Asymptotic Behaviors of a Delayed Nonautonomous Predator-Prey System Governed by Difference Equations

Abstract

Based on a predator-prey differential system with continuously distributed delays, we derive a corresponding difference version by using the method of a discretization technique. A good understanding of permanence of system and global attractivity of positive solutions of system is gained. An example and its numerical simulations are presented to substantiate our theoretical results.

1. Introduction

It is well known that the classical Lotka-Volterra differential equations for predator-prey interaction can be written as𝑥^′(𝑡)(𝑡(𝑡)(𝑡,𝑦=𝑥)𝑎−𝑏𝑥−𝑐𝑦)_′(𝑡)=𝑦(𝑡)−𝑑+𝑝𝑥(𝑡)−𝑞𝑦(𝑡), (1.1) where 𝑥(0)>0, 𝑦(0)>0, 𝑥(𝑡), and 𝑦(𝑡) are the population densities of prey and predator, respectively, at time 𝑡, and the coefficients 𝑎,𝑏,𝑐,𝑑,𝑝,𝑞 are positive constants. 𝑎 is the intrinsic growth rate of prey species, and 𝑑 represents the death rate of predator species; 𝑏,𝑞 and 𝑐,𝑝, denote their self-inhibition and interaction rates, respectively. To account for time lag effects in their interaction, Cushing [1] suggested visualizing system (1.1) as a special case of the following more general system 𝑥^′(𝑡)=𝑥(𝑡)𝑎−₀^+∞𝑥(𝑡−𝜃)𝑑𝑙₁(𝜃)−₀^+∞𝑦(𝑡−𝜃)𝑑𝑙₂(𝜃,𝑦_′)(𝑡)=𝑦(𝑡)−𝑑+₀^+∞𝑥(𝑡−𝜃)𝑑𝑙₃(𝜃)−₀^+∞𝑦(𝑡−𝜃)𝑑𝑙₄(𝜃), (1.2) where 𝑙_𝑖(𝜃) are nondecreasing functions on [0,+∞) with ∫₀^+∞𝑑𝑙_𝑖(𝜃)<+∞ for each 𝑖=1,…,4, and with lim_𝜃→0⁺[𝑙_𝑖(𝜃)−𝑙_𝑖(0)], respectively, equal to 𝑏,𝑐,𝑝,𝑞 for 𝑖=1,…,4.

As Hale [2] pointed out, one expects that the lag effects tend to diminish gradually in ever-moderating pace as one moves backward in time after the initial instantaneous effect described by the assumption lim_𝜃→0⁺[𝑙_𝑖(𝜃)−𝑙_𝑖(0)]>0. The effects are negligible after a certain length of time 𝜏>0. Mathematically, for 𝜃>0, it is reasonable that 𝑑𝑙_𝑖(𝜃) can be assumed to be of the form 𝑑𝑙_𝑖(𝜃)=𝑘_𝑖(𝜃)𝑑𝜃, 𝑖=1,…,4, where(𝑆₁)𝑘_𝑖(𝑠)≥0 and continuous for 𝑠∈[0,𝜏]; 𝑘_𝑖(𝑠)=0 for 𝑠∈[𝜏,+∞];(𝑆₂)𝑘^′_𝑖(𝑠)≤0 and continuous for 𝑠∈(0,𝜏); lim_𝑠→0⁺𝑘^′_𝑖(𝑠) and lim_{𝑠→𝜏−}𝑘^′_𝑖(𝑠) both exist;(𝑆₃)𝑘_𝑖^′′(𝑠)≥0 and continuous for 𝑠∈(0,𝜏).

The assumption (𝑆₂) represents gradual diminishing effect as one moves backward in time, and the assumption (𝑆₃) stands for the moderating pace of the decrease (see [2] or [3] for the assumptions (𝑆₁) to (𝑆₃)). Leung [4] revised system (1.2) and established the following predator-prey model with continuously distributed delays𝑥^′(𝑡)=𝑥(𝑡)𝑎−𝑏𝑥(𝑡)−𝑐𝑦(𝑡)−^𝜏₀𝑘₁(𝑠)𝑥(𝑡−𝑠)𝑑𝑠−^𝜏₀𝑘₂(𝑠)𝑦(),𝑦𝑡−𝑠𝑑𝑠_′(𝑡)=𝑦(𝑡)−𝑑+𝑝𝑥(𝑡)−𝑞𝑦(𝑡)+^𝜏₀𝑘₃(𝑠)𝑥(𝑡−𝑠)𝑑𝑠−^𝜏₀𝑘₄(𝑠)𝑦(𝑡−𝑠)𝑑𝑠. (1.3) The parameters of system (1.3) are assumed to be constant; however, in the real world the parameters are not fixed constants owing to the variation of environment. The effect of a varying environment is significant for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, it is realistic to assume that the parameters of system (1.3) are continuous functions with respect to 𝑡, and then we obtain the following form:𝑥′(𝑡)=𝑥(𝑡)𝑎(𝑡)−𝑏(𝑡)𝑥(𝑡)−𝑐(𝑡)𝑦(𝑡)−^𝜏₀𝑘₁(𝑠)𝑥(𝑡−𝑠)𝑑𝑠−^𝜏₀𝑘₂(𝑠)𝑦(),(𝑡)(𝑡(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑦(𝑡)+𝑡−𝑠𝑑𝑠𝑦′=𝑦)−𝑑+𝑝−𝑞^𝜏₀𝑘₃(𝑠)𝑥(𝑡−𝑠)𝑑𝑠−^𝜏₀𝑘₄(𝑠)𝑦(𝑡−𝑠)𝑑𝑠, (1.4) where the coefficients 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡),𝑝(𝑡),𝑞(𝑡) are positive continuous functions and 𝑘_𝑖(𝑠)𝑑𝑠 satisfy ∫^𝜏₀𝑘_𝑖(𝑠)𝑑𝑠=1 for 𝑖=1,…,4.

Although much progress on the predator-prey models with discrete or distributed delays has been made, such models are not sufficiently researched in the sense that most results are continuous time versions related (see [5–13]). Many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous time ones when populations have a short life expectancy, nonoverlapping generations in the real world (see [14–18]). Meanwhile, discrete time models can provide efficient computational models of continuous time models for numerical simulations. To the best of our knowledge, no such work has been done for the corresponding discrete version of system (1.4).

In the following, we employ the discretization technique to derive the discrete version of system (1.4). Throughout this paper, let ℤ, ℤ⁺ and ℝ² denote the sets of all integers, nonnegative integers, and two-dimensional vector space, respectively. We begin to approximate the continuous time system by replacing the integral term with discrete sums of the form^𝜏₀𝑘_𝑖(𝑠)𝑥(𝑡−𝑠)𝑑𝑠≈^𝑀_[]𝑠/ℎ=0𝑘_𝑖𝑠ℎℎ𝑥𝑡ℎ𝑠ℎ−ℎℎ𝑤(ℎ) (1.5) for 𝑡∈[𝑛ℎ,(𝑛+1)ℎ), 𝑠∈[𝑟ℎ,(𝑟+1)ℎ), 𝑛,𝑟∈ℤ⁺, 𝑖=1,…,4, where [⋅] denotes the greatest integer function. 𝑀=[𝜏/ℎ], 𝑤(ℎ)=ℎ+𝑂(ℎ) for ℎ>0, where the fixed number ℎ denotes an uniform discretization step size. We approximate system (1.4) by differential equations with piecewise constant arguments of the form(𝑡)(𝑡)𝑎𝑡𝑥′=𝑥ℎℎ𝑡−𝑏ℎℎ𝑥𝑡ℎℎ𝑡−𝑐ℎℎ𝑦𝑡ℎℎ−^𝑀_[]𝑠/ℎ=0𝑘₁𝑠ℎℎ𝑥𝑡ℎ𝑠ℎ−ℎℎℎ−^𝑀_[]𝑠/ℎ=0𝑘₂𝑠ℎℎ𝑦𝑡ℎ𝑠ℎ−ℎℎℎ,(𝑡)(𝑡)𝑡𝑦′=𝑦−𝑑ℎℎ𝑡+𝑝ℎℎ𝑥𝑡ℎℎ𝑡−𝑞ℎℎ𝑦𝑡ℎℎ+^𝑀_[]𝑠/ℎ=0𝑘₃𝑠ℎℎ𝑥𝑡ℎ𝑠ℎ−ℎℎℎ−^𝑀_[]𝑠/ℎ=0𝑘₄𝑠ℎℎ𝑦𝑡ℎ𝑠ℎ−ℎℎℎ (1.6) for 𝑡∈[𝑛ℎ,(𝑛+1)ℎ), 𝑠∈[𝑟ℎ,(𝑟+1)ℎ), 𝑛,𝑟∈ℤ⁺. Noting that [𝑡/ℎ]=𝑛, [𝑠/ℎ]=𝑟, we integrate (1.6) over [𝑛ℎ,𝑡), where 𝑡<(𝑛+1)ℎ, then (1.6) can be reformulated as𝑥′(𝑡)=𝑥(𝑡)𝑎(𝑛ℎ)−𝑏(𝑛ℎ)𝑥(𝑛ℎ)−𝑐(𝑛ℎ)𝑦(𝑛ℎ)−^𝑀_𝑟=0𝑘₁(𝑟ℎ)𝑥(𝑛ℎ−𝑟ℎ)ℎ−^𝑀_𝑟=0𝑘₂()𝑦()ℎ(𝑡)(𝑡()()𝑥()()𝑦()+𝑟ℎ𝑛ℎ−𝑟ℎ𝑦′=𝑦)−𝑑𝑛ℎ+𝑝𝑛ℎ𝑛ℎ−𝑞𝑛ℎ𝑛ℎ^𝑀_𝑟=0𝑘₃(𝑟ℎ)𝑥(𝑛ℎ−𝑟ℎ)ℎ−^𝑀_𝑟=0𝑘₄(𝑟ℎ)𝑦(𝑛ℎ−𝑟ℎ)ℎ. (1.7) Denoting 𝑥(𝑛)=𝑥(𝑛ℎ), 𝑦(𝑛)=𝑦(𝑛ℎ), 𝑎(𝑛)=𝑎(𝑛ℎ), 𝑏(𝑛)=𝑏(𝑛ℎ), 𝑐(𝑛)=𝑐(𝑛ℎ), 𝑑(𝑛)=𝑑(𝑛ℎ), 𝑝(𝑛)=𝑝(𝑛ℎ), 𝑞(𝑛)=𝑞(𝑛ℎ), 𝑘_𝑖(𝑟)=𝑘_𝑖(𝑟ℎ)ℎ, 𝑖=1,…,4, then we have𝑥(𝑡)=𝑥(𝑛)exp𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑐(𝑛)𝑦(𝑛)−^𝑀_𝑟=0𝑘₁(𝑟)𝑥(𝑛−𝑟)−^𝑀_𝑟=0𝑘₂(𝑟)𝑦(,𝑦(𝑡)(𝑛)(𝑛)(𝑛)𝑥(𝑛)(𝑛)𝑦(𝑛)+𝑛−𝑟)(𝑡−𝑛ℎ)=𝑦exp−𝑑+𝑝−𝑞^𝑀_𝑟=0𝑘₃(𝑟)𝑥(𝑛−𝑟)−^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛−𝑟)(𝑡−𝑛ℎ). (1.8) Setting 𝑡→(𝑛+1)ℎ in (1.8) and simplifying, we get a discrete time analogue of continuous time system (1.4) with the form𝑥(𝑛+1)=𝑥(𝑛)exp𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑐(𝑛)𝑦(𝑛)−^𝑀_𝑟=0𝑘₁(𝑟)𝑥(𝑛−𝑟)−^𝑀_𝑟=0𝑘₂(𝑟)𝑦(𝑦()(𝑛)(𝑛)(𝑛)𝑥(𝑛)(𝑛)𝑦(𝑛)+𝑛−𝑟)𝑛+1=𝑦exp−𝑑+𝑝−𝑞^𝑀_𝑟=0𝑘₃(𝑟)𝑥(𝑛−𝑟)−^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛−𝑟). (1.9) Our main purpose of this paper is to derive a set of easily verifiable sufficient conditions concerning the permanence and global attractivity of system (1.9). For convenience, we introduce the following definitions and notations.

Definition 1.1. It is said that system (1.9) is permanent if there exist positive constants 𝑚_𝑖,𝑀_𝑖, 𝑖=1,2 and any positive solution {(𝑥(𝑛),𝑦(𝑛))^⊤} of system (1.9) satisfies 𝑚₁≤liminf_𝑛→+∞𝑥(𝑛)≤limsup_𝑛→+∞𝑥(𝑛)≤𝑀₁,𝑚₂≤liminf_𝑛→+∞𝑦(𝑛)≤limsup_𝑛→+∞𝑦(𝑛)≤𝑀₂. (1.10)

Definition 1.2. A positive solution {(𝑥(𝑛),𝑦(𝑛))^⊤} of system (1.9) is global attractive if each other positive solution {(𝑥^∗(𝑛),𝑦^∗(𝑛))^⊤} of system (1.9) satisfies lim_𝑛→+∞𝑥(𝑛)−𝑥^∗(𝑛)=0,lim_𝑛→+∞𝑦(𝑛)−𝑦^∗(𝑛)=0. (1.11) Let 𝑓(𝑛) be a bounded sequence; we denote 𝑓^𝑢=sup_𝑛∈ℤ+𝑓(𝑛),𝑓^𝑙=inf_𝑛∈ℤ+𝑓(𝑛). (1.12) Meanwhile, we make a convention that ∏^𝑛=𝑘_𝑛=𝑚𝑓(𝑛)=1 for all 𝑚>𝑘.

This paper proceeds as follows. System (1.9) is analyzed to study the permanence and global attractivity of system (1.9) in the next two sections, respectively. In the final section, we give an example, and its numerical simulations are presented to substantiate our theoretical results.

2. Permanence of System (1.9)

In this section, we devote to investigating the permanence of system (1.9). To do so, we introduce the following lemmas.

Lemma 2.1 (see [19]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)≤𝑥(𝑛)exp[𝑟(𝑛)(1−𝑎𝑥(𝑛))] (2.1) for 𝑛∈[𝑛₁,+∞), where 𝑎 is a positive constant and 𝑛₁∈ℤ⁺. Then lim_𝑛→+∞(𝑛)≤1sup𝑥𝑎𝑟^𝑢exp𝑟^𝑢−1. (2.2)

Lemma 2.2 (see [19]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)≥𝑥(𝑛)exp[𝑟(𝑛)(1−𝑎𝑥(𝑛))] (2.3) for 𝑛∈[𝑛₂,+∞), limsup_𝑛→+∞𝑥(𝑛)≤𝐻 and 𝑥(𝑛₂)>0, where 𝑎 is a positive constant such that 𝑎𝐻>1 and 𝑛₂∈ℤ⁺. Then lim_𝑛→+∞(𝑛)≥1inf𝑥𝑎exp𝑟^𝑢(1−𝑎𝐻). (2.4)

For the simplicity of description, we denoteΔ^𝜀₁=−𝑑^𝑙+𝑝^𝑢+(𝑀+1)𝑘^𝑢₃𝑀₁,Δ+𝜀_𝜀2=𝑎^𝑙−𝑐^𝑢𝑀₂+𝜀−(𝑀+1)𝑘^𝑢₁𝑀₁+𝜀+𝑘^𝑢₂𝑀₂,Δ+𝜀_𝜀3=−𝑑^𝑢+𝑝^𝑙𝑚₁−𝜀+(𝑀+1)𝑘^𝑙₃𝑚₁−𝜀−𝑘^𝑢₄𝑀₂,Δ+𝜀_𝑖=lim_𝜀→0Δ^𝜀_𝑖,𝑖=1,2,3, (2.5) where 𝑀₁,𝑀₂,𝑚₁ are, respectively, defined in (2.8) and (2.11)–(2.14) and 𝜀 is a sufficient small positive constant. Now, we begin to search the conditions for the permanence of (1.9).

Theorem 2.3. If the conditions minΔ_𝑖,𝑖=1,2,3>0 (2.6) hold, then (1.9) is permanent.

Proof. It is easy to verify that Ω={(𝑥(𝑛),𝑦(𝑛))^⊤∣𝑥(𝑛)>0,𝑦(𝑛)>0} is a positive invariant set of (1.9) with 𝑥(0)>0, 𝑦(0)>0. Suppose that {(𝑥(𝑛),𝑦(𝑛))^⊤} is any positive solution of (1.9), we prove Theorem 2.3 by the following two steps.
Step 1. We show that {(𝑥(𝑛),𝑦(𝑛))^⊤} is uniformly ultimately upper bounded.
From the first equation of (1.9), we have 𝑥()(𝑛)[𝑎(𝑛)(𝑛)𝑥(𝑛(𝑛)𝑎(𝑛𝑏𝑛+1≤𝑥exp−𝑏)]≤𝑥exp)1−^𝑙𝑎^𝑢𝑥(𝑛). (2.7) Applying Lemma 2.1, we can obtain that lim_𝑛→+∞(𝑛)≤𝑎sup𝑥exp^𝑢−1𝑏^𝑙def=𝑀₁. (2.8) From (2.8), there exists a sufficient large 𝑛₁∈ℤ⁺ and any constant 𝜀>0 such that 𝑥(𝑛)≤𝑀₁+𝜀,𝑛≥𝑛₁. (2.9) It follows from the second equation of (1.9) that 𝑦(𝑛+1)≤𝑦(𝑛)exp−𝑑^𝑙+𝑝^𝑢+(𝑀+1)𝑘^𝑢₃𝑀₁+𝜀−𝑞^𝑙𝑦(𝑛(𝑛)Δ)=𝑦exp^𝜀₁𝑞1−^𝑙Δ^𝜀₁𝑦(𝑛). (2.10) So by Lemma 2.1 and letting 𝜀→0 in (2.10), we have lim_𝑛→+∞(𝑛)≤Δsup𝑦exp₁−1𝑞^𝑙def=𝑀₂. (2.11)Step 2. We prove that {(𝑥(𝑛),𝑦(𝑛))^⊤} is uniformly ultimately lower bounded.
For any sufficient small 𝜀>0, it follows from (2.6) that Δ^𝜀₂>0. According to (2.11), there exists an 𝑛₀>𝑛₁ such that 𝑦(𝑛)≤𝑀₂+𝜀 for 𝑛≥𝑛₀ and the above constant 𝜀. By the first equation of (1.9), it gives that 𝑥(𝑛+1)≥𝑥(𝑛)exp𝑎^𝑙−𝑐^𝑢𝑀₂+𝜀−(𝑀+1)𝑘^𝑢₁𝑀₁+𝜀+𝑘^𝑢₂𝑀₂+𝜀−𝑏^𝑢𝑥(𝑛(𝑛)Δ)=𝑥exp^𝜀₂𝑏1−^𝑢Δ^𝜀₂𝑥(𝑛). (2.12) Note that 𝑏^𝑢Δ^𝜀₂𝑀₁=𝑏^𝑢Δ^𝜀₂exp𝑎^𝑢−1𝑏^𝑙≥𝑏^𝑢𝑎^𝑙exp𝑎^𝑢−1𝑏^𝑙>exp𝑎^𝑢−1𝑎^𝑢>1, (2.13) where we employ the inequality result that exp(𝑥−1)>𝑥 for 𝑥>0. Therefore, applying Lemma 2.2 and setting 𝜀→0, one has lim_𝑛→+∞(𝑛)≥Δinf𝑥²𝑏^𝑢expΔ₂𝑏1−^𝑢Δ₂𝑀=₁^def𝑚₁. (2.14) For the above constant 𝜀>0, it follows from (2.14) that there exists a sufficient large 𝑛^∗₁∈ℤ⁺ such that 𝑥(𝑛)≥𝑚₁−𝜀,𝑛≥𝑛^∗₁. (2.15) From the second equation of (1.9), we have 𝑦(𝑛+1)≥𝑦(𝑛)exp−𝑑^𝑢+𝑝^𝑙𝑚₁−𝜀+(𝑀+1)𝑘^𝑙₃𝑚₁−𝜀−𝑘^𝑢₄𝑀₂+𝜀−𝑞^𝑢𝑦(𝑛(𝑛)Δ)=𝑦exp^𝜀₃𝑞1−^𝑢Δ^𝜀₃𝑦(𝑛). (2.16) Clearly, 𝑞^𝑢Δ^𝜀₃𝑀₂=𝑞^𝑢Δ^𝜀₃expΔ₁−1𝑞^𝑙≥𝑞^𝑙exp−𝑑^𝑢+𝑝^𝑙+(𝑀+1)𝑘^𝑙₃𝑚₁−1−𝑑^𝑢+𝑝^𝑙+(𝑀+1)𝑘^𝑙₃𝑚₁𝑞^𝑙>1. (2.17) Thus, applying Lemma 2.2 and letting 𝜀→0, one has lim_𝑛→+∞(𝑛)≥Δinf𝑦³𝑞^𝑢expΔ₃𝑞1−^𝑢Δ₃𝑀=₂^def𝑚₂. (2.18) Based on the above Step 1 with Step 2, we can see that system (1.9) is permanent. This completes the proof of Theorem 2.3.

3. Global Attractivity of Positive Solutions of System (1.9)

In this section, we investigate the global attractivity of positive solutions of system (1.9).

Theorem 3.1. Assume that there exists a positive constant 𝜂 such that min𝑏^𝑙,2𝑀₁−𝑏^𝑢−(𝑀+1)𝑘^𝑢₁+𝑘^𝑢₃−𝑝^𝑢𝑞≥𝜂,min^𝑙,2𝑀₂−𝑞^𝑢−(𝑀+1)𝑘^𝑢₂+𝑘^𝑢₄−𝑐^𝑢≥𝜂 (3.1) hold. Then any positive solution {(𝑥(𝑛),𝑦(𝑛))^⊤} of system (1.9) is global attractive.

Proof. Denote {(𝑥^∗(𝑛),𝑦^∗(𝑛))^⊤} be any other positive solution of system (1.9). Let 𝑉₁₁(𝑛)=|ln𝑥(𝑛)−ln𝑥^∗(𝑛)|, then it follows from the first equation of (1.9) that 𝑉₁₁(𝑛+1)=ln𝑥(𝑛+1)−ln𝑥^∗(≤(𝑛)𝑛+1)ln𝑥−ln𝑥^∗(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑥^∗(𝑛)+𝑐(𝑛)𝑦(𝑛)−𝑦^∗(𝑛+_𝑀𝑟=0)𝑘₁(𝑟)𝑥(𝑛−𝑟)−𝑥^∗(+𝑛−𝑟)_𝑀𝑟=0𝑘₂(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(𝑛−𝑟). (3.2) By the Mean Value theorem, we get ln𝑥(𝑛)−ln𝑥^∗(𝑛)=1𝜃₁(𝑛)𝑥(𝑛)−𝑥^∗(𝑛), (3.3) where 𝜃₁(𝑛) lies between 𝑥(𝑛) and 𝑥^∗(𝑛). Then we have ln𝑥(𝑛)−ln𝑥^∗(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑥^∗(𝑛)=ln𝑥(𝑛)−ln𝑥^∗(𝑛)−ln𝑥(𝑛)−ln𝑥^∗(𝑛)+ln𝑥(𝑛)−ln𝑥^∗(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑥^∗(𝑛)=ln𝑥(𝑛)−ln𝑥^∗(𝑛−1)𝜃₁(𝑛)𝑥(𝑛)−𝑥^∗(𝑛+1)𝜃₁(𝑛)𝑥(𝑛)−𝑥^∗(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑥^∗(𝑛)=ln𝑥(𝑛)−ln𝑥^∗(𝑛−1)𝜃₁(𝑛)−1𝜃₁(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑥^∗(𝑛). (3.4) Combining (3.2) with (3.4), we have Δ𝑉₁₁(𝑛)=𝑉₁₁(𝑛+1)−𝑉₁₁(𝑛)1≤−𝜃₁(𝑛)−1𝜃₁(𝑛)−𝑏(𝑛)𝑥(𝑛)−𝑥^∗(𝑛)+𝑐(𝑛)𝑦(𝑛)−𝑦^∗(𝑛+_𝑀𝑟=0)𝑘₁(𝑟)𝑥(𝑛−𝑟)−𝑥^∗(𝑛−𝑟)+^𝑀_𝑟=0𝑘₂(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(𝑛−𝑟). (3.5) Next, we let 𝑉₁₂(𝑛)=^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₁(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)+^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₂(𝑟)𝑦(𝑠)−𝑦^∗(𝑠). (3.6) Then we have Δ𝑉₁₂(𝑛)=𝑉₁₂(𝑛+1)−𝑉₁₂(𝑛)=^𝑀𝑛_{𝑟=0𝑠=𝑛+1−𝑟}𝑘₁(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)−^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₁(𝑟)𝑥(𝑠)−𝑥^∗(𝑠+_{𝑀𝑛𝑟=0𝑠=𝑛+1−𝑟})𝑘₂(𝑟)𝑦(𝑠)−𝑦^∗(𝑠)−^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₂(𝑟)𝑦(𝑠)−𝑦^∗(𝑠=_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟})𝑘₁(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)+^𝑀_𝑟=0𝑘₁(𝑟)𝑥(𝑛)−𝑥^∗(𝑛−_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟})𝑘₁(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)−^𝑀_𝑟=0𝑘₁(𝑟)𝑥(𝑛−𝑟)−𝑥^∗(+𝑛−𝑟)_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟}𝑘₂(𝑟)𝑦(𝑠)−𝑦^∗(𝑠)+^𝑀_𝑟=0𝑘₂(𝑟)𝑦(𝑛)−𝑦^∗(𝑛−_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟})𝑘₂(𝑟)𝑦(𝑠)−𝑦^∗(𝑠)−^𝑀_𝑟=0𝑘₂(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(=𝑛−𝑟)_𝑀𝑟=0𝑘₁(𝑟)𝑥(𝑛)−𝑥^∗(𝑛)−^𝑀_𝑟=0𝑘₁(r)𝑥(𝑛−𝑟)−𝑥^∗(+𝑛−𝑟)_𝑀𝑟=0𝑘₂(𝑟)𝑦(𝑛)−𝑦^∗(𝑛)−^𝑀_𝑟=0𝑘₂(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(𝑛−𝑟). (3.7) We set 𝑉₁(𝑛)=𝑉₁₁(𝑛)+𝑉₁₂(𝑛), (3.8) then it follows from (3.5) and (3.7) that Δ𝑉₁(𝑛)=Δ𝑉₁₁(𝑛)+Δ𝑉₁₂(𝑛)1≤−𝜃₁(𝑛)−1𝜃₁(𝑛)−𝑏(𝑛)−^𝑀_𝑟=0𝑘₁(𝑟)𝑥(𝑛)−𝑥^∗(𝑛+𝑐(𝑛)+_𝑀𝑟=0)𝑘₂(𝑟)𝑦(𝑛)−𝑦^∗(𝑛). (3.9) Similarly, we define 𝑉₂(𝑛)=𝑉₂₁(𝑛)+𝑉₂₂(𝑛), (3.10) where 𝑉₂₁(𝑛)=^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₃(𝑟)𝑥(𝑠)−𝑥^∗(𝑠+_{𝑀𝑟=0𝑛−1𝑠=𝑛−𝑟})𝑘₄(𝑟)𝑦(𝑠)−𝑦^∗(𝑠,𝑉₂₂)(𝑛)=ln𝑦(𝑛)−ln𝑦^∗(𝑛). (3.11) Then we have Δ𝑉₂₁(𝑛)=𝑉₂₁(𝑛+1)−𝑉₂₁(𝑛)=^𝑀𝑛_{𝑟=0𝑠=𝑛+1−𝑟}𝑘₃(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)−^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₃(𝑟)𝑥(𝑠)−𝑥^∗(𝑠+_{𝑀𝑛𝑟=0𝑠=𝑛+1−𝑟})𝑘₄(𝑟)𝑦(𝑠)−𝑦^∗(𝑠)−^𝑀_𝑟=0^𝑛−1_{𝑠=𝑛−𝑟}𝑘₄(𝑟)𝑦(𝑠)−𝑦^∗(𝑠=_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟})𝑘₃(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)+^𝑀_𝑟=0𝑘₃(𝑟)𝑥(𝑛)−𝑥^∗(𝑛−_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟})𝑘₃(𝑟)𝑥(𝑠)−𝑥^∗(𝑠)−^𝑀_𝑟=0𝑘₃(𝑟)𝑥(𝑛−𝑟)−𝑥^∗(+𝑛−𝑟)_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟}𝑘₄(𝑟)𝑦(𝑠)−𝑦^∗(𝑠)+^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛)−𝑦^∗(𝑛−_{𝑀𝑟=0𝑛−1𝑠=𝑛+1−𝑟})𝑘₄(𝑟)𝑦(𝑠)−𝑦^∗(𝑠)−^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(=𝑛−𝑟)_𝑀𝑟=0𝑘₃(𝑟)𝑥(𝑛)−𝑥^∗(𝑛)−^𝑀_𝑟=0𝑘₃(𝑟)𝑥(𝑛−𝑟)−𝑥^∗(+𝑛−𝑟)_𝑀𝑟=0𝑘₄(𝑟)𝑦(𝑛)−𝑦^∗(𝑛)−^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(𝑛−𝑟),Δ𝑉₂₂(𝑛)=𝑉₂₂(𝑛+1)−𝑉₂₂(𝑛)1≤−𝜃₂(𝑛)−1𝜃₂(𝑛)−𝑞(𝑛)𝑦(𝑛)−𝑦^∗(𝑛)+𝑝(𝑛)𝑥(𝑛)−𝑥^∗(𝑛+_𝑀𝑟=0)𝑘₃(𝑟)𝑥(𝑛−𝑟)−𝑥^∗(𝑛−𝑟)+^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛−𝑟)−𝑦^∗(𝑛−𝑟), (3.12) where 𝜃₂(𝑛) is between 𝑦(𝑛) and 𝑦^∗(𝑛). Consequently, one has Δ𝑉₂(𝑛)=Δ𝑉₂₁(𝑛)+Δ𝑉₂₂(𝑛)1≤−𝜃₂(𝑛)−1𝜃₂(𝑛)−𝑞(𝑛)−^𝑀_𝑟=0𝑘₄(𝑟)𝑦(𝑛)−𝑦^∗(𝑛+𝑝(𝑛)+_𝑀𝑟=0)𝑘₃(𝑟)𝑥(𝑛)−𝑥^∗(𝑛). (3.13) Now, we consider a Lyapunov-like discrete function 𝑉(𝑛) defined by 𝑉(𝑛)=𝑉₁(𝑛)+𝑉₂(𝑛). (3.14) According to (2.8) and (2.11), there exists an 𝑛^∗∈ℤ⁺ such that 𝑥(𝑛)≤𝑀₁+𝜀, 𝑦(𝑛)≤𝑀₂+𝜀 for 𝑛≥𝑛^∗ and any positive constant 𝜀. Obviously, 𝑉(𝑛)≥0 for all 𝑛∈ℤ⁺ and 𝑉(𝑛^∗+𝑀)<+∞. In view of (3.1), we choose a sufficient small 𝜀>0 such that min𝑏^𝑙,2𝑀₁+𝜀−𝑏^𝑢−(𝑀+1)𝑘^𝑢₁+𝑘^𝑢₃−𝑝^u𝑞≥𝜂,min^𝑙,2𝑀₂+𝜀−𝑞^𝑢−(𝑀+1)𝑘^𝑢₂+𝑘^𝑢₄−𝑐^𝑢≥𝜂. (3.15) Therefore, combining (3.9) with (3.13) for all 𝑛≥𝑛^∗+𝑀, we obtain Δ𝑉(𝑛)=Δ𝑉₁(𝑛)+Δ𝑉₂(𝑛)1≤−𝜃₁(𝑛)−1𝜃₁(𝑛)−𝑏(𝑛)−^𝑀_𝑟=0𝑘₁(𝑟)−^𝑀_𝑟=0𝑘₃(𝑟)−𝑝(𝑛)𝑥(𝑛)−𝑥^∗(𝑛−1)𝜃₂(𝑛)−1𝜃₂(𝑛)−𝑞(𝑛)−^𝑀_𝑟=0𝑘₄(𝑟)−^𝑀_𝑟=0𝑘₂(𝑟)−𝑐(𝑛)𝑦(𝑛)−𝑦^∗(𝑛𝑏)≤−min^𝑙,2𝑀₁+𝜀−𝑏^𝑢−(𝑀+1)𝑘^𝑢₁+𝑘^𝑢₃−𝑝^𝑢𝑥(𝑛)−𝑥^∗(𝑛−𝑞)min^𝑙,2𝑀₂+𝜀−𝑞^𝑢−(𝑀+1)𝑘^𝑢₂+𝑘^𝑢₄−𝑐^𝑢𝑦(𝑛)−𝑦^∗(𝑛𝑥(𝑛))≤−𝜂−𝑥^∗(𝑛)+𝑦(𝑛)−𝑦^∗(𝑛). (3.16) Summing both sides of (3.16) from 𝑛^∗+𝑀 to 𝑛, it derives that Δ𝑉(^𝑛_{𝑠=𝑛∗+𝑀}𝑠)=^𝑛_{𝑠=𝑛∗+𝑀}(𝑉(𝑠+1)−𝑉(𝑠))≤−𝜂^𝑛_{𝑠=𝑛∗+𝑀}𝑥(𝑠)−𝑥^∗(𝑠)+𝑦(𝑠)−𝑦^∗(𝑠), (3.17) which implies that 𝑉(𝑛+1)+𝜂^𝑛_{𝑠=𝑛∗+𝑀}𝑥(𝑠)−𝑥^∗(𝑠)+𝑦(𝑠)−𝑦^∗(𝑠)≤𝑉𝑛^∗+𝑀,forany𝑛≥𝑛^∗+𝑀,_{+∞𝑛=𝑛∗+𝑀}𝑥(𝑛)−𝑥^∗(𝑛)+𝑦(𝑛)−𝑦^∗(𝑛≤𝑉𝑛)^∗+𝑀𝜂<+∞, (3.18) from which we conclude that lim_𝑛→+∞𝑥(𝑛)−𝑥^∗(𝑛)+𝑦(𝑛)−𝑦^∗(𝑛)=0, (3.19) that is, lim_𝑛→+∞𝑥(𝑛)−𝑥^∗(𝑛)=0,lim_𝑛→+∞𝑦(𝑛)−𝑦^∗(𝑛)=0. (3.20) According to Definition 1.2, this result implies that {(𝑥(𝑛),𝑦(𝑛))^⊤} is global attractive. The proof is complete.

4. An Example and Its Numerical Simulations

In this paper, we have investigated the asymptotic behaviors of a delayed nonautonomous predator-prey difference system. Sufficient conditions which guarantee the permanence of system and global attractivity of positive solutions of system are obtained, respectively. The theoretical results are substantiated by the following example and numerical results.

Example 4.1. Consider the following system: 𝑥()(𝑛)()𝑥(𝑛)−()𝑦(𝑛)[𝑥(𝑛)(𝑦(𝑛)(,𝑦()(𝑛)()𝑥(𝑛)−()𝑦(𝑛)[𝑥(𝑛)(𝑦(𝑛)(.𝑛+1=𝑥exp0.5+0.01sin𝑛−0.38+0.02sin𝑛0.02+0.01sin𝑛−0.024+𝑥𝑛−1)]−0.001+𝑦𝑛−1)𝑛+1=𝑦exp−0.015−0.005sin𝑛+0.26+0.03sin𝑛0.97+0.02sin𝑛+0.01+𝑥𝑛−1)]−0.02+𝑦𝑛−1) (4.1) Obviously, Δ₁≈0.5175>0,Δ₂≈0.3875>0,Δ₃≈0.1347>0. (4.2) It is easy to verify that the assumptions of Theorem 2.3 are satisfied, so system (4.1) is permanent (see Figure 1). Furthermore, a calculation shows that min𝑏^𝑙,2𝑀₁−𝑏^𝑢−(𝑀+1)𝑘^𝑢₁+𝑘^𝑢₃−𝑝^𝑢𝑞≈0.0020>0.001,min^𝑙,2𝑀₂−𝑞^𝑢−(𝑀+1)𝑘^𝑢₂+𝑘^𝑢₄−𝑐^𝑢≈0.8780>0.001. (4.3) According to Theorem 3.1, any positive solution {(𝑥(𝑛),𝑦(𝑛))^⊤} of system (4.1) is global attractive (see Figure 2). From Figures 2(a1) and 2(a2), we see that 𝑥 with 𝑥(0)=1.03 tends to 𝑥^∗ with 𝑥^∗(0)=1.0. Similarly, from Figures 2(b1) and 2(b2), we see that 𝑦 with 𝑦(0)=0.31 tends to 𝑦^∗ with 𝑦^∗(0)=0.315.

(a)

(b)

Figure 1 

Permanence of (1.9) with {(𝑥(−1),𝑦(−1))⊤}={(1.0,0.29)⊤} and {(𝑥(0),𝑦(0))⊤}={(0.95,0.3)⊤}. (a) Time series of 𝑥 for 𝑛∈[0,40]. (b) Time series of 𝑦 for 𝑛∈[0,40].

Figure 2 

Dynamic behaviors of (1.9) with different initial values. (a1) and (a2) show 𝑥∗,𝑥 with 𝑥∗(−1)=1.06, 𝑥∗(0)=1.0, 𝑥(−1)=1.04, 𝑥(0)=1.03 for 𝑘∈[0,800] and 𝑘∈[700,720], respectively. (b1) and (b2) show 𝑦∗,𝑦 with 𝑦∗(−1)=0.32, 𝑦∗(0)=0.315, 𝑦(−1)=0.315, 𝑦(0)=0.31 for 𝑘∈[0,800] and 𝑘∈[700,720], respectively.

Acknowledgments

This work 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 Project of Key Laboratory of Biological Resources Protection and Utilization of Hubei Province. The authors would like to thank the anonymous referees for their helpful comments which improved the presentation of this work.