# Asymptotic Behavior of a Competition-Diffusion System with Variable Coefficients and Time Delays

Authors: Miguel Uh Zapata; Eric Avila Vales; Angel G. Estrella
Journal: Journal of Applied Mathematics
 (2008-02-03)

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## Abstract

A class of time-delay reaction-diffusion systems with variable coefficients which arise from the model of two competing ecological species is discussed. An asymptotic global attractor is established in terms of the variable coefficients, independent of the time delays and the effect of diffusion by the upper-lower solutions and iteration method.


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## Body

## 1. Introduction and Main Result

The Lotka-Volterra competition model with diffusion
and time delays has been the object of analysis by numerous authors under different approaches. For the case of two species, Ruan and Zhao [1] considered
uniform persistence and global extinction; Lu [2] studied global attractivity,
and Gourley and Ruan [3] analyzed stability and traveling fronts. The periodic
case has also been considered, Feng and Wang [4] studied asymptotic stability
and Zhou et al. [5] investigated the Hopf bifurcation. The cases of three and N-species have
also been analyzed in [6–8].In this paper, we consider the asymptotic behavior of solutions
for the competition-diffusion system
with time delays of the following two species:(1.1)∂u1∂t=AΔu1(t,x)+u1(t,x)[a1(t,x)−b1(t,x)u1(t,x)−c1(t,x)∫0∞u1(t−τ,x)dμ1(τ)−d1u2(t−r2,x)],t>0,x∈Ω,∂u2∂t=AΔu2(t,x)+u2(t,x)[a2(t,x)−b2(t,x)u2(t,x)−c2(t,x)∫0∞u2(t−τ,x)dμ2(τ)−d2u1(t−r1,x)],t>0,x∈Ω,∂u1∂η=∂u2∂η=0,t>0,x∈∂Ω,ui(t,x)=ϕi(t,x)(i=1,2)t≤0,x∈Ω¯,
where Ω⊆ℝN (N≥1) is a bounded
domain with smooth boundary ∂Ω, ∂/∂η denotes the
differentiation outward normal on ∂Ω, A≥0, di>0, ri>0, 0<ai≤ai(t,x)≤Ai, 0<bi≤bi(t,x)≤Bi, 0≤ci≤ci(t,x)≤Ci, μi(⋅) is of bounded
variation with μi(0)=0, ϕi∈C1((−∞,0])×Ω¯ is bounded and
nonnegative and 0≢ϕi(0,⋅)∈C1(Ω¯) (i=1,2), and u1 and u2 are the density
functions of two species competing for a shared limited resource.The functionsa1(x,t) and a2(x,t) denote the
intrinsic growth rate of the species, b1(x,t) and b2(x,t) represent
self-limitation rates, and c1(x,t) and c2(x,t) represent the
coefficients of the infinite continuous delay. The constants d1, d2 represent
the competition rates. The distributed time delay should be viewed as the
effects of past history.LetMi(t) denote the
total variation of μi(⋅) on [0,t] and
let Mi±(t)=(Mi(t)±μi(t))/2 , for all t∈ℝ+. Then, Mi(t) and Mi±(t) are nonnegative
and nondecreasing on ℝ+. It easy to see that (1.2)Mi+(t)+Mi−(t)=Mi(t),Mi+(t)−Mi−(t)=μi(t).Denote M0i=limt→∞Mi(t), M0i±=limt→∞Mi±(t), and μ0i=limt→∞μi(t). Then, (1.3)M0i++M0i−=M0i,M0i+−M0i−=μ0i.Our result can be stated as follows.Theorem 1.1.
Assume thatbi>ciM0i−,ai−(CiM0i+Ai)/(bi−ciM0i−)>0, and(1.4)d2A2(b2−c2M02−)(b1−c1M01−)[a2(b2−c2M02−)−C2M02+A2]<A2A1<(b2−c2M02−)[a1(b1−c1M01−)−C1M01+A1]d1A1(b1−c1M01−). Then, for any ϕi∈C1((−∞,0])×Ω¯ with ϕ(0,x)≢0, the solution of (1.1) satisfies(1.5)0<α1≤liminft→∞minx∈Ω¯u1(t,x)≤limsupt→∞maxx∈Ω¯u1(t,x)≤β1,0<α2≤liminft→∞minx∈Ω¯u2(t,x)≤limsupt→∞maxx∈Ω¯u2(t,x)≤β2, where α1, α2, β1, β2 are constants
given by the linear system
(1.6)a1−C1M01+β1−d1β2−α1(B1−c1M01−)=0,a2−C2M02+β2−d2β1−α2(B2−c2M02−)=0,A1−c1M01+α1−d1α2−β1(b1−c1M01−)=0,A2−c2M02+α2−d2α1−β2(b2−c2M02−)=0.Remark 1.2.
Ifμi(τ)=∫0τfi(t)dt with fi∈C(ℝ+)∩L1(ℝ+), A=1, and 0<ai=ai(t,x)=Ai, 0<bi=bi(t,x)=Bi, ci=ci(t,x)=Ci=1,  then by Theorem 1.1, we
get(1.7)0<α1≤liminft→∞minx∈Ω¯u1(t,x)≤limsupt→∞maxx∈Ω¯u1(t,x)≤β1,0<α2≤liminft→∞minx∈Ω¯u2(t,x)≤limsupt→∞maxx∈Ω¯u2(t,x)≤β2,where α1, α2, β1, β2 are constants
given by the linear system
(1.8)a1−M01+β1−d1β2−α1(b1−M01−)=0,a2−M02+β2−d2β1−α2(b2−M02−)=0,a1−M01+α1−d1α2−β1(b1−M01−)=0,a2−M02+α2−d2α1−β2(b2−M02−)=0. Solving this system, we have the following result:
(1.9)α1=β1=a1(b2+M02+−M02−)−a2d1(b1+M01+−M01−)(b2+M02+−M02−)−d1d2,α2=β2=a2(b1+M01+−M01−)−a1d2(b1+M01+−M01−)(b2+M02+−M02−)−d1d2,which coincides
with the result of [9], where the authors considered a system like (1.1) with
constant coefficients.Reaction-diffusion systems with delay have
been treated by many authors. There are two ways to approach
them. The first one is in the framework of
semigroup theory of dynamical systems [1, 10]. The second one is a method of
upper and lower solutions, using associated monotone iterations; several
authors have studied their dynamic properties
[2, 9, 11]. Sometimes the birth and death rates depend on both space and time,
so when we consider instantaneous and delayed interference within the species
and the diffusive effects of the species, system (1.1) will be the appropriate
model.The way we organize the paper is as follows: we first
introduce several results which play an important role in the proof of
Theorem1.1 which we will prove in Section 2. We
will provide some numerical simulations in Section 3 in order to illustrate our
theory.The following results are developed in [11]. They
considered the Volterra reaction-diffusion equations with variable
coefficients: (1.10)∂u∂t=AΔu(t,x)+u(t,x)[a(t,x)−b(t,x)u(t,x)−c(t,x)∫0∞u(t−τ,x)dμ(τ)],t>0,x∈Ω,∂u∂η=0,t>0,x∈∂Ω,u(t,x)=ϕ(t,x),t≤0,x∈Ω¯, where 0<a1≤a(t,x)≤A1, 0<b1≤b(t,x)≤B1, 0≤c1≤c(t,x)≤C1, and ϕ∈C1((−∞,0])×Ω¯ is bounded and
nonnegative and 0≢ϕ(0,⋅)∈C1(Ω¯). Let M(t) denote the
total variation of μ(⋅) and define M±, M0± of the same
form as that of Mi±, M0i±, respectively.Lemma 1.3.
Assume   thatb1>c1M0− and a1−(C1M0+A1)/(b1−c1M0−)>0, then the solution of (1.10) satisfies (1.11)α≤liminft→∞minx∈Ω¯u(t,x)≤limsupt→∞maxx∈Ω¯u(t,x)≤β,
where (1.12)α=(b1−c1M0−)a1−C1M0+A1(b1−c1M0−)(B1−c1M0−)−C12(M0+)2,β=(B1−c1M0−)A1−C1M0+a1(b1−c1M0−)(B1−c1M0−)−C12(M0+)2. If M+(t)=0,  then μ(t)=−M−(t). Thus, (1.13)∂u∂t=AΔu(t,x)+u(t,x)[a(t,x)−b(t,x)u(t,x)+c(t,x)∫0∞u(t−τ,x)dM−(τ)],t>0,x∈Ω,∂u∂η=0,t>0,x∈∂Ω,u(t,x)=ϕ(t,x),t≤0,x∈Ω¯.Lemma 1.4.
Assume thatb1>c1M0− and a1>0, then the solution of (1.13) satisfies(1.14)0<α≤liminft→∞minx∈Ω¯u(t,x)≤limsupt→∞maxx∈Ω¯u(t,x)≤β,where(1.15)α=a1B1−c1M0−,β=A1b1−c1M0−.Lemma 1.5.
Ifu(t,x) is the solution
of (1.10), then 0<u(t,x).Now, we introduce the existence-comparison result for
the competition-diffusion system (1.1) which is a particular case of Theorem 2.2
in[12].Definition 1.6.
A pair of smooth functionsu=(u1,u2) and u^=(u^1,u^2) are called
upper-lower solutions of (1.1) if ui≥u^i (i=1,2) in ℝ×Ω¯ and the
following differential inequalities hold:(1.16)∂ui∂t−AΔui≥ui(t,x)[ai(t,x)−bi(t,x)ui(t,x)−ci(t,x)∫0∞ui(t−τ,x)dμi(τ)−diu^j(t−rj,x)],j≠i,t>0,x∈Ω,∂u^i∂t−AΔu^i≤u^i(t,x)[ai(t,x)−bi(t,x)u^i(t,x)−ci(t,x)∫0∞u^i(t−τ,x)dμi(τ)−diuj(t−rj,x)],j≠i,t>0,x∈Ω,∂u^i∂η≤0≤∂ui∂η,t>0,x∈∂Ω,u^i(t,x)≤ϕi(t,x)≤ui(t,x),t≤0,x∈Ω¯(i=1,2).Lemma 1.7.
If there exists a pair of upper-lower solutionsu=(u1,u2) and u^=(u^1,u^2)
of (1.1), then
the problem (1.1) has a unique solution u∗=(u1∗,u2∗) and u^i≤ui∗≤ui, i=1,2.


## 2. Proof of the Main Result

The method of proof is via successive
improvements of upper-lower solutions of suitable systems.For givenϕ=(ϕ1,ϕ2) as initial
conditions for the system (1.1), let K1, K2 be constants
such that(2.1)K1≥max{||ϕ1||,A1b1−c1M01−},K2≥max{||ϕ2||,A2b2−c2M02−},where ||ϕi||=sup{|ϕi(t,x)|:(t,x)∈(−∞,0]×Ω¯}, i=1,2.Then,(0,0) and (K1,K2) are a pair of
lower-upper solutions of (1.1). By Lemma 1.7, there exists a unique global
nonnegative solution (u1,u2) of (1.1) and it
satisfies 0≤u1(t,x)≤K1, 0≤u2(t,x)≤K2.Defineu¯1(1)(x,t) and u¯2(1)(x,t) by (2.2)∂u¯1(1)∂t=AΔu¯1(1)+u¯1(1)[a1(t,x)−b1(t,x)u¯1(1)+c1(t,x)∫0∞u¯1(1)(t−τ,x)dM1−(τ)]t>0,x∈Ω,∂u¯2(1)∂t=AΔu¯2(1)+u¯2(1)[a2(t,x)−b2(t,x)u¯2(1)+c2(t,x)∫0∞u¯2(1)(t−τ,x)dM2−(τ)]t>0,x∈Ω,∂u¯1(1)∂η=∂u¯2(1)∂η=0,t>0,x∈∂Ω,u¯i(1)(t,x)=Ki,t≤0,x∈Ω¯(i=1,2).Then, (0,0) and (u¯1(1),u¯2(1)) are lower and
upper solutions, and by Lemma 1.7, (2.3)0≤u1≤u¯1(1),0≤u2≤u¯i(1).
By Lemma 1.4, we
can get (2.4)0<limsupt→∞maxx∈Ω¯u¯i(1)(t,x)≤βi(0),  where(2.5)βi(0)=Aibi−ciM0i−,i=1,2. Then from (2.3) and (2.4), we get(2.6)0≤limsupt→∞maxx∈Ω¯ui(t,x)≤βi(0).  From (2.6) and the definition of M0i+=limt→∞Mi+(t), we have
that for any sufficiently small ε>0, there exist t1′>0andt1>t1′such that(2.7)maxx∈Ω¯u¯i(1)(t,x)<βi(0)+ε,fort≥t1′,u¯i(1)(t−ri,x)<βi(0)+ε,fort≥t1,Ci(M0i+−Mi+(t−t1′))<ε,fort≥t1.Define u¯1(1)(t,x) and u¯2(1)(t,x) by(2.8)∂u¯1(1)∂t=AΔu¯1(1)+u¯1(1)[a1(t,x)−b1(t,x)u¯1(1)+c1(t,x)∫0∞u¯1(1)(t−τ,x)dM1−(τ)−c1(t,x)∫0∞u¯1(1)(t−τ,x)dM1+(τ)−d1u¯2(1)(t−r2,x)],t>t1,x∈Ω,∂u¯2(1)∂t=AΔu¯2(1)+u¯2(1)[a2(t,x)−b2(t,x)u¯2(1)+c2(t,x)∫0∞u¯2(1)(t−τ,x)dM2−(τ)−c2(t,x)∫0∞u¯2(1)(t−τ,x)dM2+(τ)−d2u¯1(1)(t−r1,x)],t>t1,x∈Ω,∂u¯1(1)∂η=∂u¯2(1)∂η=0,t>t1,x∈∂Ω,u¯i(1)(t,x)=12ui(t,x),(t,x)∈(−∞,t1]×Ω¯.Then, (u¯1(1),u¯2(1)) and (u¯1(1),u¯2(1)) are a pair of
lower-upper solutions of (1.1), and by Lemma 1.7,(2.9)u¯1(1)≤u1≤u¯1(1),u¯2(1)≤u2≤u¯2(1). From u¯i(1)(t,x)≤Ki, for all t∈ℝ, and (2.8), for t≥t1, we get (2.10)ci(t,x)∫0∞u¯i(1)(t−τ,x)dMi+(τ)+diu¯j(1)(t−rj,x)=ci(t,x)∫0t−t1′u¯i(1)(t−τ,x)dMi+(τ)+ci(t,x)∫t−t1′∞u¯i(1)(t−τ,x)dMi+(τ)+diu¯j(1)(t−rj,x)≤ci(t,x)∫0t−t1′(βi(0)+ε)dMi+(τ)+ci(t,x)∫t−t1′∞KidMi+(τ)+di(βj(0)+ε)≤Ci(Mi+(t−t1)−M(0))(βi(0)+ε)+KiCi(M0i+−Mi+(t−t1))+di(βj(0)+ε)≤CiM0i+(βi(0)+ε)+di(βj(0)+ε)+Kiε.It follows from (2.9) that, for t>t1, x∈Ω , (2.11)∂u¯1(1)∂t≥AΔu¯1(1)+u¯1(1)[a1(t,x)−b1(t,x)u¯1(1)+c1(t,x)∫0∞u¯1(1)(t−τ,x)dM1−(τ)−C1M01+(β1(0)+ε)−d1(β2(0)+ε)−K1ε],∂u¯2(1)∂t≥AΔu¯2(1)+u¯2(1)[a2(t,x)−b2(t,x)u¯2(1)+c2(t,x)∫0∞u¯2(1)(t−τ,x)dM2−(τ)−C2M02+(β2(0)+ε)−d2(β1(0)+ε)−K2ε].By the comparison principle, we get, for t>t1,x∈Ω,(2.12)u¯1(1)≥v1(1),u¯2(1)≥v2(1),where v1(1) and v2(1) are the
solutions of the following problem, respectively: (2.13)∂v1(1)∂t=AΔv1(1)+v1(1)[a1(t,x)−b1(t,x)v1(1)+c1(t,x)∫0∞v1(1)(t−τ,x)dM1−(τ)−C1M01+(β1(0)+ε)−d1(β2(0)+ε)−K1ε],∂v1(1)∂η=0,t>t1,x∈∂Ω,v1(1)(t,x)=12u1(t,x),(t,x)∈(−∞,t1]×Ω¯,∂v2(1)∂t=AΔv2(1)+v2(1)[a2(t,x)−b2(t,x)v2(1)+c2(t,x)∫0∞v2(1)(t−τ,x)dM2−(τ)−C2M02+(β2(0)+ε)−d2(β1(0)+ε)−K2ε],∂v2(1)∂η=0,t>t1,x∈∂Ω,v2(1)(t,x)=12u2(t,x),(t,x)∈(−∞,t1]×Ω¯.Using the three initial conditions and ε sufficiently
small, we have(2.14)a1−C1M01+(β1(0)+ε)−d1(β2(0)+ε)−K1ε>0,a2−C2M02+(β2(0)+ε)−d2(β1(0)+ε)−K2ε>0.By Lemma 1.4, we get (2.15)0<a1−C1M01+β1(0)−d1β2(0)B1−c1M01−−εC1M01++d1+K1B1−c1M01−≤liminft→∞minx∈Ω¯v1(1)(t,x),0<a2−C2M02+β2(0)−d2β1(0)B2−c2M02−−εC2M02++d2+K2B2−c2M02−≤liminft→∞minx∈Ω¯v2(1)(t,x).Then from (2.12), (2.14), and ε sufficiently
small, we can conclude that(2.16)0<α1(0)≤liminft→∞minx∈Ω¯u1(t,x)≤limsupt→∞maxx∈Ω¯u1(t,x)≤β1(0),0<α2(0)≤liminft→∞minx∈Ω¯u2(t,x)≤limsupt→∞maxx∈Ω¯u2(t,x)≤β2(0),  where(2.17)α1(0)=a1−C1M01+β1(0)−d1β2(0)B1−c1M01−,α2(0)=a2−C2M02+β2(0)−d2β1(0)B2−c2M02−. For any sufficiently small ε>0, there exist t2′>t1andt2>t2′such that(2.18)minx∈Ω¯u¯i(1)(t,x)>αi(0)−ε,fort≥t2′,u¯i(1)(t−ri,x)>αi(0)+ε,fort≥t2,M0i+−ε<Mi+(t−t2′),fort≥t2,Define u¯1(2) and u¯2(2) by(2.19)∂u¯1(2)∂t=AΔu¯1(2)+u¯1(2)[a1(t,x)−b1(t,x)u¯1(2)+c1(t,x)∫0∞u¯1(2)(t−τ,x)dM1−(τ)−c1(t,x)∫0∞u¯1(1)(t−τ,x)dM1+(τ)−d1u¯2(1)(t−r2,x)],t>t2,x∈Ω,∂u¯2(2)∂t=AΔu¯2(2)+u¯2(2)[a2(t,x)−b2(t,x)u¯2(2)+c2(t,x)∫0∞u¯2(2)(t−τ,x)dM2−(τ)−c2(t,x)∫0∞u¯2(1)(t−τ,x)dM2+(τ)−d2u¯1(1)(t−r1,x)],t>t2,x∈Ω,∂u¯1(2)∂η=∂u¯2(2)∂η=0,t>t2,x∈∂Ω,u¯i(2)(t,x)=Ki,(t,x)∈(−∞,t2]×Ω¯(i=1,2).Then (u¯1(1),u¯2(1)) and (u¯1(2),u¯2(2)) are a pair of
lower-upper solutions of (1.1). By Lemma 1.7,(2.20)u¯1(1)≤u1≤u¯1(2),u¯2(1)≤u2≤u¯2(2). From (2.19) and (2.20), for t>t2, x∈Ω, we get (2.21)∂u¯1(2)∂t≤AΔu¯1(2)+u¯1(2)[a1(t,x)−b1(t,x)u¯1(2)+c1(t,x)∫0∞u¯1(2)(t−τ,x)dM1−(τ)−c1M01+(α1(0)−ε)−d1(α2(0)−ε)+εc1α1(0)],∂u¯2(2)∂t≤AΔu¯2(2)+u¯2(2)[a2(t,x)−b2(t,x)u¯2(2)+c2(t,x)∫0∞u¯2(2)(t−τ,x)dM2−(τ)−c2M02+(α2(0)−ε)−d2(α1(0)−ε)+εc2α2(0)].By the comparison principle, for t>t2,x∈Ω, we get(2.22)u¯1(2)≤w1(1),u¯2(2)≤w2(1), where w1(1) and w2(1) are the
solutions of the following problem, respectively:(2.23)∂w1(1)∂t=AΔw1(1)+w1(1)[a1(t,x)−b1(t,x)w1(1)+c1(t,x)∫0∞w1(1)(t−τ,x)dM1−(τ)−c1M01+(α1(0)−ε)−d1(α2(0)−ε)+εc1α1(0)],t>t2,x∈Ω,∂w1(1)∂η=0,t>t2,x∈∂Ω,w1(1)(t,x)=K1,(t,x)∈(−∞,t2]×Ω¯,∂w2(1)∂t=AΔw2(1)+w2(1)[a2(t,x)−b2(t,x)w2(1)+c2(t,x)∫0∞w2(1)(t−τ,x)dM2−(τ)−c2M02+(α2(0)−ε)−d2(α1(0)−ε)+εc2α2(0)],t>t2,x∈Ω,∂w2(1)∂η=0,t>t2,x∈∂Ω,w2(1)(t,x)=K2,(t,x)∈(−∞,t2]×Ω¯.From (2.16), (2.18), and sufficiently small ε, we get(2.24)a1−c1M01+(α1(0)−ε)−d1(α2(0)−ε)>0,a2−c2M02+(α2(0)−ε)−d2(α1(0)−ε)>0.By Lemma 1.4, we get(2.25)0<limsupt→∞maxx∈Ω¯w1(1)(t,x)≤A1−c1M01+α1(0)−d1α2(0)b1−c1M01−−εc1M01++d1−c1α1(0)b1−c1M01−,0<limsupt→∞maxx∈Ω¯w2(1)(t,x)≤A2−c2M02+α2(0)−d2α1(0)b2−c2M02−−εc2M02++d2−c2α2(0)b2−c2M02−.Then from (2.22), (2.24), and
sufficiently small ε,(2.26)0<α1(0)≤liminft→∞minx∈Ω¯u1(t,x)≤limsupt→∞maxx∈Ω¯u1(t,x)≤β1(1),0<α2(0)≤liminft→∞minx∈Ω¯u2(t,x)≤limsupt→∞maxx∈Ω¯u2(t,x)≤β2(1),where(2.27)β1(1)=A1−c1M01+α1(0)−d1α2(0)b1−c1M01−,β2(1)=A2−c2M02+α2(0)−d2α1(0)b2−c2M02−.Then,(2.28)0<α1(0)≤β1(1)≤β1(0),0<α2(0)≤β2(1)≤β2(0).For any sufficiently small ε>0, there exist t3′>t2andt3>t3′such that(2.29)maxx∈Ω¯u¯i(2)(t,x)<βi(1)+ε,fort≥t3′,u¯i(1)(t−ri,x)<βi(1)+ε,fort≥t3,Ci(M0i+−Mi+(t−t3′))<ε,fort≥t3.Define u¯1(2)(t,x) and u¯2(2)(t,x) by(2.30)∂u¯1(2)∂t=AΔu¯1(2)+u¯1(2)[a1(t,x)−b1(t,x)u¯1(2)+c1(t,x)∫0∞u¯1(2)(t−τ,x)dM1−(τ)−c1(t,x)∫0∞u¯1(2)(t−τ,x)dM1+(τ)−d1u¯2(2)(t−r2,x)],t>t3,x∈Ω,∂u¯2(2)∂t=AΔu¯2(2)+u¯2(2)[a2(t,x)−b2(t,x)u¯2(2)+c2(t,x)∫0∞u¯2(2)(t−τ,x)dM2−(τ)−c2(t,x)∫0∞u¯2(2)(t−τ,x)dM2+(τ)−d2u¯1(2)(t−r1,x)],t>t3,x∈Ω,∂u¯1(2)∂η=∂u¯2(2)∂η=0,t>t3,x∈∂Ω,u¯i(2)(t,x)=12ui(t,x),(t,x)∈(−∞,t3]×Ω¯.Then, (u¯1(2),u¯2(2)) and (u¯1(2),u¯2(2)) are a pair of
lower-upper solutions of (1.1). By Lemma 1.7,(2.31)u¯1(2)≤u1≤u¯1(2),u¯2(2)≤u2≤u¯2(2).From (2.30) and (2.31), for t>t3, x∈Ω, we get(2.32)∂u¯1(2)∂t≥AΔu¯1(2)+u¯1(2)[a1(t,x)−b1(t,x)u¯1(2)+c1(t,x)∫0∞u¯1(2)(t−τ,x)dM1−(τ)−C1M01+(β1(1)+ε)−d1(β2(1)+ε)−K1ε],∂u¯2(2)∂t≥AΔu¯2(2)+u¯2(2)[a2(t,x)−b2(t,x)u¯2(2)+c2(t,x)∫0∞u¯2(2)(t−τ,x)dM2−(τ)−C2M02+(β2(1)+ε)−d2(β1(1)+ε)−K2ε]. By the comparison principle, for t>t3,x∈Ω, we get (2.33)u¯1(2)≥v1(2),u¯2(2)≥v2(2), where v1(2) and v2(2) are the
solutions of the following problem, respectively:(2.34)∂v1(2)∂t=AΔv1(2)+v1(2)[a1(t,x)−b1(t,x)v1(2)+c1(t,x)∫0∞v1(2)(t−τ,x)dM1−(τ)−C1M01+(β1(1)+ε)−d1(β2(1)+ε)−K1ε],t>t3,x∈Ω,∂v1(2)∂η=0,t>t3,x∈∂Ω,v1(2)(t,x)=12u1(t,x),(t,x)∈(−∞,t3]×Ω¯,∂v2(2)∂t=AΔv2(2)+v2(2)[a2(t,x)−b2(t,x)v2(2)+c2(t,x)∫0∞v2(2)(t−τ,x)dM2−(τ)−C2M02+(β2(1)+ε)−d2(β1(1)+ε)−K2ε],t>t3,x∈Ω,∂v2(2)∂η=0,t>t3,x∈∂Ω,v2(2)(t,x)=12u2(t,x),(t,x)∈(−∞,t3]×Ω¯.From (2.29), (2.30), and ε sufficiently
small, we get(2.35)a1−C1M01+(β1(1)+ε)−d1(β2(1)+ε)−K1ε>0,a2−C2M02+(β2(1)+ε)−d2(β1(1)+ε)−K2ε>0.By Lemma 1.4 we get(2.36)0<a1−C1M01+β1(1)−d1β2(1)B1−c1M01−−εC1M01++d1+K1B1−c1M01−≤liminft→∞minx∈Ω¯v1(1)(t,x),0<a2−C2M02+β2(1)−d2β1(1)B2−c2M02−−εC2M02++d2+K2B2−c2M02−≤liminft→∞minx∈Ω¯v2(1)(t,x).From (2.33), (2.35), and for ε, we conclude that(2.37)0<α1(1)≤liminft→∞minx∈Ω¯u1(t,x)≤limsupt→∞maxx∈Ω¯u1(t,x)≤β1(1),0<α2(1)≤liminft→∞minx∈Ω¯u2(t,x)≤limsupt→∞maxx∈Ω¯u2(t,x)≤β2(1),where(2.38)α1(1)=a1−C1M01+β1(1)−d1β2(1)B1−c1M01−,α2(1)=a2−C2M02+β2(1)−d2β1(1)B2−c2M02−. Then,(2.39)0<α1(0)≤α1(1)≤β1(1)≤β1(0),0<α2(0)≤α2(1)≤β2(1)≤β2(0).  Define the sequences α1(k), α2(k), β1(k), and β2(k) as follows:
(2.40)α1(k)=a1−C1M01+β1(k)−d1β2(k)B1−c1M01−,α2(k)=a2−C2M02+β2(k)−d2β1(k)B2−c2M02−,β1(k+1)=A1−c1M01+α1(k)−d1α2(k)b1−c1M01−,β2(k+1)=A2−c2M02+α2(k)−d2α1(k)b2−c2M02−,β1(0)=A1b1−c1M01−,β2(0)=A2b2−c2M02−.Lemma 2.1.
For the above-defined sequences, one has(2.41)[α1(k+1),β1(k+1)]⊆[α1(k),β1(k)],[α2(k+1),β2(k+1)]⊆[α2(k),β2(k)],k≥0.Proof.
Fork=0, it has been shown that [α1(1),β1(1)]⊆[α1(0),β1(0)] and [α2(1),β2(1)]⊆[α2(0),β2(1)]. Using induction, we can complete the proof.
Lemma2.1 implies that(2.42)limk→∞α1(k),limk→∞α2(k),limk→∞β1(k),limk→∞β2(k)exist, denoted
as α1, α2, β1, and β2, respectively.
From (2.41), we have the following linear system by
which we can obtain the numbers α1, α2, β1, and β2:(2.43)α1=a1−C1M01+β1−d1β2B1−c1M01−,α2=a2−C2M02+β2−d2β1B2−c2M02−,β1=A1−c1M01+α1−d1α2b1−c1M01−,β2=A2−c2M02+α2−d2α1b2−c2M02−.Lemma 2.2.
For the solutions of (1.1), one has (2.44)0<α1(k)≤liminft→∞minx∈Ω¯u1(t,x)≤limsupt→∞maxx∈Ω¯u1(t,x)≤β1(k),k≥0,(2.45)0<α2(k)≤liminft→∞minx∈Ω¯u2(t,x)≤limsupt→∞maxx∈Ω¯u2(t,x)≤β2(k),k≥0.Proof.
We have shown that (2.45) and (3.1) are valid for k=0,1. Using induction and repeating the above process, we
can complete the proof.Combining the above lemmas, we can complete the proof
of Theorem1.1Remark 2.3.
Following the same kind of proof for
Theorem1.1, it can be shown that the same
conclusions hold if instead of system (1.1) we work with solutions of (2.46)∂v1∂t=AΔv1(t,x)+v1(t,x)[a1(t,x)−b1(t,x)v1(t,x)−c1(t,x)∫0∞v1(t−τ,x)dμ1(τ)−d1v2(t−r2,x)],t>0,x∈Ω,∂v2∂t=BΔv2(t,x)+v2(t,x)[a2(t,x)−b2(t,x)v2(t,x)−c2(t,x)∫0∞v2(t−τ,x)dμ2(τ)−d2v1(t−r1,x)],t>0,x∈Ω,∂v1∂η=∂v2∂η=0,t>0,x∈∂Ω,vi(t,x)=ϕi(t,x)(i=1,2)t≤0,x∈Ω¯,as expected
since they do not include the coefficients of diffusion. We thank one of the
referees for his comments regarding this matter.


## 3. Numerical Simulations

In this section, we present some numerical results
that agree with Theorem1.1 proved above. We used
the method of upper and lower solutions as developed by Pao [12, 13] discretizing
the systems into finite difference systems. On both examples, the domain used
is Ω=(0,12).Example 3.1.
In this example, we work with coefficients that depend
ont and initial
values that depend on (t,x).
Consider the  system(3.1)∂u1∂t=∂2u1∂x2+u1[(sin(t)+2)−(11+x2+10)u1−(cos(t)+1.01)∫0∞11+τ2u1(t−τ,x)dτ−110u2(t−2,x)],t∈[0,T],x∈[0,12],∂u2∂t=∂2u2∂x2+u2[(sin2(t)+1)−(x1+x2+7.5)u2−(sin(t2)+1.01)∫0∞11+τ2u2(t−τ,x)dτ−12u1(t−1.5,x)],t∈[0,T],x∈[0,12],∂ui∂x(0,t)=∂ui∂x(12,t)=0(i=1,2),t>0,u1(t,x)=12sin(12t)+1.01,t≤0,x∈[0,12],u2(t,x)=12(sin(12t)+sin(12x))+1.01,t≤0,x∈[0,12].It is easy to see that this system satisfies the
conditions of the main theorem (Theorem 1.1),
therefore the global attractors are defined by the solutions of the linear
system(3.2)α1(11−0)=1−2.01π2β1−110β2,α2(1.5−0)=1−2.01π2β2−β1,β1(10−0)=3−0.01π2α1−110α2,β2(0.5−0)=2−0.01π2α2−α1,that is, the
global attractors for u1 and u2 are given by [0.002245,0.299884] and [0.011192,0.285680], respectively.
Figure 1 shows the numerical simulation of the solution of this system and on
it we can notice the global attractors obtained before.Numerical simulation of Example3.1.
(a)(b)Example 3.2.
According to the main theorem (Theorem1.1), the global attractors depend only on the coefficients and
not on the initial values.
In this example, we only change the initial
functions of the above example:(3.3)u1(t,x)=sin(x)+1.01,t≤0,x∈[0,12],u2(t,x)=1200(1+t2)sin(πx12)+0.01,t≤0,x∈[0,12],to obtain the
same attractors as in Example 3.1. Figure 2 shows the numerical simulation of the
solution of this system and we observe that they have the attractors expected.Numerical simulation of Example3.2.
(a)(b)



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