Stability Analysis of Three-Species Almost Periodic Competition Models with Grazing Rates and Diffusions

Abstract

Almost periodic solution of a three-species competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained by Q. C. Lin (1999), F. D. Chen and X. X. Chen (2003), and Y. Q. Liu, S. L, Xie, and Z. D. Xie (1996).

1. Introduction

In this paper, we study the following three-species competition system with grazing rates and diffusions:𝜕𝑣₁(𝑥,𝑡)𝜕𝑡=𝑘₁(𝑡)Δ𝑣₁(𝑥,𝑡)+𝑣₁(𝑥,𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁(𝑥,𝑡)−𝑐₁(𝑡)𝑣₂(𝑥,𝑡)−𝑑₁(𝑡)𝑣₃(𝑥,𝑡)+𝑓₁(𝑡),𝜕𝑣₂(𝑥,𝑡)𝜕𝑡=𝑘₂(𝑡)Δ𝑣₂(𝑥,𝑡)+𝑣₂(𝑥,𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑣₁(𝑥,𝑡)−𝑐₂(𝑡)𝑣₂(𝑥,𝑡)−𝑑₂(𝑡)𝑣₃(𝑥,𝑡)+𝑓₂(𝑡),𝜕𝑣₃(𝑥,𝑡)𝜕𝑡=𝑘₃(𝑡)Δ𝑣₃(𝑥,𝑡)+𝑣₃(𝑥,𝑡)𝑎₃(𝑡)−𝑏₃(𝑡)𝑣₁(𝑥,𝑡)−𝑐₃(𝑡)𝑣₂(𝑥,𝑡)−𝑑₃(𝑡)𝑣₃(𝑥,𝑡)+𝑓₃(𝑡), (1.1) where (𝑥,𝑡)∈Ω×𝑅⁺, Ω⊆𝑅^𝑛 is the bounded open subset of 𝑅^𝑛 with smooth boundary 𝜕Ω, which represent the habitat domain for three species. System (1.1) is supplement with boundary conditions and initial conditions:𝜕𝑣_𝑖(𝑥,𝑡)()𝜕𝑛=0,𝑖=1,2,3,𝑥,𝑡∈𝜕Ω×𝑅⁺,𝑣_𝑖(𝑥,0)=𝑣_𝑖0(𝑥)≥0,𝑣_𝑖0(𝑥)≢0,𝑖=1,2,3,𝑥∈Ω, (1.2)(1.3) where 𝜕/𝜕𝑛 denotes the outward normal derivation on 𝜕Ω, and 𝑣_𝑖(𝑥,𝑡) represent the density of 𝑖th species at point 𝑥=(𝑥₁,…,𝑥_𝑛) and the time of 𝑡. Here, 𝑘_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), 𝑑_𝑖(𝑡), and 𝑓_𝑖(𝑡) (𝑖=1,2,3) denote the diffusivity rates, competition rates, and grazing rates, respectively. They are almost periodic functions in real number field 𝑅. Δ is a Laplace operator on Ω.

System (1.1)–(1.3) describes the interaction among three species and is an important model in biomathcmatics, which has been intensively investigated, and much attention is carried to the problem [1–8]. When there is no diffusion, Jiang [1] and Lin [2] studied the existence, uniqueness, and stability on periodic solution and almost periodic solution for two-species competition system under the condition that the coefficients are the periodic function and almost periodic function, respectively; F. D. Chen and X. X. Chen [3] extended the results in [2] to n-species case. When there are no diffusion and grazing rates, Zhang and Wang [4, 5] investigated the existence of a positive periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay and the existence of multiple positive periodic solutions for a generalized delayed population model with exploited term by using the continuation theorem of coincidence degree theory; Hu and Zhang [6] established criteria for the existence of at least four positive periodic solutions for a discrete time-delayed predator-prey system with nonmonotonic functional response and harvesting by employing the continuation theorem of coincidence degree theory. When there are no grazing rates, Pao and Wang [7] proved the stability for invariable coefficient case by utilizing the method of upper and lower solutions. Liu et al. [8] showed the stability on the periodic solution for n-species competition system with grazing rates and diffusions. Nevertheless, generally speaking, the system does not always change strictly according to periodic laws, sometimes it changes according to almost periodic laws, and it is important to survey almost periodic solution for the multispecies competition system with grazing rates and diffusions. To sum up, we pay more attention to almost periodic solution of a three-species competition system (1.1)–(1.3) with grazing rates and diffusions; in this paper, by using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we obtain sufficient conditions which ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained in [2, 3, 8]. Many other results on the periodic solution and almost periodic solution can be found in [9–16].

2. Preliminary

Firstly, we give out some definitions and lemmas.

Definition 2.1. Suppose that 𝑓(𝑡) is a continuous function in 𝑅. Then 𝑓(𝑡) is said to be almost periodic in 𝑡∈𝑅 if for every 𝜀>0 corresponds 𝑇(𝜀)>0 such that for any interval 𝐼 whose length is equal to 𝑇(𝜀) there is at least one 𝜏∈𝐼 such that 𝑓(𝑡+𝜏)−𝑓(𝑡)≤𝜀,∀𝑡∈𝑅. (2.1)

Definition 2.2. If a smooth function 𝑉(𝑡)=(𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) satisfies (1.1) in 𝑅⁺, and every component of 𝑉(𝑡) is the almost periodic function, we called that 𝑉(𝑡) is a spatial homogeneity almost periodic solution for (1.1), which is denoted by 𝑉(𝑡,𝑇(𝜀)).

Definition 2.3. For any nonnegative smooth initial data 𝑉(𝑥,0)=𝑣₁(𝑥,0),𝑣₂(𝑥,0),𝑣₃(𝑥,0)=𝑣₁₀(𝑥),𝑣₂₀(𝑥),𝑣₃₀(𝑥)≥0,𝑉(𝑥,0)≢0,𝑥∈Ω, (2.2) if there exists a unique positive solution 𝑉(𝑥,𝑡)=(𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)) for the system (1.1) with boundary conditions (1.2), and lim_𝑡→∞(𝑉_𝑖(𝑥,𝑡)−𝑉_𝑖(𝑡,𝑇(𝜀)))=0, 𝑖=1,2,3, uniformly for 𝑥∈Ω, we called that spatial homogeneity almost periodic solution 𝑉(𝑡,𝑇(𝜀)) is globally asymptotically stable.

Definition 2.4. Suppose that 𝑉(𝑥,𝑡)≡(𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)), 𝑉(𝑥,𝑡)≡(𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)); if 𝑉(𝑥,𝑡)≥𝑉(𝑥,𝑡) and 𝜕𝑣₁(𝑥,𝑡)𝜕𝑡≥𝑘₁(𝑡)Δ𝑣₁(𝑥,𝑡)+𝑣₁(𝑥,𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁(𝑥,𝑡)−𝑐₁(𝑡)𝑣₂(𝑥,𝑡)−𝑑₁(𝑡)𝑣₃(𝑥,𝑡)+𝑓₁(𝑡),(𝑥,𝑡)∈Ω×𝑅⁺,𝜕𝑣₂(𝑥,𝑡)𝜕𝑡≥𝑘₂(𝑡)Δ𝑣₂(𝑥,𝑡)+𝑣₂(𝑥,𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑣₁(𝑥,𝑡)−𝑐₂(𝑡)𝑣₂(𝑥,𝑡)−𝑑₂(𝑡)𝑣₃(𝑥,𝑡)+𝑓₂(𝑡),(𝑥,𝑡)∈Ω×𝑅⁺,𝜕𝑣₃(𝑥,𝑡)𝜕𝑡≥𝑘₃(𝑡)Δ𝑣₃(𝑥,𝑡)+𝑣₃(𝑥,𝑡)𝑎₃(𝑡)−𝑏₃(𝑡)𝑣₁(𝑥,𝑡)−𝑐₃(𝑡)𝑣₂(𝑥,𝑡)−𝑑₃(𝑡)𝑣₃(𝑥,𝑡)+𝑓₃(𝑡),(𝑥,𝑡)∈Ω×𝑅⁺,𝜕𝑣_𝑖(𝑥,𝑡)()𝜕𝑛≥0,𝑖=1,2,3,𝑥,𝑡∈𝜕Ω×𝑅⁺,𝑣_𝑖(𝑥,0)≥𝑣_𝑖0(𝑥),𝑖=1,2,3,𝑥∈Ω,𝜕𝑣₁(𝑥,𝑡)𝜕𝑡≤𝑘₁(𝑡)Δ𝑣₁(𝑥,𝑡)+𝑣₁(𝑥,𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁(𝑥,𝑡)−𝑐₁(𝑡)𝑣₂(𝑥,𝑡)−𝑑₁(𝑡)𝑣₃(𝑥,𝑡)+𝑓₁(𝑡),(𝑥,𝑡)∈Ω×𝑅⁺,𝜕𝑣₂(𝑥,𝑡)𝜕𝑡≤𝑘₂(𝑡)Δ𝑣₂(𝑥,𝑡)+𝑣₂(𝑥,𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑣₁(𝑥,𝑡)−𝑐₂(𝑡)𝑣₂(𝑥,𝑡)−𝑑₂(𝑡)𝑣₃(𝑥,𝑡)+𝑓₂(𝑡),(𝑥,𝑡)∈Ω×𝑅⁺,𝜕𝑣₃(𝑥,𝑡)𝜕𝑡≤𝑘₃(𝑡)Δ𝑣₃(𝑥,𝑡)+𝑣₃(𝑥,𝑡)𝑎₃(𝑡)−𝑏₃(𝑡)𝑣₁(𝑥,𝑡)−𝑐₃(𝑡)𝑣₂(𝑥,𝑡)−𝑑₃(𝑡)𝑣₃(𝑥,𝑡)+𝑓₃(𝑡),(𝑥,𝑡)∈Ω×𝑅⁺,𝜕𝑣_𝑖(𝑥,𝑡)()𝜕𝑛≤0,𝑖=1,2,3,𝑥,𝑡∈𝜕Ω×𝑅⁺,𝑣_𝑖(𝑥,0)≤𝑣_𝑖0(𝑥),𝑖=1,2,3,𝑥∈Ω, (2.3) we called 𝑉(𝑥,𝑡) and𝑉(𝑥,𝑡) a pair of ordered upper and lower solutions for systems (1.1)–(1.3).

Lemma 2.5 (see [12, 17]). Suppose that 𝑉(𝑥,𝑡)and𝑉(𝑥,𝑡) are a pair of ordered upper and lower solution for systems (1.1)–(1.3), then there exists a unique solution 𝑉(𝑥,𝑡) for systems (1.1)–(1.3). Moreover, one has 𝑉(𝑥,𝑡)≥𝑉(𝑥,𝑡)≥𝑉(𝑥,𝑡). (2.4)

For the almost periodic function 𝐹(𝑡) in 𝑅, one denotes 𝐹=sup{𝐹(𝑡),𝑡∈𝑅}, 𝐹_∼=inf{𝐹(𝑡),𝑡∈𝑅}, and 𝑀[𝐹]=lim_{(𝑡−𝑠)→∞}{∫^𝑡_𝑠𝐹(𝜏)𝑑𝜏/(𝑡−𝑠)}. When 𝐹(𝑡) is 𝑇-periodic function, one denotes 𝑀[𝐹]=∫^𝑇₀𝐹(𝑠)𝑑𝑠/𝑇.

3. Main Results and Proofs

Now we are in a position to state our main results and give our proofs.

Theorem 3.1. If 𝑎_∼𝑖,𝑏_∼𝑖,𝑐_∼𝑖,𝑑_∼𝑖,𝑓_∼𝑖 are positive numbers, and 𝑏̃_𝑖+𝑐_𝑖̃+𝑑_𝑖𝑎_∼𝑖≤𝐿=min𝑏_∼1𝑓₁,_∼𝑐₂𝑓₂,𝑑_∼3𝑓₃,𝑑_∼1+𝑐_∼1𝑎₁,𝑏_∼2+𝑑_∼2𝑎₂,𝑏_∼3+𝑐_∼3𝑎₃ (3.1) are satisfied for 𝑖=1,2,3, then there exists a strictly positive spatial homogeneity almost periodic solution 𝑉(𝑡)=(𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) for (1.1).

Proof. By the conditions in Theorem 3.1, we have 𝑐̃0<¹+𝑑₁𝐿𝑎_∼1−𝑏̃₁𝑏̃≤1,0<²+𝑑₂𝐿𝑎_∼2−𝑐₂̃𝑏̃≤1,0<³+𝑐₃̃𝐿𝑎_∼3−𝑑₃≤1. (3.2)
Let 𝑐̃𝑚=𝐿⋅max¹+𝑑₁𝐿𝑎_∼1−𝑏̃₁,𝑏̃₂+𝑑₂𝐿𝑎_∼2−𝑐₂̃,𝑏̃₃+𝑐₃̃𝐿𝑎_∼3−𝑑₃. (3.3) Then we have 0<𝑚≤𝐿, and 𝑐₁̃+𝑑₁𝐿𝑚≤𝐿𝑎_∼1−𝑏̃₁,𝑏̃₂+𝑑₂𝐿𝑚≤𝐿𝑎_∼2−𝑐₂̃,𝑏̃₃+𝑐₃̃𝐿𝑚≤𝐿𝑎_∼3−𝑑₃. (3.4) Therefore 𝑏̃₁+𝑐₁̃+𝑑₁𝐿𝑚−𝑓_∼1𝑚²≤𝐿𝑎_∼1,𝑐₂̃+𝑏̃₂+𝑑₂𝐿𝑚−𝑓_∼2𝑚²≤𝐿𝑎_∼2,𝑑₃+𝑏̃₃+𝑐₃̃𝐿𝑚−𝑓_∼3𝑚²≤𝐿𝑎_∼3. (3.5) Furthermore, by the given conditions in Theorem 3.1, one has 𝑏_∼1−𝑓₁𝐿²≥0,𝑐_∼2−𝑓₂𝐿²≥0,𝑑_∼3−𝑓₃𝐿²≥0,𝑐_∼1+𝑑_∼1𝑚𝐿≥𝑚𝑎₁,𝑏_∼2+𝑑_∼2𝑚𝐿≥𝑚𝑎₂,𝑏_∼3+𝑐_∼3𝑚𝐿≥𝑚𝑎₃. (3.6) Thus 𝑏_∼1+𝑐_∼1+𝑑_∼1𝑚𝐿−𝑓₁𝐿²≥𝑚𝑎₁,𝑐_∼2+𝑏_∼2+𝑑_∼2𝑚𝐿−𝑓₂𝐿²≥𝑚𝑎₂,𝑑_∼3+𝑏_∼3+𝑐_∼3𝑚𝐿−𝑓₃𝐿²≥𝑚𝑎₃. (3.7) Combining (3.5) and (3.7), we have 𝑏̃₁+𝑐₁̃+𝑑₁𝐿𝑚−𝑓_∼1𝑚²≤𝐿𝑎_∼1,𝑏_∼1+𝑐_∼1+𝑑_∼1𝑚𝐿−𝑓₁𝐿²≥𝑚𝑎₁,𝑐₂̃+𝑏̃₂+𝑑₂𝐿𝑚−𝑓_∼2𝑚²≤𝐿𝑎_∼2,𝑐_∼2+𝑏_∼2+𝑑_∼2𝑚𝐿−𝑓₂𝐿²≥𝑚𝑎₂,𝑑₃+𝑏̃₃+𝑐₃̃𝐿𝑚−𝑓_∼3𝑚²≤𝐿𝑎_∼3,𝑑_∼3+𝑏_∼3+𝑐_∼3𝑚𝐿−𝑓₃𝐿²≥𝑚𝑎₃. (3.8)
Let 𝐻^𝑚_𝐿=𝜙(𝑡),𝜑(𝑡),𝛾(𝑡)∶𝜙,𝜑,𝛾arealmostperiodicin𝑡∈𝑅,0<𝑚≤𝜙,𝜑,𝛾≤𝐿. (3.9) We consider the following system corresponding to (1.1): 𝑣̇₁=𝑣₁𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁−𝑐₁(𝑡)𝑣₂−𝑑₁(𝑡)𝑣₃+𝑓₁(𝑡),𝑡∈𝑅⁺,𝑣̇₂=𝑣₂𝑎₂(𝑡)−𝑏₂(𝑡)𝑣₁−𝑐₂(𝑡)𝑣₂−𝑑₂(𝑡)𝑣₃+𝑓₂(𝑡),𝑡∈𝑅⁺,𝑣̇₃=𝑣₃𝑎₃(𝑡)−𝑏₃(𝑡)𝑣₁−𝑐₃(𝑡)𝑣₂−𝑑₃(𝑡)𝑣₃+𝑓₃(𝑡),𝑡∈𝑅⁺. (3.10) Let 𝑧_𝑖=1/𝑣_𝑖, 𝑖=1,2,3; then (3.10) becomes 𝑧̇₁=𝑏₁(𝑡)−𝑎₁(𝑡)𝑧₁+𝑐₁(𝑡)𝑧₁𝑧₂+𝑑₁(𝑡)𝑧₁𝑧₃−𝑓₂(𝑡)𝑧²₁,𝑧̇₂=𝑐₂(𝑡)−𝑎₂(𝑡)𝑧₂+𝑏₂(𝑡)𝑧₂𝑧₁+𝑑₂(𝑡)𝑧₂𝑧₃−𝑓₂(𝑡)𝑧²₂,𝑧̇₃=𝑑₃(𝑡)−𝑎₃(𝑡)𝑧₃+𝑏₃(𝑡)𝑧₃𝑧₁+𝑐₃(𝑡)𝑧₃𝑧₂−𝑓₃(𝑡)𝑧²₃. (3.11) For any (𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿, by 𝑀[𝑏₁]>0, 𝑀[𝑐₂]>0, 𝑀[𝑑₃]>0, we observe [18] that 𝑧̇₁=𝑏₁(𝑡)−𝑎₁(𝑡)𝑧₁+𝑐₁(𝑡)𝜙(𝑡)𝜑(𝑡)+𝑑₁(𝑡)𝜙(𝑡)𝛾(𝑡)−𝑓₁(𝑡)𝜙²(𝑡),𝑧̇₂=𝑐₂(𝑡)−𝑎₂(𝑡)𝑧₂+𝑏₂(𝑡)𝜑(𝑡)𝜙(𝑡)+𝑑₂(𝑡)𝜑(𝑡)𝛾(𝑡)−𝑓₂(𝑡)𝜑²(𝑡),𝑧̇₃=𝑑₃(𝑡)−𝑎₃(𝑡)𝑧₃+𝑏₃(𝑡)𝛾(𝑡)𝜙(𝑡)+𝑐₃(𝑡)𝛾(𝑡)𝜑(𝑡)−𝑓₃(𝑡)𝛾²(𝑡), (3.12) have almost periodic solution: 𝑧₁(𝑡)=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎1(𝑟)𝑑𝑟}𝑏₁(𝑠)+𝑐₁(𝑠)𝜙(𝑠)𝜑(𝑠)+𝑑₁(𝑠)𝜙(𝑠)𝛾(𝑠)−𝑓₁(𝑠)𝜙²(𝑠𝑧)𝑑𝑠,₂(𝑡)=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎2(𝑟)𝑑𝑟}𝑐₂(𝑠)+𝑏₂(𝑠)𝜑(𝑠)𝜙(𝑠)+𝑑₂(𝑠)𝜑(𝑠)𝛾(𝑠)−𝑓₂(𝑠)𝜑²(𝑠𝑧)𝑑𝑠,₃(𝑡)=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎2(𝑟)𝑑𝑟}𝑑₃(𝑠)+𝑏₃(𝑠)𝛾(𝑠)𝜙(𝑠)+𝑐₃(𝑠)𝛾(𝑠)𝜑(𝑠)−𝑓₃(𝑠)𝛾²(𝑠)𝑑𝑠. (3.13) By the system (3.13), we define a mapping 𝐴: 𝐴𝜙(𝑡),𝜑(𝑡),𝛾(𝑡)=𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡),∀𝜙(𝑡),𝜑(𝑡),𝛾(𝑡)∈𝐻^𝑚_𝐿. (3.14) Combining (3.8) and (3.13), we have 𝑧₁(𝑡)≥^𝑡_−∞𝑒⁻𝑎_{1(𝑡−𝑠)}𝑏_∼1+𝑐_∼1+𝑑_∼1𝑚𝐿−𝑓₁𝐿²1𝑑𝑠=𝑎₁𝑏_∼1+𝑐_∼1+𝑑_∼1𝑚𝐿−𝑓₁𝐿²𝑧≥𝑚,₁(𝑡)≤^𝑡_−∞𝑒^{−𝑎∼1(𝑡−𝑠)}𝑏̃₁+𝑐₁̃+𝑑₁𝐿𝑚−𝑓_∼1𝑚²1𝑑𝑠=𝑎_∼1𝑏̃₁+𝑐₁̃+𝑑₁𝐿𝑚−𝑓_∼1𝑚²𝑧≤𝐿,₂(𝑡)≥^𝑡_−∞𝑒⁻𝑎_{2(𝑡−𝑠)}𝑐_∼2+𝑏_∼2+𝑑_∼2𝑚𝐿−𝑓₂𝐿²1𝑑𝑠=𝑎_2∼𝑐₂+𝑏_∼2+𝑑_∼2𝑚𝐿−𝑓₂𝐿²𝑧≥𝑚,₂(𝑡)≤^𝑡_−∞𝑒^{−𝑎∼2(𝑡−𝑠)}𝑐₂̃+𝑏̃₂+𝑑₂𝐿𝑚−𝑓_∼2𝑚²1𝑑𝑠=𝑎_∼2𝑐₂̃+𝑏̃₂+𝑑₂𝐿𝑚−𝑓_∼2𝑚²𝑧≤𝐿,₃(𝑡)≥^𝑡_−∞𝑒⁻𝑎₃^{(𝑡−𝑠)}𝑑_∼3+𝑏_∼3+𝑐_∼3𝑚𝐿−𝑓₃𝐿²1𝑑𝑠=𝑎₃𝑑_∼3+𝑏_∼3+𝑐_∼3𝑚𝐿−𝑓₃𝐿²𝑧≥𝑚,₃(𝑡)≤^𝑡_−∞𝑒^{−𝑎∼3(𝑡−𝑠)}𝑑₃+𝑏̃₃+𝑐₃̃𝐿𝑚−𝑓_∼3𝑚²1𝑑𝑠=𝑎_∼3𝑑₃+𝑏̃₃+𝑐₃̃𝐿𝑚−𝑓_∼3𝑚²≤𝐿. (3.15) Therefore, (𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡))∈𝐻^𝑚_𝐿, that is, 𝐴𝐻^𝑚_𝐿⊂𝐻^𝑚_𝐿. If 𝐴 is uniformly boundness and equicontinuous, by Ascoli-Arzela theorem, 𝐴 is compact mapping.
It is obvious to obtain uniformly boundedness. In fact, for any (𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿, by (3.15) we have (𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡))=𝐴(𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿; that is, it satisfies 0<(𝑚,𝑚,𝑚)≤𝐴𝜙(𝑡),𝜑(𝑡),𝛾(𝑡)=𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡)≤(𝐿,𝐿,𝐿). (3.16)
Next we prove equicontinuous. For any (𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿, we denote (𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡))=𝐴(𝜙(𝑡),𝜑(𝑡),𝛾(𝑡)),𝑡₁<𝑡₂, and then 𝑧₁𝑡₁−𝑧₁𝑡₂=^𝑡1_−∞𝑒^{−∫𝑡1𝑠𝑎1(𝑟)𝑑𝑟}𝑏₁(𝑠)+𝑐₁(𝑠)𝜙(𝑠)𝜑(𝑠)+𝑑₁(𝑠)𝜙(𝑠)𝛾(𝑠)−𝑓₁(𝑠)𝜙²(𝑠−_𝑡2−∞)𝑒^{−∫𝑡2𝑠𝑎1(𝑟)𝑑𝑟}𝑏₁(𝑠)+𝑐₁(𝑠)𝜙(𝑠)𝜑(𝑠)+𝑑₁(𝑠)𝜙(𝑠)𝛾(𝑠)−𝑓₁(𝑠)𝜙²(𝑠). (3.17)
Let ℎ₁(𝑡)=𝑏₁(𝑡)+𝑐₁(𝑡)(𝜙(𝑡)/𝜑(𝑡))+𝑑₁(𝑡)(𝜙(𝑡)/𝛾(𝑡))−𝑓₁(𝑡)𝜙²(𝑡); we obtain 𝑧₁𝑡₂−𝑧₁𝑡₁=^𝑡2_−∞𝑒^{−∫𝑡2𝑠𝑎1(𝑟)𝑑𝑟}ℎ₁(𝑠)𝑑𝑠−^𝑡1_−∞𝑒^{−∫𝑡1𝑠𝑎1(𝑟)𝑑𝑟}ℎ₁(𝑠)≤𝑑𝑠_𝑡2𝑡1𝑒^{−∫𝑡2𝑠𝑎1(𝑟)𝑑𝑟}ℎ₁(𝑠)𝑑𝑠+^𝑡1_−∞𝑒^{−∫𝑡1𝑠𝑎1(𝑟)𝑑𝑟}𝑒^{−∫𝑡2𝑡1𝑎1(𝑟)𝑑𝑟}−1ℎ₁(𝑠)𝑑𝑠. (3.18) Recalling (𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿, we deduce that there exists a positive number 𝑀 such that |ℎ₁(𝑠)|≤𝑀; then (3.18) becomes 𝑧₁𝑡₂−𝑧₁𝑡₁≤𝑀𝑒^{−∫𝑡2𝜉1𝑎1(𝑟)𝑑𝑟}𝑡₂−𝑡₁+1𝑎_∼1𝑀1−𝑒^{−∫𝑡2𝑡1𝑎1(𝑟)𝑑𝑟}, (3.19) where 𝜉₁∈(𝑡₁,𝑡₂).
Similarly, we have 𝑧₂𝑡₂−𝑧₂𝑡₁≤𝑁𝑒^{−∫𝑡2𝜉2𝑎2(𝑟)𝑑𝑟}𝑡₁−𝑡₂+1𝑎_∼2𝑁1−𝑒^{−∫𝑡2𝑡1𝑎2(𝑟)𝑑𝑟}, (3.20) where 𝜉₂∈(𝑡₁,𝑡₂), and 𝑁 is a positive number.
By a completely analogous argument, we obtain 𝑧₃𝑡₂−𝑧₃𝑡₁≤𝑃𝑒^{−∫𝑡2𝜉3𝑎3(𝑟)𝑑𝑟}𝑡₁−𝑡₂+1𝑎_∼3𝑃1−𝑒^{−∫𝑡2𝑡1𝑎3(𝑟)𝑑𝑟}, (3.21) where 𝜉₃∈(𝑡₁,𝑡₂), and 𝑃 is a positive number.
By (3.19)–(3.21), for any (𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿, we derive lim_𝜉→0sup_{𝑡1−𝑡2≤𝜉}𝐴𝜙,𝜑,𝛾𝑡₁−𝐴𝜙,𝜑,𝛾𝑡₂=0,uniformlyfor𝑥∈Ω. (3.22)
Thus, 𝐴 is a compact mapping which maps 𝐻^𝑚_𝐿 into itself; by Schauder fixed point theorem, there exists a fixed point (𝜙(𝑡),𝜑(𝑡),𝛾(𝑡))∈𝐻^𝑚_𝐿 for 𝐴; namely, (3.11) has a solution; therefore there exists a strictly positive almost periodic solution (𝑣^∗₁(𝑡),𝑣^∗₂(𝑡),𝑣^∗₃(𝑡))=(1/𝜙(𝑡),1/𝜑(𝑡),1/𝛾(𝑡)),𝑡∈𝑅⁺ for system (3.10). It is obvious that (𝑣^∗₁(𝑡),𝑣^∗₂(𝑡),𝑣^∗₃(𝑡)),𝑡∈𝑅⁺ is also the spatial homogeneity almost periodic solution for (1.1).

Theorem 3.2. Under the conditions of Theorem 3.1, suppose that system (1.1) satisfies the following conditions: sup_𝑡≥0𝑏₃(𝑡)+𝑏₂(𝑡)−𝑏₁(𝑡)=−𝜀₁<0,sup_𝑡≥0𝑐₃(𝑡)+𝑐₁(𝑡)−𝑐₂(𝑡)=−𝜀₂<0,sup_𝑡≥0𝑑₁(𝑡)+𝑑₂(𝑡)−𝑑₃(𝑡)=−𝜀₃<0. (3.23) Then there exists a strictly positive spatial homogeneity almost periodic solution (𝑣^∗₁(𝑡),𝑣^∗₂(𝑡),𝑣^∗₃(𝑡)) for (1.1), and the corresponding solution for systems (1.1)–(1.3) is globally asymptotically stable; that is, the solution (𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)),(𝑥,𝑡)∈Ω×𝑅⁺ satisfies lim_𝑡→∞𝑣_𝑖(𝑥,𝑡)−v^∗_𝑖(𝑡)=0,𝑖=1,2,3,uniformlyfor𝑥∈Ω. (3.24)

Proof. We have obtained the existence by Theorem 3.1; next we pay more attention to the stability. Concerning (3.24), we have two cases on initial data 𝑣_𝑖0(𝑥),𝑖=1,2,3. (1) 𝑣_𝑖0(𝑥)>0,𝑥∈Ω. (2)There exists a point 𝑥₀∈Ω, such that 𝑣₁₀(𝑥₀)=0,𝑣₂₀(𝑥₀)=0 or 𝑣₃₀(𝑥₀)=0.
For the case (1), let 𝑙_𝑖=min_𝑥∈Ω𝑣_𝑖0(𝑥), 𝑟_𝑖=max_𝑥∈Ω𝑣_𝑖0(𝑥), 𝑖=1,2,3; then 0<𝑙_𝑖≤𝑣_𝑖0(𝑥)≤𝑟_𝑖. Suppose that (𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) and (𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) are the solution for (3.10) corresponding to initial datum (𝑣₁(0),𝑣₂(0),𝑣₃(0))=(𝑟₁,𝑟₂,𝑟₃) and (𝑣₁(0),𝑣₂(0),𝑣₃(0))=(𝑙₁,𝑙₂,𝑙₃), respectively; then there are a pair of ordered upper and lower solutions (𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) and (𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) for (1.1)–(1.3); by Lemma 2.5, there exists a unique solution (𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)),(𝑥,𝑡)∈Ω×𝑅⁺ for system (1.1)–(1.3), which satisfies 𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)≤𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)≤𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡). (3.25)
If we have lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣^∗_𝑖(𝑡)=lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣^∗_𝑖(𝑡)=0,𝑖=1,2,3, (3.26) then (3.24) holds. Therefore, if we want to obtain (3.26), we only need to prove that the solution (𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) for (3.10) with arbitrary positive initial data (𝑣₁(0),𝑣₂(0),𝑣₃(0))=(𝑣₁₀,𝑣₂₀,𝑣₃₀) satisfies lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣^∗_𝑖(𝑡)=0,𝑖=1,2,3. (3.27)
Because of the initial datum (𝑣₁₀,𝑣₂₀,𝑣₃₀)>0 and grazing rates (𝑓_∼1,𝑓_∼2,𝑓_∼3)>0, by the practical meaning in biology, we know that (𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡))>0. Now let 𝑃_𝑖(𝑡)=ln𝑣_𝑖(𝑡),𝑄_𝑖(𝑡)=ln𝑣^∗_𝑖(𝑡),𝑖=1,2,3. (3.28) Then one has 𝑑𝑃𝑑𝑡¹(𝑡)−𝑄₁(𝑡)=−𝑏₁(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₁(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₁(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)+1𝑣₁(𝑡)−1𝑣^∗₁(𝑡)𝑓₁(𝑡),𝑑𝑃𝑑𝑡²(𝑡)−𝑄₂(𝑡)=−𝑏₂(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₂(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₂(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)+1𝑣₂(𝑡)−1𝑣^∗₂(𝑡)𝑓₂(𝑡),𝑑𝑃𝑑𝑡³(𝑡)−𝑄₃(𝑡)=−𝑏₃(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₃(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₃(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)+1𝑣₃(𝑡)−1𝑣^∗₃(𝑡)𝑓₃(𝑡). (3.29) Namely, 𝑑𝑃𝑑𝑡¹(𝑡)−𝑄₁(𝑡)=−𝑏₁(𝑡)+𝑓₁(𝑡)𝑣₁(𝑡)𝑣^∗₁(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₁(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₁(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡),𝑑𝑃𝑑𝑡²(𝑡)−𝑄₂(𝑡)=−𝑏₂(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₂(𝑡)+𝑓₂(𝑡)𝑣₂(𝑡)𝑣^∗₂(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₂(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡),𝑑𝑃𝑑𝑡³(𝑡)−𝑄₃(𝑡)=−𝑏₃(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₃(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₃(𝑡)+𝑓₃(𝑡)𝑣₃(𝑡)𝑣^∗₃(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡). (3.30) Consider the following Lyapunov function: 𝑈(𝑡)=³_𝑖=1𝑃_𝑖(𝑡)−𝑄_𝑖(𝑡),𝑡≥0. (3.31)
Let 𝐷⁺𝑈 represent the right derivation on function 𝑈(𝑡); we have 𝐷⁺𝑈(𝑡)=³_𝑖=1𝐷⁺𝑃_𝑖(𝑡)−𝑄_𝑖(𝑡)=³_𝑖=1sgn𝑃_𝑖(𝑡)−𝑄_𝑖(𝑡)𝑑𝑃𝑑𝑡^𝑖(𝑡)−𝑄_𝑖(𝑡)=sgn𝑃₁(𝑡)−𝑄₁(𝑡)×−𝑏₁(𝑡)+𝑓₁(𝑡)𝑣₁(𝑡)𝑣^∗₁(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₁(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₁(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)+sgn𝑃₂(𝑡)−𝑄₂(𝑡×)−𝑏₂(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₂(𝑡)+𝑓₂(𝑡)𝑣₂(𝑡)𝑣^∗₂(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₂(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)+sgn𝑃₃(𝑡)−𝑄₃(𝑡)×−𝑏₃(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)−𝑐₃(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)−𝑑₃(𝑡)+𝑓₃(𝑡)𝑣₃(𝑡)𝑣^∗₃(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)≤𝑏₃(𝑡)+𝑏₂(𝑡)−𝑏₁(𝑡)𝑒^𝑃1(𝑡)−𝑒^𝑄1(𝑡)+𝑐₁(𝑡)+𝑐₃(𝑡)−𝑐₂(𝑡)𝑒^𝑃2(𝑡)−𝑒^𝑄2(𝑡)+𝑑₁(𝑡)+𝑑₂(𝑡)−𝑑₃(𝑡)𝑒^𝑃3(𝑡)−𝑒^𝑄3(𝑡)≤−𝜀₁𝑣₁(𝑡)−𝑣^∗₁(𝑡)−𝜀₂𝑣₂(𝑡)−𝑣^∗₂(𝑡)−𝜀₃𝑣₃(𝑡)−𝑣^∗₃(𝑡). (3.32) Integrated by the time, we have 𝑈(𝑡)+³_𝑖=1𝜀_𝑖^𝑡₀𝑣_𝑖(𝑠)−𝑣^∗_𝑖(𝑠)𝑑𝑠≤𝑈(0). (3.33) By the nonnegative of 𝑈(𝑡) and the boundedness of 𝑈(0), we obtain that the 𝑈(𝑡) is bounded, and ^𝑡₀𝑣_𝑖(𝑡)−𝑣^∗_𝑖(𝑡)𝑑𝑠,𝑖=1,2,3, (3.34) convergences, by (3.32) we get 𝐷⁺𝑈(𝑡)<0, then the limit lim_𝑡→∞𝑈(𝑡)=𝑙 (3.35) exists, and 𝑈(𝑡)≥𝑙. If 𝑙>0, then at least one of the following three inequalities 𝑃₁(𝑡)−𝑄₁(𝑡>𝑙)4,𝑃₂(𝑡)−𝑄₂(𝑡>𝑙)4,𝑃₃(𝑡)−𝑄₃(𝑡>𝑙)4 (3.36) holds. Without loss of generality, we assume that |𝑃₁(𝑡)−𝑄₁(𝑡)|>𝑙/4. Thus there is no point of intersection between 𝑃₁(𝑡) and 𝑄₁(𝑡). Suppose that 𝑃₁(𝑡)>𝑄₁(𝑡); then we have 𝑃₁(𝑡)−𝑄₁(𝑡)>𝑙/4. Thus ^𝑡₀𝑣₁(𝑡)−𝑣^∗₁(𝑡)𝑑𝑠=^𝑡₀𝑒^𝑃1(𝑠)−𝑒^𝑄1(𝑠)𝑑𝑠=^𝑡₀𝑒^𝑄1(𝑠)𝑒^{𝑃1(𝑠)−𝑄1(𝑠)}−1𝑑𝑠≥𝑚^𝑡₀𝑒^{𝑃1(𝑠)−𝑄1(𝑠)}−1𝑑𝑠>𝑚^𝑡₀𝑒^𝑙/4−1𝑑𝑠=𝑚𝑒^𝑙/4−1𝑡⟶+∞, (3.37) which contradicts with the convergence of ∫^𝑡₀|𝑣_𝑖(𝑠)−𝑣^∗_𝑖(𝑠)|𝑑𝑠. Therefore 𝑙=0; consequently lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣^∗_𝑖(𝑡)=0,𝑖=1,2,3. (3.38) Then we obtain (3.27).
For the case (2), firstly, choose three sufficient large positive numbers 𝑀₁,𝑀₂,and𝑀₃, such that 𝑓₁(𝑡)≤−𝑀₁𝑎₁(𝑡)−𝑏₁(𝑡)𝑀₁𝑓,𝑡>0,₂(𝑡)≤−𝑀₂𝑎₂(𝑡)−𝑐₂(𝑡)𝑀₂𝑓,𝑡>0,₃(𝑡)≤−𝑀₃𝑎₃(𝑡)−𝑑₃(𝑡)𝑀₃,𝑡>0, (3.39) and 𝑀_𝑖≥max_𝑥∈Ω𝑣_𝑖0(𝑥), 𝑖=1,2,3. Let 𝑣_∼𝑖=0, 𝑣_𝑖=𝑀_𝑖, 𝑖=1,2,3; then we have 𝜕𝑣₁𝜕𝑡−𝑘₁(𝑡)Δ𝑣₁−𝑣₁𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁−𝑐₁(𝑡)𝑣_∼2−𝑑₁(𝑡)𝑣_∼3−𝑓₁(𝑡)≥0,𝜕𝑣_∼1𝜕𝑡−𝑘₁(𝑡)Δ𝑣_∼1−𝑣_∼1𝑎₁(𝑡)−𝑏₁(𝑡)𝑣_∼1−𝑐₁(𝑡)𝑣₂−𝑑₁(𝑡)𝑣₃−𝑓₁(𝑡)𝜕𝑣≤0,₂𝜕𝑡−𝑘₂(𝑡)Δ𝑣₂−𝑣₂𝑎₂(𝑡)−𝑏₂(𝑡)𝑣_∼1−𝑐₂(𝑡)𝑣₂−𝑑₂(𝑡)𝑣_∼3−𝑓₂(𝑡)≥0,𝜕𝑣_∼2𝜕𝑡−𝑘₂(𝑡)Δ𝑣_∼2−𝑣_∼2𝑎₂(𝑡)−𝑏₂(𝑡)𝑣₁−𝑐₂(𝑡)𝑣_∼2−𝑑₂(𝑡)𝑣₃−𝑓₂(𝑡)𝜕𝑣≤0,₃𝜕𝑡−𝑘₃(𝑡)Δ𝑣₃−𝑣₃𝑎₃(𝑡)−𝑏₃(𝑡)𝑣_∼1−𝑐₃(𝑡)𝑣_∼2−𝑑₃(𝑡)𝑣₃−𝑓₃(𝑡)≥0,𝜕𝑣_∼3𝜕𝑡−𝑘₃(𝑡)Δ𝑣_∼3−𝑣_∼3𝑎₃(𝑡)−𝑏₃(𝑡)𝑣₁−𝑐₃(𝑡)𝑣₂−𝑑₃(𝑡)𝑣_∼3−𝑓₃(𝑡)≤0. (3.40) Namely, 𝑣_∼𝑖=0, and 𝑣_𝑖=𝑀_𝑖, 𝑖=1,2,3 are a pair of ordered upper and lower solutions for systems (1.1)–(1.3). By Lemma 2.5, there exists a unique solution (𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)) for systems (1.1)–(1.3), which satisfy 0≤𝑣_𝑖(𝑥,𝑡)≤𝑀_𝑖,𝑖=1,2,3;(𝑥,𝑡)∈Ω×[0,∞). (3.41)
Secondly, we choose positive numbers 𝛿₁,𝛿₂,and𝛿₃ such that 𝛿₁+𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁(𝑥,𝑡)−𝑐₁(𝑡)𝑣₂(𝑥,𝑡)−𝑑₁(𝑡)𝑣₃(𝑥,𝑡)>0,(𝑥,𝑡)∈[),𝛿Ω×0,∞₂+𝑎₂(𝑡)−𝑏₂(𝑡)𝑣₁(𝑥,𝑡)−𝑐₂(𝑡)𝑣₂(𝑥,𝑡)−𝑑₂(𝑡)𝑣₃(𝑥,𝑡)>0,(𝑥,𝑡)∈[),𝛿Ω×0,∞₃+𝑎₃(𝑡)−𝑏₃(𝑡)𝑣₁(𝑥,𝑡)−𝑐₃(𝑡)𝑣₂(𝑥,𝑡)−𝑑₃(𝑡)𝑣₃(𝑥,𝑡)>0,(𝑥,𝑡)∈Ω×[0,∞). (3.42) Accordingly, we have 𝜕𝑣₁𝜕𝑡−𝑘₁(𝑡)Δ𝑣₁+𝛿₁𝑣₁=𝑣₁𝛿₁+𝑎₁(𝑡)−𝑏₁(𝑡)𝑣₁−𝑐₁(𝑡)𝑣₂−𝑑₁(𝑡)𝑣₃+𝑓₁(𝑡)≥0,𝜕𝑣₂𝜕𝑡−𝑘₂(𝑡)Δ𝑣₂+𝛿₂𝑣₂=𝑣₂𝛿₂+𝑎₂−𝑏₂(𝑡)𝑣₁−𝑐₂(𝑡)𝑣₂−𝑑₂(𝑡)𝑣₃+𝑓₂(𝑡)≥0,𝜕𝑣₃𝜕𝑡−𝑘₃(𝑡)Δ𝑣₃+𝛿₃𝑣₃=𝑣₃𝛿₃+𝑎₃−𝑏₃(𝑡)𝑣₁−𝑐₃(𝑡)𝑣₂−𝑑₃(𝑡)𝑣₃+𝑓₃(𝑡)≥0. (3.43) Next, we prove 𝑣_𝑖(𝑥,𝑡)>0 in Ω×(0,∞) for 𝑖=1,2,3. Firstly, we show 𝑣_𝑖(𝑥,𝑡)>0 in Ω×(0,∞). If there exists one point (𝑥₀,𝑡₀)∈Ω×(0,∞) such that 𝑣_𝑖(𝑥₀,𝑡₀)=0, by extremum principle, we have 𝑣_𝑖(𝑥,𝑡)≡0 in Ω×[0,𝑡₀). However 𝑣_𝑖(𝑥,0)=𝑣_𝑖0(𝑥)≥0, and not being constant zero, we obtain a contradiction. Therefore we have 𝑣_𝑖(𝑥,𝑡)>0 in Ω×(0,∞). Then we show 𝑣_𝑖(𝑥,𝑡)>0 in 𝜕Ω×(0,∞). If there exists a point (𝑥₀,𝑡₀)∈𝜕Ω×(0,∞) such that 𝑣_𝑖(𝑥₀,𝑡₀)=0, by the extremum principle, we have 𝜕𝑣_𝑖(𝑥,𝑡)/𝜕𝑛<0, where (𝑥,𝑡)∈𝜕Ω×(0,∞), which is contrary with boundary conditions (1.2). Thus we have 𝑣_𝑖(𝑥,𝑡)>0 in Ω×(0,∞).
For a fixed number 𝜆>0, by (3.41), we have 0<𝑣_𝑖(𝑥,𝜆)≤𝑀_𝑖,𝑖=1,2,3,𝑥∈Ω. (3.44) Because 𝑣_𝑖(𝑥,𝑡+𝜆) satisfy system (1.1) in Ω×(0,∞) and the conditions (1.2) in 𝜕Ω×(0,∞), thereby (𝑣₁(𝑥,𝑡+𝜆),𝑣₂(𝑥,𝑡+𝜆),𝑣₃(𝑥,𝑡+𝜆)) is regarded as a solution for system (1.1) under initial data (𝑣₁₀(𝑥),𝑣₂₀(𝑥),𝑣₃₀(𝑥))=(𝑣₁(𝑥,𝜆),𝑣₂(𝑥,𝜆),𝑣₃(𝑥,𝜆)), nevertheless, we have 𝑣_𝑖0>0 in Ω,𝑖=1,2,3; combining the conclusions in case (1), we have lim_𝑡⟶∞𝑣_𝑖(𝑥,𝑡+𝜆)−𝑣^∗_𝑖(𝑡)=0,𝑖=1,2,3,uniformlyfor𝑥∈Ω. (3.45) By the arbitrariness of 𝜆, we obtain lim_𝑡⟶∞𝑣_𝑖(𝑥,𝑡)−𝑣^∗_𝑖(𝑡)=0,𝑖=1,2,3,uniformlyfor𝑥∈Ω. (3.46)
If 𝑘_𝑖(𝑡),𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖(𝑡),𝑑_𝑖(𝑡),and𝑓_𝑖(𝑡)(𝑖=1,2,3) of (1.1) are periodic functions in real number field 𝑅, respectively, then we have the following results.

Corollary 3.3. If 𝑎_∼𝑖,𝑏_∼𝑖,𝑐_∼𝑖,𝑑_∼𝑖,𝑓_∼𝑖 are positive numbers, and 𝑏̃_𝑖+𝑐_𝑖̃+𝑑_𝑖𝑎_∼𝑖≤𝐿=min𝑏_∼1𝑓₁,_∼𝑐₂𝑓₂,𝑑_∼3𝑓₃,𝑑_∼1+𝑐_∼1𝑎₁,𝑏_∼2+𝑑_∼2𝑎₂,𝑏_∼3+𝑐_∼3𝑎₃ (3.47) are satisfied for 𝑖=1,2,3, then there exists a strictly positive spatial homogeneity periodic solution 𝑉(𝑡)=(𝑣₁(𝑡),𝑣₂(𝑡),𝑣₃(𝑡)) for (1.1).

Corollary 3.4. Under the conditions of Corollary 3.3, suppose that system (1.1) satisfies the following conditions: sup_𝑡≥0𝑏₃(𝑡)+𝑏₂(𝑡)−𝑏₁(𝑡)=−𝜀₁<0,sup_𝑡≥0𝑐₃(𝑡)+𝑐₁(𝑡)−𝑐₂(𝑡)=−𝜀₂<0,sup_𝑡≥0𝑑₁(𝑡)+𝑑₂(𝑡)−𝑑₃(𝑡)=−𝜀₃<0. (3.48) Then there exists a strictly positive spatial homogeneity periodic solution (𝑣^∗₁(𝑡),𝑣^∗₂(𝑡),𝑣^∗₃(𝑡)) for (1.1), and the corresponding solution for systems (1.1)–(1.3) is globally asymptotically stable; that is, the solution (𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),𝑣₃(𝑥,𝑡)),(𝑥,𝑡)∈Ω×𝑅⁺ satisfies lim_𝑡→∞𝑣_𝑖(𝑥,𝑡)−𝑣^∗_𝑖(𝑡)=0,𝑖=1,2,3,uniformlyfor𝑥∈Ω. (3.49)

4. Conclusion

This paper presents the use of upper and lower solutions method for systems of nonlinear reaction-diffusion equations. This method is a powerful tool for solving nonlinear differential equations in mathematical physics, chemistry, and engineering, and so forth. The technique constructing a pair of upper and lower solutions and Lyapunov function provides a new efficient method to handle the nonlinear structure.

We have dealt with the problem of almost periodic solution for a three-species competition system with grazing rates and diffusions. The general sufficient conditions have been obtained to ensure the existence and stability of the strictly positive space homogenous almost periodic solution for the nonlinear reaction-diffusion equations. These criteria generalize and improve some known results. In particular, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear three-species competition system.

Acknowledgments

The authors are grateful to the reviewers for their comments which helped in improving the paper. This work is supported by the Science and Technology Project of Chongqing Municipal Education Committee (Grant no. kJ110501) of China, the NSFC (Grant nos. 51005264, 40801214) of China, and Natural Science Foundation Project of CQ CSTC (Grant no. 2010BB9401) of China.