Feedback Control in a Periodic Delay Single-Species Difference System

Abstract

A periodic delay single-species difference system with feedback control is established. With the help of analysis method and Lyapunov function, a good understanding of the permanence and global attractivity of the system is gained. Numerical simulations are presented to verify the validity of the proposed criteria. Our results show that feedback control has no influence on the permanence while it has influence on the global attractivity of the system.

1. Introduction

In 1978, Ludwig et al. [1] considered a single-species system which is modeled by 𝑥^′(𝑡)=𝑥(𝑡)[𝑎−𝑏𝑥(𝑡)]−ℎ(𝑥), (1.1) where 𝑥(𝑡) is the density of species 𝑥 at time 𝑡, 𝑎 is the intrinsic growth rate, 𝑏 is the competing rate, and ℎ(𝑥)-term represents predation. To be specific, Murray [2] took ℎ(𝑥) in the form of 𝑐𝑥²(𝑡)/(𝑑+𝑥²(𝑡)), and the dynamic behavior of 𝑥(𝑡) is then governed by 𝑥^′(𝑡)=𝑥(𝑡)[𝑎−𝑏𝑥(𝑡)]−𝑐𝑥²(𝑡)𝑑+𝑥²(𝑡), (1.2) where 𝑐𝑥²(𝑡)/(𝑑+𝑥²(t))is an 𝑆-shaped function and 𝑎,  𝑏, 𝑐,  and𝑑 are positive constants. For the relevant ecology sense of system (1.2), we refer the readers to [1, 2] and the references cited therein.

It seems reasonable to assume that the reproduction of species 𝑥 will not be instantaneous, but mediated by a delay required for gestation of species 𝑥. Thus, a revised version is 𝑥^′(𝑡)=𝑥(𝑡)[𝑎−𝑏𝑥(𝑡−𝜏)]−𝑐𝑥²(𝑡)𝑑+𝑥²(𝑡), (1.3) where the constant delay 𝜏 is positive. It is evident that all the coefficients of system (1.3) are assumed to be constant. However, in the real world, the coefficients are not fixed constants owing to the variation of environments. The influence of a varying environment is important for evolutionary theory as the selective forces on systems in such a fluctuating medium differ from those in a stable environment. In addition, as we know, ecosystems are often disturbed by unpredictable forces and, so, species population may experience changes. In ecology, an interesting issue is whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control, we call the disturbance functions control variables. For more discussion on this direction, we refer the readers to [3–8].

Considering the possible effects of fluctuating environment and feedback control on system (1.3), we obtain the following periodic system: 𝑥^′(𝑡)(𝑡𝑎(𝑡)(𝑡)𝑥(−𝑐(𝑡)𝑥=𝑥)[−𝑏𝑡−𝜏)]²(𝑡)𝑑(𝑡)+𝑥²(𝑡)(𝑡)𝑥(𝑡)𝑢(𝑡),𝑢−𝑔_′(𝑡)=−𝑒(𝑡)𝑢(𝑡)+ℎ(𝑡)𝑥(𝑡−𝜏), (1.4) where 𝑢(𝑡) is the control variable. We assume that coefficients 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑔(𝑡), 𝑒(𝑡), and ℎ(𝑡) are continuous and bounded above and below by positive constants and 𝑒(𝑡)∈(0,1).

Following the same idea and method in [9–12], one can easily derive the discrete analogue of system (1.4), which takes the form of 𝑥()(𝑘)𝑎(𝑘)(𝑘)𝑥()−𝑐(𝑘)𝑥(𝑘)𝑘+1=𝑥exp−𝑏𝑘−𝑙𝑑(𝑘)+𝑥²(𝑘)(𝑘)𝑢(𝑘,(𝑘)(𝑘)𝑢(𝑘)(𝑘)𝑥(),−𝑔)Δ𝑢=−𝑒+ℎ𝑘−𝑙 (1.5) where 𝑘∈{0,1,2,…}, 𝑙 is a positive integer, all the coefficients 𝑎(𝑘), 𝑏(𝑘), 𝑐(𝑘), 𝑑(𝑘), 𝑔(𝑘), 𝑒(𝑘), and ℎ(𝑘) are positive bounded sequences and 𝑒(𝑘)∈(0,1). Δ is the first forward difference operator Δ𝑢(𝑘)=𝑢(𝑘+1)−𝑢(𝑘). For biological reasons, we only consider the solution (𝑥(𝑘),𝑢(𝑘)) of system (1.5) with initial value 𝑥(−𝑙),𝑥(−𝑙+1),…,𝑥(−1)≥0, (𝑥(0),𝑢(0))>0. The principle aim of this paper is to explore the permanence and global attractivity of system (1.5). To the best of our knowledge, no work has been done for system (1.5).

For the sake of simplicity and convenience, the notations and definitions below will be used through this paper: 𝑍⁺ and ⌈⌉  denote the set of nonnegative integers and the greatest integer function, respectively. We denote 𝑓^𝑈=sup_𝑘∈𝑍⁺𝑓(𝑘),  𝑓^𝐿=inf_𝑘∈𝑍⁺𝑓(𝑘) for any bounded nonnegative sequence {𝑓(𝑘)}. Meanwhile, we denote the product of 𝑓(𝑘) from 𝑘=𝛼 to 𝑘=𝛽 by ∏^𝑘=𝛽_𝑘=𝛼𝑓(𝑘) with the understanding that ∏^𝑘=𝛽_𝑘=𝛼𝑓(𝑘)=1 for all 𝛼>𝛽.

Definition 1.1. System (1.5) is said to be permanent if there exist positive constants  𝑥, 𝑥  and  𝑢, 𝑢  such that 𝑥⩽lim_𝑘→+∞inf𝑥(𝑘)⩽lim_𝑘→+∞sup𝑥(𝑘)⩽𝑢𝑥,⩽lim_𝑘→+∞inf𝑢(𝑘)⩽lim_𝑘→+∞sup𝑢(𝑘)⩽𝑢. (1.6)

Definition 1.2. The positive solutions of system (1.5) are globally attractive if any two positive solutions (𝑥(𝑘),𝑢(𝑘)) and (𝑥^∗(𝑘),𝑢^∗(𝑘)) of system (1.5) satisfy lim_𝑘→+∞𝑥(𝑘)−𝑥^∗(𝑘)=0,lim_𝑘→+∞𝑢(𝑘)−𝑢^∗(𝑘)=0. (1.7)

The organization of this paper is as follows. In the next Sections 2 and 3, two main results on permanence and global attractivity of system (1.5) are given, respectively. Numerical simulations are present to illustrate the validity of our main results in Section 4, and a brief conclusion is provided to summarize the paper in the final section.

2. Permanence

This section is concerned with the permanence of system (1.5). We first introduce the following lemmas which are useful for establishing our result.

Lemma 2.1 (see [4]). Assume the constant A>0 and y(0)>0, and further suppose that(1)if 𝑦(𝑘+1)≤𝐴𝑦(𝑘)+𝐵(𝑘),𝑘=1,2,…, (2.1) then for any integer 𝑝≤𝑘, 𝑦(𝑘)≤𝐴^𝑝𝑦𝑘−𝑝+^𝑝−1_𝑖=0𝐴^𝑖𝐵(𝑘−𝑖−1). (2.2) Especially, if 𝐴<1 and 𝐵(𝑘) is bounded above with respect to 𝑀, then lim_𝑘→+∞(𝑘)≤𝑀sup𝑦1−𝐴. (2.3)(2)If 𝑦(𝑘+1)≥𝐴𝑦(𝑘)+𝐵(𝑘),𝑘=1,2,…, (2.4) then for any integer 𝑝≤𝑘, 𝑦(𝑘)≥𝐴^𝑝𝑦𝑘−𝑝+^𝑝−1_𝑖=0𝐴^𝑖𝐵(𝑘−𝑖−1). (2.5) Especially, if 𝐴<1 and 𝐵(𝑘) is bounded below with respect to 𝑚, then lim_𝑘→+∞(𝑘)≥𝑚inf𝑦1−𝐴. (2.6)

Lemma 2.2 (see [10]). Assume that y(k) satisfies y(n₁)>0 and 𝑦(𝑘+1)≤𝑦(𝑘)exp𝑟(𝑘)1−𝛼𝑦(𝑘) (2.7) for 𝑘∈[𝑛₁,+∞), where 𝛼 is a positive constant and 𝑛₁∈𝑍⁺. Then lim_𝑘→+∞(𝑘)≤1sup𝑦𝛼𝑟^𝑈exp𝑟^𝑈−1. (2.8)

Lemma 2.3 (see [10]). Assume that 𝑦(𝑘) satisfies 𝑦(𝑛₂)>0 and 𝑦(𝑘+1)≥𝑦(𝑘)exp𝑟(𝑘)1−𝛼𝑦(𝑘) (2.9) for 𝑘∈[𝑛₂,+∞),lim_𝑘→+∞sup𝑦(𝑘)≤𝑀, where 𝛼 is a positive constant such that 𝛼𝑀>1 and 𝑛₂∈𝑍⁺. Then lim_𝑘→+∞(𝑘)≥1inf𝑦𝛼exp𝑟^𝑈(1−𝛼𝑀). (2.10)

Now, we state the permanence of system (1.5).

Theorem 2.4. System (1.5) is permanent provided that 2𝑎^𝐿𝑑^𝐿>𝑐^𝑈. (2.11)

Proof. It follows from the first equation of system (1.5) that 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)}. (2.12) Let 𝑥(𝑘)=exp{𝑦(𝑘)}, then (2.12) can be rewritten as 𝑦(𝑘+1)−𝑦(𝑘)≤𝑎(𝑘). (2.13) Summing both sides of (2.13) from 𝑘−𝑙 to 𝑘−1 yields ^𝑘−1_{𝑖=𝑘−𝑙}𝑦(𝑖+1)−𝑦(𝑖)≤^𝑘−1_{𝑖=𝑘−𝑙}𝑎(𝑖)≤𝑎^𝑈𝑙, (2.14) which implies that 𝑦(𝑘−𝑙)≥𝑦(𝑘)−𝑎^𝑈𝑙, (2.15) and hence 𝑥(𝑘−𝑙)≥𝑥(𝑘)exp−𝑎^𝑈𝑙. (2.16) Following the above result, again from the first equation of system (1.5), we have 𝑥(𝑘+1)≤𝑥(𝑘)exp𝑎(𝑘)−𝑏(𝑘)exp−𝑎^𝑈𝑙𝑥(𝑘). (2.17) By applying Lemma 2.2 to (2.17), we can obtain that lim_𝑘→+∞(𝑘)≤𝑎sup𝑥exp^𝑈(𝑙+1)−1𝑏^𝐿def=𝑥. (2.18) For any constant 𝜀>0, it follows from (2.18) that there exists a 𝑛₁∈𝑍⁺ large enough such that 𝑥(𝑘)≤𝑥+𝜀,for𝑘≥𝑛₁, (2.19) which, together with the second equation of system (1.5), leads to 𝑢(𝑘)≤−𝑒(𝑘)𝑢(𝑘)+ℎ(𝑘)(𝑥+𝜀),for𝑘≥𝑛₁+𝑙, (2.20) that is, 𝑢(𝑘+1)≤1−𝑒^𝐿𝑢(𝑘)+ℎ^𝑈(𝑥+𝜀),for𝑘≥𝑛₁+𝑙. (2.21) Combing (2.21) with Lemma 2.1(1) and setting 𝜀→0 in (2.21), one has lim_𝑘→+∞(𝑘)≤ℎsup𝑢^𝑈𝑥𝑒^𝐿def=𝑢. (2.22) For any sufficient small constant 𝜀>0, it follows from (2.18) and (2.22) that there exists 𝑛₂>𝑛₁+𝑙 such that 𝑥(𝑘)≤𝑥+𝜀,𝑢(𝑘)≤𝑢+𝜀,for𝑘≥𝑛₂. (2.23) Thus, by (2.23) and the first equation of system (1.5), we have 𝑥()(𝑘)𝑎(𝑘)(𝑘)𝑥()−𝑐(𝑘)𝑥(𝑘)𝑘+1=𝑥exp−𝑏𝑘−𝑙𝑑(𝑘)+𝑥²(𝑘)(𝑘)𝑢(𝑘(𝑘)𝑎(𝑘)(𝑘)−𝑔)≥𝑥exp−𝑏𝑐(𝑘)𝑥−𝑑(𝑘)(𝑘)(𝑘)(𝑘)/𝑥+𝑥−𝑔𝑢≥𝑥(𝑘)exp𝑎(𝑘)−𝑏(𝑘)𝑐(𝑘)𝑥−2𝑑(𝑘)−𝑔(𝑘)𝑢 (2.24) for 𝑘≥𝑛₂. Let 𝑥(𝑘)=exp{𝑦(𝑘)}, then (2.24) is equivalent to 𝑦(𝑘+1)−𝑦(𝑘)≥𝑎(𝑘)−𝑏(𝑘)𝑐(𝑘)𝑥−2𝑑(𝑘)−𝑔(𝑘)𝑢. (2.25) For convenience of exposition, we set 𝑅(𝑘)={𝑎(𝑘)−𝑏(𝑘)𝑥−𝑐(𝑘)/2𝑑(𝑘)−𝑔(𝑘)𝑢}. Summing both sides of (2.25) from 𝑘−𝑙 to 𝑘−1 leads to ^𝑘−1_{𝑖=𝑘−𝑙}𝑦(𝑖+1)−𝑦(𝑖)≥^𝑘−1_{𝑖=𝑘−𝑙}𝑅(𝑖). (2.26) This implies that 𝑦(𝑘−𝑙)≤𝑦(𝑘)−^𝑘−1_{𝑖=𝑘−𝑙}𝑅(𝑖), (2.27) and hence, 𝑥(𝑘−𝑙)≤𝑥(𝑘)exp−^𝑘−1_{𝑖=𝑘−𝑙}𝑅(𝑖). (2.28) From the second equation of system (1.5), we have 𝑢()=((𝑘𝑢(𝑘)(𝑘)𝑥()≤𝑘+11−𝑒))+ℎ𝑘−𝑙1−𝑒^𝐿𝑢(𝑘)+ℎ(𝑘)𝑥(𝑘−𝑙_def)=𝐴𝑢(𝑘)+𝐵(𝑘), (2.29) where 𝐴=1−𝑒^𝐿 and 𝐵(𝑘)=ℎ(𝑘)𝑥(𝑘−𝑙). Then, for any integer 𝑝≤𝑘−𝑙, combing (2.28), (2.29) with Lemma 2.1 (1), we have 𝑢(𝑘)≤𝐴^𝑝𝑢𝑘−𝑝+^𝑝−1_𝑖=0𝐴^𝑖𝐵(𝑘−𝑖−1)=𝐴^𝑝𝑢𝑘−𝑝+^𝑝−1_𝑖=0𝐴^𝑖ℎ(𝑘−𝑖−1)𝑥(𝑘−𝑖−1−𝑙)≤𝐴^𝑝𝑢𝑘−𝑝+^𝑝−1_𝑖=0𝐴^𝑖ℎ^𝑈𝑥(𝑘)exp−^𝑘−1_{()𝑗=𝑘−𝑖+1+𝑙}𝑅𝑗. (2.30) Since 𝑒(𝑘)∈(0,1), we obtain that 0<𝐴<1. So 0≤𝐴^𝑝𝑢𝑘−𝑝≤𝑢𝐴^𝑝⟶0,for𝑝⟶+∞. (2.31) By the assumption of Theorem 2.4, for any solution (𝑥(𝑘),𝑢(𝑘)) of system (1.5), there exists an integer 𝑞>0 such that 𝑔^𝑈𝐴^𝑝𝑢(𝑘−𝑝)<(1/2)(𝑎^𝐿−𝑐^𝑈/2𝑑^𝐿)  for  𝑝>𝑞. In fact, we can choose 𝑞=max{1,⌈log_𝐴((𝑎^𝐿−𝑐^𝑈)/(2𝑑^𝐿)/2𝑔^𝑈𝑢)⌉+1}. Then 𝑢(𝑘)≤𝐴^𝑞𝑢𝑘−𝑞+^𝑞−1_𝑖=0𝐴^𝑖ℎ^𝑈𝑥(𝑘)exp−^𝑘−1_{()𝑗=𝑘−𝑖+1+𝑙}𝑅𝑗≤𝑢𝐴^𝑞+^𝑞−1_𝑖=0𝐴^𝑖ℎ^𝑈exp−(𝑖+1+𝑙)𝑅^𝐿𝑥(𝑘),for𝑘>𝑞. (2.32) Since ∑^𝑞−1_𝑖=0𝐴^𝑖ℎ^𝑈exp[−(𝑖+1+𝑙)𝑅^𝐿] is bounded above, let 𝐾₁={∑^𝑞−1_𝑖=0𝐴^𝑖ℎ^𝑈exp[−(𝑖+1+𝑙)𝑅^𝐿]}, then we have 𝑢(𝑘)≤𝑢𝐴^𝑞+𝐾₁𝑥(𝑘),for𝑘>𝑞, (2.33) which, together with (2.24) and (2.28), leads to 𝑥()(𝑘)𝑎(𝑘)(𝑘)𝑥()−𝑐(𝑘)𝑥(𝑘)𝑘+1=𝑥exp−𝑏𝑘−𝑙𝑑(𝑘)+𝑥²(𝑘)(𝑘)𝑢(𝑘(𝑘)𝑎(𝑘)−𝑐(𝑘)−𝑔)≥𝑥exp2𝑑(𝑘)−𝑢𝐴^𝑞𝑔(𝑘)−𝐾₁𝑔(𝑘)+exp−𝑙𝑅^𝐿𝑏(𝑘𝑥(𝑘(𝑘)𝑆))≥𝑥exp₁(𝑘𝑆)1−²(𝑘)𝑆₁(𝑘)𝑥(𝑘), (2.34) where 𝑆₁(𝑘)=𝑎(𝑘)−𝑐(𝑘)/(2𝑑(𝑘))−𝑢𝐴^𝑞𝑔(𝑘), 𝑆₂(𝑘)=𝐾₁𝑔(𝑘)+exp(−𝑙𝑅^𝐿)𝑏(𝑘). Note that 𝑆^𝑈₂𝑆^𝐿₁𝑔𝑥=^𝑈𝐾₁+𝑏^𝑈exp−𝑙𝑅^𝐿𝑎^𝐿−𝑐^𝑈/2𝑑^𝐿−𝑢𝐴^𝑞𝑔^𝑈×𝑥≥𝑏^𝑈exp−𝑙𝑅^𝐿𝑎^𝐿×exp𝑎^𝑈(𝑙+1)−1𝑏^𝐿≥𝑏^𝑈exp−𝑙𝑎^𝐿𝑎^𝐿×exp𝑎^𝐿(𝑙+1)−1𝑏^𝑈≥exp𝑎^𝐿−1𝑎^𝐿>1, (2.35) where we use the inequality exp(𝑥−1)>𝑥 for 𝑥>0. Therefore, by Lemma 2.3, it follows that lim_𝑘→+∞(𝑘)≥𝑆inf𝑥^𝐿1𝑆^𝑈₂exp𝑆^𝑈₁𝑆1−^𝑈2𝑆^𝐿₁𝑥=^def𝑥. (2.36) For any constant 𝜀>0, by (2.36) we know that there exists a sufficiently large integer 𝑛₃>𝑛₂+𝑙 such that 𝑥(𝑘)≥𝑥−𝜀,for𝑘≥𝑛₃. (2.37) Hence, it follows from (2.37) and the second equation of system (1.5) that 𝑢(𝑘+1)≥1−𝑒^𝑈𝑢(𝑘)+ℎ^𝐿𝑥−𝜀,for𝑘≥𝑛₃. (2.38) Applying Lemma 2.1(2) and setting 𝜀→0 in (2.38), we have lim_𝑘→+∞(𝑘)≥ℎinf𝑢^𝐿𝑒^𝑈𝑥=^def𝑢. (2.39) Consequently, combing (2.18), (2.22), and (2.36) with (2.39), system (1.5) is permanent. This completes the proof.

3. Global Attractivity

On the basis of permanence, in this section we further provide sufficient conditions that guarantee the positive solutions of system (1.5) are globally attractive. To do so, we first give the following lemma.

Lemma 3.1. For any two positive solutions (x(k),u(k)) and (x^∗(k),u^∗(k)) of system (1.5), one has 𝑥()ln𝑘+1𝑥^∗()𝑥(𝑘)𝑘+1=ln𝑥^∗(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)−𝑐(𝑘)𝐽(𝑘)−𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘(𝑘))+𝑏^𝑘−1_{𝑠=𝑘−𝑙}𝑃(𝑠)𝑎(𝑠)−𝑏(𝑠)𝑥^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠×𝑥(𝑠))−𝑥^∗(𝑠)−𝑄(𝑠)𝑥(𝑠)𝑏(𝑠)𝑥(𝑠−𝑙)−𝑥^∗((𝑠)𝐽(𝑠)(𝑠𝑢(𝑠)𝑠−𝑙)+𝑐+𝑔)−𝑢^∗(𝑠), (3.1) where 𝐽(𝑠)=𝑑(𝑠)−𝑥(𝑠)𝑥^∗(𝑠)×𝑥(𝑠)−𝑥^∗(𝑠)𝑑(𝑠)+𝑥(𝑠)²×𝑑(𝑠)+𝑥^∗(𝑠)²,𝑃(𝑠)=exp𝜂(𝑠)𝑎(𝑠)−𝑏(𝑠)𝑥^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠𝑄(𝑠)𝜉(𝑠𝑎(𝑠)(𝑠)𝑥()−𝑐(𝑠)𝑥(𝑠))=exp)−𝑏𝑠−𝑙𝑑(𝑠)+𝑥(𝑠)²(𝑠)𝑢(𝑠+((𝑠𝑎(𝑠)(𝑠)𝑥−𝑔)1−𝜉))−𝑏^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠𝜂(𝑠)(𝑠)∈().),𝜉0,1 (3.2)

Proof. It follows from system (1.5) that we have 𝑥()ln𝑘+1𝑥^∗()𝑥(k)𝑘+1−ln𝑥^∗(𝑘)𝑥()=ln𝑘+1𝑥(𝑘)𝑥−ln^∗(𝑘+1)𝑥^∗(𝑘)=𝑎(𝑘)(𝑘)𝑥()−𝑐(𝑘)𝑥(𝑘)−𝑏𝑘−𝑙𝑑(𝑘)+𝑥(𝑘)²(𝑘)𝑢(𝑘−𝑎(𝑘)(𝑘)𝑥−𝑔)−𝑏^∗()−𝑐(𝑘)𝑥𝑘−𝑙^∗(𝑘)𝑑(𝑘)+𝑥^∗(𝑘)²−𝑔(𝑘)𝑢^∗(𝑘(𝑘𝑥())=−𝑏)𝑘−𝑙−𝑥^∗(𝑘−𝑙)−𝑐(𝑘)𝐽(𝑘)−𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘). (3.3) Hence, 𝑥()ln𝑘+1𝑥^∗()𝑥(𝑘)𝑘+1=ln𝑥^∗(𝑘)−𝑐(𝑘)𝐽(𝑘)−𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘(𝑘𝑥(𝑘))−𝑏)−𝑥^∗(𝑘)−[𝑥(𝑘)−𝑥(𝑘−𝑙)]+𝑥^∗(𝑘)−𝑥^∗(𝑥(𝑘)𝑘−𝑙)=ln𝑥^∗(𝑘)−𝑐(𝑘)𝐽(𝑘)−𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘(𝑘𝑥(𝑘))−𝑏)−𝑥^∗(𝑘)+𝑏(𝑘)[𝑥(𝑘)−𝑥(𝑘−𝑙)]−𝑥^∗(𝑘)−𝑥^∗(𝑘−𝑙). (3.4) Note that [𝑥(𝑘)−𝑥(𝑘−𝑙)]−𝑥^∗(𝑘)−𝑥^∗(𝑘−𝑙)=^𝑘−1_{𝑠=𝑘−𝑙}[𝑥(𝑠+1)−𝑥(𝑠)]−^𝑘−1_{𝑠=𝑘−𝑙}𝑥^∗(𝑠+1)−𝑥^∗(𝑠=_{𝑘−1𝑠=𝑘−𝑙})𝑥(𝑠+1)−𝑥^∗(𝑠+1)−𝑥(𝑠)−𝑥^∗(𝑠), (3.5)𝑥(𝑠+1)−𝑥^∗(𝑠+1)−𝑥(𝑠)−𝑥^∗(𝑠(𝑠)𝑎(𝑠)(𝑠)𝑥()−𝑐(𝑠)𝑥(𝑠))=𝑥exp−𝑏𝑠−𝑙𝑑(𝑠)+𝑥(𝑠)²−𝑔(𝑠)𝑢(𝑠)−𝑥^∗(𝑠)exp𝑎(𝑠)−𝑏(𝑠)𝑥^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠)−𝑥(𝑠)−𝑥^∗(𝑠(𝑠𝑎(𝑠)(𝑠)𝑥()−𝑐(𝑠)𝑥(𝑠))=𝑥)exp−𝑏𝑠−𝑙𝑑(𝑠)+𝑥(𝑠)²(𝑠)𝑢(𝑠𝑎(𝑠)(𝑠)𝑥−𝑔)−exp−𝑏^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠+𝑥(𝑠))−𝑥^∗(𝑠)exp𝑎(𝑠)−𝑏(𝑠)𝑥^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠)−1. (3.6) By the mean value theorem, one has 𝑥(𝑠+1)−𝑥^∗(𝑠+1)−𝑥(𝑠)−𝑥^∗(𝑠(𝑠𝑎(𝑠)(𝑠)𝑥)=𝑃)−𝑏^∗()−𝑐(𝑠)𝑥𝑠−𝑙^∗(𝑠)𝑑(𝑠)+𝑥^∗(𝑠)²−𝑔(𝑠)𝑢^∗(𝑠)𝑥(𝑠)−𝑥^∗(𝑠(𝑠)𝑥(𝑠𝑏(𝑠𝑥())−𝑄))𝑠−𝑙−𝑥^∗(𝑠−𝑙)+𝑐(𝑠)𝐽(𝑠)+𝑔(𝑠)𝑢(𝑠)−𝑢^∗(𝑠). (3.7) Then we can easily obtain (3.1) by substituting (3.5) and (3.7) into (3.4). The proof is complete.

Now, we state our main result on the global attractivity of system (1.5).

Theorem 3.2. If the assumption of Theorem 2.4 holds and, further, suppose there exist positive constants 𝜌 and 𝛿 such that 𝜆=^defmin𝜌𝜙−𝛿ℎ^𝑈,𝛿𝑒^𝐿−𝜌𝜑>0, (3.8) where 𝜙, 𝜑 are defined by (3.27) and (3.28), respectively. Then the positive solutions of system (1.5) are globally attractive.

Proof. Let (𝑥(𝑘),𝑢(𝑘)) and (𝑥^∗(𝑘),𝑢^∗(𝑘)) be any two positive solutions of system (1.5). To prove Theorem 3.2, we first consider the following three steps for the first equation of system (1.5).
Step 1. Let 𝑉₁₁(𝑘)=|ln𝑥(𝑘)−ln𝑥^∗(𝑘)|. It follows from (3.1) that 𝑥()ln𝑘+1𝑥^∗()≤𝑥(𝑘)𝑘+1ln𝑥^∗(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)+𝑐(𝑘)|𝐽(𝑘)|+𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘(𝑘))+𝑏^𝑘−1_{𝑠=𝑘−𝑙}𝑃(𝑠)𝛽(𝑠)𝑥(𝑠)−𝑥^∗(𝑠(𝑠)𝑄(𝑠)𝑥(𝑠𝐽(𝑠(𝑠)𝑥(𝑠𝑏(𝑠𝑥())+𝑐)|)|+𝑄))𝑠−𝑙−𝑥^∗(𝑠−𝑙)+𝑔(𝑠)𝑢(𝑠)−𝑢^∗(𝑠≤𝑥(𝑘))ln𝑥^∗(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)+𝛾(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)+𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘(𝑘))+𝑏^𝑘−1_{𝑠=𝑘−𝑙}𝑃(𝑠)𝛽(𝑠)+𝛾(𝑠)𝑄(𝑠)𝑥(𝑠)𝑥(𝑠)−𝑥^∗(𝑠(𝑠)𝑥(𝑠𝑏(𝑠𝑥())+𝑄))𝑠−𝑙−𝑥^∗(𝑠−𝑙)+𝑔(𝑠)𝑢(𝑠)−𝑢^∗(𝑠), (3.9) where 𝛽(𝑠)=𝑎(𝑠)+𝑏(𝑠)𝑥^∗()+𝑐(𝑠)𝑠−𝑙2𝑑(𝑠)+𝑔(𝑠)𝑢^∗(𝑠),𝛾(𝑠)=𝑐(𝑠)𝑑(𝑠)+𝑥(𝑠)𝑥^∗(𝑠)𝑑(𝑠)². (3.10) By the mean value theorem, we get 𝑥(𝑘)−𝑥^∗(𝑘)=exp[ln𝑥(𝑘)]−expln𝑥^∗(𝑘(𝑘)𝑥(𝑘))=𝜃ln𝑥^∗(𝑘), (3.11) that is, 𝑥(𝑘)ln𝑥^∗(𝑘)=1𝜃(𝑘)𝑥(𝑘)−𝑥^∗(𝑘), (3.12) where 𝜃(𝑘) lies between 𝑥(𝑘) and 𝑥^∗(𝑘), then 𝑥(𝑘)ln𝑥^∗(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘=𝑥(𝑘))ln𝑥^∗(𝑘)−𝑥(𝑘)ln𝑥^∗(𝑘)+𝑥(𝑘)ln𝑥^∗(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘=𝑥(𝑘))ln𝑥^∗(𝑘)−1𝜃(𝑘)𝑥(𝑘)−𝑥^∗(𝑘+1)𝜃(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘=𝑥(𝑘))ln𝑥^∗(𝑘)−1𝜃(𝑘)𝑥(𝑘)−𝑥^∗(𝑘+1)𝜃(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘=𝑥(𝑘))ln𝑥^∗(𝑘)−1𝜃(𝑘)−1𝜃(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘). (3.13) According to Theorem 2.4, there exists a 𝑛₀∈𝑍⁺ such that {𝑥(𝑘),𝑥^∗(𝑘)}≤𝑥  and {𝑢(𝑘),𝑢^∗(𝑘)}≤𝑢  for all 𝑘≥𝑛₀. Therefore, for all 𝑘≥𝑛₀, we can obtain that Δ𝑉₁₁=𝑉₁₁(𝑘+1)−𝑉₁₁(𝑘)1≤−𝜃(𝑘)−1𝜃(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)+𝛾(𝑘)𝑥(𝑘)−𝑥^∗(𝑘(𝑘𝑢(𝑘))+𝑔)−𝑢^∗(𝑘(𝑘))+𝑏^𝑘−1_{𝑠=𝑘−𝑙}𝑃(𝑠)𝛽(𝑠)+𝛾(𝑠)𝑄(𝑠)𝑥𝑥(𝑠)−𝑥^∗(𝑠(𝑠)𝑄(𝑠))+𝑏𝑥𝑥(𝑠−𝑙)−𝑥^∗(𝑠−𝑙)+𝑔(𝑠)𝑄(𝑠)𝑥𝑢(𝑠)−𝑢^∗(𝑠). (3.14)Step 2. Let 𝑉₁₂(𝑘)=^{𝑘−1+𝑙}_𝑠=𝑘𝑏(𝑠)^𝑘−1_{𝑖=𝑠−𝑙}𝑃(𝑖)𝛽(𝑖)+𝛾(𝑖)𝑄(𝑖)𝑥𝑥(𝑖)−𝑥^∗(𝑖(𝑖)𝑄(𝑖))+𝑏𝑥𝑥(𝑖−𝑙)−𝑥^∗(𝑖−𝑙)+𝑔(𝑖)𝑄(𝑖)𝑥𝑢(𝑖)−𝑢^∗(𝑖). (3.15) Then Δ𝑉₁₂=𝑉₁₂(𝑘+1)−𝑉₁₂(𝑘)=^𝑘+𝑙_𝑠=𝑘+1𝑏(𝑠)𝑃(𝑘)𝛽(𝑘)+𝛾(𝑘)𝑄(𝑘)𝑥𝑥(𝑘)−𝑥^∗(𝑘(𝑘)𝑄(𝑘))+𝑏𝑥𝑥(𝑘−𝑙)−𝑥^∗(𝑘−𝑙)+𝑔(𝑘)𝑄(𝑘)𝑥𝑢(𝑘)−𝑢^∗(𝑘(𝑘))−𝑏^𝑘−1_{𝑖=𝑘−𝑙}𝑃(𝑖)𝛽(𝑖)+𝛾(𝑖)𝑄(𝑖)𝑥𝑥(𝑖)−𝑥^∗(𝑖(𝑖)𝑄(𝑖))+𝑏𝑥𝑥(𝑖−𝑙)−𝑥^∗(𝑖−𝑙)+𝑔(𝑖)𝑄(𝑖)𝑥𝑢(𝑖)−𝑢^∗(𝑖). (3.16)Step 3. Let 𝑉₁₃(𝑘)=𝑥^𝑘−1_{𝑤=𝑘−𝑙}𝑏(𝑤+𝑙)𝑄(𝑤+𝑙)𝑥(𝑤)−𝑥^∗(𝑤)^𝑤+2𝑙_{𝑠=𝑤+𝑙+1}𝑏(𝑠). (3.17) By a simple calculation, it derives that 𝑉₁₃=𝑉₁₃(𝑘+1)−𝑉₁₃(𝑘)=^𝑘+2𝑙_{𝑠=𝑘+𝑙+1}𝑏(𝑠)𝑏(𝑘+𝑙)𝑄(𝑘+𝑙)𝑥𝑥(𝑘)−𝑥^∗(𝑘−_{𝑘+𝑙𝑠=𝑘+1})𝑏(𝑠)𝑏(𝑘)𝑄(𝑘)𝑥𝑥(𝑘−𝑙)−𝑥^∗(𝑘−𝑙). (3.18) Now, we can define 𝑉₁(𝑘)=𝑉₁₁(𝑘)+𝑉₁₂(𝑘)+𝑉₁₃(𝑘). (3.19) Then for all 𝑘≥𝑛₀, it follows from (3.14)–(3.18) that 𝑉₁=𝑉₁(𝑘+1)−𝑉₁(𝑘)1≤−𝜃(𝑘)−1𝜃(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)+𝛾(𝑘)𝑥(𝑘)−𝑥^∗(𝑘)+𝑔(𝑘)𝑢(𝑘)−𝑢^∗(𝑘+_{𝑘+𝑙𝑠=𝑘+1})𝑏(𝑠)𝑔(𝑘)𝑄(𝑘)𝑥𝑢(𝑘)−𝑢^∗(𝑘)+^𝑘+2𝑙_{𝑠=𝑘+𝑙+1}𝑏(𝑠)𝑏(𝑘+𝑙)𝑄(𝑘+𝑙)𝑥𝑥(𝑘)−𝑥^∗(𝑘+_{𝑘+𝑙𝑠=𝑘+1})𝑏(𝑠)𝑃(𝑘)𝛽(𝑘)+𝛾(𝑘)𝑄(𝑘)𝑥𝑥(𝑘)−𝑥^∗(𝑘). (3.20)For the second equation of system (1.5), we will consider the following two steps.Step 1. Let 𝑉₂₁(𝑘)=|𝑢(𝑘)−𝑢^∗(𝑘)|. Then 𝑉₂₁=𝑉₂₁(𝑘+1)−𝑉₂₁(𝑘)=𝑢(𝑘+1)−𝑢^∗(𝑘+1)−𝑢(𝑘)−𝑢^∗(𝑘=(𝑘𝑢(𝑘))(1−𝑒))−𝑢^∗(𝑘)+ℎ(𝑘)𝑥(𝑘−𝑙)−𝑥^∗(𝑘−𝑙)−𝑢(𝑘)−𝑢^∗(𝑘≤((𝑘𝑢(𝑘))1−𝑒))−𝑢^∗(𝑘)−𝑢(𝑘)−𝑢^∗(𝑘)+ℎ(𝑘)𝑥(𝑘−𝑙)−𝑥^∗((𝑘𝑢(𝑘)𝑘−𝑙)≤−𝑒)−𝑢^∗(𝑘)+ℎ(𝑘)𝑥(𝑘−𝑙)−𝑥^∗(𝑘−𝑙). (3.21)Step 2. Let 𝑉₂₂(𝑘)=∑^𝑘−1_{𝑤=𝑘−𝑙}ℎ(𝑤+𝑙)|𝑥(𝑤)−𝑥^∗(𝑤)|. Then 𝑉₂₂=𝑉₂₂(𝑘+1)−𝑉₂₂(𝑘)=^𝑘_{𝑤=𝑘+1−𝑙}ℎ(𝑤+𝑙)𝑥(𝑤)−𝑥^∗(𝑤)−^𝑘−1_{𝑤=𝑘−𝑙}ℎ(𝑤+𝑙)𝑥(𝑤)−𝑥^∗(𝑤(𝑥(𝑘))=ℎ𝑘+𝑙)−𝑥^∗(𝑘)−ℎ(𝑘)𝑥(𝑘−𝑙)−𝑥^∗(𝑘−𝑙). (3.22) So we can define 𝑉₂(𝑘)=𝑉₂₁(𝑘)+𝑉₂₂(𝑘). (3.23) Then it follows from (3.21) and (3.22) that 𝑉₂=𝑉₂(𝑘+1)−𝑉₂(𝑘)≤ℎ(𝑘+𝑙)𝑥(𝑘)−𝑥^∗(𝑘)−𝑒(𝑘)𝑢(𝑘)−𝑢^∗(𝑘). (3.24) Now, we could define 𝑉(𝑘)=𝜌𝑉₁(𝑘)+𝛿𝑉₂(𝑘), (3.25) where 𝜌, 𝛿 are mentioned in (3.8).
Obviously, 𝑉(𝑘)≥0 for all 𝑘∈𝑍⁺ and 𝑉(𝑛₀+𝑙)<+∞. Therefore, combining (3.20) and (3.24), for all 𝑘≥𝑛₀+𝑙, we have Δ𝑉=𝜌Δ𝑉₁+𝛿Δ𝑉₂−1≤𝜌𝜃(𝑘)−1𝜃(𝑘)−𝑏(𝑘)+𝛾(𝑘)+^𝑘+2𝑙_{𝑠=𝑘+𝑙+1}𝑏(𝑠)𝑏(𝑘+𝑙)𝑄(𝑘+𝑙)𝑥+^𝑘+𝑙_𝑠=𝑘+1𝑏(𝑠)𝑃(𝑘)𝛽(𝑘)+𝛾(𝑘)𝑄(𝑘)𝑥𝑥(𝑘)−𝑥^∗(𝑘)+𝛿ℎ(𝑘+𝑙)𝑥(𝑘)−𝑥^∗(𝑘𝑔(𝑘)+)+𝜌^𝑘+𝑙_𝑠=𝑘+1𝑏(𝑠)𝑔(𝑘)𝑄(𝑘)𝑥𝑢(𝑘)−𝑢^∗(𝑘)−𝛿𝑒(𝑘)𝑢(𝑘)−𝑢^∗(𝑘𝑏)≤−𝜌min^𝐿,2𝑥−𝑏^𝑈−𝛾^𝑈−𝑙𝑏^𝑈ϝ𝑥(𝑘)−𝑥^∗(𝑘)+𝛿ℎ^𝑈𝑥(𝑘)−𝑥^∗(𝑘−)𝛿𝑒^𝐿−𝜌𝑔^𝑈+𝑙𝑏^𝑈𝑔^𝑈𝑄^𝑈𝑥𝑢(𝑘)−𝑢^∗(𝑘)≤−𝜌𝜙−𝛿ℎ^𝑈𝑥(𝑘)−𝑥^∗(𝑘)−𝛿𝑒^𝐿−𝜌𝜑𝑢(𝑘)−𝑢^∗(𝑘𝑥(𝑘))≤−𝜆−𝑥^∗(𝑘)+u(𝑘)−𝑢^∗(𝑘), (3.26) where ϝ=𝑏^𝑈𝑄^𝑈𝑥+𝑃^𝑈𝛽^𝑈+𝛾^𝑈𝑄^𝑈𝑏𝑥,𝜙=min^𝐿,2𝑥−𝑏^𝑈−𝛾^𝑈−𝑙𝑏^𝑈ϝ, (3.27)𝜑=𝑔^𝑈+𝑙𝑏^𝑈𝑔^𝑈𝑄^𝑈𝑥. (3.28) Summing both sides of (3.26) from 𝑛₀+𝑙 to 𝑘, it derives that ^𝑘_{𝑠=𝑛0+𝑙}[𝑉(𝑠+1)−𝑉(𝑠)]≤−𝜆^𝑘_{𝑠=𝑛0+𝑙}𝑥(𝑠)−𝑥^∗(𝑠)+𝑢(𝑠)−𝑢^∗(𝑠), (3.29) which implies 𝑉(𝑘+1)+𝜆^𝑘_{𝑠=𝑛0+𝑙}𝑥(𝑠)−𝑥^∗(𝑠)+𝑢(𝑠)−𝑢^∗(𝑠)≤𝑉𝑛₀+𝑙≤𝑉(0). (3.30) It follows from the above inequality that ^+∞_{𝑠=𝑛0+𝑙}𝑥(𝑠)−𝑥^∗(𝑠)+𝑢(𝑠)−𝑢^∗(𝑠≤𝑉(0))𝜆<+∞, (3.31) that is, lim_𝑘→+∞𝑥(𝑘)−𝑥^∗(𝑘)+𝑢(𝑘)−𝑢^∗(𝑘)=0, (3.32) and we can easily obtain that lim_𝑘→+∞𝑥(𝑘)−𝑥^∗(𝑘)=0,lim_𝑘→+∞𝑢(𝑘)−𝑢^∗(𝑘)=0, (3.33) which implies that the positive solutions of system (1.5) are globally attractive, this completes the proof.

4. Numerical Simulations

To verify the feasibilities of our main results, we consider a specific example:𝑥()(𝑘)(𝑘−[(𝑘𝑥()−[(𝑘𝑥(𝑘)𝑘+1=𝑥exp[0.0069+0.00069sin)]0.25−0.0025sin)]𝑘−10.0011+0.00011sin)][4.02+0.402sin(𝑘)]+𝑥²(𝑘)−[(𝑘𝑢(𝑘,𝑢()={[(𝑘𝑢(𝑘)+[(𝑘𝑥().0.034−0.0034sin)])𝑘+11−0.99+0.0099sin)]}0.042−0.0042sin)]𝑘−1 (4.1)

Obviously, 2𝑎^𝐿𝑑^𝐿≈0.0236, 𝑐^𝑈≈0.0012, that is, 2𝑎^𝐿𝑑^𝐿>𝑐^𝑈. Then the assumption of Theorem 2.4 is satisfied, which indicates that system (4.1) is permanent (see Figure 1).

(a)

(b)

(c)

(d)

Figure 1 

Permanence of system (4.1) with 𝑥(−1)=0.04 and {𝑥(0),𝑢(0)}={0.0265,0.0008}. (a) and (b) show 𝑥 for 𝑘∈[0,1000] and 𝑥 for 𝑘∈[980,1000], respectively. (c) and (d) show 𝑢 for 𝑘∈[0,1000] and 𝑢 for 𝑘∈[980,1000], respectively.

We assume 𝜌=20 and 𝛿=1.5, by a simple computation, we have 𝜙≈0.0049, 𝜑≈0.0584, 𝜆≈0.0286, then the sufficient conditions of Theorem 3.2 are satisfied. Thus, the positive solutions of system (4.1) are globally attractive. From Figure 2, we can see that 𝑥(𝑘) and 𝑢(𝑘) tend to 𝑥^∗(𝑘) and 𝑢^∗(𝑘), respectively.

(a)

(b)

(c)

(d)

Figure 2 

Global attractivity of system (4.1) with different initial values. 𝑥 with 𝑥(−1)=0.04, {𝑥(0),𝑢(0)}={0.0265,0.0008} and 𝑥∗ with 𝑥∗(−1)=0.05, {𝑥∗(0),𝑢∗(0)}={0.0274,0.0014}. (a) and (b) show (𝑥,𝑥∗) for 𝑘∈[0,1000] and for 𝑘∈[980,1000], respectively. (c) and (d) show (𝑢, 𝑢∗) for 𝑘∈[0,1000] and for 𝑘∈[980,1000], respectively.

5. Conclusion

We conclude with a brief discussion of our results. The sufficient condition required for the result of Theorem 2.4 does not depend on the size of the feedback control. However, the sufficient conditions required for the result of Theorem 3.2 show that the feedback control has an effect on the global attractivity of system (1.5).

Acknowledgments

The paper is supported by the Key Project of Chinese Ministry of Education(no. #210134), the Innovation Term of Educational Department of Hubei Province of China (no. #T200804), and the Scientific Research Foundation of the Education Department of Hubei Province of China (no. #D20101902).