞𝑗(𝑘)−𝛼_𝑡2𝐴_𝑡𝑗𝛼(𝑘)<0,_𝑡4𝑒_𝑡(𝑘)−𝛼_𝑡2𝐷_𝑡𝛼(𝑘)<0,_𝑡1𝐴_𝑞𝑞(𝑘)−𝛼_𝑡2𝐴_𝑡𝑞(𝑘)+𝛼_𝑡3𝑅_𝑞(𝑘)<0,−𝛼_𝑡3𝑒_𝑞(𝑘)+𝛼_𝑡1𝐷_𝑞𝛼(𝑘)<0_𝑡1𝐴_𝑞𝑡(𝑘)−𝛼_𝑡2𝐴_𝑡𝑡(𝑘)−𝛼_𝑡4𝑅_𝑡(𝑘)<0 (3.25)(3.26)(3.27)(3.28)(3.29)(3.30) for all 𝑘≥𝐾_𝑡 and 𝑗=1,2,…,𝑡. Constructing the following discrete Lyapunov functional 𝑉_𝑡(𝑘)=𝑥^−𝛼𝑡1_𝑞(𝑘)𝑥^𝛼𝑡2_𝑡(𝑘)×exp{𝛼_𝑡3𝑢_𝑞(𝑘)−𝛼_𝑡4𝑢_𝑡(𝑘)+𝛼^𝑛_𝑡1𝑗=1^𝑘−1_{𝑠=𝑘−𝜏𝑞𝑗}𝑎_𝑞𝑗(𝑘+𝜏_𝑞𝑗)𝑥_𝑗(𝑠)+𝛼^𝑛_𝑡1^𝜎_𝑗=1^𝑞𝑗_𝑙=0^𝑘−1_{𝑠=𝑘−𝑙}𝑏_𝑞𝑗𝑙(𝑘+𝑙)𝑥_𝑗(𝑠)+𝛼_𝑡1^𝑘−1_{𝑠=𝑘−𝜏𝑞}𝑑_𝑞(𝑘+𝜏_𝑞)𝑢_𝑞(𝑠)−𝛼^𝑛_𝑡2𝑗=1^𝑘−1_{𝑠=𝑘−𝜏𝑡𝑗}𝑎_𝑡𝑗(𝑘+𝜏_𝑡𝑗)𝑥_𝑗(𝑠)−𝛼^𝑛_𝑡2^𝜎_𝑗=1^𝑡𝑗_𝑙=0^𝑘−1_{𝑠=𝑘−𝑙}𝑏_𝑡𝑗𝑙(𝑘+𝑙)𝑥_𝑗(𝑠)−𝛼_𝑡2^𝑘−1_{𝑠=𝑘−𝜏𝑡}𝑑_𝑡(𝑘+𝜏_𝑡)𝑢_𝑡(𝑠)+𝛼_𝑡3^𝑘−1_{𝑠=𝑘−𝛿𝑞}𝑔_𝑞(𝑘+𝛿_𝑞)𝑥_𝑞(𝑠)−𝛼_𝑡4^𝑘−1_{𝑠=𝑘−𝛿𝑡}𝑔_𝑡(𝑘+𝛿_𝑡)𝑥_𝑡(𝑠)}. (3.31) Calculating 𝑉_𝑡(𝑘+1)/𝑉_𝑡(𝑘), similarly to 𝑉_𝑛(𝑘), we can obtain 𝑉_𝑡(𝑘+1)𝑉_𝑡(𝑘)=exp{−𝛼_𝑡1𝑟_𝑞(𝑘)+𝛼_𝑡2𝑟_𝑡(𝑘)−(𝛼_𝑡3𝑒_𝑞(𝑘)−𝛼_𝑡1𝐷_𝑞(𝑘))𝑢_𝑞(𝑘)+(−𝛼_𝑡2𝐷_𝑡(𝑘)+𝛼_𝑡4𝑒_𝑡(𝑘))𝑢_𝑡+(𝑘)_{𝑛−1𝑗≠𝑞,𝑡}(𝛼_𝑡1𝐴_𝑞𝑗(𝑘)−𝛼_𝑡2𝐴_𝑡𝑗(𝑘))𝑥_𝑗(𝑘)+(𝛼_𝑡1𝐴_𝑞𝑞(𝑘)−𝛼_𝑡2𝐴_𝑡𝑝(𝑘)−𝛼_𝑡2𝑅_𝑞(𝑘))𝑥_𝑞(𝑘)+(𝛼_𝑡1𝐴_𝑞𝑡(𝑘)−𝛼_𝑡2𝐴_𝑡𝑡(𝑘)−𝛼_𝑡4𝑅_𝑡(𝑘))𝑥_𝑡(𝑘)} (3.32) for all 𝑘≥𝐾_𝑡. From inequalities (3.26)–(3.30), we further obtain 𝑉_𝑡(𝑘+1)≤𝑉_𝑡(𝑘)exp{−𝛼_𝑡1𝑟_𝑞(𝑘)+𝛼_𝑡2𝑟_𝑡(𝑘)+𝑃(𝑘)} (3.33) for all 𝑘≥𝐾_𝑡, where 𝑃(𝑘)=^𝑛_𝑗=𝑡+1(𝛼_𝑡1𝐴_𝑞𝑗(𝑘)+𝛼_𝑡2𝐴_𝑡𝑗(𝑘))𝑥_𝑗(𝑘). (3.34) From (3.19) we have 𝑃(𝑘)→0 as 𝑘→∞. Hence, lim_𝑘→∞^{𝑘+𝜔−1}_𝑠=𝑘𝑃(𝑠)=0. (3.35) Thus, from (3.22) we can obtain that there exists an integer 𝐾^∗_𝑡≥𝐾_𝑡 such that ^{𝑘+𝜔−1}_𝑠=𝑘(−𝛼_t1𝑟_𝑞(𝑠)+𝛼_𝑡2𝑟_𝑡1(𝑠)+𝑃(𝑠))<−2𝜀_𝑡𝛼_𝑛2𝜂₀ (3.36) for all 𝑘>𝐾^∗_𝑡. By calculating, from (3.33) we obtain 𝑉_𝑡(𝑘)≤𝑉_𝑡(𝐾^∗_𝑡)exp{^𝑘−1_{𝑠=𝐾∗𝑡}(−𝛼_𝑡1𝑟_𝑞(𝑠)+𝛼_𝑡2𝑟_𝑡(𝑠)+𝑃(𝑠))}. (3.37) From this, a similar argument as in the proof of (3.15)–(3.19), we further can obtain 𝑥_𝑡(𝑘)≤𝑄^∗_𝑡exp{−𝜆^∗_𝑡𝑘}∀𝑘≥𝐾^∗_𝑡, (3.38) where 𝑄^∗_𝑡 and 𝜆^∗_𝑡 are two positive constants. From (3.38), we finally obtain lim_𝑘→∞𝑥_𝑡(𝑘)=0,^∞_𝑘=0𝑥_𝑡(𝑘)<∞. (3.39)
Next, we consider the second equation of system (1.8), applying Lemma 2.3 we can easily obtain 𝑢_𝑡(𝑘)→0 as 𝑘→∞.
Finally, according to the induction principle, we have
lim_𝑘→∞𝑥_𝑖(𝑘)=0,lim_𝑘→∞𝑢_𝑖(𝑘)=0,^∞_𝑘=0𝑥_𝑖(𝑘)<∞∀𝑖>𝑟. (3.40) This completes the proof of Theorem 3.1.

As consequences of Theorem 3.1, we consider the following two special cases of system (1.8).

Case 1. Nondelayed nonautonomous discrete 𝑛-species Lotka-Volterra competitive systems with feedback controls 𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−∑^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘)−𝑑_𝑖(𝑘)𝑢_𝑖𝑢(𝑘)},_𝑖(𝑘+1)=(1−𝑒_𝑖(𝑘))𝑢_𝑖(𝑘)+𝑔_𝑖(𝑘)𝑥_𝑖(𝑘),𝑖=1,2,…,𝑛. (3.41) For system (3.41), assumptions (𝐻₁) and (𝐻₃) become into the following form(𝐻^′₁)𝑟_𝑖(𝑘) is a bounded sequence defined on 𝑍; 𝑎_𝑖𝑗(𝑘), 𝑑_𝑖(𝑘) and 𝑔_𝑖(𝑘)(𝑖,𝑗=1,2,…,𝑛) are nonnegative bounded sequences defined on 𝑍.(𝐻^′₃)There exist positive integers 𝜔 and 𝜆 such that for each 𝑖=1,2,…,𝑛liminf_𝑛→∞^{𝑛+𝜔−1}_𝑠=𝑛𝑟_𝑖(𝑠)>0,liminf_𝑛→∞^{𝑛+𝜆−1}_𝑠=𝑛𝑎_𝑖𝑖(𝑠)>0. (3.42)

Directly from Theorem 3.1, we have the following corollary.

Corollary 3.2. Assume that assumptions (𝐻^′₁), (𝐻₂), (𝐻^′₃) and (𝐻₄) hold and there exists a integer 1≤𝑟<𝑛 such that for any ℎ>𝑟 there exists an integer 𝑖_ℎ<ℎ such that limsup_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ(𝑠)<liminf_𝑘→∞𝑎_ℎ𝑗(𝑘)𝑎_𝑖ℎ𝑗(𝑘)∀𝑗≤ℎ,liminf_𝑘→∞𝑑_ℎ(𝑘)𝑒_ℎ(𝑘)>limsup_𝑘→∞(𝑎_𝑖ℎℎ(𝑘)𝑔_ℎ(𝑘)limsup_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ−𝑎(𝑠)^ℎℎ(𝑘)𝑔_ℎ(𝑘)),limsup_𝑘→∞𝑑_𝑖ℎ(𝑘)𝑒_𝑖ℎ(𝑘)<liminf_𝑘→∞(𝑎_ℎ𝑖ℎ(𝑘)𝑔_𝑖ℎ(𝑘)liminf_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ−𝑎(𝑠)^{𝑖ℎ𝑖ℎ}(𝑘)𝑔_𝑖ℎ(𝑘)), (3.43) then for each 𝑖=𝑟+1,…,𝑛 one has lim_𝑘→∞𝑥_𝑖(𝑘)=0,lim_𝑘→∞𝑢_𝑖(𝑘)=0,^∞_𝑘=0𝑥_𝑖(𝑘)<∞ (3.44) for any positive solution (𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(𝑘)) of system (3.41).

Case 2. Pure delayed nonautonomous discrete 𝑛-species Lotka-Volterra competitive systems without feedback controls 𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−∑^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑖𝑗)−∑^𝑛∑^𝜎_𝑗=1^𝑖𝑗_𝑙=0𝑏_𝑖𝑗𝑙(𝑘)𝑥_𝑗(𝑘−𝑙)},𝑖=1,2,…,𝑛. (3.45) For system (3.41), assumption (𝐻₁) becomes into the following form.(𝐻₁^′′)𝑟𝑖(𝑘) is a bounded sequence defined on 𝑍; 𝑎_𝑖𝑗(𝑘) and 𝑏_𝑖𝑗𝑙(𝑘)(𝑖,𝑗=1,2,…,𝑛,𝑙=0,1,…,𝜎_𝑖𝑗) are nonnegative bounded sequences defined on 𝑍; 𝜏_𝑖𝑗 and 𝜎_𝑖𝑗(𝑖,𝑗=1,2,…,𝑛) are nonnegative integers.

Directly from Theorem 3.1, we have the following corollary.

Corollary 3.3. Assume that assumptions (𝐻₁^′′) and (𝐻₃) hold and there exists an integer 1≤𝑟<𝑛 such that for any ℎ>𝑟 there exists an integer 𝑖_ℎ<ℎ such that limsup_𝑘→∞∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_ℎ(𝑠)∑^{𝑘+𝜔−1}_𝑠=𝑘𝑟_𝑖ℎ(𝑠)<liminf_𝑘→∞𝐴_ℎ𝑗(𝑘)𝐴_𝑖ℎ𝑗(𝑘)∀𝑗≤ℎ, (3.46) then for each 𝑖=𝑟+1,…,𝑛 one has lim_𝑘→∞𝑥_𝑖(𝑘)=0,^∞_𝑘=0𝑥_𝑖(𝑘)<∞ (3.47) for any positive solution (𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)) of system (3.45).

Remark 3.4. By comparison, we easily see that the results obtained in this paper are a very good extension of the corresponding results obtained in [9] on the extinction of species for nonautonomous continuous Lotka-Volterra competitive system with pure-delays and feedback controls to discrete ones.

4. An Example

We consider the following periodic discrete three-species Lotka-Volterra competitive system with pure-delays and feedback controls:

𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp{𝑟_𝑖(𝑘)−∑³_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑖𝑗)−𝑑_𝑖(𝑘)𝑢_𝑖(𝑘−𝜏_𝑖𝑢)},_𝑖(𝑘+1)=(1−𝑒_𝑖(𝑘))𝑢_𝑖(𝑘)+𝑔_𝑖(𝑘)𝑥_𝑖(𝑘−𝛿_𝑖),𝑖=1,2,3. (4.1) where

𝑟₁1(𝑘)=21+sin(3+𝜋2𝑘),𝑟₂1(𝑘)=21+sin(4+𝜋2𝑟𝑘),₃1(𝑘)=3(41+sin(5+𝜋2𝑘)),𝑎₁₁1(𝑘)=3(1+cos²(13+𝜋2𝑎𝑘)),₁₂1(𝑘)=6+sin²(13+𝜋2𝑘),𝑎₁₃1(𝑘)=4+cos²(13+𝜋2𝑎𝑘),₂₁(𝑘)=1+cos²(13+𝜋2𝑘),𝑎₂₂1(𝑘)=2+cos²(13+𝜋2𝑎𝑘),₃₁(𝑘)=4+sin²(13+𝜋2𝑘),𝑎₃₂9(𝑘)=2+sin²(13+𝜋2𝑎𝑘),₃₃(𝑘)=8+sin²(13+𝜋2𝑘),𝑑₁1(𝑘)=10(1+sin²(13+𝜋2𝑑𝑘)),₂1(𝑘)=2+sin²(13+𝜋2𝑘),𝑑₃(𝑘)=2+cos²(13+𝜋2𝑒𝑘),₁1(𝑘)=3(1+cos²(13+𝜋2𝑘)),𝑒₂1(𝑘)=4(1+sin²(13+𝜋2𝑒𝑘)),₃1(𝑘)=3(1+sin²(13+𝜋2𝑘)),𝑔₁1(𝑘)=3(1+cos²(13+𝜋2𝑔𝑘)),₂(𝑘)=1+cos²(13+𝜋2𝑘),𝑔₃(𝑘)=1+cos²(13+𝜋2𝑘). (4.2)

Clearly, in system (4.2), 𝑟=2,ℎ=3,and𝑖_ℎ=1. By calculating, we obtain

limsup_𝑘→∞∑^𝑘+3_𝑠=𝑘𝑟₃(𝑠)∑^𝑘+3_𝑠=𝑘𝑟₁(𝑠)=limsup_𝑘→∞∑^𝑘+3_𝑠=𝑘3(1/4+sin(1/5+(𝜋/2)𝑠))∑^𝑘+3_𝑠=𝑘=3[1/2+sin(1/3+(𝜋/2)𝑠)]2,liminf_𝑘→∞∑^𝑘+3_𝑠=𝑘𝑟₁(𝑠)∑^𝑘+3_𝑠=𝑘𝑟₃(𝑠)=liminf_𝑘→∞∑^𝑘+3_𝑠=𝑘[1/2+sin(1/3+(𝜋/2)𝑠)]∑^𝑘+3_𝑠=𝑘=23(1/4+sin(1/5+(𝜋/2)𝑠))3,liminf_𝑘→∞𝐴₃₃(𝑘)𝐴₁₃(𝑘)=liminf_𝑘→∞8+sin²(1/3+(𝜋/2)(𝑘+𝜏₃₃))1/4+cos²(1/3+(𝜋/2)(𝑘+𝜏₁₃≥))325,liminf_𝑘→∞𝐴₃₂(𝑘)𝐴₁₂(𝑘)=liminf_𝑘→∞9/2+sin²(1/3+(𝜋/2)(𝑘+𝜏₃₂))1/6+sin²(1/3+(𝜋/2)(𝑘+𝜏₁₃≥))54,14liminf_𝑘→∞𝐴₃₁(𝑘)𝐴₁₁(𝑘)=liminf_𝑘→∞4+sin²(1/3+(𝜋/2)(𝑘+𝜏₃₁))1/3(1+cos²(1/3+(𝜋/2)(𝑘+𝜏₁₁)))≥6,limsup_𝑘→∞(𝐴₁₃(𝑘)𝑅₃(𝑘)limsup_𝑘→∞∑^𝑘+3_𝑠=𝑘𝑟₃(𝑠)∑^𝑘+3_𝑠=𝑘𝑟₁−𝐴(𝑠)³³(𝑘)𝑅₃)(𝑘)=limsup_𝑘→∞3/2(1/4+cos²(1/3+(𝜋/2)(𝑘+𝜏₁₃)))−(8+sin²(1/3+(𝜋/2)(𝑘+𝜏₃₃)))1+cos²(1/3+(𝜋/2)(𝑘+𝛿₃≤))3/2(1/4+1)−82<0,liminf_𝑘→∞(𝐴₃₁(𝑘)𝑅₁(𝑘)liminf_𝑘→∞∑^𝑘+3_𝑠=𝑘𝑟₁(𝑠)∑^𝑘+3_𝑠=𝑘𝑟₃−𝐴(𝑠)¹¹(𝑘)𝑅₁)(𝑘)=liminf_𝑘→∞2/3(4+sin²(1/3+(𝜋/2)(𝑘+𝜏₃₁)))−(1/3(1+cos²(1/3+(𝜋/2)(𝑘+𝜏₁₁))))1/3(1+cos²(1/3+(𝜋/2)(𝑘+𝛿₁≥)))(2/3)4−(1/3)22/3=3,liminf_𝑘→∞𝐷₃(𝑘)𝑒₃(𝑘)=liminf_𝑘→∞2+cos²(1/3+(𝜋/2)(𝑘+𝜏₃))1/3(1+sin²(1/3+(𝜋/2)𝑘))≥3,limsup_𝑘→∞𝐷₁(𝑘)𝑒₁(𝑘)=limsup_𝑘→∞1/10(1+sin²(1/3+(𝜋/2)(𝑘+𝜏₁)))1/3(1+cos²≤3(1/3+(𝜋/2)𝑘))5. (4.3) From these inequalities we see that all conditions (3.2) in Theorem 3.1 hold. In addition, we also see that assumptions (𝐻₁)-(𝐻₄) obviously hold. Therefore, by Theorem 3.1 we obtain that species 𝑥₃ in system (4.3) is extinct.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (10961022, 10901130), the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2007G01, XJEDU2006I05, XJEDU2008S10), and the National Key Technologies R & D Program of China (2008BAI68B01) and the Foundation of Education Committee of Jiangxi (GJJ0931).