# Global Asymptotic Stability of 3-Species Mutualism Models with  Diffusion and Delay Effects

Authors: Chang-you Wang; Shu Wang; Xiang-ping Yan
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

In this paper, the Lotka-Volterra 3-species mutualism models
with diffusion and delay effects is investigated. A simple and easily
verifiable condition is given to ensure the global asymptotic stability of
the unique positive steady-state solution of the corresponding steady-state
problem in a bounded domain with Neumann boundary condition. Our
approach to the problem is based on inequality skill and the method of the
upper and lower solutions for a more general reaction—diffusion system.
Finally, some numerical simulations are given to illustrate our results.


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## Body

## 1. Introduction

Modeling and analysis of the dynamics of biological populations by means of differential equations is one of the primary concerns in population growth problems. A well-known and extensively studied class of models in population dynamics is the Lotka-Volterra system which models the interaction among various species. In the earlier literature most of the discussions are devoted to coupled systems of two equations (cf. [1–11]). In the recent years, attention has been given to reaction-diffusion systems with three population species, and the main concerns in these works are the prey-predator and competition models with or without time delays (cf. [12–24]). The purpose of this paper is to investigate the asymptotic behavior of the time-dependent solution for a 3-species mutualism model, where the effect of diffusion and time delays is both taken into consideration by obtaining existence of positive solution for the corresponding inequalities. The system of equations under consideration is the Lotka-Volterra 3-species mutualism model, where the population densities do not move across the boundary of a habitat, and time delays may appear in the opposing species; the densities of three populations u,v,and w are governed by the following coupled equations:(1.1)∂u∂t-d1L1u=u(a1-c1u+e1v+β1J2*v),∂v∂t-d2L2v=v(a2+b2u-c2v+e2w+α2J1*u+β2J3*w),∂w∂t-d3L3w=w(a3+e3v-c3w+β3J2*v)(x∈Ω,t>0),
with the boundary and initial conditions(1.2)∂u(x,t)∂n=∂v(x,t)∂n=∂w(x,t)∂n=0(x∈∂Ω,t>0),u(x,t)=η1(x,t)(t∈I1),v(x,t)=η2(x,t)(t∈I2),w(x,t)=η3(x,t)(t∈I3,x∈Ω).
Here Ω is a bounded domain in Rn with boundary ∂Ω, ∂/∂n  denotes the outward normal derivative on ∂Ω, and  ηi(x,t) is a smooth function. The above problem (1.1)-(1.2) arises in a simple food chain describing three interacting species in a spatial habitat Ω. For each i=1,2,3, ai,ci are positive constants. bi,ei,αi,  and βi are nonnegative constants with ei*≡ei+βi>0, b2*≡b2+α2>0, and its respective diffusion rate is denoted by di. The real number ai is the net birth rate of the ith species, and ci is its respective intraspecific competition. The parameters bi,ei,αi and βi are interspecific cooperation. Li  is a uniformly elliptic operator in the form(1.3)Liu=-∑j,k=1Naj,k(i)(x,t)∂2u∂xj∂xk+∑j=1Nbj(i)(x,t)∂u∂xj(i=1,2,3)
(cf. [1]). The functions Ji*ui,i=1,2,3, with (u1,u2,u3)=(u,v,w) are given either by the discrete time delay(1.4)Ji*ui=ui(t-τi,x),
or by the continuous time delay
(1.5)Ji*ui=∫-τitJi(t-s)ui(s,x)ds,
and the interval Ii is given by Ii=[-τi,0],  where τi>0 is a constant representing the time delay. It is allowed that the type of time delays and the values of τi may be different for different ui. This consideration includes various combination of discrete and continuous time delays for the species u,v, and w.Throughout the paper we assumed that the functionJi(t) is piecewise continuous in R+≡[0,+∞) and possesses the property(1.6)Ji(t)≥0fort≥0,Ji(t)≡0fort>τi,∫0τiJi(t)dt=1(i=1,2,3).
The above property implies that for any constant function ui(t,x)=ρi, we have(1.7)∫-τitJi(t-s)ui(s,x)ds=ρi(t≥0).
The same is obviously true if Ji*ui is given by (1.4). It is also assumed that the domain Ω is smooth and the coefficients aj,k(i),bj(i) of Li are smooth functions in Ω (cf. [12]). In the special case of the diffusion operator Liui=Di(x)D2ui, it suffices to assume that Di(x) is strictly positive on Ω̅ and in C1+α(Ω̅) for some α∈(0,1). We allow Li=0 (and without the corresponding boundary condition) for some or all i. In particular, if Liui≡0 for all i then the equations in (1.1)-(1.2) are reduced to the ordinary differential system (with time delays)(1.8)dudt=u(a1-c1u+e1v+β1J2*v)(t>0),dvdt=v(a2+b2u-c2v+e2w+α2J1*u+β2J3*w)(t>0),dwdt=w(a3+e3v-c3w+β3J2*v)(t>0),u(t)=η1(t)(t∈I1),v(t)=η2(t)(t∈I2),w(x,t)=η3(t)(t∈I3).
Problem (1.8) and various similar problems have been investigated by many investigators in the framework of ordinary differential systems (cf. [25–28] and references therein). It is to be noted that if Δui≠0 and αi=βi=0  for every  i  then problem (1.1) is reduced to the following 3-species mutualism models without time delays:
(1.9)u1t-d1Δu1=u1(a1-c1u1+e1u2),inΩT,u2t-d2Δu2=u2(a2+b2u1-c2u2+e2u3),inΩT,u3t-d3Δu3=u3(a3+b3u2-c3u3),inΩT,u1(x,t)=u2(x,t)=u3(x,t)=0,on∂ΩT,ui(x,0)=ui,0(x),fori=1,2,3,inΩ,
where the domains  ΩT  and  ∂ΩT  are defined as, respectively,  Ω×(0,T)  and  ∂Ω×(0,T).  Here  Ω is a bounded domain in RN with smooth boundary ∂Ω. T is the maximal existence time of the solution.  ui,0 is a smooth function satisfying the compatibility condition ui,0(x)=0 for x∈∂Ω. The meaning of the parameters of systems (1.9) is as same as those of (1.1)-(1.2). Kim and Lin [22] proved that global solutions of (1.9) exist if the intraspecific competitions are strong; whereas blow-up solutions exist under certain conditions of that the intraspecific competitions are weak, and [23] obtained the upper bound of blow-up rate for any N and the lower bound of blow-up rate for N=1. On the other hand, if e2=β2=0  then it reduces to the 2-species cooperating models, which have been extensively investigated in the current literature (cf. [2, 10, 11] and references therein). Our conclusion about the global asymptotic stability of a positive steady-state solution to system (1.1)-(1.2) is directly applicable to the above special cases (see Corollary 3.2).This work is motivated from the following two-prey one-predator model:(1.10)∂u∂t-L1u=a1(x)u(1-u-b1v-c1w-β1J2*v-γ1J3*w),∂v∂t-L2v=a2(x)v(1-v-b2u-c2w-β2J1*u-γ2J3*w),∂w∂t-L3w=a3(x)w(1-w+b3u+c3v+β3J1*u+γ3J2*v)(x∈Ω,t>0),∂u∂n=∂v(x,t)∂n=∂w(x,t)∂n=0(x∈∂Ω,t>0),u(x,t)=η1(x,t)(t∈I1),v(x,t)=η2(x,t)(t∈I2),w(x,t)=η3(x,t)(t∈I3,x∈Ω),
and one-prey two-predator model
(1.11)∂u∂t-L1u=a1(x)u(1-u-b1v-c1w-β1J2*v-γ1J3*w),∂v∂t-L2v=a2(x)v(1-v+b2u-c2w+β2J1*u-γ2J3*w),∂w∂t-L3w=a3(x)w(1-w+b3u-c3v+β3J1*u-γ3J2*v)(x∈Ω,t>0),∂u∂n=∂v(x,t)∂n=∂w(x,t)∂n=0(x∈∂Ω,t>0),u(x,t)=η1(x,t)(t∈I1),v(x,t)=η2(x,t)(t∈I2),w(x,t)=η3(x,t)(t∈I3,x∈Ω)
as well as three-species food-chain model(1.12)∂u∂t-L1u=a1(x)u(1-u-b1v-c1w-β1J2*v-γ1J3*w),∂v∂t-L2v=a2(x)v(1-v+b2u-c2w+β2J1*u-γ2J3*w),∂w∂t-L3w=a3(x)w(1-w+b3u+c3v+β3J1*u+γ3J2*v)(x∈Ω,t>0),∂u∂n=∂v(x,t)∂n=∂w(x,t)∂n=0(x∈∂Ω,t>0),u(x,t)=η1(x,t)(t∈I1),v(x,t)=η2(x,t)(t∈I2),w(x,t)=η3(x,t)(t∈I3,x∈Ω).
Pao [29] obtained some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-steady solution for each of the above three-model problems.Based on the above results, we are mainly interested in studying the asymptotic behavior of the solution of (1.1)-(1.2). The corresponding steady-state problem is given by(1.13)-d1L1u=u(a1-c1u+e1v+β1v)(x∈Ω),-d2L2v=v(a2+b2u-c2v+e2w+α2u+β2w)(x∈Ω),-d3L3w=w(a3+e3v-c3w+β3v)(x∈Ω),∂u∂n=∂v∂n=∂w∂n=0(x∈∂Ω).
It is clear that the above system has always the trivial solution (0, 0, 0) and various forms of constant semitrivial solutions, that is, constant nonnegative solutions with at least one component zero and one component positive. Our aim is to prove the existence and global asymptotic stability of a positive constant steady-state solution in the mutualism model (with respect to nonnegative initial perturbations). This global asymptotic stability result implies that system (1.1)-(1.2) is permanent, the trivial and all forms of semitrivial solutions are unstable, and the nonuniform steady-state solution does not exist.This paper is arranged as follows. In Section2, we give some preliminary results for a more general time-delayed reaction-diffusion system. The main results and their proofs are given in Section 3. Some numerical simulations are shown in Section 4 to illustrate our theoretical analysis.


## 2. Some Preliminary Results

To prove the main results in this paper we first give some preliminary results for a more general time-delayed parabolic system in the form(2.1)∂ui∂t-Liui=fi(x,U,J*U)(x∈Ω,t>0),∂ui∂n=0(x∈∂Ω,t>0),ui(x,t)=ηi(x,t)(x∈Ω,t∈Ii),i=1,2,…,N,
where U=(u1,u2,…,uN),J*U=(J1*u1,J2*u2,…,JN*uN), and for each i=1,2,…,N,fi(·,U,J*U)  is, in general, a nonlinear function of U and J*U, and Li is a uniformly elliptic operator in the form of (1.3). The components  Ji*ui of J*U are given by (1.4) for i=1,2,…,n0 and by (1.5) for i=n0+1,…,N,  where n0 is a nonnegative integer. (The case n0=0 corresponds to (1.5) for every  i=1,2,…,N.)By writingU and V in the split forms U≡(ui,[u]ai,[u]bi) and V≡([v]ci,[v]di), respectively, where  ai,bi,ci,  and di are nonnegative integers satisfying
(2.2)ai+bi=N-1,ci+di=N(i=1,2,…,N),
and [w]δi denotes a vector with δi components of w, we write fi(x,U,V) in the form(2.3)fi(x,U,V)=fi(x,ui,[u]ai,[u]bi,[v]ci,[v]di)foreveryi=1,2,…,N.Definition 2.1.
We say that the vector functionF(·,U,V)≡(f1(·,U,V),f2(·,U,V),…,fN(·,U,V))  possesses a mixed quasimonotone property in a subset S of  RN, if for each  i  there exist nonnegative integers  ai,bi,ci, and di satisfying  (2.2)  such that  fi(·,ui,[u]ai,[u]bi,[v]ci,[v]di) is nondecreasing in [u]ai,[v]ci and is nonincreasing in  [u]bi,[v]di for every  U,V  in  S  (cf. [1, 12]). In particular, if   bi=di=0  for all  i then F(·,U,V) is said to be quasimonotone nondecreasing in S. It is easily seen that the reaction functions in the models  (1.1) are quasimonotone nondecreasing in R+N.Definition 2.2.
For quasimonotone nondecreasing functionsF(·,U,V) we call a pair of smooth functions  Ũ≡(ũ1,ũ2,…,ũN), Û≡(û1,û2,…,ûN) coupled upper and lower solutions of (2.1) if Ũ>Û and if
(2.4)∂ũi∂t-Liũi≥fi(x,Ũ,J*Ũ)(x∈Ω,t>0),∂ûi∂t-Liûi≤fi(x,Û,J*Û)(x∈Ω,t>0),∂ũi∂n≥0≥∂ûi∂n(x∈∂Ω,t>0),ũi(x,t)≥ηi(x,t)≥ûi(x,t)(x∈Ω,t∈Ii,i=1,2,…,N),
where inequality between vectors is in the componentwise sense. It is obvious that if Ũ≡M≡(M1,M2,…,MN)  and Û≡δ≡(δ1,δ2,…,δN) are constant vectors, then the above inequalities become
(2.5)fi(x,M,M)≤0(x∈Ω),fi(x,δ,δ)≥0(x∈Ω),Mi≥ηi(x,t)≥δi(x∈Ω,t∈Ii),i=1,2,…,N.
Notice that the boundary inequalities in (2.4) are trivially satisfied. For a given pair of constant vectors  M,δ  satisfying (2.5) we set
(2.6)S≡{U∈C(Ω):δ≤U≤M},
where C(Ω) denotes the set of continuous functions on Ω. In the following we give our basic hypotheses on  F(·,U,V) with V≡J*U.(H)  The function F(·,U,V) is quasimonotone nondecreasing for U,V∈S, and for each i,fi(·,U,V)  satisfies the Lipschitz condition(2.7)|fi(x,U,V)-fi(x,U′,V′)|≤Ki(∥U-U′∥+∥V-V′∥)(U,U′,V,V′∈S),
where Ki is a positive constant, and ∥w∥=|w1|+|w2|+⋯+|wN| for w∈RN(|·| is a maximum norm).It is clear that if there exists a positive constant vectorM=(M1,M2,…,MN) and a zero vector o=(0,0,…,0) such that(2.8)fi(x,M,M)≤0(x∈Ω),fi(x,o,o)≥0(x∈Ω),i=1,2,…,N,
then the pairsŨ=M,Û=o are coupled upper and lower solutions of  (2.1) whenever Mi≥ηi(x,t)≥0. By an application of [12,Theorem  2.1] we have the following global existence result.Lemma 2.3.
LetŨ≡M≡(M1,M2,…,MN),   Û≡δ≡(δ1,δ2,…,δN)  be a pair of constant vectors satisfying (2.5) and M≥δ,  and let hypothesis (H)  hold. Then for any η(x,t) in S,  problem (2.1) has a unique global solution U(x,t)∈S for all t≥0.  In particular, if condition (2.8) holds for every M≥o then problem (2.1) has a unique bounded nonnegative global solution whenever η(x,t)≥o.In system (2.1) if fi is given in the form(2.9)fi(x,U,V)=uifi*(x,U,V),i=1,2,…,N,
then the pairs Ũ=M,Û=o satisfy (2.8) whenever(2.10)fi*(x,M,M)≤0(x∈Ω),i=1,2,…,N.
Moreover, by the positivity lemma for the linear scalar parabolic problem(2.11)∂Wi∂t-LiWi=Wifi*(x,U,V)(x∈Ω,t>0),∂Wi∂n=0(x∈∂Ω,t>0),Wi(x,0)=ηi(x,0)x∈Ω,i=1,2,…,N,
where U≡U(x,t) is the solution of (2.1), the solution Wi≡Wi(x,t) is positive in (0,+∞)×Ω  whenever  ηi(x,0)≥0 and not identical 0 (cf. [1]). Since for each i the solution component ui≡ui(x,t) is also a solution of (2.11) the uniqueness property of Wi ensures that ui(x,t)=Wi(x,t)>0 in (0,+∞)×Ω. This observation leads to the following.Lemma 2.4.
Letfi(x,U,V) be given by (2.9)  and satisfy hypothesis  (H),   and  let  M≡(M1,M2,…,MN) be a constant vector satisfying (2.10). Then for any nontrivial nonnegative ηi(x,t)≤Mi problem (2.1)  has a unique bounded positive solution u(x,t) in (0,+∞)×Ω.To investigate the asymptotic behavior of the solution of (2.1), we consider the special case
(2.12)fi(x,U,V)=gi(U,V),i=1,2,…,N,
where gi(U,V)  is not explicitly dependent on x. The corresponding steady-state problem of (2.1) is given by(2.13)-Liui=gi(U,V)(x∈Ω),∂ui∂n=0(x∈∂Ω,),i=1,2,…,N,
Assume that G(U,V)≡(g1(U,V),…,gN(U,V)) is quasimonotone nondecreasing in S. Then by using U̅(0)=M,U̲(0)=δ as a pair of coupled initial iterations we construct two sequences {U̅(m)}≡{u̅1,…,u̅N},{U̲(m)}≡{u̲1,…,u̲N} from the linear iteration process(2.14)-Liu̅i(m)+Kiu̅i(m)=Kiu̅i(m-1)+gi(u̅1(m-1),…,u̅N(m-1),J*u̅1(m-1),…,J*u̅N(m-1)),-Liu̲i(m)+Kiu̲i(m)=Kiu̲i(m-1)+gi(u̲1(m-1),…,u̲N(m-1),J*u̲1(m-1),…,J*u̲N(m-1)),∂u̅i(m)∂n=∂u̲i(m)∂n=0(i=1,2,…,N),
where Ki is the Lipschitz constant in (2.7). It is clear that the sequences {U̅(m)},{U̲(m)} are well defined. Since the initial iterations in (2.14) are the constant vectors M,δ and gi(U,V) is independent of x whenever U and V are constants we conclude from the uniqueness property of solution for linear boundary-value problems that the solutions u̅i(m),u̲i(m) of (2.14) are constants and are given by
(2.15)u̅i(m)=u̅i(m-1)+1Kigi(u̅1(m-1),…,u̅N(m-1),u̅1(m-1),…,u̅N(m-1)),u̲i(m)=u̲i(m-1)+1Kigi(u̲1(m-1),…,u̲N(m-1),u̲1(m-1),…,u̲N(m-1))(i=1,2,…,N).
It is easy to show from the quasimonotone nondecreasing property of G(U,V) that the sequences u̅i(m),u̲i(m) possess the monotone property(2.16)δ≤U̲(m)≤U̲(m+1)≤U̅(m+1)≤U̅(m)≤M(m=1,2,…)
(cf. [12]). The above property implies that the constant limits
(2.17)limm→∞U̅(m)=ρ̅,limm→∞U̲(m)=ρ̲
exist and satisfy the relation δ≤ρ̲≤ρ̅≤M. Letting m→∞ in (2.15) shows that ρ̲ and ρ̅ satisfy the equations(2.18)gi(ρ̅,ρ̅)=0,gi(ρ̲,ρ̲)=0(i=1,2,…,N).
It is easy to verify that the limits ρ̅ and ρ̲ are maximal and minimal solutions of systems (2.13) in S, respectively, furthermore, if ρ̅=ρ̲(≡ρ*) then ρ* is the unique solution in S (cf. [1, 12]). In the latter case, the time-dependent solution of (2.1) (with fi(x,U,V) given by (2.12) converges to ρ* as t→∞ (cf. [12]). To summarize the above conclusions we have the following.Lemma 2.5.
Let the conditions in Lemma2.3 be satisfied with respect to the function given by (2.12).Then (i)
the sequences{U̅(m)},{U̲(m)} given by (2.14) with U̅(0)=M,U̲(0)=δ are constant functions and converge monotonically to their respective constant limitsρ̅ and ρ̲ that satisfy (2.18);(ii)
for anym∈N,  if U̲(m)≤η(x,t)≤U̅(m),  then the constant pairs Ũ=U̅(m),Û=U̲(m) are also coupled upper and lower solutions of (2.1),  and the problem (2.1) has a unique positive solution U(x,t) such that U̲(m)≤U(x,t)≤U̅(m) and ρ̲≤u(x,t)≤ρ̅,ast→∞(x∈Ω̅);(iii)
ifρ̅=ρ̲(≡ρ*) then ρ* is the unique solution of (2.13) in S,  and for any initial function η(x,t)∈S the corresponding solution U(x,t) of (2.1) converges to ρ*as  t→∞.In Lemma2.5 the convergence of the time-dependent solution U(x,t) to ρ* is for the class of initial functions in S. For arbitrary nontrivial nonnegative initial functions we have the following result from [12].Lemma 2.6.
Let the conditions in Lemma2.5 be  satisfied,  and  let U≡(u1,u2,…,uN) be the solution of (2.1) corresponding to an arbitrary nontrivial nonnegative η(x,t).  Assume that ρ̅=ρ̲(≡ρ*)  and there existst*>0 such that
(2.19)δi≤ui(x,t)≤Mifort*-τi≤t≤t*,x∈Ω̅,(i=1,2,…,N).
Then  U(x,t)converges to  ρ*as t→∞.To prove the existence and uniqueness of constant positive solution of the steady-state problem (1.13) we also need the following result.Lemma 2.7.
Letai>0,bi>0,ci>0(i=1,2,3). If a1b2c3-a2b1c3-a1b3c2>0 then the inequalities
(2.20)a1x-b1y>0,-a2x+b2y-c2z>0,-b3y+c3z>0
have a positive solution(ρ1,ρ2,ρ3).Proof.
With the inequalities (2.20) exists a positive solution if and only if there exist positive constants  λ1,λ2,λ3  such that the following equations:
(2.21)a1x-b1y=λ1,-a2x+b2y-c2z=λ2,-b3y+c3z=λ3
have a positive solution. Let  A=(a1-b10-a2b2-c20-b3c3), we have |A|>0. From the Cramer theorem, the solutions of (2.21) are as follows:
(2.22)x=λ1|b2-c2-b3c3|-λ2|-b10-b3c3|+λ3|-b10b2-c2||A|=λ1(b2c3-b3c2)+λ2b1c3+λ3b1c3|A|,y=-λ1|-a2-c20c3|+λ2|a100c3|-λ3|a10-a2-c2||A|=λ1a2c3+λ2a1c3+λ3a1c2|A|,z=λ1|-a2b20-bc3|-λ2|a1-b10-b3|+λ3|a1-b10-b3||A|=λ1a2b3+λ2a1b3+λ3(a1b2-a2b1)|A|.
Let λ1=λ3=1,  λ2=max{(1-b1c3-b2c3+b3c2)/b1c3,(1-a1b2+a2b1-a2b3)/a1b3,1},  then with the equations (2.21) exists a positive solution (ρ1,ρ2,ρ3). Therefore the inequalities (2.20) have a positive solution.


## 3. The Main Theorems and Proof

To prove the main results in this paper, we apply Lemmas2.3–2.7 for the models in  (1.1)-(1.2) with U=(u,v,w) by constructing a suitable pair of constant upper and lower solutions Ũ=M and Û=δ. This is equivalent to show that M and δ satisfy condition (2.5) for the reaction functions fi(x,U,V).Theorem 3.1.
Letei*≡ei+βi>0(i=1,2,3)  and  b2*≡b2+α2>0.   If c1c2c3-c1e2*e3*-e1*b2*c3>0, then the following statements hold true. (i)
The steady-state problem (1.9) has a unique constant positive solution.(ii)
For any nontrivial nonnegative(η1,η2,η3),  problems (1.1)-(1.2) have a unique nonnegative global solution (u,v,w),  and the solution is uniformly boundary in Ω̅×[0,∞).(iii)
The positive constant solutionρ*of the steady-state problem (1.9) is globally asymptotic stable.Proof.
(i)Since the boundary conditions areNeumann type, we seek a pair of positive constant vectors M=(M1,M2,M3),  δ=(δ1,δ2,δ3)satisfying
(3.1)M1(a1-c1M1+e1*M2)≤0≤δ1(a1-c1δ1+e1*δ2),M2(a2+b2*M1-c2M2+e2*M3)≤0≤δ2(a2+b2*δ1-c2δ2+e2*δ3),M3(a3+e3*M2-c3M3)≤0≤δ3(a3+e3*δ2-c3δ3),∂δi∂n≤0≤∂Mi∂n(i=1,2,3).
It is clear that the boundary conditions are satisfied, and from the positive property of Miand δi that condition (3.1) is satisfied if
(3.2)(a1-c1M1+e1*M2)≤0≤(a1-c1δ1+e1*δ2),(a2+b2*M1-c2M2+e2*M3)≤0≤(a2+b2*δ1-c2δ2+e2*δ3),(a3+e3*M2-c3M3)≤0≤(a3+e3*δ2-c3δ3).
We construct the following inequalities:
(3.3)c1x-e1*y>0,-b2*x+c2y-e2*z>0,-e3*y+c3z>0.
Let A=(c1-e1*0-b2*c2-e2*0-e3*c3), we have |A|>0. Since ei*≡ei+βi>0(i=1,2,3) and b2*≡b2+α2>0,  from the Lemma 2.7 the inequalities (3.3) have a positive solution (λ1,λ2,λ3). By choosing M=(M1,M2,M3)=(Pλ1,Pλ2,Pλ3), δ=(δ1,δ2,δ3)=(γλ1,γλ2,γλ3),  where P and γ are a sufficiently large and small positive constant, respectively, we see that all the inequalities in (3.2) are satisfied. That is,  M and δ is a pair of positive constant upper-lower solution of (1.1)-(1.2). By Lemma 2.5 the sequences governed by (2.14) with the reaction function (f1,f2,f3) given by (1.1) converge monotonically, respectively, to some constant limits ρ̅≡(ρ̅1,ρ̅2,ρ̅3),ρ̲≡(ρ̲1,ρ̲2,ρ̲3) that satisfy
(3.4)ρ̅1(a1-c1ρ̅1+e1*ρ̅2)=0,ρ̲1(a1-c1ρ̲1+e1*ρ̲2)=0,ρ̅2(a2+b2*ρ̅1-c2ρ̅2+e2*ρ̅3)=0,ρ̲2(a2+b2*ρ̲1-c2ρ̲2+e2*ρ̲3)=0,ρ̅3(a3+e3*ρ̅2-c3ρ̅3)=0,ρ̲3(a3+e3*ρ̲2-c3ρ̲3)=0.
Let ρi=ρ̅i-ρ̲i(i=1,2,3).  Then by the positivity of ρ̅i and ρ̲i, a subtraction of the above pairs of equations, leads to
(3.5)c1ρ1-e1*ρ2=0,-b2*ρ1+c2ρ2-e2*ρ3=0,-e3*ρ2+c3ρ3=0.
Since these equations are equivalent to Aρ=0, where A is coefficient matrix and ρ  is the column vector (ρ1,ρ2,ρ3)T, we conclude from the nonsingular property of A that ρ=o. This proves ρ̅=ρ̲(≡ρ*) and ρ* is the unique positive solution of (1.13) in S.
(ii)It is obvious that the pairsM=(M1,M2,M3)=(Pλ1,Pλ2,Pλ3),δ=(δ1,δ2,δ3)=(0,0,0)  satisfy all the equalities in (3.1). By Lemma 2.3 and the arbitrariness of P, problems (1.1)-(1.2) have a unique bounded nonnegative global solution (u,v,w) for any nontrivial nonnegative (η_1,η_2,η_3).
(iii)Lemma2.5 ensures that for any initial function η(x,t)≡(η1(x,t),η2(x,t),η3(x,t)) with γ(λ1,λ2,λ3)≡δ≤η(x,t)≤M≡P(λ1,λ2,λ3), where P and γare a sufficiently large and small positiveconstants, respectively, the corresponding solution U(x,t)≡(u(x,t),v(x,t),w(x,t)) of (1.1)-(1.2) converges to ρ* as t→∞. To show this convergence property for an arbitrary nontrivial nonnegative  η(x,t), we observe from Lemma 2.4, the arbitrary smallness of γ and the arbitrary largeness of P that there exists t0>0 such that  δ≤U(x,t)≤M on [t0,∞)×Ω̅. The upper and lower bounds of  U(x,t)show that there exists t*>0 such that condition (2.19) is satisfied for U≡(u1,u2,u3)=(u,v,w). The conclusion of U(x,t)→ρ* as t→∞ follows from Lemma 2.6. We next consider the stability of U(x,t). We observe from Lemma 2.5 that for any U(ρ*,ε) there exists δ>0 such that U(x,t)∈U*≡U(ρ*,δ)⊆(U̲(m),U¯(m))⊂U(ρ*,ε) as initial function η(x,t)∈U*. This completes the proof.Corollary 3.2.
All the results of Theorem3.1 hold true if di=0 for some or all i. In particular, these results hold true for the ordinary differential system (1.8). They are also true if the reaction functions involve no time delays.Proof.
This follows from the argument in the proof of Theorem3.1 by letting di=0 (and without the corresponding boundary condition) and by letting τi=0(i=1,2,3), respectively.Remark 3.3.
Our model and result are different from the existence ones such as those of Pao [29] and Kim and Lin [22, 23]. In some sense, we enrich the results of the 3-specics Lotka-Volterra reaction-diffusion systems.


## 4. Numerical Simulations

In this section, we give numerical simulations supporting our theoretical analysis. As an example, we consider system (1.1) with different diffusion rates di, birth rate ai, and time delays τi, that is, the following systems:
(4.1)∂u∂t-uxx=u(t,x)[1-u(t,x)+12v(t,x)+12v(t-0.1,x)](2π>x>0,t>0),∂v∂t-vxx=v(t,x)[1+12u(t,x)-3v(t,x)+12w(t,x),+12u(t-0.1,x)+12w(t-0.1,x)](2π>x>0,t>0),∂w∂t-wxx=w(t,x)[1+12v(t,x)-w+(12v(t-0.1,x))](2π>x>0,t>0),∂u∂n=∂v∂n=∂w∂n=0(t>0,x=0,2π),u(t,x)=v(t,x)=w(t,x)=(1+10t)[1-sin(x+π2)](t,x)∈[-0.1,0]×[0,2π].(4.2)∂u∂t-uxx=u(t,x)[1-u(t,x)+12v(t,x)+12v(t-0.01,x)](2π>x>0,t>0),∂v∂t-vxx=v(t,x)[1+12u(t,x)-3v(t,x)+12w(t,x),+12u(t-0.01,x)+12w(t-0.01,x)](2π>x>0,t>0),∂w∂t-wxx=w(t,x)[1+12v(t,x)-w+12v(t-0.01,x)](2π>x>0,t>0),∂u∂n=∂v∂n=∂w∂n=0(t>0,x=0,2π),u(t,x)=v(t,x)=w(t,x)=(1+t)[1-sin(x+π2)](t,x)∈[-0.01,0]×[0,2π].(4.3)∂u∂t-uxx=u(t,x)[1-u(t,x)+12v(t,x)+12v(t-0.1,x)](2π>x>0,t>0),∂v∂t-vxx=v(t,x)[2+12u(t,x)-3v(t,x)+12w(t,x),+12u(t-0.1,x)+12w(t-0.1,x)](2π>x>0,t>0),∂w∂t-wxx=w(t,x)[3+12v(t,x)-w+12v(t-0.1,x)](2π>x>0,t>0),∂u∂n=∂v∂n=∂w∂n=0(t>0,x=0,2π),u(t,x)=v(t,x)=w(t,x)=(1+10t)[1-sin(x+π2)](t,x)∈[-0.1,0]×[0,2π].(4.4)∂u∂t-uxx=u(t,x)[1-u(t,x)+12v(t,x)+12v(t-0.1,x)](2π>x>0,t>0),∂v∂t-2vxx=v(t,x)[1+12u(t,x)-3v(t,x)+12w(t,x),+12u(t-0.1,x)+12w(t-0.1,x)](2π>x>0,t>0),∂w∂t-3wxx=w(t,x)[1+12v(t,x)-w+12v(t-0.1,x)](2π>x>0,t>0),∂u∂n=∂v∂n=∂w∂n=0(t>0,x=0,2π),u(t,x)=v(t,x)=w(t,x)=(1+10t)[1-sin(x+π2)](t,x)∈[-0.1,0]×[0,2π].
By using the classical implicit format solving the partial differential equations and the method of steps for differential difference equations and employing the software package MATLAB7.0, we can solve the numerical solutions of systems (4.1), (4.2), (4.3), and (4.4) which are shown respectively in Figures 1, 2, 3, and 4.The solution of system (4.1).
(a)(b)(c)The solution of system (4.2).
(a)(b)(c)The solution of system (4.3).
(a)(b)(c)The solution of system (4.4).
(a)(b)(c)


## 5. Conclusions

It is well known that the analysis of stability for a system of delay reaction-diffusion multispecies model is quite difficult since the reaction among multispecies is more complex. Therefore, the works on this subject are very rare. A detailed analysis on the stability for a two-prey one-predator model, one-prey two-predator model, and three-species food-chain model with delay and diffusion was given by Pao [29], and he obtained some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems.In this paper, based on the ideas of Pao [12], we have considered a delay cooperative three-species system with Neumann boundary condition. It is shown that the system has a positive equilibrium under some certain conditions. We have obtained the similar conclusions to those of Pao [29]. More precisely, we have obtained the following results.(a) The steady-state problem (1.9) has a unique constant positive solution if  c1c2c3-c1e2*e3*-e1*b2*c3>0.(b) For any nontrivial nonnegativ(η1,η2,η3), problems (1.1)-(1.2) have a unique nonnegative global solution  (u,v,w), and the solution is uniformly boundary in Ω̅×[0,∞).(c) The positive constant solutionρ* of the steady-state problem (1.9) is globally asymptotic stable.The conditionc1c2c3-c1e2*e3*-e1*b2*c3>0 involves only the reaction rate constants, which shows that the diffusion rates di, the birth rate ai, and time delays τi do not bring effect on permanence of one species as well as contribution to its extinction. The result of global asymptotic stability implies that the three-species model system coexists, is permanent, and the trivial and all semitrivial solutions are unstable.



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*Source: 317298-2009-08-20.xml*