Feedback Control Variables Have No Influence on the Permanence of a Discrete 𝑛-Species Schoener Competition System with Time Delays

Abstract

We consider a discrete 𝑛-species Schoener competition system with time delays and feedback controls. By using difference inequality theory, a set of conditions which guarantee the permanence of system is obtained. The results indicate that feedback control variables have no influence on the persistent property of the system. Numerical simulations show the feasibility of our results.

1. Introduction

The Schoener's competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated and many excellent results have been derived [1–5].

Liu et al. [1] propose and study the global stability of the following continuous Schoener's competition model with delays:𝑥̇₁(𝑡)=𝑥₁𝑎(𝑡)[¹⁰(𝑡)𝑥₁(𝑡−𝜏₁₀)+𝑘₁(𝑡)−𝑎₁₁(𝑡)𝑥₁(𝑡−𝜏₁₁)−𝑎₁₂(t)𝑥₂(𝑡−𝜏₁₂)−𝑐₁𝑥̇(𝑡)],₂(𝑡)=𝑥₂𝑎(𝑡)[²⁰(𝑡)𝑥₂(𝑡−𝜏₂₀)+𝑘₂(𝑡)−𝑎₂₁(𝑡)𝑥₁(𝑡−𝜏₂₁)−𝑎₂₂(𝑡)𝑥₂(𝑡−𝜏₂₂)−𝑐₂(𝑡)], (1.1) where {𝑘_𝑖(𝑡)}, {𝑎_𝑖𝑗(𝑡)}, and {𝑐_𝑖(𝑡)} are all positive bounded and continuous functions, 𝜏_𝑖𝑗(𝑖=1,2;𝑗=0,1,2) are positive integers, 𝜏=max_{0≤𝑖,𝑗≤2}{𝜏_𝑖𝑗},𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠)≥0, 𝑠∈[−𝜏,0]∩𝑍, 𝜙_𝑖(0)>0, 𝑖=1,2.

As we all know, though most dynamic behaviors of population models are based on the 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. It has been found that the dynamics behaviors of the discrete system are rather complex and contain more rich dynamics than the continuous ones [6]. Recently, more and more scholars paid attention to study the discrete population models (see [4–13] and the references cited therein). For example, [5] considered the permanence and global attractivity of the following discrete Schoener's competition model with delays: 𝑥₁(𝑛+1)=𝑥₁𝑎(𝑛)exp{¹⁰(𝑛)𝑥₁(𝑛−𝜏₁₀)+𝑘₁(𝑛)−𝑎₁₁(𝑛)𝑥₁(𝑛−𝜏₁₁)−𝑎₁₂(𝑛)𝑥₂(𝑛−𝜏₁₂)−𝑐₁𝑥(𝑛)},₂(𝑛+1)=𝑥₂𝑎(𝑛)exp{²⁰(𝑛)𝑥₂(𝑛−𝜏₂₀)+𝑘₂(𝑛)−𝑎₂₁(𝑛)𝑥₁(𝑛−𝜏₂₁)−𝑎₂₂(𝑛)𝑥₂(𝑛−𝜏₂₂)−𝑐₂(𝑛)}, (1.2) where {𝑘_𝑖(𝑛)}, {𝑎_𝑖𝑗(𝑛)}, and {𝑐_𝑖(𝑛)} are real positive bounded sequences, 𝜏_𝑖𝑗(𝑖=1,2;𝑗=0,1,2) are positive integers, 𝜏=max_{0≤𝑖,𝑗≤2}{𝜏_𝑖𝑗},𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠)≥0, 𝑠∈[−𝜏,0]∩𝑍, 𝜙_𝑖(0)>0, 𝑖=1,2.

On the other hand, as was pointed out by Huo and Li [14], ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. 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. During the last decade, many scholars did excellent works on the feedback control ecosystems (see [14–21] and the references cited therein); however, most of those works are concerned with the continuous model and seldom did scholars considered the discrete ecosystem with feedback controls ([13, 15, 21]).

Recently, Li and Yang [15] proposed the following discrete 𝑛-species Schoener competition system with time delays and feedback controls: 𝑥_𝑖(𝑘+1)=𝑥_𝑖𝑟(𝑘)exp{^𝑖(𝑘)𝑥_𝑖(𝑘−𝜏_𝑖)+𝑎_𝑖−(𝑘)^𝑛_𝑗=1𝑏_𝑖𝑗(𝑘)𝑥_𝑗(𝑘−𝜏_𝑗)−𝑐_𝑖(𝑘)−𝑑_𝑖(𝑘)𝑢_𝑖(𝑘)−𝑒_𝑖(𝑘)𝑢_𝑖(𝑘−𝜂_𝑖)},Δ𝑢_𝑖(𝑘)=−𝛼_𝑖(𝑘)𝑢_𝑖(𝑘)+𝛽_𝑖(𝑘)𝑥_𝑖(𝑘)+𝛾_𝑖(𝑘)𝑥_𝑖(𝑘−𝜎_𝑖), (1.3) where 𝑥_𝑖(𝑘)(𝑖=1,2,…,𝑛) is the density of competitive species at 𝑘th generations; 𝑢_𝑖(𝑘) is the control variable; Δ is the first-order forward difference operator Δ𝑢_𝑖(𝑘)=𝑢_𝑖(𝑘+1)−𝑢_𝑖(𝑘), 𝑖=1,2,…,𝑛.

Throughout this paper, we assume the following.(𝐻₁)𝛼_𝑖(𝑘), 𝛽_𝑖(𝑘), 𝛾_𝑖(𝑘), 𝑎_𝑖(𝑘), 𝑏_𝑖𝑗(𝑘), 𝑟_𝑖(𝑘), 𝑐_𝑖(𝑘), 𝑑_𝑖(𝑘), 𝑒_𝑖(𝑘), 𝑖=1,2,…,𝑛, are all bounded nonnegative sequences such that 0<𝛼^𝑙_𝑖≤𝛼^𝑢_𝑖<1,0<𝛽^𝑙_𝑖≤𝛽^𝑢_𝑖,0<𝛾^𝑙_𝑖≤𝛾^𝑢_𝑖,0<𝑎^𝑙_𝑖≤𝑎^𝑢_𝑖,0<𝑏^𝑙_𝑖𝑗≤𝑏^𝑢_𝑖𝑗,0<𝑟^𝑙_𝑖≤𝑟^𝑢_𝑖,0<𝑐^𝑙_𝑖≤𝑐^𝑢_𝑖,0<𝑑^𝑙_𝑖≤𝑑^𝑢_𝑖,0<𝑒^𝑙_𝑖≤𝑒^𝑢_𝑖. (1.4)

Here, for any bounded sequence {𝑎(𝑘)}, 𝑎^𝑢=sup_𝑘∈𝑁𝑎(𝑘), 𝑎^𝑙=inf_𝑘∈𝑁𝑎(𝑘).(𝐻₂)𝜏_𝑖, 𝜂_𝑖, 𝜎_𝑖, 𝑖=1,2,…,𝑛, are all nonnegative integers.

Let 𝜏=max{𝜏_𝑖,𝜂_𝑖,𝜎_𝑖,𝑖=1,2,…,𝑛}; we consider (1.3) together with the following initial conditions: 𝑥_𝑖(𝜃)=𝜑_𝑖(𝜃),𝜃∈𝑁[−𝜏,0]={−𝜏,−𝜏+1,…,0},𝜑_𝑖𝑢(0)>0,_𝑖(𝜃)=𝜙_𝑖(𝜃),𝜃∈𝑁[−𝜏,0]={−𝜏,−𝜏+1,…,0},𝜙_𝑖(0)>0. (1.5) It is not difficult to see that solutions of (1.3) and (1.5) are well defined for all 𝑘≥0 and satisfy 𝑥_𝑖(𝑘)>0,𝑢_𝑖(𝑘)>0,𝑘∈𝑍,𝑖=1,2,…,𝑛. (1.6)

By applying the comparison theorem of difference equation, they obtained a set of sufficient conditions which guarantee the permanence of the system (1.3). Their result shows that feedback control variables play important roles on the persistent property of the system (1.3). But the question is whether or not the feedback control variables have influence on the permanence of the system. The aim of this paper is to apply the analysis technique of Chen et al. [18] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system.

The organization of this paper is as follow. In Section 2, we will introduce several lemmas. The permanence of system (1.3) is then studied in Section 3. In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results.

2. Preliminaries

In this section, we will introduce several useful lemmas.

Lemma 2.1 (see [11]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘)>0 and 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)} (2.1) for 𝑘∈𝑁, where 𝑎(𝑘) and 𝑏(𝑘) are nonnegative sequences bounded above and below by positive constants. Then limsup_𝑘→+∞1𝑥(𝑘)≤𝑏^𝑙exp(𝑎^𝑢−1). (2.2)

Lemma 2.2 (see [12]). Assume that {𝑥(𝑘)} satisfies 𝑥(k+1)≥𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)},𝑘≥𝑁₀, (2.3)limsup_𝑘→+∞𝑥(𝑘)≤𝑥^∗ and 𝑥(𝑁₀)>0, where 𝑎(𝑘) and 𝑏(𝑘) are nonnegative sequences bounded above and below by positive constants and 𝑁₀∈𝑁. Then liminf_𝑘→+∞𝑎𝑥(𝑘)≥min{^𝑙𝑏^𝑢exp(𝑎^𝑙−𝑏^𝑢𝑥^∗𝑎),^l𝑏^𝑢}. (2.4)

Lemma 2.3 (see [13]). Assume that 𝐴>0, 𝑦(0)>0. Suppose that 𝑦(𝑘+1)≤𝐴𝑦(𝑘)+𝐵(𝑘),𝑘=1,2,…. (2.5) Then for any integer 𝑚≤𝑘,𝑦(𝑘)≤𝐴^𝑚𝑦(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗𝐵(𝑘−𝑗−1). (2.6) If 𝐴<1 and 𝐵 is bounded above with respect to 𝑀, then limsup_𝑘→+∞𝑀𝑦(𝑘)≤.1−𝐴 (2.7)

Lemma 2.4 (see [13]). Assume that 𝐴>0, 𝑦(0)>0. Suppose that 𝑦(𝑘+1)≥𝐴𝑦(𝑘)+𝐵(𝑘),𝑘=1,2,…. (2.8) Then for any integer 𝑚≤𝑘,𝑦(𝑘)≥𝐴^m𝑦(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗𝐵(𝑘−𝑗−1). (2.9) If 𝐴<1 and 𝐵 is bounded below with respect to 𝑃, then liminf_𝑘→+∞𝑃𝑦(𝑘)≥.1−𝐴 (2.10)

3. Permanence

In this section, we establish the following permanence result for system (1.3).

Theorem 3.1. Assume that (𝐻₁) and (𝐻₂) hold. Then there exist positive constants 𝑀_𝑖, 𝑁_𝑖 which are independent of the solutions of system (1.3) such that limsup_𝑘→∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2,…,𝑛,limsup_𝑘→∞𝑢_𝑖(𝑘)≤𝑁_𝑖,𝑖=1,2,…,𝑛. (3.1)

Proof. Let 𝑥(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘)⋯𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘)⋯𝑢_𝑛(𝑘))^𝑇 be any positive solution of system (1.3) with initial condition (1.5). From the first equation of system (1.3), it follows that 𝑥_𝑖(𝑘+1)≤𝑥_𝑖𝑟(𝑘)exp{^𝑖(𝑘)𝑎_𝑖(𝑘)−𝑏_𝑖𝑖(𝑘)𝑥_𝑖(𝑘−𝜏_𝑖)}≤𝑥_𝑖𝑟(𝑘)exp{^𝑢𝑖𝑎^𝑙_𝑖}. (3.2) By using (3.2), one can easily obtain that ^𝑘−1_{𝑗=𝑘−𝜏𝑖}𝑥_𝑖(𝑗+1)≤^𝑘−1_{𝑗=𝑘−𝜏𝑖}𝑥_𝑖𝑟(𝑗)exp{^𝑢𝑖𝑎^𝑙_𝑖}, (3.3) that is, 𝑥_𝑖(𝑘)≤𝑥_𝑖(𝑘−𝜏_𝑖𝜏)exp{^𝑖𝑟^𝑢_𝑖𝑎^𝑙_𝑖}. (3.4) Substituting (3.4) into (3.2), it follows that 𝑥_𝑖(𝑘+1)≤𝑥_𝑖𝑟(𝑘)exp{^𝑖(𝑘)𝑎_𝑖(𝑘)−𝑏_𝑖𝑖𝜏(𝑘)exp{−^𝑖𝑟^𝑢_𝑖𝑎^𝑙_𝑖}𝑥_𝑖(𝑘)}. (3.5) By applying Lemma 2.1, it follows that limsup_𝑘→∞𝑥_𝑖(𝑘)≤exp{(𝑟^𝑢_𝑖/𝑎^𝑙_𝑖)(1+𝜏_𝑖)−1}𝑏^𝑙_𝑖𝑖=∶𝑀_𝑖,𝑖=1,2,…,𝑛. (3.6) Thus, for all 𝜀>0, there exists a 𝐾₁>0, 𝐾₁∈𝑁, for all 𝑘>𝐾₁,𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀, 𝑖=1,2,…,𝑛, and so 𝑢_𝑖(𝑘+1)=(1−𝛼_𝑖(𝑘))𝑢_𝑖(𝑘)+𝛽_𝑖(𝑘)𝑥_𝑖(𝑘)+𝛾_𝑖(𝑘)𝑥_𝑖(𝑘−𝜎_𝑖)≤(1−𝛼^𝑙_𝑖)𝑢_𝑖(𝑘)+(𝛽^𝑢_𝑖+𝛾^𝑢_𝑖)(𝑀_𝑖+𝜀)(𝑘>𝐾₁+𝜎_𝑖). (3.7) For 0<1−𝛼^𝑙_𝑖<1, Lemma 2.3 implies that limsup_𝑘→∞𝑢_𝑖(𝑘)≤(𝛽^𝑢_𝑖+𝛾^𝑢_𝑖)(𝑀_𝑖+𝜀)𝛼^𝑙_𝑖. (3.8) Letting 𝜀→0 in the above inequality leads to limsup_𝑘→∞𝑢_𝑖(𝑘)≤(𝛽^𝑢_𝑖+𝛾^𝑢_𝑖)𝑀_𝑖𝛼^𝑙_𝑖∶=𝑁_𝑖. (3.9) The proof of Theorem 3.1 is completed.

Theorem 3.2. Assume that (𝐻₁) and (𝐻₂) hold; assume further that 𝑅_𝑖𝑟∶=^𝑙𝑖𝑀_𝑖+𝑎^𝑢_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏^𝑢_𝑖𝑗𝑀_𝑗−𝑐^𝑢_𝑖>0,𝑖=1,2,…,𝑛. (𝐻) Then there exist positive constants 𝑚_𝑖, 𝑛_𝑖 which are independent of the solution of system (1.3), such that liminf_𝑘→∞𝑥_𝑖(𝑘)≥𝑚_𝑖,𝑖=1,2,…,𝑛,liminf_𝑘→∞𝑢_𝑖(𝑘)≥𝑛_𝑖,𝑖=1,2,…,𝑛. (3.10)

Proof. Let 𝑥(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘)⋯𝑥_𝑛(𝑘),𝑢₁(𝑘),𝑢₂(𝑘)⋯𝑢_𝑛(𝑘))^𝑇 be any positive solution of system (1.3) with initial condition (1.5). From Theorem 3.1, for all 0<𝜀<1, there exists a 𝐾₂>𝐾₁, for all 𝑘>𝐾₂,𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀,𝑢_𝑖(𝑘)≤𝑁_𝑖+𝜀,𝑖=1,2,…,𝑛,𝑅_𝑖𝜀𝑟∶=^𝑙𝑖(𝑀_𝑗+𝜀)+𝑎^𝑢_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)−𝑐^𝑢_𝑖>0,𝑖=1,2,…,𝑛. (3.11)(3.12) From (3.12), we have 𝑥_𝑖(𝑘+1)≥𝑥_𝑖𝑟(𝑘)exp{^𝑙𝑖(𝑀_𝑖+𝜀)+𝑎^𝑢_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)−𝑐^𝑢_𝑖−𝑑^𝑢_𝑖𝑢_𝑖(𝑘)−𝑒^𝑢_𝑖𝑢_𝑖(𝑘−𝜂_𝑖)−𝑏^𝑢_𝑖𝑖𝑥_𝑖(𝑘−𝜏_𝑖)}≥𝑥_𝑖(𝑘)exp{−^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)−𝑐^𝑢_𝑖−(𝑑^𝑢_𝑖+𝑒^𝑢_𝑖)(𝑁_𝑖+𝜀)}=∶𝑥_𝑖(𝑘)exp{𝐷_𝑖𝜀}. (3.13) for all 𝑘>𝐾₂+𝜏, where 𝐷_𝑖𝜀=−∑^𝑛_𝑗=1𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀)−𝑐^𝑢_𝑖−(𝑑^𝑢_𝑖+𝑒^𝑢_𝑖)(𝑁_𝑖+𝜀)<0.
Thus, for 𝜂<𝐾, by using (3.13) we obtain 𝑥_𝑖(𝜂)≤𝑥_𝑖(𝑘)exp{−(𝑘−𝜂)𝐷_𝑖𝜀}. (3.14) From the second equation of (1.3), we obtain 𝑢_𝑖(𝑘+1)=(1−𝛼_𝑖(𝑘))𝑢_𝑖(𝑘)+𝛽_𝑖(𝑘)𝑥_𝑖(𝑘)+𝛾_𝑖(𝑘)𝑥_𝑖(𝑘−𝜎_𝑖)≤(1−𝛼^𝑙_𝑖)𝑢_𝑖(𝑘)+𝛽^𝑢_𝑖𝑥_𝑖(𝑘)+𝛾^𝑢_𝑖𝑥_𝑖(𝑘−𝜎_𝑖)≜𝐴_𝑖𝑢_𝑖(𝑘)+𝐵_𝑖(𝑘), (3.15) where 0<𝐴_𝑖=1−𝛼^𝑙_𝑖<1 and 𝐵_𝑖(𝑘)=𝛽^𝑢_𝑖𝑥_𝑖(𝑘)+𝛾^𝑢_𝑖𝑥_𝑖(𝑘−𝜎_𝑖).
Then, Lemma 2.3, (3.14), and (3.15) imply that for any 𝑚≤𝐾, 𝑢_𝑖(𝑘)≤𝐴^𝑚_𝑖𝑢_𝑖(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗_𝑖𝐵_𝑖(𝑘−𝑗−1)=𝐴^𝑚_𝑖𝑢_𝑖(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗_𝑖{𝛽^𝑢_𝑖𝑥_𝑖(𝑘−𝑗−1)+𝛾^𝑢_𝑖𝑥_𝑖(𝑘−𝑗−1−𝜎_𝑖)}≤𝐴^𝑚_𝑖𝑢_𝑖+(𝑘−𝑚)_{𝑚−1𝑗=0}𝐴^𝑗_𝑖{𝛽^𝑢_𝑖exp{−(𝑗+1)𝐷_𝑖𝜀}+𝛾^𝑢_𝑖exp{−(𝑗+1+𝜎_𝑖)𝐷_𝑖𝜀}}𝑥_𝑖(𝑘)≤𝐴^𝑚_𝑖𝑢_𝑖(𝑘−𝑚)+𝑥_𝑖(𝑘)^𝑚−1_𝑗=0𝐴^𝑗_𝑖(𝛽^𝑢_𝑖+𝛾^𝑢_𝑖)exp{−(𝑗+1)𝐷_𝑖𝜀}. (3.16) For 0<𝐴_𝑖<1,lim_𝑚→∞(𝑑^𝑢_𝑖+𝑒^𝑢_𝑖)(𝑁_𝑖+1)𝐴^𝑚_𝑖=0. That is, there exists an 𝑀>0, for all 𝑚>𝑀(𝑁_𝑖+𝜀)(𝑑^𝑢_𝑖+𝑒^𝑢_𝑖)𝐴^𝑚_𝑖<(𝑁_𝑖+1)(𝑑^𝑢_𝑖+𝑒^𝑢_𝑖)𝐴^𝑚_𝑖≤𝑅_𝑖𝜀02≤𝑅_𝑖𝜀2. (3.17) For enough small 𝜀₀>0, we have 𝑅_𝑖𝜀0𝑟∶=^𝑙𝑖(𝑀_𝑖+𝜀₀)+𝑎^𝑢_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏^𝑢_𝑖𝑗(𝑀_𝑗+𝜀₀)−𝑐^𝑢_𝑖<𝑅_𝑖𝜀<𝑅_𝑖,𝑖=1,2,…,𝑛. (3.18) We choose 𝑀≥max{1,log_𝐴𝑖(𝑅_𝑖𝜀0/2(𝑁_𝑖+1)(𝑑^𝑢_𝑖+𝑒^𝑢_𝑖))}, 𝑖=1,2,…,𝑛. For fixed 𝑀, we get 𝑢_𝑖(𝑘)≤𝐴^𝑀_𝑖𝑢_𝑖(𝑘−𝑀)+𝑥_𝑖(𝑘)^𝑀−1_𝑗=0𝐴^𝑗_𝑖(𝛽^𝑢_𝑖+𝛾^𝑢_𝑖)exp{−(𝑗+1)𝐷_𝑖𝜀}≤𝐴^𝑀_𝑖(𝑁_𝑖+𝜀)+𝐸_𝑖𝜀𝑥_𝑖(𝑘), (3.19) for all 𝑘≥𝐾₂+𝑀 where 𝐸_𝑖𝜀=∑^𝑀−1_𝑗=0𝐴^𝑗_𝑖(𝛽^𝑢_𝑖+𝛾^𝑢_𝑖)exp{−(𝑗+1)𝐷_𝑖𝜀}.
Substituting (3.14) (3.17) and (3.19) into (3.13), for 𝑘≥𝐾₂+𝑀+𝜏, one has 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp{𝑅_𝑖𝜀−𝑏^𝑢_𝑖𝑖𝑥_𝑖(𝑘−𝜏_𝑖)−𝑑^𝑢_𝑖𝑢_𝑖(𝑘)−𝑒^𝑢_𝑖𝑢_𝑖(𝑘−𝜂_𝑖)}≥𝑥_𝑖(𝑘)exp{𝑅_𝑖𝜀−(𝑒^𝑢_𝑖+𝑑^𝑢_𝑖)𝐴^𝑀_𝑖(𝑁_𝑖+𝜀)−𝑏^𝑢_𝑖𝑖𝑥_𝑖(𝑘−𝜏_𝑖)−𝑑^𝑢_𝑖𝐸_𝑖𝜀𝑥_𝑖(𝑘)−𝑒^𝑢_𝑖𝐸_𝑖𝜀𝑥_𝑖(𝑘−𝜂_𝑖)}≥𝑥_𝑖𝑅(𝑘)exp[^𝑖𝜀2−𝑏^𝑢_𝑖𝑖exp{−𝜏_𝑖𝐷_𝑖𝜀}−𝑑^𝑢_𝑖𝐸_𝑖𝜀−𝑒^𝑢_𝑖𝐸_𝑖𝜀exp{−𝜂_𝑖𝐷_𝑖𝜀}]𝑥_𝑖(𝑘)∶=𝑥_𝑖𝑅(𝑘)exp{^𝑖𝜀2−𝑄_𝑖𝜀𝑥_𝑖(𝑘)}, (3.20) where 𝑄_𝑖𝜀∶=𝑏^𝑢_𝑖𝑖exp{−𝜏_𝑖𝐷_𝑖𝜀}+𝑑^𝑢_𝑖𝐸_𝑖𝜀+𝑒^𝑢_𝑖𝐸_𝑖𝜀exp{−𝜂_𝑖𝐷_𝑖𝜀}.
By applying Lemma 2.2, it follows that liminf_𝑘→∞𝑅𝑥(𝑘)≥min{^𝑖𝜀2𝑄_𝑖𝜀𝑅exp{^𝑖𝜀2−𝑄_𝑖𝜀𝑀_𝑖𝑅},^𝑖𝜀2𝑄_𝑖𝜀}. (3.21) Letting 𝜀→0 in the above inequality leads to liminf_𝑘→∞𝑅𝑥(𝑘)≥min{^𝑖2𝑄_𝑖𝑅exp{^𝑖2−𝑄_𝑖𝑀_𝑖𝑅},^𝑖2𝑄_𝑖}∶=𝑚_𝑖. (3.22) For the above 𝜀>0, there exists a 𝐾₃>𝐾₂, for all 𝑘>𝐾₃, we have 𝑥_𝑖(𝑘)≥𝑚_𝑖−𝜀, 𝑖=1,2,…,𝑛. Then for 𝑘>𝐾₃+𝜎_𝑖, 𝑢_𝑖(𝑘+1)=(1−𝛼_𝑖(𝑘))𝑢_𝑖(𝑘)+𝛽_𝑖(𝑘)𝑥_𝑖(𝑘)+𝛾_𝑖(𝑘)𝑥_𝑖(𝑘−𝜎_𝑖)≥(1−𝛼^𝑢_𝑖)𝑢_𝑖(𝑘)+(𝛽^𝑙_𝑖+𝛾^𝑙_𝑖)(𝑚_𝑖−𝜀). (3.23) By applying Lemma 2.4, it follows that liminf_𝑘→∞𝑢_𝑖(𝑘)≥(𝛽^𝑙_𝑖+𝛾^𝑙_𝑖)(𝑚_𝑖−𝜀)𝛼^𝑢_𝑖. (3.24) Letting 𝜀→0 in the above inequality leads to liminf_𝑘→∞𝑢_𝑖(𝑘)≥(𝛽^𝑙_𝑖+𝛾^𝑙_𝑖)𝑚_𝑖𝛼^𝑢_𝑖∶=𝑛_𝑖,𝑖=1,2,…,𝑛. (3.25) The proof of Theorem 3.2 is completed.

Remark 3.3. Theorems 3.1 and 3.2 show that under the assumption (𝐻₁), (𝐻₂) and (𝐻) hold, and system (1.3) is permanent.

4. Example and Numeric Simulations

The following example lends credence to the plausibility of Theorem 3.2.

Example 4.1. Consider the following system: 𝑥₁(𝑘+1)=𝑥₁(𝑘)exp{0.5𝑥₁(𝑘−1)+1−𝑥₁(𝑘−1)−0.02(sin(𝑘)+2)𝑥₂(𝑘−2)−0.01−(cos(𝑘)+2.5)𝑢₁(𝑘)−(cos(𝑘)+1.5)𝑢₁𝑥(𝑘−1)},₂(𝑘+1)=𝑥₂1(𝑘)exp{𝑥₂(𝑘−2)+3−0.05(cos(𝑘)+2)𝑥₁(𝑘−1)−2𝑥₂(𝑘−2)−0.05−(sin(𝑘)+2)𝑢₂(𝑘)−(sin(𝑘)+1.1)𝑢₂(𝑘−1)},Δ𝑢₁(𝑘)=−0.5𝑢₁(𝑘)+0.1(sin(10+𝑘)+1.5)𝑥₁(𝑘)+0.1(sin(𝑘)+1.5)𝑥₁(𝑘−1),Δ𝑢₂1(𝑘)=−3𝑢₂(𝑘)+0.1(cos(10+𝑘)+1.5)𝑥₂(𝑘)+0.1(cos(𝑘+5)+1.5)𝑥₂(𝑘−1). (4.1) Here corresponding to system (1.3), we assume that 𝑟₁(𝑘)=0.5,𝑎₁(𝑘)=1,𝑏₁₁(𝑘)=1,𝑏₁₂𝑐(𝑘)=0.02(sin(𝑘)+2),₁(𝑘)=0.01,𝜂₁=1,𝑑₁(𝑘)=cos(𝑘)+2.5,𝑒₁𝛼(𝑘)=cos(𝑘)+1.5,₁(𝑘)=0.5,𝛽₁𝛾(𝑘)=0.1(sin(𝑘+10)+1.5),₁(𝑘)=0.1(sin(𝑘)+1.5),𝜎₁𝑟=1,₂(𝑘)=1,𝜏₂=2,𝑎₂(𝑘)=3,𝑏₂₂(𝑘)=2,𝑏₂₁𝑐(𝑘)=0.05(cos(𝑘)+2),₂(𝑘)=0.05,𝜂₂=1,𝑑₂(𝑘)=sin(𝑘)+2,𝑒₂𝛼(𝑘)=sin(𝑘)+1.1,₂1(𝑘)=3,𝛽₂𝛾(𝑘)=0.1(cos(𝑘+10)+1.5),₂(𝑘)=0.1(cos(𝑘+5)+1.5),𝜎₂=1. (4.2)
It is easy to see that 𝑀₁=1,𝑀₂=0.5, and 𝑟^𝑙₁𝑀₁+𝑎^𝑢₁−𝑏^𝑢₁₂𝑀₂−𝑐^𝑢₁𝑟=0.21>0,_𝑙2𝑀₂+𝑎^𝑢₂−𝑏^𝑢₂₁𝑀₁−𝑐^𝑢₂=335>0. (4.3) Inequalities (4.3) show that all the conditions of Theorem 3.2 are satisfied; thus, system (1.3) is permanent. Numerical simulation from Figure 1 supports this conclusion.

(a)

(b)

(c)

(d)

Figure 1 

Dynamic behaviors of system (4.1) with initial conditions (𝑥1(𝑠),𝑥2(𝑠),𝑢1(𝑠),𝑢2(𝑠))𝑇=(0.73,0.1,0.1,0.1)𝑇, (0.33,0.24,0.25,0.15)𝑇, (0.52,0.18,0.5,0.2)𝑇, 𝑠=−1,0.

5. Discussion

In this paper we have attempted to understand the effect of feedback control variables on the permanence of system (1.3). The present work is an extension of an earlier work by Li and Yang [15]. By developing the analysis technique of Chen et al. [18], a set of sufficient conditions are established for the permanence of system (1.3). Theorems 3.1-3.2 and the numerical simulations indicate that feedback control variables have no influence on the permanence of system (1.3).

We would like to mention here that an interesting but challenging problem associated with the study of system (1.3) should be the global attractivity. We leave this for future work.

Acknowledgments

The authors would like to thank the two anonymous referees for their constructive suggestions on improving the presentation of the paper. Also, this work is supported by the Technology Innovation Platform Project of Fujian Province (2009J1007).