Almost Periodic Solution of a Diffusive Mixed System with Time Delay and Type III Functional Response

Abstract

A delayed predator-prey model with diffusion and competition is proposed. Some sufficient conditions on uniform persistence of the model have been obtained. By applying Liapunov-Razumikhin technique, we will point out, under almost periodic circumstances, a set of sufficient conditions that assure the existence and uniqueness of the positive almost periodic solution which is globally asymptotically stable.

1. Introduction

In the nature world, diffusion often occurs in an ecological environment; that is, species can diffuse between patches. The works about autonomous systems in this field were pioneered by Levin, after Levin [1], Kishimoto [2], and Takeuchi [3] studied this kind of model. But all the coefficients in the system they studied are constants. Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a varying environment are considered as important selective forces on systems in a fluctuating environment. More realistic and interesting models should take into account both the seasonality of the changing environment and the effects of time delays [4–7]. This motivated Chen et al. [8–11], and others to consider nonautonomous predator-prey models with almost periodic coefficients and diffusion. In this paper, we study the almost periodic solution of the delayed predator-prey model with diffusion and competition so as to obtain some conditions under which three species are uniformly persistent. In addition, we obtain that for the almost periodic system there exists a unique almost positive periodic solution which is globally asymptotically stable.

The organization of this paper is as follows. In the next section, we develop our model, establish its important properties, and give several lemmas, which will be a key for our proofs and discussions. In Section 3, sufficient conditions are given for uniform persistence of three species. In Section 4, by applying Liapunov-Razumikhin technique, we prove the existence and uniqueness of the positive almost periodic solution which is globally asymptotically stable. Finally, we give a discussion of our results.

2. Model and Preliminaries

It is assumed that the ecosystem is composed of two isolated patches, and the prey population can disperse among the patches instantaneously. The state variables of the models, 𝑥_𝑖=𝑥_𝑖(𝑡)(𝑖=1,2), describe the densities of the prey population in Patch 1 and Patch 2, respectively. We suppose that the net exchange of the prey population from Patch 𝑗 to Patch 𝑖 is proportional to the difference of the concentration between 𝑥_𝑗−𝑥_𝑖 with 𝐷_𝑖(𝑡),𝑖,𝑗=1,2. The state variables of the models, 𝑥_𝑖=𝑥_𝑖(𝑡)(𝑖=3,4), describe the densities of the predator population in Patch 1 with competition.

Let us consider the following delayed diffusive predator-prey system with competition and functional response:𝑥^′₁=𝑥₁(𝑎₁₀(𝑡)−𝑎₁₁(𝑡)𝑥₁𝛼)−¹(𝑡)𝑥²₁𝑥₃1+𝛽₁(𝑡)𝑥²₁−𝛼₂(𝑡)𝑥²₁𝑥₄1+𝛽₂(𝑡)𝑥²₁+𝐷₁(𝑡)(𝑥₂−𝑥₁𝑥),_′2=𝑥₂(𝑎₂₀(𝑡)−𝑎₂₁(𝑡)𝑥₂)+𝐷₂(𝑡)(𝑥₁−𝑥₂𝑥),_′3=𝑥₃(−𝑎₃₀(𝑡)+𝑎₃₁𝛼(𝑡)¹(𝑡)𝑥²₁(𝑡−𝜏₁)1+𝛽₁(𝑡)𝑥²₁(𝑡−𝜏₁)−𝑎₃₂(𝑡)𝑥₃−𝑎₃₄(𝑡)𝑥₄𝑥),_′4=𝑥₄(−𝑎₄₀(𝑡)+𝑎₄₁𝛼(𝑡)²(𝑡)𝑥²₁(𝑡−𝜏₂)1+𝛽₂(𝑡)𝑥²₁(𝑡−𝜏₂)−𝑎₄₂(𝑡)𝑥₄−𝑎₄₃(𝑡)𝑥₃), (2.1)with the initial condition 𝑥₁(𝑠)=𝜙₁(𝑠)∈𝐶([−𝜏,0],𝐑₊),𝑠∈[−𝜏,0],𝜙₁(0)≥0,𝑥_𝑖(0)=𝜙_𝑖≥0(constants),𝑖=2,3,4. (2.2) Here, 𝑎_𝑖0(𝑡) and 𝑎_𝑖1(𝑡)(𝑖=1,2) represent the intrinsic growth rate and the intraspecific interference coefficient of the prey population 𝑥_𝑖(𝑖=1,2), respectively. We then assume that the death rate of the predator population 𝑥_𝑖(𝑖=3,4) in Patch 1 is proportional to both the existing predator population with the proportional functions 𝑎₃₀(𝑡) and, respectively, 𝑎₄₀(𝑡) and to its square with the proportional functions 𝑎₃₂(𝑡) and, respectively, 𝑎₄₂(𝑡). The predator consumes the prey according to Holling type III functional response [12, 13], that is, 𝛼₁(𝑡)𝑥²₁𝑥₃/(1+𝛽₁(𝑡)𝑥²₁) and 𝛼₂(𝑡)𝑥²₁𝑥₄/(1+𝛽₂(𝑡)𝑥²₁). 𝜏_𝑖(𝑖=1,2) is the time to digest food in the predator organism. 𝜏=max{𝜏₁,𝜏₂}. 𝐑₊≐{𝑧∶𝑧≥0}.

We introduce some notations and definitions, and state some preliminary lemmas which will be useful for establishing our main results.

Let 𝐑⁴₊={𝑋∈𝑅⁴∶𝑋=(𝑥₁,𝑥₂,𝑥₃,𝑥₄),𝑥_𝑖>0,𝑖=1,2,3,4}. 𝐶=𝐶([−𝜏,∞)×𝐑₊×𝐑₊×𝐑₊,𝐑⁴₊). Assume Ω is a subset of 𝐑₊×𝐶([−𝜏,0],𝐑₊)×𝐑₊×𝐑₊×𝐑₊. Denote by 𝑓=(𝑓₁,𝑓₂,𝑓₃,𝑓₄)^𝑇∶Ω→𝐑⁴₊ the map defined by the right-hand side of system (2.1). If 𝑉∶𝐑₊×𝐶→𝐑₊ is a continuous function, then the upper right derivative of 𝑉(𝑡,𝑥) with respect to system (2.1) is defined as𝐷⁺𝑉(𝑡,𝑋)=lim_ℎ→0+1supℎ[𝑉(𝑡+ℎ,𝑋+ℎ𝑓(𝑡,𝑋))−𝑉(𝑡,𝑋)]. (2.3)Obviously, the global existence and uniqueness of solutions of system (2.1) are guaranteed by the smoothness properties of 𝑓 (see [14, 15] for details on fundamental properties of retarded functional differential equations).

For convenience, we introduce the following notations:𝜓=sup_𝑡≥0{𝜓(𝑡)},𝜓=inf_𝑡≥0{𝜓(𝑡)}. (2.4)In this paper, we need all the coefficients to satisfymin_{𝑖=1,2,3,4,𝑗=0,1,2,3,}{𝑎_𝑖𝑗,𝛼_𝑖,𝛽_𝑖,𝐷_𝑖}>0,max_{𝑖=1,2,3,4,𝑗=0,1,2,3.}{𝑎_𝑖𝑗,𝛼_𝑖,𝛽_𝑖,𝐷_𝑖}<∞. (2.5)

Definition 2.1. System (2.1) is said to be uniformly persistent if there exists a compact region 𝐷⊂𝑅⁴₊ such that every solution (𝑥₁,𝑥₂,𝑥₃,𝑥₄) of system (2.1) with initial conditions (2.2) eventually enters and remains in the region 𝐷.

For convenience, the set CIP={𝑢∶[0,∞)→[0,∞)∣𝑢(𝑠)is positive and nondecreasing for 𝑠>0,𝑢(0)=0}.

Lemma 2.2 (See [16, 17]). Consider the following almost periodic equation: 𝑥^′(𝑡)=𝑔(𝑡,𝑥_𝑡). (2.6)Let 𝐶_𝐻^∗={𝑥_𝑡∈𝐶∶‖𝑥_𝑡‖=sup_{𝜃∈[−𝜏,0]}|𝑥_𝑡(𝜃)|<𝐻^∗}, 𝑆_𝐻^∗={𝑥∈𝐑^𝑛∶|𝑥|<𝐻^∗}, 𝐻^∗∈𝐑₊ or 𝐻^∗=+∞, 𝑔∶𝐑×𝐶_𝐻^∗→𝐑^𝑛, and 𝑔 is uniformly almost periodic with respect to 𝑡. Let 𝑉∶𝐑₊×𝑆_𝐻^∗×𝑆_𝐻^∗→𝐑₊. Assume that the following conditions hold:
(i)𝑎(‖𝑥−𝑦‖)≤𝑉(𝑡,𝑥,𝑦)≤𝑏(‖𝑥−𝑦‖),𝑎(⋅),𝑏(⋅)∈𝐶𝐼𝑃,𝑏(0)>0;(ii)|𝑉(𝑡,𝑥₁,𝑦₁)−𝑉(𝑡,𝑥₂,𝑦₂)|≤𝐿(‖𝑥₁−𝑥₂‖+‖𝑦₁−𝑦₂‖), where 𝐿 is a positive constant;(iii) there exists a continuous nondecreasing function 𝑃(𝑆) such that𝐷𝑃(𝑆)>𝑆if𝑆>0,₊(𝑉(𝑡,𝑥₁(𝑡),𝑥₂(𝑡)))≤−𝒞𝑉(𝑡,𝑥₁(𝑡),𝑥₂(𝑡)),𝒞∈𝐑₊,if𝑃(𝑉(𝑡,𝑥₁(𝑡),𝑥₂(𝑡)))≥𝑉(𝑡+𝜃,𝑥₁(𝑡+𝜃),𝑥₂(𝑡+𝜃)),𝜃∈[−𝜏,0]. (2.7) If system (2.6) has a solution 𝜉(𝑡)∶‖𝜉_𝑡‖≤𝐻<𝐻^∗,𝑡≥𝑡₀, then system (2.6) has a unique positive almost periodic solution 𝜂(𝑡) which is uniformly asymptotically stable, and mod(𝜂)⊂mod(𝑔). Furthermore, if 𝑔 is 𝜔-periodic with respect to 𝑡, then system (2.6) has a positive 𝜔-periodic solution which is globally asymptotically stable.

Here, mod(𝜙) denotes the module of 𝜙(𝑡) which is the set consisting of all real numbers which are finite linear combinations of elements of the setΛ={𝛼∈𝐑∣lim_𝑇→∞1𝑇^𝑇₀𝜙(𝑡)exp(−𝑖𝛼𝑡)𝑑𝑡≠0} (2.8)with integer coefficients.

Lemma 2.3. 𝐑⁴₊={(𝑥₁,𝑥₂,𝑥₃,𝑥₄)∣𝑥_𝑖>0,𝑖=1,2,3,4} is a positive invariant set of system (2.1).

Proof. Let (𝑥₁,𝑥₂,𝑥₃,𝑥₄) be a solution of system (2.1) with initial conditions (2.2). Hence, for 𝑡∈𝐑 and (𝑥₁,𝑥₂,𝑥₃,𝑥₄)∈𝐑⁴₊, we can derive𝑥^′₁|_𝑥1=0=𝐷₁(𝑡)𝑥₂>0for𝑥₂𝑥>0,_′2|_𝑥2=0=𝐷₂(𝑡)𝑥₁>0for𝑥₁𝑥>0,₃>𝑥₃(0)exp(^𝑡₀(−𝑎₃₀(𝑠)−𝑎₃₂(𝑠)𝑥₃(𝑠)−𝑎₃₄(𝑠)𝑥₄𝑥(𝑠))𝑑𝑠)>0,₄>𝑥₄(0)exp(^𝑡₀(−𝑎₄₀(𝑠)−𝑎₄₂(𝑠)𝑥₄(𝑠)−𝑎₄₃(𝑠)𝑥₃(𝑠))𝑑𝑠)>0. (2.9)Therefore, we obtain the positive invariance of 𝐑⁴₊. This completes the proof.

We will focus our discussion on 𝐑⁴₊ with respect to a biological meaning. This also ensures the solution with positive initial value to be positive all the time.

3. Uniform Persistence

In what follows, we want to construct an ultimately bounded region of system (2.1).

Theorem 3.1. There exist three constants 𝑀_𝑖>𝑀^∗_𝑖(𝑖=1,2,3) such that 𝑥_𝑗(𝑡)≤𝑀₁(𝑗=1,2), 𝑥₃(𝑡)≤𝑀₂, and 𝑥₄(𝑡)≤𝑀₃ for each positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)) of system (2.1) with 𝑡 large enough, where𝑀^∗₁=max{𝑎₁₀𝑎₁₁,𝑎₂₀𝑎₂₁},𝑀^∗₂=𝐴𝑎₃₂,𝑀^∗₃=𝐵𝑎₄₂,𝐴≐𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₀>0,𝐵≐𝑎₄₁(𝛼₂𝛽₂)−𝑎₄₀>0. (3.1)(3.2)

Proof. Suppose that (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)) is a solution of system (2.1) with initial conditions (2.2). According to the first two equations of (2.1), we have𝑥^′₁≤𝑎₁₀𝑥₁−𝑎₁₁𝑥²₁+𝐷₁(𝑡)[𝑥₂−𝑥₁𝑥],_′2≤𝑎₂₀𝑥₂−𝑎₂₁𝑥²₂+𝐷₂(𝑡)[𝑥₁−𝑥₂]. (3.3)We define the following lines in 𝑥₁-𝑥₂ plane: Line𝐿₁:𝑥₁=𝑀₁,0≤𝑥₂≤𝑀₁,Line𝐿₂:𝑥₂=𝑀₁,0≤𝑥₁≤𝑀₁. (3.4)Then, we have𝑥^′₁|_𝐿1<0,𝑥^′₂|_𝐿2<0. (3.5)Hence, it follows frommax{𝑥₁(0),𝑥₂(0)}≤𝑀₁ (3.6)thatmax{𝑥₁(𝑡),𝑥₂(𝑡)}≤𝑀₁for𝑡≥0. (3.7)Ifmax{𝑥₁(0),𝑥₂(0)}>𝑀₁, (3.8)we only consider what follows. If 𝑥_𝑖>𝑀₁,𝑖=1,2, from the given condition we get𝑎_𝑖0𝑥_𝑖−𝑎_𝑖1𝑥²_𝑖<𝑀₁(𝑎_𝑖0−𝑎_𝑖1𝑀₁)<0,𝑖=1,2. (3.9)Let−𝛼≐max_𝑖=1,2{𝑀₁(𝑎_𝑖0−𝑎_𝑖1𝑀₁)},𝑔(𝑡)=max{𝑥₁(𝑡),𝑥₂(𝑡)}. (3.10)Next, we consider the following three cases.
Case 1. 𝑥₁(0)>𝑥₂(0), 𝑔(0)=𝑥₁(0)>𝑀₁. Then, there exists 𝜀>0 such that 𝑔(𝑡)=𝑥₁(𝑡)>𝑀₁ for 𝑡∈[0,𝜀). We also derive that𝑥^′₁≤𝑎₁₀𝑥₁−𝑎₁₁𝑥²₁<−𝛼<0. (3.11)Hence, if 𝑡₂>𝑡₁ and 𝑡₁,𝑡₂∈[0,𝜀), we get𝑔(𝑡₂)−𝑔(𝑡₁)<−𝛼(𝑡₂−𝑡₁). (3.12)
Case 2. 𝑥₂(0)>𝑥₁(0),𝑔(0)=𝑥₂(0)>𝑀₁. Similarly, we could obtain that there exists [0,𝜀). If 𝑡₂>𝑡₁ and 𝑡₁,𝑡₂∈[0,𝜀), we get𝑔(𝑡₂)−𝑔(𝑡₁)<−𝛼(𝑡₂−𝑡₁). (3.13)
Case 3. 𝑥₂(0)=𝑥₁(0)=𝑔(0)>𝑀₁. We can also find an interval [0,𝜀) such that 𝑔(𝑡)=𝑥₁(𝑡)>𝑀₁ or 𝑔(𝑡)=𝑥₂(𝑡)>𝑀₁. In the same way, if 𝑡₂>𝑡₁ and 𝑡₁,𝑡₂∈[0,𝜀), we can obtain𝑔(𝑡₂)−𝑔(𝑡₁)<−𝛼(𝑡₂−𝑡₁). (3.14)Now, we can know that if 𝑔(0)>𝑀₁, g(t) will monotonously decrease by speed 𝛼. So, there exists 𝑇₁>0. If 𝑡≥𝑇₁, we have𝑔(𝑡)<𝑀₁. (3.15)
According to the third equation of (2.1), we have𝑥^′₃≤𝑥₃(−𝑎₃₀+𝑎₃₁𝛼₁(𝑡)𝑥²₁(𝑡−𝜏₁)1+𝛽₁(𝑡)𝑥²₁(𝑡−𝜏₁)−𝑎₃₂𝑥₃)<𝑥₃(−𝑎₃₀+𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₂𝑥₃𝑥),_′3|_𝑥3=𝑀2<𝑥₃(−𝑎₃₀+𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₂𝑥₃). (3.16)Hence, it follows from 𝑥₃(0)≤𝑀₂ that 𝑥₃(𝑡)≤𝑀₂ for 𝑡≥0.
If𝑥₃(0)>𝑀₂, (3.17)we only consider what follows. If 𝑥₃>𝑀₂, from the given condition we obtain𝑥₃(−𝑎₃₀+𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₂𝑥₃)<𝑀₂[−𝑎₃₀+𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₂𝑀₂]<0. (3.18)Let−𝛽=𝑀₂[−𝑎₃₀+𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₂𝑀₂]. (3.19)We also derive that𝑥^′₃<𝑀₂[−𝑎₃₀+𝑎₃₁(𝛼₁𝛽₁)−𝑎₃₂𝑀₂]=−𝛽<0. (3.20)Hence, if 𝑡₂>𝑡₁ and 𝑡₁,𝑡₂∈[0,𝜀), we get𝑥₃(𝑡₂)−𝑥₃(𝑡₁)<−𝛽(𝑡₂−𝑡₁). (3.21)Now, we can know that if 𝑥₃(0)>𝑀₂, 𝑥₃(𝑡) will monotonously decrease by speed 𝛽. So, there exists 𝑇₂ such that 𝑥₃(𝑡)<𝑀₂ for 𝑡≥𝑇₂. Similarly, we also get𝑥^′₄≤𝑥₄(−𝑎₄₀+𝑎₄₁𝛼₂(𝑡)𝑥²₁(𝑡−𝜏₂)1+𝛽₂(𝑡)𝑥²₁(𝑡−𝜏₂)−𝑎₄₂𝑥₄)<𝑥₄(−𝑎₄₀+𝑎₄₁(𝛼₂𝛽₂)−𝑎₄₂𝑥₄). (3.22)We can also choose the same 𝑀₃. There exists 𝑇₃>0 such that 𝑥₄(𝑡)<𝑀₃ for 𝑡>𝑇₃. This completes the proof.

Theorem 3.2. Suppose that system (2.1) satisfies the following conditions:𝑎₁₀−𝐷₁>0,𝑎₂₀−𝐷₂>0,𝐸=(𝑎₁₀−𝐷₁)²−4𝑎₁₁(𝛼₁𝛽₁+𝛼₂𝛽₂)𝑀_𝑥𝑎>0,₃₁𝛼₁𝑚²₁1+𝛽₁𝑚²₁−𝑎₃₀−𝑎₃₄𝑀_𝑥𝑎>0,₄₁𝛼₂𝑚²₁1+𝛽₂𝑚²₁−𝑎₄₀−𝑎₄₃𝑀_𝑥>0 (3.23)in which𝑚₁=𝑎₁₀−𝐷₁+√𝐸2𝑎₁₁. (3.24)Then, system (2.1) is uniformly persistent.

Proof. Suppose (𝑥₁,𝑥₂,𝑥₃,𝑥₄) is a solution of system (2.1) with the initial condition (2.2). According to the first equation of (2.1), we get𝑥^′₁(𝑡)≥𝑥₁(𝑡)((𝑎₁₀(𝑡)−𝐷₁𝛼(𝑡))−¹(𝑡)𝑥²₁(𝑡)𝑥₃(𝑡)1+𝛽₁(𝑡)𝑥²₁−𝛼(𝑡)²(𝑡)𝑥²₁(𝑡)𝑥₄(𝑡)1+𝛽₂(𝑡)𝑥²₁)(𝑡)≥−𝑎₁₁(𝑡)𝑥²₁(𝑡)+(𝑎₁₀(𝑡)−𝐷₁(𝑡))𝑥₁𝛼(𝑡)−¹(𝑡)𝑀_𝑥𝛽₁−𝛼(𝑡)²(𝑡)𝑀_𝑥𝛽₂.(𝑡) (3.25)So,liminf_𝑡→∞𝑥₁(𝑡)≥𝑚₁>0. (3.26)Then, there exists a 𝑇₅>0 such that𝑥₁(𝑡)≥𝑚₁for𝑡≥𝑇₅. (3.27)Similarly,liminf_𝑡→∞𝑥₂(𝑡)≥𝑚₂≐𝑎₂₀−𝐷₂𝑎₂₁>0. (3.28)Then, there exists a 𝑇₆>0 such that𝑥₂(𝑡)≥𝑚₂for𝑡≥𝑇₆. (3.29)From the third equation of (2.1), we obtain𝑥^′₃(𝑡)≥𝑥₃(𝑡)(−𝑎₃₀(𝑡)+𝑎₃₁𝛼(𝑡)¹(𝑡)𝑚²₁1+𝛽₁(𝑡)𝑚²₁−𝑎₃₂(𝑡)𝑥₃(𝑡)−𝑎₃₄(𝑡)𝑀_𝑥). (3.30)So,liminf_𝑡→∞𝑥₃(𝑡)≥𝑚₃≐𝑎₃₁(𝛼₁𝑚²₁/(1+𝛽₁𝑚²₁))−𝑎₃₀−𝑎₃₄𝑀_𝑥𝑎₃₂>0. (3.31)Then, there exists a 𝑇₇>0 such that𝑥₃(𝑡)≥𝑚₃for𝑡≥𝑇₇. (3.32)Similarly, we also getliminf_𝑡→∞𝑥₄(𝑡)≥𝑚₄≐𝑎₄₁(𝛼₂𝑚²₁/(1+𝛽₂𝑚²₁))−𝑎₄₀−𝑎₄₃𝑀_𝑥𝑎₄₂>0. (3.33)Then, there exists a 𝑇₈>0 such that𝑥₄(𝑡)≥𝑚₄for𝑡≥𝑇₈. (3.34)
Finally, let𝐷={(𝑥₁,𝑥₂,𝑥₃,𝑥₄)∣𝑚_𝑥<𝑥_𝑖<𝑀_𝑥,𝑖=1,2,3,4}, (3.35)where 𝑚_𝑥=min_𝑖=1,2,3,4{𝑚_𝑖} and 𝑀_𝑥=max{𝑀^∗₁,𝑀^∗₂,𝑀^∗₃}; 𝑀^∗_𝑖(𝑖=1,2,3) is given in Theorem 3.1. From Theorem 3.1 and the above analysis, we see that 𝐷 is a bounded compact region in 𝐑⁴₊ which has positive distance from coordinate hyperplanes. Let 𝑇=max{𝑇_𝑖,𝑖=1,…,8}, then we obtain that if 𝑡>𝑇, then every positive solution of system (2.1) with initial conditions (2.2) eventually enters and remains in the region 𝐷. This completes the proof.

4. Almost Periodic Solution

In this section, we derive sufficient conditions which guarantee that the periodic solution of periodic system (2.2) is globally attractive.

Theorem 4.1. In addition to (2.5), (3.2), and (3.23), assume further that all the coefficients of system (2.1) are continuous and positive almost periodic functions and(𝑎₁₁+𝐷₁𝑚₂𝑀²₁+𝛼₁𝑚₃1+𝛽₁𝑀²₁+𝛼₂𝑚₄1+𝛽₂𝑀²₁)𝑚₁2>(𝛼₁𝛽₁𝑀²₁𝑀₃(1+𝛽₁𝑚²₁)²+2𝛼₂𝛽₂𝑀²₁𝑀₄(1+𝛽₂𝑚²₁)²+𝐷₂𝑚₂)𝑀₁+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²),(𝑎₂₁+𝐷₂𝑚₁𝑀²₂)𝑚₂>𝐷₁𝑚₁𝑀₂+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²𝑎),₃₂𝑚₃>(𝛼₁𝑀₁1+𝛽₁𝑚²₁+𝑎₄₃)𝑀₃+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²𝑎),₄₂𝑚₄>(𝛼₂𝑀₁1+𝛽₂𝑚²₁+𝑎₃₄)𝑀₃+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²). (4.1)Then, system (2.1) has a unique positive almost periodic solution which is globally asymptotically stable. Furthermore, if system (2.1) is an 𝜔-periodic system, then system (2.1) has a positive 𝜔-periodic solution which is globally asymptotically stable.

Proof. Consider the product system of (2.1):𝑥^′₁=𝑥₁(𝑎₁₀(𝑡)−𝑎₁₁(𝑡)𝑥₁𝛼)−¹(𝑡)𝑥²₁𝑥₃1+𝛽₁(𝑡)𝑥²₁−𝛼₂(𝑡)𝑥²₁𝑥₄1+𝛽₂(𝑡)𝑥²₁+𝐷₁(𝑡)(𝑥₂−𝑥₁𝑥),_′2=𝑥₂(𝑎₂₀(𝑡)−𝑎₂₁(𝑡)𝑥₂)+𝐷₂(𝑡)(𝑥₁−𝑥₂𝑥),_′3=𝑥₃(−𝑎₃₀(𝑡)+𝑎₃₁𝛼(𝑡)¹(𝑡)𝑥²₁(𝑡−𝜏₁)1+𝛽₁(𝑡)𝑥²₁(𝑡−𝜏₁)−𝑎₃₂(𝑡)𝑥₃−𝑎₃₄(𝑡)𝑥₄𝑥),_′4=𝑥₄(−𝑎₄₀(𝑡)+𝑎₄₁𝛼(𝑡)²(𝑡)𝑥²₁(𝑡−𝜏₂)1+𝛽₂(𝑡)𝑥²₁(𝑡−𝜏₂)−𝑎₄₂(𝑡)𝑥₄−𝑎₄₃(𝑡)𝑥₃𝑦),_′1=𝑦₁(𝑎₁₀(𝑡)−𝑎₁₁(𝑡)𝑦₁𝛼)−¹(𝑡)𝑦²₁𝑦₃1+𝛽₁(𝑡)𝑦²₁−𝛼₂(𝑡)𝑦²₁𝑦₄1+𝛽₂(𝑡)𝑦²₁+𝐷₁(𝑡)(𝑦₂−𝑦₁𝑦),_′2=𝑦₂(𝑎₂₀(𝑡)−𝑎₂₁(𝑡)𝑦₂)+𝐷₂(𝑡)(𝑦₁−𝑦₂𝑦),_′3=𝑦₃(−𝑎₃₀(𝑡)+𝑎₃₁𝛼(𝑡)¹(𝑡)𝑦²₁(𝑡−𝜏₁)1+𝛽₁(𝑡)𝑦²₁(𝑡−𝜏₁)−𝑎₃₂(𝑡)𝑦₃−𝑎₃₄(𝑡)𝑦₄𝑦),_′4=𝑦₄(−𝑎₄₀(𝑡)+𝑎₄₁𝛼(𝑡)²(𝑡)𝑦²₁(𝑡−𝜏₂)1+𝛽₂(𝑡)𝑦²₁(𝑡−𝜏₂)−𝑎₄₂(𝑡)𝑦₄−𝑎₄₃(𝑡)𝑦₃). (4.2)It is easily noted that the existence and uniqueness of the positive almost periodic solution of system (2.1) are equivalent to the existence and uniqueness of the positive almost periodic solution of system (4.2). Then, choose the following function:𝑉(𝑡)=𝑉(𝑡,𝑥_𝑖,𝑦_𝑖)=⁴_𝑖=1|ln𝑥_𝑖(𝑡)−ln𝑦_𝑖(𝑡)|. (4.3)Obviously, 𝑉(𝑡) satisfies conditions (i) and (ii) of Lemma 2.2. Next, we will prove that 𝑉(𝑡) satisfies condition (iii) of Lemma 2.2. It follows that𝑥^′₁𝑥₁−𝑦^′₁𝑦₁=−𝑎₁₁(𝑥₁−𝑦₁)−𝐷₁(𝑥₂𝑥₁−𝑦₂𝑦₁𝛼)−(¹𝑥₁𝑥₃1+𝛽₁𝑥²₁−𝛼₁𝑦₁𝑦₃1+𝛽₁𝑦²₁𝛼)−(²𝑥₁𝑥₄1+𝛽₂𝑥²₁−𝛼₂𝑦₁𝑦₄1+𝛽₂𝑦²₁) (4.4)in which𝛼₁𝑥₁𝑥₃1+𝛽₁𝑥²₁−𝛼₁𝑦₁𝑦₃1+𝛽₁𝑦²₁𝛼=(¹𝑥₃1+𝛽₁𝑥²₁−𝛼₁𝛽₁𝑦₁𝑦₃(𝑥₁+𝑦₁)(1+𝛽₁𝑥²₁)(1+𝛽₁𝑦²₁))(𝑥₁−𝑦₁𝛼)+¹𝑦₁1+𝛽₁𝑥²₁(𝑥₃−𝑦₃); (4.5)also,𝑥^′₂𝑥₂−𝑦^′₂𝑦₂=(−𝑎₂₁−𝐷₂𝑦₁𝑥₂𝑦₂)(𝑥₂−𝑦₂𝐷)+²𝑥₁(𝑥₁−𝑦₁𝑥),_′3𝑥₃−𝑦^′₃𝑦₃=𝑎₃₁𝛼₁[𝑥₁(𝑡−𝜏₁)+𝑦₁(𝑡−𝜏₁)][1+𝛽₁𝑥²₁(𝑡−𝜏₁)][1+𝛽₁𝑦²₁(𝑡−𝜏₁)][𝑥₁(𝑡−𝜏₁)−𝑦₁(𝑡−𝜏₁)]−𝑎₃₂(𝑥₃−𝑦₃)−𝑎₃₄(𝑥₄−𝑦₄𝑥),_′4𝑥₄−𝑦^′₄𝑦₄=𝑎₄₁𝛼₂[𝑥₁(𝑡−𝜏₂)+𝑦₁(𝑡−𝜏₂)][1+𝛽₁𝑥²₁(𝑡−𝜏₂)][1+𝛽₁𝑦²₁(𝑡−𝜏₂)][𝑥₁(𝑡−𝜏₂)−𝑦₁(𝑡−𝜏₂)]−𝑎₄₂(𝑥₄−𝑦₄)−𝑎₄₃(𝑥₃−𝑦₃). (4.6)In this regard, after few computations, it is noted that 𝐷⁺𝑉(𝑡,𝑥_𝑖,𝑦_𝑖)=⁴_𝑖=1sgn(𝑥_𝑖(𝑡)−𝑦_𝑖𝑥(𝑡))(^′𝑖(𝑡)𝑥_𝑖−𝑦(𝑡)^′𝑖(𝑡)𝑦_𝑖)(𝑡)=[−𝑎₁₁−𝐷₁𝑦₂𝑥₁𝑦₁−𝛼₁𝑥₃(1+𝛽₁𝑥²₁)−𝛼₂𝑥₄(1+𝛽₂𝑥²₁)+𝛼₁𝛽₁𝑦₁𝑦₃(𝑥₁+𝑦₁)(1+𝛽₁𝑥²₁)(1+𝛽₁𝑦²₁)+𝛼₂𝛽₁𝑦₁𝑦₄(𝑥₁+𝑦₁)(1+𝛽₂𝑥²₁)(1+𝛽₂𝑦²₁)]|𝑥₁−𝑦₁|+[−𝑎₂₁−𝐷₂𝑦₁𝑥₂𝑦₂]|𝑥₂−𝑦₂|−𝑎₃₂|𝑥₃−𝑦₃|−𝑎₄₂|𝑥₄−𝑦₄|+sgn(𝑥₁−𝑦₁)𝐷₁(𝑥₂−𝑦₂)𝑥₁−sgn(𝑥₁−𝑦₁)𝛼₁𝑦₁(1+𝛽₁𝑥²₁)(𝑥₃−𝑦₃)−sgn(𝑥₁−𝑦₁)𝛼₂𝑦₁(1+𝛽₂𝑥²₁)(𝑥₄−𝑦₄)+sgn(𝑥₂−𝑦₂)𝐷₂(𝑥₁−𝑦₁)𝑥₂+sgn(𝑥₃−𝑦₃)𝑎₃₁𝛼₁[𝑥₁(𝑡−𝜏₁)+𝑦₁(𝑡−𝜏₁)][1+𝛽₁𝑥²₁(𝑡−𝜏₁)][1+𝛽₁𝑦²₁(𝑡−𝜏₁)][𝑥₁(𝑡−𝜏₁)−𝑦₁(𝑡−𝜏₁)]−𝑎₃₄sgn(𝑥₃−𝑦₃)(𝑥₄−𝑦₄)+sgn(𝑥₄−𝑦₄)𝑎₄₁𝛼₂[𝑥₁(𝑡−𝜏₂)+𝑦₁(𝑡−𝜏₂)][1+𝛽₁𝑥²₁(𝑡−𝜏₂)][1+𝛽₁𝑦²₁(𝑡−𝜏₂)][𝑥₁(𝑡−𝜏₂)−𝑦₁(𝑡−𝜏₂)]−𝑎₄₃sgn(𝑥₄−𝑦₄)(𝑥₃−𝑦₃)≤−(𝑎₁₁+𝐷₁𝑚₂𝑀²₁+𝛼₁𝑚₃1+𝛽₁𝑀²₁+𝛼₂𝑚₄1+𝛽₂𝑀²₁−2𝛼₁𝛽₁𝑀²₁𝑀₃(1+𝛽₁𝑚²₁)²−2𝛼₂𝛽₂𝑀²₁𝑀₄(1+𝛽₂𝑚²₁)²−𝐷₂𝑚₂)|𝑥₁−𝑦₁|+(−𝑎₂₁−𝐷₂𝑚₁𝑀²₂+𝐷₁𝑚₁)|𝑥₂−𝑦₂|+(−𝑎₃₂+𝛼₁𝑀₁1+𝛽₁𝑚²₁+𝑎₄₃)|𝑥₃−𝑦₃|+(−𝑎₄₂+𝛼₂𝑀₁1+𝛽₂𝑚²₁+𝑎₃₄)|𝑥₄−𝑦₄|+2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²|𝑥₁(𝑡−𝜏₁)−𝑦₁(𝑡−𝜏₁2)|+𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²|𝑥₁(𝑡−𝜏₂)−𝑦₁(𝑡−𝜏₂)|. (4.7) It follows from (4.1) that 1𝑀_𝑥⁴_𝑖=1‖𝑥_𝑖−𝑦_𝑖‖≤𝑉(𝑡,𝑥_𝑖,𝑦_𝑖1)≤𝑚_𝑥⁴_𝑖=1‖𝑥_𝑖−𝑦_𝑖‖. (4.8) Choose 𝑃(𝑠)=(𝑀_𝑥/𝑚_𝑥)𝑠>𝑠>0,𝑎(𝑠)=(1/𝑀_𝑥)𝑠>0,𝑏(𝑠)=(1/𝑚_𝑥)𝑠>0. When𝑃(𝑉(𝑡,𝑥_𝑖(𝑡),𝑦_𝑖(𝑡)))≥𝑉(𝑡+𝜃,𝑥_𝑖(𝑡+𝜃),𝑦_𝑖(𝑡+𝜃)),𝜃∈[−𝜏,0],𝑖=1,2,3,4,|𝑥₁(𝑡−𝜏)−𝑦₁(𝑡−𝜏)|≤𝑀_𝑥|ln𝑥₁(𝑡−𝜏)−ln𝑦₁(𝑡−𝜏)|≤𝑀_𝑥𝑉(𝑡−𝜏,𝑥_𝑖(𝑡−𝜏),𝑦_𝑖(𝑡−𝜏))≤𝑀_𝑥𝑀_𝑥𝑚_𝑥𝑉(𝑡,𝑥_𝑖(𝑡),𝑦_𝑖(𝑡)); (4.9)then2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²|𝑥₁(𝑡−𝜏₁)−𝑦₁(𝑡−𝜏₁𝑀)|≤^2𝑥𝑚_𝑥2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²𝑉(𝑡,𝑥_𝑖(𝑡),𝑦_𝑖2(𝑡)),𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²|𝑥₁(𝑡−𝜏₂)−𝑦₁(𝑡−𝜏₂𝑀)|≤^2𝑥𝑚_𝑥2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²𝑉(𝑡,𝑥_𝑖(𝑡),𝑦_𝑖(𝑡)). (4.10)Hence,𝐷⁺𝑉(𝑡,𝑥_𝑖,𝑦_𝑖)≤[−(𝑎₁₁+𝐷₁𝑚₂𝑀²₁+𝛼₁𝑚₃1+𝛽₁𝑀²₁+𝛼₂𝑚₄1+𝛽₂𝑀²₁)𝑚₁2+(𝛼₁𝛽₁𝑀²₁𝑀₃(1+𝛽₁𝑚²₁)²+2𝛼₂𝛽₂𝑀²₁𝑀₄(1+𝛽₂𝑚²₁)²+𝐷₂𝑚₂)𝑀_𝑥]|ln𝑥₁−ln𝑦₁|+[−(𝑎₂₁+𝐷₂𝑚₁𝑀²₂)𝑚₂+𝐷₁𝑚₁𝑀₂]|ln𝑥₂−ln𝑦₂|+[−𝑎₃₂𝑚₃+(𝛼₁𝑀₁1+𝛽₁𝑚²₁+𝑎₄₃)𝑀₃]|ln𝑥₃−ln𝑦₃|+[−𝑎₄₂𝑚₄+(𝛼₂𝑀₁1+𝛽₂𝑚²₁+𝑎₃₄)𝑀₄]|ln𝑥₄−ln𝑦₄|+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²)𝑉(𝑡,𝑥_𝑖(𝑡),𝑦_𝑖(𝑡))≤−𝒞𝑉(𝑡,𝑥_𝑖(𝑡),𝑦_𝑖(𝑡)), (4.11)where−𝒞=max{−(𝑎₁₁+𝐷₁𝑚₂𝑀²₁+𝛼₁𝑚₃1+𝛽₁𝑀²₁+𝛼₂𝑚₄1+𝛽₂𝑀²₁)𝑚₁2+(𝛼₁𝛽₁𝑀²₁𝑀₃(1+𝛽₁𝑚²₁)²2𝛼₂𝛽₂𝑀²₁𝑀₄(1+𝛽₂𝑚²₁)²+𝐷₂𝑚₂)𝑀_𝑥+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²)−(𝑎₂₁+𝐷₂𝑚₁𝑀²₂)𝑚₂+𝐷₁𝑚₁𝑀₂+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²)−𝑎₃₂𝑚₃+(𝛼₁𝑀₁1+𝛽₁𝑚²₁+𝑎₄₃)𝑀₃+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²)−𝑎₄₂𝑚₄+(𝛼₂𝑀₁1+𝛽₂𝑚²₁+𝑎₃₄)𝑀₄+𝑀²_𝑥𝑚_𝑥(2𝛼₁𝑎₃₁𝑀₁(1+𝛽₁𝑚²₁)²+2𝛼₂𝑎₄₁𝑀₁(1+𝛽₂𝑚²₁)²)}. (4.12)This completes the proof.

5. Discussion

In this work, we consider a nonautonomous delayed predator-prey model with competition and diffusion. Some sufficient conditions on uniform persistence of the model have been given. By means of the Liapunov-Razumikhin technique, it is also seen that, under almost periodic circumstances, the existence and uniqueness of the positive almost periodic solution which is globally asymptotically stable are governed by several inequalities.

Acknowledgments

The author is thankful to the learned referees for their valuable comments which have helped to present a better exposition of the paper. This work is supported by the first project proposals of Guangxi education teaching reform in the 11th five-year plan (2005240), and the project of qualified course reform and establishment of the new century teaching reform in the 11th five-year plan (2006072).