# Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers

Authors: Yujuan Jiao; Shengmao Fu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a strongly coupled predator-prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.


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## Body

## 1. Introduction

The principle of competitive exclusion asserts that two or more consumer species cannot coexist indefinitely upon a single limiting resource, which dates back to the pioneering work of Volterra [1] in the 1920s. Subsequently, Ayala [2] in 1969 demonstrated experimentally that two species of Drosophila can coexist upon a single limiting resource. Ayala's experiments have received much attention (see the comprehensive survey by Cantrell and Cosner [3]). Schoener [4] in 1976 found that intraspecific interference among consumers may lead to coexistence of multiple consumer species upon a single resource. To examine more closely the implications of feeding interference among conspecific consumers on consumer-resource dynamics,  Cantrell et al. [5] in 2004 proposed the following predator-prey system:
(1.1)dudt=ru(1-uK)-auv1+bu-Auw1+Bu+Dw,dvdt=v(-l+eu1+bu),dwdt=w(-L+Eu1+Bu+Dw),
where r,a,b,e,l,A,B,D,E,L, and K are positive constants, u(t) represents the density of the limiting resource at time t,v(t) and w(t) denote two consumers species. The first consumer species feeds upon the resource according to the Holling II functional response, while the second consumer species feeds upon the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. For more details on the backgrounds about this system see [5].The system (1.1) has a positive equilibrium E~=(u~,v~,w~) under the suitable conditions, where
(1.2)u~=le-bl,v~=eal{ru~(1-u~K)-ADE[(E-BL)u~-L]},w~=(E-BL)u~-LDL.
The Jacobian matrix of the system (1.1) at E~ can be written as
(1.3)J(E~)=(a11a12a13a21a22a23a31a32a33),
where
(1.4)a11=u~[-ru~K+abv~(1+bu~)2+ABw~(1+Bu~+Dw~)2],a12=-au~1+bu~<0,a13=-Au~(1+Bu~)(1+Bu~+Dw~)2<0,a21=ev~(1+bu~)2>0,a31=Ew~(1+Dw~)(1+Bu~+Dw~)2>0,a33=-DEu~w~(1+Bu~+Dw~)2<0,a22=a23=a32=0.
The following results were proved in [5]:(1)
the system (1.1) is dissipative;(2)
the positive equilibriumE~ of (1.1) is locally stable if -(a11+a33)>0 and a11(a11a33-a13a31-a12a21)-a33(a11a33-a31a31)<0; and(3)
the positive equilibriumE~ of (1.1) is globally stable if max{b,B}(K-u~)<1.Rescaling the system (1.1) such that
(1.5)uK⟼u,v⟼v,w⟼w,rt⟼t,ar⟼a,bK⟼b,Ar⟼A,BK⟼B,D⟼D,lr⟼l,(eK)l⟼e,Lr⟼L,(EK)L⟼E
yields
(1.6)dudt=u(1-u)-auv1+bu-Auw1+Bu+Dw,dvdt=lv(-1+eu1+bu),dwdt=Lw(-1+Eu1+Bu+Dw).
The corresponding weakly coupled reaction-diffusion system for (1.6) is as follows:
(1.7)ut=d1Δu+u(1-u)-auv1+bu-Auw1+Bu+Dw,x∈Ω,t>0,vt=d2Δv+lv(-1+eu1+bu),x∈Ω,t>0,wt=d3Δw+Lw(-1+Eu1+Bu+Dw),x∈Ω,t>0,∂νu=∂νv=∂νw=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω,
where Ω⊂ℝN is a bounded smooth domain, ν is the outward unit normal vector of the boundary ∂Ω,∂ν=∂/∂ν. The constants d1,d2, and d3, called diffusion coefficients, are positive, and u0(x),v0(x), and w0(x) are nonnegative functions which are not identically zero. The system (1.7) has a constant positive steady-state solution E*=(u*,v*,w*) if and only if
(1.8)e>b,E-B>e-b,DE(e-b-1)-A(e-b)[(E-B)-(e-b)]>0,
where
(1.9)u*=1e-b,v*=e{DE(e-b-1)-A(e-b)[(E-B)-(e-b)]}aDE(e-b)2,w*=(E-B)-(e-b)D(e-b).In [6], Hei and Yu proved the following main results.(1) The equilibrium(u*,v*,w*) of (1.7) is locally asymptotically stable if (1.8) and
(1.10)DE2[b(e-b-1)-e]+A(e-b)(Be-bE)[(E-B)-(e-b)]<0
hold.(2) Letd2* and d3* be fixed positive constants which satisfy d2*μ1≥l(e-b-1)/(1+b) and d3*μ1≥L(E-B-1)/(1+B).Then there exists a positive constantD1, such that (1.7) has no nonconstant positive solution if d1>D1, d2≥d2* and d3≥d3*, where 0=μ0<μ1<μ2<⋯ are the eigenvalues of the operator -Δ on Ω with the homogeneous Neumann boundary condition.(3) Letdi(i=1,3) be fixed positive constants. Assume that a11*>0,
(1.11)min⁡{4d2-4eld1-ed24d2,[D(E-B)-A(E-B-1)]D(E-B)}>1e-b
and (1.8) hold. Furthermore, assume that one of the following conditions is satisfied:(i)
a11*a33*-a31*a13*<0, μˇ∈(μn,μn+1) for some n≥1, and the sum σn=∑i=1n‍dimE(μi) is odd;(ii)
a11*a33*-a31*a13*>0, μ^∈(μk,μk+1), μˇ∈(μn,μn+1) for some n≥k≥1, and the sum σn=∑i=k+1n‍dimE(μi) is odd.Then there exists a positive constant D2, such that (1.7) has at least one nonconstant positive solution if d2≥D2, where
(1.12)a11*=-u*+abv*e2u*+ABw*E2u*,a33*=-DELu*w*(1+Bu*+Dw*)2,a13*=-Aw*Eu*,a31*=ELw*(1+Dw*)(1+Bu*+Dw*)2,μˇ=a11*d3+a33*d1+(a11*d3-a33*d1)2+4a31*a13*d1d32d1d3,μ^=a11*d3+a33*d1-(a11*d3-a33*d1)2+4a31*a13*d1d32d1d3
and E(μi) is the eigenspace corresponding to μi in H1(Ω).(4) The bifurcation of nonconstant positive solutions for (1.7) was studied.In recent years, the SKT type cross-diffusion systems have attracted the attention of a great number of investigators and have been successfully developed on the theoretical backgrounds. The above work mainly concentrate on (1) the instability and stability induced by cross-diffusion, and the existence of nonconstant positive steady-state solutions [7–14]; (2) the global existence of strong solutions [15–23]; (3) the global existence of weak solutions based on semidiscretization or finite element approximation [24–30]; and (4) the dynamical behaviors [18, 19, 31, 32], and so forth. The corresponding SKT type cross-diffusion system for (1.7) is as follows:
(1.13)ut=Δ(d1u+α11u2+α12uv+α13uw)+u(1-u)-auv1+bu-Auw1+Bu+Dw,x∈Ω,t>0,vt=Δ(d2v+α21uv+α22v2+α23vw)+lv(-1+eu1+bu),x∈Ω,t>0,wt=Δ(d3w+α31uw+α32vw+α33w2)+Lw(-1+Eu1+Bu+Dw),x∈Ω,t>0,∂νu=∂νv=∂νw=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω,
where αij(i,j=1,2,3) are positive constants, αii(i=1,2,3) are referred as self-diffusion pressures, and αij(i,j=1,2,3,i≠j) are cross-diffusion pressures. The self-diffusion implies the movement of individuals from a higher to lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. The value of cross-diffusion coefficient may be positive, negative, or zero. The positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species (e.g., [33]).The local existence of solutions for the system (1.13) is an immediate consequence of a series of important papers [34–36] by Amann. Roughly speaking, if u0(x),v0(x), and w0(x) in Wp1(Ω) with p>n, then (1.13) has a unique nonnegative solution u,v,w∈C([0,T),Wp1(Ω))∩C∞((0,T),C∞(Ω)), where T∈(0,∞] is the maximal existence time for the solution. If the solution (u,v,w) satisfies the estimate
(1.14)sup⁡{‖u(⋅,t)‖Wp1(Ω),‖v(⋅,t)‖Wp1(Ω),‖w(⋅,t)‖Wp1(Ω):0<t<T}<∞,
then T=+∞. Moreover, if u0(x),v0(x),w0(x)∈Wp2(Ω), then u,v,w∈C([0,∞),Wp2(Ω)).For the following SKT system(P)ut=d1Δ[(1+αv+γu)u]+au(1-u-cv),x∈Ω,t>0,vt=d2Δ[(1+δv)v]+bv(1-du-v),x∈Ω,t>0,∂νu=∂νv=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,
Yamada in [23] proposed four open problems:(1)
the global existence of solutions of(P) in the case δ>0 and the space dimension N≥6;(2)
the global existence in the caseγ=0;(3)
in order to study the asymptotic behavior ofu,v as t→∞ need to establish the uniform boundedness of global solutions; and(4)
the global existence of solutions for the following full SKT system:(1.15)ut=d1Δ[(1+αv+γu)u]+au(1-u-cv),x∈Ω,t>0,vt=d2Δ[(1+βu+δv)v]+bv(1-du-v),x∈Ω,t>0,∂νu=∂νv=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ωwith α,γ,β,δ>0.Very few global existence results for (1.13) are known. The main purpose of this paper is to establish the uniform boundedness of global solutions for the system (1.13) in one space dimension. For convenience, we consider the following system:
(1.16)ut=(d1u+α11u2+α12uv+α13uw)xx+u(1-u)-auv1+bu-Auw1+Bu+Dw,0<x<1,t>0,vt=(d2v+α21uv+α22v2+α23vw)xx+lv(-1+eu1+bu),0<x<1,t>0,wt=(d3w+α31uw+α32vw+α33w2)xx+Lw(-1+Eu1+Bu+Dw),0<x<1,t>0,ux(x,t)=vx(x,t)=wx(x,t)=0,x=0,1,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),0<x<1.
We firstly investigate the global existence and the uniform boundedness of the solutions for (1.16), then prove the global asymptotic stability of the positive equilibrium (u*,v*,w*) of (1.16) by an important lemma from [37]. The proof is complete and complement to the uniform convergence theorems in papers [38–40].It is obvious that(u*,v*,w*) is the unique positive equilibrium of the system (1.16) if (1.8) holds.For simplicity, we denote∥·∥Wpk(0,1) by |·|k,p and ∥·∥Lp(0,1) by |·|p. Our main results are as follows.Theorem 1.1.
Letu0(x),v0(x),w0(x)∈W22(0,1), (u,v,w) be the unique nonnegative solution of the system (1.16) in the maximal existence interval [0,T). Assume that
(1.17)8α11α21α31>α21α132+α122α31,8α12α22α32>α32α212+α232α12,8α13α23α33>α23α312+α322α13.
Then there exist t0>0 and positive constants M′,M which depend on di,αij(i,j=1,2,3),a,b,e,l,A,B,D,E and L, such that
(1.18)sup⁡{|u(⋅,t)|1,2,|v(⋅,t)|1,2,|w(⋅,t)|1,2:t∈(t0,T)}≤M′,(1.19)max⁡{u(x,t),v(x,t),w(x,t):(x,t)∈[0,1]×(t0,T)}≤M,
and T=+∞. Moreover, in the case that d1,d2,d3≥1,d2/d1,d3/d1∈[d_,d¯], where d_ and d¯ are positive constants, M′,M depend on d_,d¯, but do not depend on d1,d2,d3.Remark 1.2.
Since the continuous embeddingH1(Ω)↪L∞(Ω) holds only in one space dimension, we can only establish the uniform maximum-norn estimates about time for the solution in one space dimension.Theorem 1.3.
Assume that all conditions in Theorem1.1 are satisfied. Assume further that
(1.20)4αβu*v*w*d1d2d3>u*M2(αα23v*+βα32w*)2(d1+2α11M+α12M+α13M)+αv*M2(α13u*+βα31w*)2(d2+α21M+2α22M+α23M)+βw*M2(α12u*+αα21v*)2(d3+α31M+α32M+2α33M),(1.21)DE[(e-b)-b(e-b-1)]>A(e-b)[(E-B)-(e-b)](B-b),
and (1.8) hold, where α=a(1+bu*)/el,β=A(1+Bu*)/EL(1+Dw*),M is given by (1.19). Then the unique positive equilibrium (u*,v*,w*) of (1.16) is globally asymptotically stable.Remark 1.4.
The system (1.16) has no nonconstant positive steady-state solution if all conditions of Theorem 1.3 hold.Examples. The following two examples satisfy all conditions of Theorem 1.3:(1.22)α11=14,α12=1,α13=1,α21=52,α22=1,α23=14,α31=1,α32=1,α33=2,b=1,e=3,A=1,B=12,D=23,E=3,d1,d2,d3≫1;α11=2,α12=1,α13=2,α21=32,α22=12,α23=1,α31=2,α32=2,α33=3,b=12,e=52,A=23,B=12,D=34,E=4,d1,d2,d3≫1.


## 2. Global Solutions

In order to establish the uniformW21-estimates of the solutions for the system (1.16), the following Gagliardo-Nirenberg-type inequalities and the corresponding corollary play important roles (see [38, 41]).Theorem 2.1.
LetΩ⊂ℝn be a bounded domain with ∂Ω∈Cm. For every function u∈Wrm(Ω),1≤q,r≤∞, the derivative Dju(0≤j<m) satisfies the inequality
(2.1)|Dju|p≤C(|Dmu|ra|u|q1-a+|u|q),
provided one of the following three conditions is satisfied: (1)  r≤q,  (2)  0<n(r-q)/(mrq)<1, or (3)  n(r-q)/(mrq)=1 and m-n/q is not a nonnegative integer, where 1/p=j/n+a(1/r-m/n)+(1-a)/q for all a∈[j/m,1), and the positive constant C depends on n,m,j,q,r,a.Corollary 2.2.
There exists a universal constantC such that
(2.2)|u|2≤C(|ux|21/3|u|12/3+|u|1),∀u∈W21(0,1),(2.3)|u|4≤C(|ux|21/2|u|11/2+|u|1),∀u∈W21(0,1),(2.4)|u|5/2≤C(|ux|22/5|u|13/5+|u|1),∀u∈W21(0,1),(2.5)|ux|2≤C(|uxx|23/5|u|12/5+|u|1),∀u∈W22(0,1).
Throughout this paper, we always denote thatC is a Sobolev embedding constant or other kind of universal constant,   Aj,Bj,Cj are some positive constants which depend only on αij(i,j=1,2,3),a,b,e,l,A,B,D,E and L,Kj are positive constants depending on di,αij(i,j=1,2,3),a,b,e,l,A,B,D,E and L. When d1,d2,d3≥1,d2/d1,d3/d1∈[d_,d¯],   Kj depend on d_,d¯, but do not depend on d1,d2,d3.Proof of Theorem1.1.
Taking integration of the three equations in (1.16) over (0,1), respectively, and combining the three integration equalities linearly, we have
(2.6)ddt∫01‍[(el+EL)u+av+Aw]dx≤∫01‍[(el+EL)u(1-u)-alv-ALw]dx.
It follows from the Young inequality and the Hölder inequality that
(2.7)ddt∫01‍[(el+EL)u+av+Aw]dx≤C1-k∫01‍[(el+EL)u+av+Aw]dx,
where k=min{l,L},C1=(k+1)2(el+EL)/4. So there exist positive constants M0 and τ0 depending on a,b,e,l,A,B,D,E, and L, such that
(2.8)∫01‍udx,∫01‍vdx,∫01‍wdx≤M0,t≥τ0.
Moreover, there exists a positive constant M0′ which depends on a,b,e,l,A,B,D,E,L and L1-norm of u0,v0,w0, such that
(2.8)′∫01‍udx,∫01‍vdx,∫01‍wdx≤M0′,t≥0.
Multiplying the first three equations in the system (1.16) by u,v,w, respectively, and integrating over (0,1), we have
(2.9)12ddt∫01‍u2dx≤-d1∫01‍ux2dx-∫01‍[(2α11u+α12v+α13w)ux2+(α12vx+α13wx)uux]dx+∫01‍u2dx,12ddt∫01‍v2dx≤-d2∫01‍vx2dx-∫01‍[(α21u+2α22v+α23w)vx2+α21(ux+α23wx)vvx]dx+elb∫01‍v2dx,12ddt∫01‍w2dx≤-d3∫01‍wx2dx-∫01‍[(α31u+α32v+2α33w)wx2+(α31ux+α32vx)wwx]dx+ELB∫01‍w2dx,
from which it follows that
(2.10)12ddt∫01‍(u2+v2+w2)dx≤-d∫01‍(ux2+vx2+wx2)dx+∫01‍u2dx+elb∫01‍v2dx+ELB∫01‍w2dx-∫01‍q(ux,vx,wx)dx≤-d∫01‍(ux2+vx2+wx2)dx+(1+elb+ELB)∫01‍(u2+v2+w2)dx-∫01‍q(ux,vx,wx)dx,
where d=min{d1,d2,d3},
(2.11)q(ux,vx,wx)=(2α11u+α12v+α13w)ux2+(α12u+α21v)uxvx+(α21u+2α22v+α23w)vx2+(α13u+α31w)uxwx+(α31u+α32v+2α33w)wx2+(α23v+α32w)vxwx.
It is obvious that q(ux,vx,wx) is a positive definite quadratic form of ux,vx,wx if (1.17) holds. So (1.17) implies that
(2.12)12ddt∫01‍(u2+v2+w2)dx≤-d∫01‍(ux2+vx2+wx2)dx+(1+elb+ELB)∫01‍(u2+v2+w2)dx.
Now, we proceed in the following two cases.
(i) One hast≥τ0. The inequality (2.2) implies that |u|26≤C(|ux|22|u|14+|u|16)≤CM04(|ux|22+M02). So we have ∫01‍ux2dx≥(1/CM04)(∫01‍u2dx)3-M02, and
(2.13)-∫01‍(ux2+vx2+wx2)dx≤-19CM04[∫01‍(u2+v2+w2)dx]3+3M02.
It follows from (2.12) and (2.13) that (2.14)12ddt∫01‍(u2+v2+w2)dx≤d{-C2[∫01‍(u2+v2+w2)dx]3+3M02+1d(1+elb+ELB)∫01‍(u2+v2+w2)dx}.
This means that there exist positive constants τ1 and M1 depending on di(i=1,2,3),a,b,e,l,A,B,D,E, and L, such that
(2.15)∫01‍u2dx,∫01‍v2dx,∫01‍w2dx≤M1,t≥τ1.
When d≥1,M1 is independent of d because the zero point of the right-hand side in (2.14) can be estimated by positive constants independent of d.
(ii) One hast≥0. Repeating estimates in (i) by (2.8)′, we can obtain that there exists a positive constant M1′ depending on di(i=1,2,3),a,b,e,l,A,B,D,E,L and the L1,L2-norm of u0,v0,w0, such that
(2.15)′∫01‍u2dx,∫01‍v2dx,∫01‍w2dx≤M1′,t≥0.
When d≥1,M1′ is independent of d.
To estimate|ux|2,|vx|2,|wx|2, we introduce the following scaling:
(2.16)u^=ud1,v^=vd1,w^=wd1,t^=d1t.
Denoting ξ=d2/d1,η=d3/d1, and using u,v,w,t instead of u^,v^,w^,t^, respectively, then the system (1.16) reduces to
(2.17)ut=Pxx+f(u,v,w),0<x<1,t>0,vt=Qxx+g(u,v,w),0<x<1,t>0,wt=Rxx+h(u,v,w),0<x<1,t>0,ux(x,t)=vx(x,t)=wx(x,t)=0,x=0,1,t>0,u(x,0)=u^0(x),v(x,0)=v^0(x),w(x,0)=w^0(x),0<x<1,
where P=u+α11u2+α12uv+α13uw,Q=ξv+α21uv+α22v2+α23vw,R=ηw+α31uw+α32vw+α33w2,f(u,v,w)=d1-1u-u2-(auv/(1+bd1u))-(Auw/(1+Bd1u+Dd1w)),g(u,v,w)=lv(-d1-1+(eu/(1+bd1u))),h(u,v,w)=Lw(-d1-1+(Eu/(1+Bd1u+Dd1w))).
We still proceed in the following two cases.(i)
One hast≥τ1*=d1τ1. It is not hard to verify that
(2.18)∫01‍udx,∫01‍vdx,∫01‍wdx≤M0d1-1,∫01‍u2dx,∫01‍v2dx,∫01‍w2dx≤M1d1-2,|P|1,|Q|1,|R|1≤A1K1d1-1,where K1=(1+ξ+η)+M1d1-1,A1=max{M0,α11+α12+α13,α21+α22+α23,α31+α32+α33}.
Multiplying the first three equations in (2.17) by Pt,Qt,Rt, integrating them over the domain (0,1), respectively, and then adding up the three integration equalities, we have
(2.19)12y¯′(t)=-∫01‍ut2dx-ξ∫01‍vt2dx-η∫01‍wt2dx-∫01‍q(ut,vt,wt)dx+∫01‍[(1+2α11u+α12v+α13w)utf+α12uvtf+α13uwtf]dx+∫01‍[(ξ+α21u+2α22v+α23w)vtg+α21vutg+α23vwtg]dx+∫01‍[(η+α31u+α32v+2α33w)wth+α31wuth+α32wvth]dx,
where y¯(t)=∫01‍(Px2+Qx2+Rx2)dx. Notice by (1.17) that there exists a positive constant C3 depending only on αij(i,j=1,2,3), such that
(2.20)q(ut,vt,wt)≥C3(u+v+w)(ut2+vt2+wt2).
Thus,
(2.21)12y¯′(t)≤-∫01‍ut2dx-ξ∫01‍vt2dx-η∫01‍wt2dx-C3∫01‍(u+v+w)(ut2+vt2+wt2)dx+∫01‍[(1+2α11u+α12v+α13w)utf+α12uvtf+α13uwtf]dx+∫01‍[(ξ+α21u+2α22v+α23w)vtg+α21vutg+α23vwtg]dx+∫01‍[(η+α31u+α32v+2α33w)wth+α31wuth+α32wvth]dx.
Using the Young inequality, Hölder inequality, and (2.18), we can obtain the following estimates:
(2.22)∫01‍u4dx≤(∫01‍u2dx)1/3(∫01‍u5dx)2/3≤M11/3d1-2/3(∫01‍u5dx)2/3,∫01‍uv2dx≤(∫01‍u2dx)1/2(∫01‍v2dx)1/6(∫01‍v5dx)1/3≤M12/3d1-4/3(∫01‍v5dx)1/3,∫01‍u3dx≤(∫01‍u2dx)2/3(∫01‍u5dx)1/3≤M12/3d1-4/3(∫01‍u5dx)1/3,∫01‍u4vdx≤45∫01‍u5dx+15∫01‍v5dx≤45∫01‍(u5+v5)dx.
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of (2.21), we have(2.23)-∫01‍ut2dx≤-12∫01‍Pxx2dx+∫01‍f2dx,-ξ∫01‍vt2dx≤-ξ2∫01‍Qxx2dx+ξ∫01‍g2dx,-η∫01‍wt2dx≤-η2∫01‍Rxx2dx+η∫01‍h2dx,∫01‍f2dx≤∫01‍(d1-2u2+u4+a2v2(bd1)2+A2w2(Bd1)2+2au2vbd1+2Au2wBd1+2aAvwbBd12)dx∫01‍f2dx≤[1+2(ab)2+2(AB)2]M1d1-4+M11/3d1-2/3(∫01‍u5dx)2/3+2(ab+AB)M12/3d1-7/3(∫01‍u5dx)1/3,ξ∫01‍g2dx≤ξ∫01‍(l2d1-2v2+e2l2v2(bd1)2)dx≤ξl2[1+(eb)2]M1d1-4,η∫01‍h2dx≤η∫01‍(L2d1-2w2+E2L2w2(Bd1)2)dx≤ηL2[1+(EB)2]M1d1-4,(2.24)-∫01‍ut2dx-ξ∫01‍vt2dx-η∫01‍wt2dx≤-12∫01‍Pxx2dx-ξ2∫01‍Qxx2dx-η2∫01‍Rxx2dx+{1+2[(ab)2+(AB)2]+ξl2[1+(eb)2]+ηL2[1+(EB)2]}M1d1-4+M11/3d1-2/3(∫01‍u5dx)2/3+2(ab+AB)M12/3d1-7/3(∫01‍u5dx)1/3.
Similarly, we can obtain
(2.25)∫01‍utfdx≤∫01‍ut(d1-1u+u2+abd1v+ABd1w)dx≤d1-22ε∫01‍udx+ε2∫01‍uut2dx+12ε∫01‍u3dx+ε2∫01‍uut2dx+(a/bd1)22ε∫01‍vdx+ε2∫01‍vut2dx+(A/Bd1)22ε∫01‍wdx+ε2∫01‍wut2dx≤12ε[1+(ab)2+(AB)2]M0d1-3+12εM12/3d1-4/3(∫01‍u5dx)1/3+ε∫01‍uut2dx+ε2∫01‍vut2dx+ε2∫01‍wut2dx,2α11∫01‍uutfdx≤2α11∫01‍uut(d1-1u+u2+abd1v+ABd1w)dx≤α112d1-2ε∫01‍u3dx+ε∫01‍uut2dx+α112ε∫01‍u5dx+ε∫01‍uut2dx+α112a2εb2d12∫01‍u2vdx+ε∫01‍vut2dx+α112A2εB2d12∫01‍u2wdx+ε∫01‍wut2dx≤α112ε[1+(ab)2+(AB)2]M12/3d1-10/3(∫01‍u5dx)1/3+α112ε∫01‍u5dx+2ε∫01‍uut2dx+ε∫01‍vut2dx+ε∫01‍wut2dx,α12∫01‍vutfdx≤α12∫01‍vut(d1-1u+u2+abd1v+ABd1w)dx≤α122d1-22ε∫01‍uv2dx+ε2∫01‍uut2dx+α1222ε∫01‍u4vdx+ε2∫01‍vut2dx+α122a22εb2d12∫01‍v3dx+ε2∫01‍vut2dx+α122A22εB2d12∫01‍v2wdx+ε2∫01‍wut2dx≤α1222ε[1+(ab)2+(AB)2]M12/3d1-10/3(∫01‍v5dx)1/3+2α1225ε∫01‍(u5+v5)dx+ε2∫01‍uut2dx+ε∫01‍vut2dx+ε2∫01‍wut2dx,α13∫01‍wutfdx≤α13∫01‍wut(d1-1u+u2+abd1v+ABd1w)dx≤α132d1-22ε∫01‍uw2dx+ε2∫01‍uut2dx+α1322ε∫01‍u4wdx+ε2∫01‍wut2dx+α132a22εb2d12∫01‍w2vdx+ε2∫01‍vut2dx+α132A22εB2d12∫01‍w3dx+ε2∫01‍wut2dx≤α1322ε[1+(ab)2+(AB)2]M12/3d1-10/3(∫01‍w5dx)1/3+2α1325ε∫01‍(u5+w5)dx+ε2∫01‍uut2dx+ε∫01‍wut2dx+ε2∫01‍vut2dxα12∫01‍uvtfdx≤α12∫01‍uvt(d1-1u+u2+abd1v+ABd1w)dx≤α122d1-22ε∫01‍u3dx+ε2∫01‍uvt2dx+α1222ε∫01‍u5dx+ε2∫01‍uvt2dx+α122a22εb2d12∫01‍u2vdx+ε2∫01‍vvt2dx+α122A22εB2d12∫01‍u2wdx+ε2∫01‍wvt2dx≤α1222ε[1+(ab)2+(AB)2]M12/3d1-10/3(∫01‍u5dx)1/3+α1222ε∫01‍u5dx+ε∫01‍uvt2dx+ε2∫01‍vvt2dx+ε2∫01‍wvt2dx,α13∫01‍uwtfdx≤α13∫01‍uwt(d1-1u+u2+abd1v+ABd1w)dx≤α132d1-22ε∫01‍u3dx+ε2∫01‍uwt2dx+α1322ε∫01‍u5dx+ε2∫01‍uwt2dx+α132a22εb2d12∫01‍u2vdx+ε2∫01‍vwt2dx+α132A22εB2d12∫01‍u2wdx+ε2∫01‍wwt2dx,≤α1322ε[1+(ab)2+(AB)2]M12/3d1-10/3(∫01‍u5dx)1/3+α1322ε∫01‍u5dx+ε∫01‍uwt2dx+ε2∫01‍vwt2dx+ε2∫01‍wwt2dx,ξ∫01‍vtgdx≤ξ∫01‍vtlv(d1-1+ebd1)dx≤ξ2l2d1-2(1+(e/b))22ε∫01‍vdx+ε2∫01‍vvt2dx≤ξ2l2(1+(e/b))22εM0d1-3+ε2∫01‍vvt2dx,α21∫01‍uvtgdx≤α21∫01‍uvtlv(d1-1+ebd1)dx≤α212l2d1-2(1+(e/b))22ε∫01‍u2vdx+ε2∫01‍vvt2dx≤α212l2(1+(e/b))22εM12/3d1-10/3(∫01‍u5dx)1/3+ε2∫01‍vvt2dx,2α22∫01‍vvtgdx≤2α22∫01‍v2vtl(d1-1+ebd1)dx≤α222l2d1-2(1+(e/b))2ε∫01‍v3dx+ε∫01‍vvt2dx≤α212l2(1+(e/b))2εM12/3d1-10/3(∫01‍v5dx)1/3+ε∫01‍vvt2dx,α23∫01‍wvtgdx≤α23∫01‍wvtlv(d1-1+ebd1)dx≤α232l2d1-2(1+(e/b))22ε∫01‍v2wdx+ε2∫01‍wvt2dx≤α232l2(1+(e/b))22εM12/3d1-10/3(∫01‍v5dx)1/3+ε2∫01‍wvt2dx,α21∫01‍vutgdx≤α21∫01‍v2vtl(d1-1+ebd1)dx≤α212l2d1-2(1+(e/b))22ε∫01‍v3dx+ε2∫01‍vut2dx≤α212l2(1+(e/b))22εM12/3d1-10/3(∫01‍v5dx)1/3+ε2∫01‍vut2dx,α23∫01‍vwtgdx≤α23∫01‍v2wtl(d1-1+ebd1)dx≤α232l2d1-2(1+(e/b))22ε∫01‍v3dx+ε2∫01‍vwt2dx≤α232l2(1+(e/b))22εM12/3d1-10/3(∫01‍v5dx)1/3+ε2∫01‍vwt2dx,η∫01‍wthdx≤η∫01‍wtLw(d1-1+EBd1)dx≤η2L2d1-2(1+(E/B))22ε∫01‍wdx+ε2∫01‍wwt2dx≤η2L2(1+(E/B))22εM0d1-3+ε2∫01‍wwt2dx,α31∫01‍uwthdx≤α31∫01‍uwtLw(d1-1+EBd1)dx≤α312L2d1-2(1+(E/B))22ε∫01‍w2udx+ε2∫01‍uwt2dx≤α312L2(1+(E/B))22εM12/3d1-10/3(∫01‍w5dx)1/3+ε2∫01‍uwt2dx,α32∫01‍vwthdx≤α32∫01‍vwtLw(d1-1+EBd1)dx≤α322L2d1-2(1+(E/B))22ε∫01‍w2vdx+ε2∫01‍vwt2dx≤α322L2(1+(E/B))22εM12/3d1-10/3(∫01‍w5dx)1/3+ε2∫01‍vwt2dx,2α33∫01‍wwthdx≤2α33∫01‍w2wtL(d1-1+EBd1)dx≤α332L2d1-2(1+(E/B))2ε∫01‍w3dx+ε∫01‍wwt2dx≤α332L2(1+(E/B))2εM12/3d1-10/3(∫01‍w5dx)1/3+ε∫01‍wwt2dx,α31∫01‍wuthdx≤α31∫01‍w2utL(d1-1+EBd1)dx≤α312L2d1-2(1+(E/B))22ε∫01‍w3dx+ε2∫01‍wut2dx≤α312L2(1+(E/B))22εM12/3d1-10/3(∫01‍v5dx)1/3+ε2∫01‍wut2dx,α32∫01‍wvthdx≤α32∫01‍w2vtL(d1-1+EBd1)dx≤α322L2d1-2(1+(E/B))22ε∫01‍w3dx+ε2∫01‍wvt2dx≤α322L2(1+(E/B))22εM12/3d1-10/3(∫01‍w5dx)1/3+ε2∫01‍wvt2dx.
By the above inequalities and the condition (1.17), we have
(2.26)∫01‍[(1+2α11u+α12v+α13w)utf+α12uvtf+α13uwtf]dx+∫01‍[(ξ+α21u+2α22v+α23w)vtg+α21vutg+α23vwtg]dx+∫01‍[(η+α31u+α32v+2α33w)wth+α31wuth+α32wvth]dx≤λ¯ε∫01‍(u+v+w)(ut2+vt2+wt2)dx+C4ε(1+ξ2+η2)M0d1-3+C5ε(1+d1-2)M12/3d1-4/3[∫01‍(u5+v5+w5)dx]1/3+C6ε∫01‍(u5+v5+w5)dx,
where λ¯ is a constant depending only on ε(αij)(i,j=1,2,3). Choose a small enough positive number ε which depends on αij(i,j=1,2,3),a,b,e,l,A,B,D,E, and L, such that λ¯ε<C3. Substituting inequalities (2.24) and (2.26) into (2.21), one can obtain
(2.27)12y¯′(t)≤-12∫01‍Pxx2dx-ξ2∫01‍Qxx2dx-η2∫01‍Rxx2dx+B1K2d1-3+B2Y+B3K3d1-2/3Y2/3+B4K4d1-4/3Y1/3,
where Y=∫01‍(u5+v5+w5)dx, K2=(1+ξ2+η2)M0+(1+ξ+η)M1d1-1, K3=M11/3,  K4=M12/3(1+d1-2).
Note that(2.28)P≥α11u2,Q≥α22v2,R≥α33w2.
It follows from (2.18) and (2.4) to functions P,Q,R that
(2.29)Y≤B5∫01‍(P5/2+Q5/2+R5/2)dx≤B6K13/2d1-3/2y¯1/2+B6K15/2d1-5/2,Y1/3≤B7K11/2d1-1/2y¯1/6+B7K15/6d1-5/6,Y2/3≤B8K1d1-1y¯1/3+B8K15/3d1-5/3.
Moreover, one can obtain by (2.5) and (2.18) that
(2.30)-12∫01‍Pxx2dx-ξ2∫01‍Qxx2dx-η2∫01‍Rxx2dx≤-B9min⁡{1,ξ,η}K1-4/3d14/3y¯5/3+(1+ξ+η)K12d1-2.
Combining (2.27),(2.29), and (2.30), we have
(2.31)12y¯′(t)≤-A1min⁡{1,ξ,η}K1-4/3d14/3y¯5/3+A2[(1+ξ+η)K12d1-2+K2d1-3+K15/2d1-5/2+K15/3K3d1-7/3+K15/6K4d1-13/6]+A3K13/2d1-3/2y¯1/2+A4K1K3d1-5/3y¯1/3+A5K11/2K4d1-11/6y¯1/6.
Multiplying inequality (2.31) by d12, we have
(2.32)12y′(t)≤-A1min⁡{1,ξ,η}K1-4/3y5/3+A2[(1+ξ+η)K12+K2d1-1+K15/2d1-1/2+K15/3K3d1-1/3+K15/6K4d1-1/6]+A3K13/2d1-1/2y1/2+A4K1K3d1-1/3y1/3+A5K11/2K4d1-1/6y1/6,where y=∫01‍[(d1Px)2+(d1Qx)2+(d1Rx)2]dx. The inequality (2.32) implies that there exist τ~2>0 and positive constant M~2 depending on di,αij(i,j=1,2,3),a,b,e,l,A,B,D,E, and L, such that
(2.33)∫01‍(d1Px)2dx,∫01‍(d1Qx)2dx,∫01‍(d1Rx)2dx≤M~2,t≥τ~2.
In the case that d1,d2,d3≥1, ξ,η∈[d_,d¯], the coefficients of inequality (2.31) can be estimated by some constants which depend on d_,d¯, but do not depend on d1,d2,d3. So M~2 depends on αij(i,j=1,2,3),a,b,e,l,A,B,D,E, L,d_, and d¯, but it is irrelevant to d1,d2,d3, when d1,d2,d3≥1 and ξ,η∈[d_,d¯]. Since
(2.34)(PxQxRx)=(PuPvPwQuQvQwRuRvRw)(uxvxwx),
we can transform the formulations of ux,vx,wx into fraction representations, then distribute the denominators of the absolute value of the fractions to the numerators item and enlarge the term concerning with ux,vx or wx to obtain
(2.35)|d1ux|+|d1vx|+|d1wx|≤C′(|d1Px|+|d1Qx|+|d1Rx|),0<x<1,t>0,
where C′ is a constant depending only on ξ,η,αij(i,j=1,2,3). After scaling back and contacting estimates (2.33) and (2.35), there exist positive constant M2 depending on di,αij(i,j=1,2,3),a,b,e,l,A,B,D,E,L, and τ2>0, such that
(2.36)∫01‍ux2dx,∫01‍vx2dx,∫01‍wx2dx≤M2,t≥τ2.
When d1,d2,d3≥1 and ξ,η∈[d_,d¯],  M2 is independent of d1,d2,d3.(ii)
One hast≥0. Modifying the dependency of the coefficients in inequalities (2.17)–(2.22), namely, replacing M0,M1 with M0′,M1′, there exists a positive constant M2′ depending on di,αij(i,j=1,2,3),a,b,e,l,A,B, D,E,L, and the W21-norm of u0,v0,w0, such that
(2.36)′∫01‍ux2dx,∫01‍vx2dx,∫01‍wx2dx≤M2′,t≥0.Furthermore, in the case that d1,d2,d3≥1, ξ,η∈[d_,d¯], M2′ depends on d_,d¯, but does not depend on d1,d2,d3.
Summarizing estimates (2.8), (2.15), (2.36) and Sobolev embedding theorem, there exist positive constants M,M′ depending only on di,αij(i,j=1,2,3),a,b,e,l,A,B,D,E, and L, such that (1.18) and (1.19) hold. In particular, M,M′ depend only on αij(i,j=1,2,3),a,b,e,l,A,B,D,E,L,d_, and d¯, but do not depend on d1,d2,d3, when d1,d2,d3≥1 and ξ,η∈[d_,d¯].
Similarly, according to (2.8)′, (2.15)′, (2.36)′, there exists a positive constant M′′ depending on di,αij(i,j=1,2,3),a,b,e,l,A,B,D,E,L and the initial functions u0,v0,w0, such that
(2.37)|u(⋅,t)|1,2,|v(⋅,t)|1,2,|w(⋅,t)|1,2≤M′′,t≥0.
Further, in the case that d1,d2,d3≥1, ξ,η∈[d_,d¯], M′′ depends only on d_,d¯, but do not depend on d1,d2,d3. Thus, T=+∞. This completes the proof of Theorem 1.1.


## 3. Global Stability

In order to obtain the uniform convergence of the solution for the system (1.16), we recall the following result which can be found in [37, 42].Lemma 3.1.
Leta and b be positive constants. Assume that φ,ψ∈C1([a,+∞)),ψ(t)≥0 and φ is bounded from below. If φ′(t)≤-bψ(t) and ψ′(t) is bounded from above in [a,+∞), then limt→∞ψ(t)=0.Proof of Theorem1.3.
Let(u,v,w) be a solution for the system (1.16) with initial functions u0(x),v0(x),w0(x)≥(≢)0. From the strong maximum principle for parabolic equations, it is not hard to verify that u,v,w>0 for t>0. Define the function
(3.1)H(u,v,w)=∫01‍(u-u*-u*ln⁡uu*)dx+α∫01‍(v-v*-v*ln⁡vv*)dx+β∫01‍(w-w*-w*ln⁡ww*)dx.
Then the time derivative of H(u,v,w) for the system (1.16) satisfies
(3.2)dHdt=∫01‍u-u*uutdx+α∫01‍v-v*vvtdx+β∫01‍w-w*wwtdx=-∫01‍[u*u2(d1+2α11u+α12v+α13w)ux2+u*u(α12vx+α13wx)ux+αv*v2(d2+α21u+2α22v+α23w)vx2+αv*v(α21ux+α23wx)vx+βw*w2(d3+α31u+α32v+2α33w)wx2+βw*w(α31ux+α13vx)wx]dx+∫01‍(u-u*)(1-u-av1+bu-Aw1+Bu+Dw)dx+α∫01‍(v-v*)m(-1+eu1+bu)dx+β∫01‍(w-w*)M(-1+Eu1+Bu+Dw)dx.
The first integrand in the right hand of (3.2) is positive definite if
(3.3)4αβu*v*w*(d1+2α11u+α12v+α13w)(d2+α21u+2α22v+α23w)(d3+α31u+α32v+2α33w)>u*(αα23v*w+βα32w*v)2(d1+2α11u+α12v+α13w)+αv*(α13u*w+βα31w*u)2(d2+α21u+2α22v+α23w)+βw*(α12u*v+αα21v*u)2(d3+α31u+α32v+2α33w).
By the maximum-norm estimate in Theorem 1.1, the condition (1.20) implies (3.3). Therefore, we have
(3.4)dHdt≤-∫01‍[1-abv*(1+bu*)(1+bu)-ABw*(1+Bu*+Dw*)(1+Bu+Dw)](u-u*)2dx-β∫01‍MDEu*(1+Bu*+Dw*)(1+Bu+Dw)(w-w*)2dx≤-∫01‍[1-abv*1+bu*-ABw*1+Bu*+Dw*](u-u*)2dx-β∫01‍MDEu*1+Bu*+Dw*(w-w*)2dx≜-l1∫01‍(u-u*)2dx-βl2∫01‍(w-w*)2dx,
where
(3.5)l1=1-abv*1+bu*-ABw*1+Bu*+Dw*,l2=MDEu*1+Bu*+Dw*>0.
The condition (1.21) implies l1>0. Using the similar argument in the proof of Theorem 4.2 in [42], by the maximum-norm estimate in Theorem 1.1 and some tedious calculations, we can prove
(3.6)lim⁡t→∞∫01‍(u-u*)2dx=lim⁡t→∞∫01‍(v-v*)2dx=lim⁡t→∞∫01‍(w-w*)2dx=0.
It follows from (3.6) and Gagliardo-Nirenberg-type inequality |u|∞≤C|u|1,21/2|u|21/2 that (u,v,w) converges uniformly on (u*,v*,w*). By the fact that H(u,v,w) is decreasing for t≥0, it is obvious that (u*,v*,w*) is globally asymptotically stable. So the proof of Theorem 1.3 is completed.



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*Source: 183285-2012-08-07.xml*