# Global Behavior of Solutions in a Predator-Prey Cross-Diffusion Model with Cannibalism

Authors: Meijun Chen; Shengmao Fu; Xiaoli Yang
Journal: Complexity
 (2020)

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## Abstract

The global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paper. Firstly, the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed. It is shown that the equilibria have the same stability properties for the corresponding ODE model and semilinear reaction-diffusion model, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs. Secondly, the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one. Finally, the global stability of the positive equilibrium is established by constructing a Lyapunov function. The result indicates that, under certain conditions on reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.


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## Body

## 1. Introduction

In 1999, Magnússon [1] proposed the following predator-prey model:(1)dXdt=−MmatX+AY+BXY+CXZ,dYdt=RX−AY−MjuvY−SXY,dZdt=TZ−UZ2−VXZ,where X,Y, and Z represent the biomasses of adult predators, juvenile predators, and prey, respectively, Mmat and Mjuv represent the death rates of adult predators and juvenile predators, respectively, A denotes the specific rate of juveniles recruited to the adults class, R denotes the birth rate, T and U are logistic coefficients, S represents the cannibalism attack rate, B=γS, where γ is the conversion efficiency of eaten juveniles into adult biomass, and the biological meanings of V and C are similar to S and B. For more details on model (1), please refer to [1].Let(2)u=SMmatX,v=SAMmat2Y,w=CMmatZ,t˜=Mmatt,and still denote t˜ by t; then, (1) becomes(3)dudt=−u+v+auv+uw,dvdt=bu−gv−uv,dwdt=sw−rw2−euw,where(4)a=BMmatSA=γMmatA,b=RAMmat2,g=A+MjuvMmat,s=TMmat,r=UC,e=VS.We can see thate is the only parameter depending on S in system (3).Most works on cannibalism adopt a McKendrick–von Foerster age-structured model [2–4], but the model is usually rewritten as an ODE system or nonlinear Volterra integral equations [4, 5]. In 1995, Kohlmeier and Ebenhöh [6] studied a two-dimensional ODE model without any structure, and they obtained that, in some cases, cannibalism can lead to a higher long term predator stock size. Then, van den Bosch and Gabriel [7] found that cannibalism can stabilize a predator-prey system in a structured model where the oscillations are due to age structure.Magnússon [1] discussed model (1) which is in some ways a simplification of the system studied by van den Bosch and Gabriel in [7]. Magnússon made two simplifications. Firstly, it is assumed that all juveniles are vulnerable to predation by the adults. Secondly, instantaneous maturation into the adult class is proportional to the present juvenile biomass, i.e., a constant per capita rate of maturation, and making this simplifying assumption means that any oscillations that may occur are not caused by a delay inherent in the system. Moreover, the stability and Hopf bifurcation of solutions for system (3) were studied in [1]. The author obtained the following conclusions. On one hand, if the mortality rate of juveniles is low and/or the recruitment rate to the mature population is high, then there is a stable equilibrium with all three population sizes as positive. On the other hand, if the mortality rate of juveniles is high and/or the recruitment rate to the mature population is low, then the equilibrium will be stable for low levels of cannibalism, but a loss of stability by a Hopf bifurcation will take place as the level of cannibalism increases.As we all know, the growth of biological population depends not only on time, but also on spatial distribution. Spatial species interaction includes the self-diffusion, which is the natural dispersive force of movement of individual, and the cross-diffusion, that is, the population fluxes of one species due to the presence of other species [8–16]. Therefore, taking into account the effect of competition for resources on species growth law, we are naturally led to the following weakly coupled reaction-diffusion model:(5)ut=d1Δu−u+v+auv+uw,x∈Ω,t>0,vt=d2Δv+bu−gv−uv,x∈Ω,t>0,wt=d3Δw+sw−rw2−euw,x∈Ω,t>0,∂ηu=∂ηv=∂ηw=0,x∈∂Ω,t>0,ux,0=u0x,vx,0=v0x,wx,0=w0x,x∈Ω,and the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion(6)ut=Δd1u+α11u2+α12uv+α13uw−u+v+auv+uw,x∈Ω,t>0,vt=Δd2v+α21uv+α22v2+α23vw+bu−gv−uv,x∈Ω,t>0,wt=Δd3w+α31uw+α32vw+α33w2+sw−rw2−euw,x∈Ω,t>0,∂ηu=∂ηv=∂ηw=0,x∈∂Ω,t>0,ux,0=u0x,vx,0=v0x,wx,0=w0x,x∈Ω,where Ω⊂ℝn is a bounded domain with smooth boundary ∂Ω and η is the outward unit normal vector of the boundary ∂Ω, ∂η=∂/∂η. The coefficients dii=1,2,3 are the diffusion rates of u,v,andw, αiii=1,2,3 are referred as self-diffusion pressures, and αiji≠j,i,j=1,2,3 are cross-diffusion coefficients which may be positive, negative, or zero [17]. The functions u0,v0,andw0 are nonnegative functions which are not identically zero.Very recently, Zhang et al. [18] found that the positive equilibrium of (3) with r=0 can undergo stability switch (from stable to unstable to stable, or from unstable to stable to unstable) with the change of the cannibalization rate. Their results indicate that large cannibalization rate can make the positive equilibrium globally stable although its stability would change with the increase of the rate. In fact, an important research subject for cannibalism models is the stabilizing/destabilizing effect of cannibalism [1, 6, 19]. For this, we need to compare the effect of cannibalism on the dynamic behavior for the ODE system, the semilinear reaction-diffusion system, and the quasi-linear cross-diffusion system.In this paper, we first prove that stability properties of equilibria for the ODE model (3) and the semilinear reaction-diffusion model (5) is similar, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability in the quasi-linear model (6) occurs. It is found that the cross diffusion rate α13 is a decisive factor of destabilizing positive steady state, that is, cannibalism has no longer a stabilizing effect. Then, the uniform boundedness and the global existence of time-varying solutions for the cross-diffusion system (6) are investigated by using the energy estimates and Gagliardo–Nirenberg-type inequalities when the space dimension is one. Finally, some criteria on the global asymptotic stability of the positive equilibrium point for (6) are given by Lyapunov function. The obtained results indicate that, for any cannibalism rate e, under certain conditions on the other reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.Moreover, we, in [14], discussed another cross-diffusion model:(7)ut−d1Δu=−u+v+auv+uw,x∈Ω,t>0,vt−Δd2v+d4vε+u2=bu−gv−uv,x∈Ω,t>0,wt−d3Δw=sw−rw2−euw,x∈Ω,t>0,∂ηu=∂ηv=∂ηw=0,x∈∂Ω,t>0,ux,0=u0x,vx,0=v0x,wx,0=w0x,x∈Ω,where d4 is a cross-diffusion coefficient and ε is a positive constant (please refer to [12]). We proved that if a,b,g,s,r,e,ε,d1,d2, and d3 are fixed such that 0<ε<ε0,ag<1,r>s and (9) (see Section 2) hold, then there exists a positive constant d4∗ such that (7) has at least one nonconstant positive steady state for d4≥d4∗. This implies that the cross-diffusion rate is the decisive factor of destabilization on the constant positive steady state and therefore cannibalism is an auxiliary destabilizing force. Recently, in another paper, we detailedly describe the local structure of the nonconstant steady states and discuss the stability and instability of the steady state bifurcation.


## 2. Linearization Analysis

In this section, according to the linearization analysis and Routh–Hurwitz criterion, we focus on discussing the stability of the reaction-diffusion system (5) and cross-diffusion system (6). To this end, we first introduce the stability of the ODE system (3).


### 2.1. ODE Model (3)

Obviously, problem (3) has trivial equilibrium E00,0,0, semitrivial equilibrium E10,0,s/r, and E2g−b/ab−1,g−b/ag−1,0 if g<b<1/a or g>b>1/a. Moreover, if r=0,b<g+s1−ag/e+as, then a positive equilibrium point is given by E3u¯,v¯,w¯, where(8)u¯=se,v¯=bseg+s,w¯=eg−b+s1−abeg+s.Ifr≠0 and(9)br+gs−gr>0,s+eg+r1−ab>0,s1−ab+eg−b>0,then a positive equilibrium point is uniquely given by u˜=u˜,v˜,w˜, where(10)u˜=α+α2+4eβ2e,v˜=bu˜g+u˜,w˜=s−eu˜r,α=abr+s−r−eg,β=br+gs−gr.In fact, the stability of these nonnegative equilibria in (3) has been obtained in [1] as follows:(1)
E0 is unconditionally unstable.(2)
E1 is locally asymptotically stable if s<r and is unstable if s>r.(3)
E2 is locally asymptotically stable if g<b<1/a and is unstable if b<g+s1−ag/e+as.(4)
(a)
Ifag<1, then E3 is locally asymptotically stable.(b)
Ifag>1, S=0 (i.e., e⟶∞) and b<g, then E3 is locally asymptotically stable.(c)
Ifag>1, S≠0 (i.e., e≠∞), b<g and ab<1, then there exists e∗ such that E3 is locally asymptotically stable when e>e∗, and is unstable when e<e∗.Now, we give the stability ofu˜ with respect to model (3).Theorem 1.
The positive equilibriumu˜ of (3) is locally asymptotically stable if ag<1.Proof.
The Jacobian matrix of (3) to the generic us=us,vs,ws reads(11)Jus=j11j12j13j21j22j23j31j32j33,where(12)j11=−1+avs+ws,j12=1+aus,j13=us,j21=b−vs,j22=−g−us,j23=0,j31=−ews,j32=0,j33=s−2rws−eus.
A direct calculation yields that the characteristic equation ofJu˜ is(13)φλ=λI−J=λ3+α1λ2+α2λ+α3=0,with(14)α1=1u˜gu˜+v˜++u˜2+ru˜w˜>0,α2=1u˜1−agu˜v˜+rw˜v˜+gu˜+e+ru˜2w˜,α3=1u˜ew˜u˜2g+u˜+r1−agu˜v˜w˜,α1α2−α3=rw˜u˜2+1−agv˜+2grw˜+rew˜2+r2w˜2u˜+rg2w˜+ev˜w˜+2rv˜w˜+gr2w˜2+1−aggv˜+rw˜v˜2u˜2+1u˜1−agv˜2+r2w˜2v˜+2grv˜w˜.
It is easy to see thatα3>0 and α1α2−α3>0, thanks to ag<1. Therefore, by Routh–Hurwitz criterion, we know that u˜,v˜,w˜ is locally asymptotically stable.
Notice that the unstable equilibrium points of (3) are also unstable for (5) and (6). Therefore, for systems (5) and (6), we only discuss the stability of the equilibrium points which are stable for (3).


### 2.2. Weakly Coupled Reaction-Diffusion System (5)

Let0=μ1<μ2<μ3<⋯ be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition, and let Eμi be the eigenspace corresponding to μi in H1Ω. Let X be the closure of C1Ω¯3 in H1Ω3, ϕij:j=1,2,…,dimEμi be an orthonormal basis of Eμi, and let Xij=cϕij:c∈ℝ3. Then,(15)X=⊕i=1+∞Xi,Xi=⊕j=1dimEμiXij.For system (5), by the linearization analysis and some similar arguments to the proof of Theorem 2 and Theorem 3 in [12], we can obtain the following two theorems.Theorem 2.
(1)
The semitrivial equilibriumE1 of (5) is locally asymptotically stable if s<r(2)
The semitrivial equilibriumE2 of (5) is locally asymptotically stable if g<b<1/a(3)
The positive equilibriumE3 of (5) is locally asymptotically stable either ag<1, or ag>1, S=0 (i.e., e⟶∞) and b<gProof.
We only prove (3). The proof of (1) and (2) is similar to (3), so we omit it here. Letu=u,v,w,D=diagd1,d2,d3, and ℒ=DΔ+JE3, where j11=−be/eg+s<0,j12=e+as/e>0. j13=u¯>0,j21=beg/eg+s>0,j22=−g+u¯<0,j23=0,j31=−ew¯<0,andj32=j33=0. The linearization of (5) at E3 is ut=ℒu. For each i≥1, Xi is invariant under the operator ℒ, and λ is an eigenvalue of ℒ if and only if it is an eigenvalue of the matrix −μiD+JE3 for some i≥1, in which case there is an eigenvector in Xi. Denote the characteristic polynomial of −μiD+JE3 by φiλ=λ3+Aiλ2+Biλ+Ci, where(16)Ai=μid1+d2+d3−j11+j22,Bi=μi2d1d2+d2d3+d1d3−μid1j22+d2j11+d3j11+d3j22+j11j22−j12j21−j13j31,Ci=μi3d1d2d3−μi2d1d3a22+d2d3j11+μid3j11j22−d3j12j21−d2j13j31+j22j13j31.
A straightforward computation givesHi=AiBi−Ci=I1μi3+I2μi2+I3μi+I4, where(17)I1=d12d2+d3+d22d1+d3+d32d1+d2+2d1d2d3,I2=−j11d22+d32−j22d12+d32−2j11+j22d1d2+d2d3+d1d3,I3=j11+j22d1j22+d2j11+d3j11+d3j22+d1+d2j11j22−j12j21−d1+d3j13j31,I4=−j11+j22j11j22−j12j21+j11j13j31.
Obviously, ifj11j22>j12j21 (i.e., ag<1), then Hi>0,Ai>0,andCi>0. It follows from the Routh–Hurwitz criterion that the three roots λi1,λi2,andλi3 of φiλ=0 all have negative real parts. Moreover, if S=0, then e=V/S⟶∞,w¯⟶1−b/g, j11⟶−b/g,j12⟶1,j13⟶0,j21⟶b,j22⟶−g,andj31⟶−e1−b/g. Therefore, if b<g, then Hi>0,Ai>0,andCi>0.
We claim that, for eachi≥1, there exists a positive constant δ such that(18)Reλi1,Reλi,2,Reλi,3≤−δ,i≥1.
Letλ=μiξ. Then, φiλ=μi3ξ3+Aiμi2ξ2+Biμiξ+Ci≜φi˜ξ. Since μi⟶∞ as i⟶∞, it follows that(19)limi⟶∞φi˜ξμi3=ξ3+d1+d2+d3ξ2+d1d2+d2d3+d1d3ξ+d1d2d3≜φ˜ξ.
Clearly,φ˜ξ=0 has three negative roots −d1,−d2, and −d3. Let d∗=mind1,d2,d3. By continuity, one can see that there exists a i0 such that the three roots ξi1,ξi2, and ξi3 of φi˜ξ=0 satisfy Reξi1,Reξi2,Reξi3≤−d∗/2 when i≥i0. Therefore, if i≥i0, then Reλi1,Reλi2,Reλi3≤−μid∗/2≤−d∗/2. Let δ˜=−max0≤i≤i0Reλi1,Reλi2,Reλi3. Then, δ˜>0, and (18) holds for δ=minδ˜,d∗/2. So, the positive equilibrium point E3 of (5) is locally asymptotically stable.In order to obtain the global asymptotic behavior of the solutions to (5) or (6), we need the following result which can be found in [12].Lemma 1.
Leta and b be positive constants. Assume that φ,ψ∈C1a,+∞, ψt≥0, and φ is bounded from below. If φ′t≤−bψt and ψ′t is bounded from above in a,+∞, then limt⟶∞ψt=0.Theorem 3.
(1)
The positive equilibriumE3 of (5) is globally asymptotically stable if ag<1 and eg+ags<egs<eg+s.(2)
The semitrivial equilibriumE2 of (5) is globally asymptotically stable if 1<1−abb−g<b/g.Proof.
Letu,v,w be the unique positive solution of (5). Estimates similar to Theorem 2.1 in [15] and TheoremA2 in [20] show(20)u⋅,tC2+αΩ¯,v⋅,tC2+αΩ¯,u⋅,tC2+αΩ¯≤C,∀t≥1,where α∈0,1 and C does not depend on t.(1)
Define(21)Vu,v,w=∫Ωu−u¯−u¯lnuu¯+λv−v¯−v¯lnvv¯+ρw−w¯−w¯lnww¯dx,whereλ=e+as/egandρ=1/e. Obviously, Vu,v,w is nonnegative and Vu,v,w=0 if and only if u=u¯,v=v¯,andw=w¯. The time derivative of Vu,v,w for (5) satisfies that(22)dVu,v,wdt=−∫Ωd1u¯u2∇u2+λd2v¯v2∇v2+ρd3w¯w2∇w2dx−1s∫Ωes−e−assvuu−u¯2+e+aseg+s−egseguvv−v¯2+e+assvuu−u¯−se+asuvv−v¯2dx≤−∫Ωd1u¯u2∇u2+λd2v¯v2∇v2+ρd3w¯w2∇w2dx−1s∫Ωes−e−assvuu−u¯2+e+aseg+s−egseguvv−v¯2dx.From Theorem2, Lemma 1, and (20), we know that(23)limt⟶∞∫Ω∇u2+∇v2+∇w2dx=0,(24)limt⟶∞∫Ωu−u¯2dx=0,limt⟶∞∫Ωv−v¯2dx=0.It follows from the Poincaré inequality that(25)limt⟶∞∫Ωu−u^2dx=0,limt⟶∞∫Ωv−v^2dx=0,limt⟶∞∫Ωw−w^2dx=0,wheref^=∫01fdx. As u^−u¯2=∫Ωu^−u¯2dx≤2∫Ωu^−u2dx+2∫Ωu−u¯2dx, from (24) and (25), we have(26)limt⟶∞u^t=u¯,limt⟶∞v^t=v¯.Therefore, there exists a sequencetm such that u^′tm⟶0 as tm⟶∞. Since w^tm is bounded, so there exists a subsequence, still denoted by w^tm, such that w^tm⟶w˜ as tm⟶∞.From the first equation of (5), we have(27)u^′tm=∫Ωav+w−1u−u¯+au¯+1v−v¯+u¯w−w¯dxtm.Letm⟶∞ in the above equation, and from (25), we have(28)limtm⟶∞w^t=w¯.On the contrary, (20) implies that there exists a subsequence, still denoted by tm, and nonnegative functions u∗,v∗,w∗∈C2Ω¯ such that(29)u⋅,tm−u∗C2Ω¯⟶0,v⋅,tm−v∗C2Ω¯⟶0,w⋅,tm−w∗C2Ω¯⟶0,m⟶∞.Combining this with (25)–(28), one can obtain that u∗=u¯,v∗=v¯,w∗=w¯, and(30)u⋅,tm−u¯C2Ω¯⟶0,v⋅,tm−v¯C2Ω¯⟶0,w⋅,tm−w¯C2Ω¯⟶0,m⟶∞.The global asymptotic stability ofE3 follows from this together with Theorem 2.(2)
Define(31)Vu,v,w=∫Ωu−u2−u2lnuu2+lv−v2−v2lnvv2+1ewdx,wherel=1−ag/g1−ab and u2=g−b/ab−1,v2=g−b/ag−1. Then,(32)dVu,v,wdt=−∫Ωd1u2u2∇u2+λd2v2v2∇v2dx−∫Ω1−abb−g−1b−g2vuu−u22+1−ag2bgb−g1−ab−1uvv−v22+rew2+1b−gvuu−u2−1−aguvv−v22dx.From this, (2) can be proved using similar arguments as in (1).Remark 1.
The stability ofu˜ is demonstrated specifically in [14], that is, u˜ is locally asymptotically stable if ag<1.


### 2.3. Strongly Coupled Cross-Diffusion System (6)

Comparing Sections2.1 and 2.2, we find that Ei,i=1,2,3, and u˜ have the same stability properties in systems (3) and (5). For the sake of convenience, we denote nonnegative equilibria Ei,i=1,2,3, and u˜ by us=us,vs,ws. Now, we show that the destabilization effect of cross-diffusion on us=us,vs,ws.Linearizing system (6) at an equilibrium us,vs,ws, we can obtain(33)∂u∂t=D+PΔu+Ju,x∈Ω,t>0,∂u∂ν=0,x∈∂Ω,t>0,ux,0=u0x,x∈Ω,where u=u,v,wT, D=diagd1,d2,d3, and J is given in (11).(34)P=p11p12p13p21p22p23p31p33p33,with(35)p11=2α11us+α12vs+α13ws,p12=α12us,p13=α13us,p21=α21vs,p22=α21us+2α22vs+α23ws,p23=α23vs,p31=α31ws,p32=α32ws,p33=α31us+α32vs+2α33ws.We denote(36)ℒ≔−μiD+P+J,then the corresponding characteristic polynomial of ℒ is(37)ϕ=λ3+γ1λ2+γ2λ+γ3,where(38)γ1=trD+trPμi−trJ,γ2=β1μi2+β2μi+β3,γ3=detμiP−J=β4μi3+β5μi2+β6μi+β7,with(39)β1=d1d2+d1d3+d2d3+d1p22+p33+d2p11+p33+d3p11+p22+M11+M22+M33,β2=−j11d2+d3+p22+p33−j22d1+d3+p11+p33−j33d1+d2+p11+p22+p21j12+p12j21+p31j13+p13j31,β3=M¯22+M¯33+j22j33,β4=detD+detP+d1d2p33+d3p22+M11+d2M22+d3M33+d2d3P11,β5=−j11M11+d2p33+d3p22+d2d3+j12M12+d3p21+j13d2p31−M13+j21d3p12+M21−j22M22+d1p33+d3p11+d1d3+j31d2p13−M31−j33M33+d1d2+d1p22+d2p11,β6=p11+d1j22j33−p12j21j33−p13j22j31−p21j12j33+p22j11j33−p22j13j31+p23j12j31−p31j13j22+p32j13j21+p33+d3M¯33+d2M¯22,β7=−j33M¯33+j13j22j31,where Mij and M¯ij,i,j=1,2,3, are cofactors of matrix P and J, respectively.Therefore, according to the principle of the linearized stability ((21], Th 8.6), (22], Th 5.2)), the local stability of nonnegative equilibria of model (6) is given below.Theorem 4.
Letdi and αij, i,j=1,2,3, be positive constants. Then, the following statements for system (6) hold.(1)
us is locally asymptotically stable if and only if for every i∈ℕ, all the eigenvalues of the linearization matrix ℒ have negative real part.(2)
us is unstable if and only if there exists an i∈ℕ, such that the linearization matrix ℒ has at least one eigenvalue with positive real part.
By applying the Routh–Hurwitz criterion or Corollary 2.2 in [23], we have the following stability and instability results.Corollary 1.
(1)
E1 is locally asymptotically stable if s<r and b<gm, here m=mins,r−s/r,rs2/r−s.(2)
E2 is locally asymptotically stable if g<b<1/a, 1−ab/1−ag<s<eb−g/1−ab, and ab−1ag−1d3+ab−1g−bα32+g−bag−1α31>0.(3)
E3 is locally asymptotically stable if ag<1, eα13<α31, and g+u¯2w¯α33+d3+u¯v¯α32>su¯α21+2v¯α22+w¯α23+d2.(4)
u˜ is locally asymptotically stable if s<δ≔r1−ab−eg+abr−r+eg2+4erg−b, b<g,ag<1, and e/rα23≤α21≤α13≤α31/e≤α21α33/eα23.(5)
Ifag<1 and s>f, then there exists a positive constant α˜13 such that u˜ is unstable when α13>α˜13.(6)
Ifag>1, then there exists a positive constant d˜3 such that u˜ is unstable when d3>d˜3.Proof.
We only prove the case of (4) and (5), other cases can be treated similarly. With the help of the Maple application, we can evaluate thatγi>0,i=1,2,3 and γ1γ2−γ3>0 due to b<g,ag<1, s<δ≔r1−ab−eg+abr−r+eg2+4erg−b and e/rα23<α21<α13<α31/e<α21α33/eα23. Therefore, Routh–Hurwitz criterion shows that u˜ is locally asymptotically stable.
On the contrary, the constant equilibrium is unstable if and only if there exists ani∈ℕ such that γ3<0 or γ1γ2<γ3. Thanks to calculate γ1γ2−γ3 is complicated, we now only discuss the case of γ3<0.
In fact,γ3=detμiP−J=β4μi3+β5μi2+β6μi+β7≔Cα13;μi, where β4>0. In order to study whether cross-diffusion has unstable effect, we assume that u˜ is stable in the corresponding self-diffusion system, i.e., ag<1, which implies that β7=−j33v˜1−ag+j13j22j31>0. Let μ˜1,μ˜2,andμ˜3 be the three roots of Cα13;μi with Reμ˜1≤Reμ˜2≤Reμ˜3. Then, μ˜1μ˜2μ˜3=−β4/β1<0, at least one of μ˜1,μ˜2,andμ˜3 is real and negative, and the product of the other two is positive.
Now, we consider the following limits:(40)C1≔limα13⟶∞β1α13=2w˜3α23α33+2u˜α21α33+4v˜α22α33+2d3α33+d3α23w˜2+2v˜α32u˜α21+v˜α22+d2d3+v˜α32+d3u˜α21+2v˜α22,C2≔limα13⟶∞β2α13=rw˜−eu˜u˜w˜α21+2v˜w˜α22+w˜2α23+2u˜v˜+gv˜−bu˜w˜α32+2w˜2g+uα33,C3≔limα13⟶∞β3α13=s−2eu˜g+u˜.
Note that(41)limα13⟶∞Cα13;νiα13=μiC1μi2+C2μi+C3,and C1>0, C3<0 since s<2eu˜, i.e. s>r1−ab−eg+abr−r+eg2+4erg−b. A continuity argument yields that μ˜1 is real and negative. Moreover, as μ˜2μ˜3>0, μ˜2 and μ˜3 are real and positive and(42)limα13⟶∞μ˜1=−C2−C2−4C1C32C1<0,limα13⟶∞μ˜2=0,limα13⟶∞μ˜3=−C2+C2−4C1C32C1>0.
Thus, there exists a positive constantα˜13 such that, for all α13>α˜13, the three roots μ˜1,μ˜2,andμ˜3 of Cα13;μi are all real and satisfy(43)−∞<μ˜1<0<μ˜2<μ˜3,α3=Cα13;μi<0,whenμi∈−∞,μ˜1∪μ˜2,μ˜3,α3=Cα13;μi>0,whenμi∈μ˜1,μ˜2∪μ˜3,+∞.
Therefore, the characteristic equationϕ=0 has at least one positive eigenvalue when μi∈μ˜2,μ˜3.Remark 2.
Remark1 and conclusion (5) of Corollary 1 indicate that if ag<1 and s>f, the large cross-diffusion rate α13 has destabilizing effect for positive equilibrium point u˜,v˜,w˜. This means that under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs if the cross-diffusion rate is sufficiently large.
To demonstrate the results of stability and instability foru˜, we give the following examples.Example 1.
Assume that the spatial domainΩ=0,10π.(1)
Let(44)α11=α12=α21=α22=α23=α32=1,α13=1.2,α31=3,α33=4,a=0.5,b=1,g=1.5,s=3,e=2,r=4,d1=3,d2=3,d3=2.Then, the equilibrium pointu˜,v˜,w˜=0.725,0.326,0.388, γ1=5.809>0, γ2=10.163>0, γ3=5.082>0, and γ1γ2−γ3=53.959>0. From Corollary 1 (4), u˜ is locally asymptotically stable.(2)
Let(45)α11=α12=α21=α22=α23=α31=α32=α33=1,d1=3,d2=3,d3=2,a=0.5,b=1.3,g=1.5,s=3,e=5,r=4,and chooseα13=300. Then, the equilibrium point u˜,v˜,w˜=0.453,0.302,0.184, ag=0.75<1, s=3>f=1.195, and γ3=−0.663<0. Therefore, u˜ is unstable thanks to Corollary 1 (5).(3)
Letαij=0.1,i,j=1,2,3, a=2.8,b=1,g=3,s=3,e=3,r=4.5,d1=3,andd2=3, and choose d3=200. Then, the equilibrium point u˜,v˜,w˜=0.7,0.189,0.2, ag=8.4>1 and γ3=−1.643<0. Thus, u˜ is unstable due to Corollary 1 (6).


## 2.1. ODE Model (3)

Obviously, problem (3) has trivial equilibrium E00,0,0, semitrivial equilibrium E10,0,s/r, and E2g−b/ab−1,g−b/ag−1,0 if g<b<1/a or g>b>1/a. Moreover, if r=0,b<g+s1−ag/e+as, then a positive equilibrium point is given by E3u¯,v¯,w¯, where(8)u¯=se,v¯=bseg+s,w¯=eg−b+s1−abeg+s.Ifr≠0 and(9)br+gs−gr>0,s+eg+r1−ab>0,s1−ab+eg−b>0,then a positive equilibrium point is uniquely given by u˜=u˜,v˜,w˜, where(10)u˜=α+α2+4eβ2e,v˜=bu˜g+u˜,w˜=s−eu˜r,α=abr+s−r−eg,β=br+gs−gr.In fact, the stability of these nonnegative equilibria in (3) has been obtained in [1] as follows:(1)
E0 is unconditionally unstable.(2)
E1 is locally asymptotically stable if s<r and is unstable if s>r.(3)
E2 is locally asymptotically stable if g<b<1/a and is unstable if b<g+s1−ag/e+as.(4)
(a)
Ifag<1, then E3 is locally asymptotically stable.(b)
Ifag>1, S=0 (i.e., e⟶∞) and b<g, then E3 is locally asymptotically stable.(c)
Ifag>1, S≠0 (i.e., e≠∞), b<g and ab<1, then there exists e∗ such that E3 is locally asymptotically stable when e>e∗, and is unstable when e<e∗.Now, we give the stability ofu˜ with respect to model (3).Theorem 1.
The positive equilibriumu˜ of (3) is locally asymptotically stable if ag<1.Proof.
The Jacobian matrix of (3) to the generic us=us,vs,ws reads(11)Jus=j11j12j13j21j22j23j31j32j33,where(12)j11=−1+avs+ws,j12=1+aus,j13=us,j21=b−vs,j22=−g−us,j23=0,j31=−ews,j32=0,j33=s−2rws−eus.
A direct calculation yields that the characteristic equation ofJu˜ is(13)φλ=λI−J=λ3+α1λ2+α2λ+α3=0,with(14)α1=1u˜gu˜+v˜++u˜2+ru˜w˜>0,α2=1u˜1−agu˜v˜+rw˜v˜+gu˜+e+ru˜2w˜,α3=1u˜ew˜u˜2g+u˜+r1−agu˜v˜w˜,α1α2−α3=rw˜u˜2+1−agv˜+2grw˜+rew˜2+r2w˜2u˜+rg2w˜+ev˜w˜+2rv˜w˜+gr2w˜2+1−aggv˜+rw˜v˜2u˜2+1u˜1−agv˜2+r2w˜2v˜+2grv˜w˜.
It is easy to see thatα3>0 and α1α2−α3>0, thanks to ag<1. Therefore, by Routh–Hurwitz criterion, we know that u˜,v˜,w˜ is locally asymptotically stable.
Notice that the unstable equilibrium points of (3) are also unstable for (5) and (6). Therefore, for systems (5) and (6), we only discuss the stability of the equilibrium points which are stable for (3).


## 2.2. Weakly Coupled Reaction-Diffusion System (5)

Let0=μ1<μ2<μ3<⋯ be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition, and let Eμi be the eigenspace corresponding to μi in H1Ω. Let X be the closure of C1Ω¯3 in H1Ω3, ϕij:j=1,2,…,dimEμi be an orthonormal basis of Eμi, and let Xij=cϕij:c∈ℝ3. Then,(15)X=⊕i=1+∞Xi,Xi=⊕j=1dimEμiXij.For system (5), by the linearization analysis and some similar arguments to the proof of Theorem 2 and Theorem 3 in [12], we can obtain the following two theorems.Theorem 2.
(1)
The semitrivial equilibriumE1 of (5) is locally asymptotically stable if s<r(2)
The semitrivial equilibriumE2 of (5) is locally asymptotically stable if g<b<1/a(3)
The positive equilibriumE3 of (5) is locally asymptotically stable either ag<1, or ag>1, S=0 (i.e., e⟶∞) and b<gProof.
We only prove (3). The proof of (1) and (2) is similar to (3), so we omit it here. Letu=u,v,w,D=diagd1,d2,d3, and ℒ=DΔ+JE3, where j11=−be/eg+s<0,j12=e+as/e>0. j13=u¯>0,j21=beg/eg+s>0,j22=−g+u¯<0,j23=0,j31=−ew¯<0,andj32=j33=0. The linearization of (5) at E3 is ut=ℒu. For each i≥1, Xi is invariant under the operator ℒ, and λ is an eigenvalue of ℒ if and only if it is an eigenvalue of the matrix −μiD+JE3 for some i≥1, in which case there is an eigenvector in Xi. Denote the characteristic polynomial of −μiD+JE3 by φiλ=λ3+Aiλ2+Biλ+Ci, where(16)Ai=μid1+d2+d3−j11+j22,Bi=μi2d1d2+d2d3+d1d3−μid1j22+d2j11+d3j11+d3j22+j11j22−j12j21−j13j31,Ci=μi3d1d2d3−μi2d1d3a22+d2d3j11+μid3j11j22−d3j12j21−d2j13j31+j22j13j31.
A straightforward computation givesHi=AiBi−Ci=I1μi3+I2μi2+I3μi+I4, where(17)I1=d12d2+d3+d22d1+d3+d32d1+d2+2d1d2d3,I2=−j11d22+d32−j22d12+d32−2j11+j22d1d2+d2d3+d1d3,I3=j11+j22d1j22+d2j11+d3j11+d3j22+d1+d2j11j22−j12j21−d1+d3j13j31,I4=−j11+j22j11j22−j12j21+j11j13j31.
Obviously, ifj11j22>j12j21 (i.e., ag<1), then Hi>0,Ai>0,andCi>0. It follows from the Routh–Hurwitz criterion that the three roots λi1,λi2,andλi3 of φiλ=0 all have negative real parts. Moreover, if S=0, then e=V/S⟶∞,w¯⟶1−b/g, j11⟶−b/g,j12⟶1,j13⟶0,j21⟶b,j22⟶−g,andj31⟶−e1−b/g. Therefore, if b<g, then Hi>0,Ai>0,andCi>0.
We claim that, for eachi≥1, there exists a positive constant δ such that(18)Reλi1,Reλi,2,Reλi,3≤−δ,i≥1.
Letλ=μiξ. Then, φiλ=μi3ξ3+Aiμi2ξ2+Biμiξ+Ci≜φi˜ξ. Since μi⟶∞ as i⟶∞, it follows that(19)limi⟶∞φi˜ξμi3=ξ3+d1+d2+d3ξ2+d1d2+d2d3+d1d3ξ+d1d2d3≜φ˜ξ.
Clearly,φ˜ξ=0 has three negative roots −d1,−d2, and −d3. Let d∗=mind1,d2,d3. By continuity, one can see that there exists a i0 such that the three roots ξi1,ξi2, and ξi3 of φi˜ξ=0 satisfy Reξi1,Reξi2,Reξi3≤−d∗/2 when i≥i0. Therefore, if i≥i0, then Reλi1,Reλi2,Reλi3≤−μid∗/2≤−d∗/2. Let δ˜=−max0≤i≤i0Reλi1,Reλi2,Reλi3. Then, δ˜>0, and (18) holds for δ=minδ˜,d∗/2. So, the positive equilibrium point E3 of (5) is locally asymptotically stable.In order to obtain the global asymptotic behavior of the solutions to (5) or (6), we need the following result which can be found in [12].Lemma 1.
Leta and b be positive constants. Assume that φ,ψ∈C1a,+∞, ψt≥0, and φ is bounded from below. If φ′t≤−bψt and ψ′t is bounded from above in a,+∞, then limt⟶∞ψt=0.Theorem 3.
(1)
The positive equilibriumE3 of (5) is globally asymptotically stable if ag<1 and eg+ags<egs<eg+s.(2)
The semitrivial equilibriumE2 of (5) is globally asymptotically stable if 1<1−abb−g<b/g.Proof.
Letu,v,w be the unique positive solution of (5). Estimates similar to Theorem 2.1 in [15] and TheoremA2 in [20] show(20)u⋅,tC2+αΩ¯,v⋅,tC2+αΩ¯,u⋅,tC2+αΩ¯≤C,∀t≥1,where α∈0,1 and C does not depend on t.(1)
Define(21)Vu,v,w=∫Ωu−u¯−u¯lnuu¯+λv−v¯−v¯lnvv¯+ρw−w¯−w¯lnww¯dx,whereλ=e+as/egandρ=1/e. Obviously, Vu,v,w is nonnegative and Vu,v,w=0 if and only if u=u¯,v=v¯,andw=w¯. The time derivative of Vu,v,w for (5) satisfies that(22)dVu,v,wdt=−∫Ωd1u¯u2∇u2+λd2v¯v2∇v2+ρd3w¯w2∇w2dx−1s∫Ωes−e−assvuu−u¯2+e+aseg+s−egseguvv−v¯2+e+assvuu−u¯−se+asuvv−v¯2dx≤−∫Ωd1u¯u2∇u2+λd2v¯v2∇v2+ρd3w¯w2∇w2dx−1s∫Ωes−e−assvuu−u¯2+e+aseg+s−egseguvv−v¯2dx.From Theorem2, Lemma 1, and (20), we know that(23)limt⟶∞∫Ω∇u2+∇v2+∇w2dx=0,(24)limt⟶∞∫Ωu−u¯2dx=0,limt⟶∞∫Ωv−v¯2dx=0.It follows from the Poincaré inequality that(25)limt⟶∞∫Ωu−u^2dx=0,limt⟶∞∫Ωv−v^2dx=0,limt⟶∞∫Ωw−w^2dx=0,wheref^=∫01fdx. As u^−u¯2=∫Ωu^−u¯2dx≤2∫Ωu^−u2dx+2∫Ωu−u¯2dx, from (24) and (25), we have(26)limt⟶∞u^t=u¯,limt⟶∞v^t=v¯.Therefore, there exists a sequencetm such that u^′tm⟶0 as tm⟶∞. Since w^tm is bounded, so there exists a subsequence, still denoted by w^tm, such that w^tm⟶w˜ as tm⟶∞.From the first equation of (5), we have(27)u^′tm=∫Ωav+w−1u−u¯+au¯+1v−v¯+u¯w−w¯dxtm.Letm⟶∞ in the above equation, and from (25), we have(28)limtm⟶∞w^t=w¯.On the contrary, (20) implies that there exists a subsequence, still denoted by tm, and nonnegative functions u∗,v∗,w∗∈C2Ω¯ such that(29)u⋅,tm−u∗C2Ω¯⟶0,v⋅,tm−v∗C2Ω¯⟶0,w⋅,tm−w∗C2Ω¯⟶0,m⟶∞.Combining this with (25)–(28), one can obtain that u∗=u¯,v∗=v¯,w∗=w¯, and(30)u⋅,tm−u¯C2Ω¯⟶0,v⋅,tm−v¯C2Ω¯⟶0,w⋅,tm−w¯C2Ω¯⟶0,m⟶∞.The global asymptotic stability ofE3 follows from this together with Theorem 2.(2)
Define(31)Vu,v,w=∫Ωu−u2−u2lnuu2+lv−v2−v2lnvv2+1ewdx,wherel=1−ag/g1−ab and u2=g−b/ab−1,v2=g−b/ag−1. Then,(32)dVu,v,wdt=−∫Ωd1u2u2∇u2+λd2v2v2∇v2dx−∫Ω1−abb−g−1b−g2vuu−u22+1−ag2bgb−g1−ab−1uvv−v22+rew2+1b−gvuu−u2−1−aguvv−v22dx.From this, (2) can be proved using similar arguments as in (1).Remark 1.
The stability ofu˜ is demonstrated specifically in [14], that is, u˜ is locally asymptotically stable if ag<1.


## 2.3. Strongly Coupled Cross-Diffusion System (6)

Comparing Sections2.1 and 2.2, we find that Ei,i=1,2,3, and u˜ have the same stability properties in systems (3) and (5). For the sake of convenience, we denote nonnegative equilibria Ei,i=1,2,3, and u˜ by us=us,vs,ws. Now, we show that the destabilization effect of cross-diffusion on us=us,vs,ws.Linearizing system (6) at an equilibrium us,vs,ws, we can obtain(33)∂u∂t=D+PΔu+Ju,x∈Ω,t>0,∂u∂ν=0,x∈∂Ω,t>0,ux,0=u0x,x∈Ω,where u=u,v,wT, D=diagd1,d2,d3, and J is given in (11).(34)P=p11p12p13p21p22p23p31p33p33,with(35)p11=2α11us+α12vs+α13ws,p12=α12us,p13=α13us,p21=α21vs,p22=α21us+2α22vs+α23ws,p23=α23vs,p31=α31ws,p32=α32ws,p33=α31us+α32vs+2α33ws.We denote(36)ℒ≔−μiD+P+J,then the corresponding characteristic polynomial of ℒ is(37)ϕ=λ3+γ1λ2+γ2λ+γ3,where(38)γ1=trD+trPμi−trJ,γ2=β1μi2+β2μi+β3,γ3=detμiP−J=β4μi3+β5μi2+β6μi+β7,with(39)β1=d1d2+d1d3+d2d3+d1p22+p33+d2p11+p33+d3p11+p22+M11+M22+M33,β2=−j11d2+d3+p22+p33−j22d1+d3+p11+p33−j33d1+d2+p11+p22+p21j12+p12j21+p31j13+p13j31,β3=M¯22+M¯33+j22j33,β4=detD+detP+d1d2p33+d3p22+M11+d2M22+d3M33+d2d3P11,β5=−j11M11+d2p33+d3p22+d2d3+j12M12+d3p21+j13d2p31−M13+j21d3p12+M21−j22M22+d1p33+d3p11+d1d3+j31d2p13−M31−j33M33+d1d2+d1p22+d2p11,β6=p11+d1j22j33−p12j21j33−p13j22j31−p21j12j33+p22j11j33−p22j13j31+p23j12j31−p31j13j22+p32j13j21+p33+d3M¯33+d2M¯22,β7=−j33M¯33+j13j22j31,where Mij and M¯ij,i,j=1,2,3, are cofactors of matrix P and J, respectively.Therefore, according to the principle of the linearized stability ((21], Th 8.6), (22], Th 5.2)), the local stability of nonnegative equilibria of model (6) is given below.Theorem 4.
Letdi and αij, i,j=1,2,3, be positive constants. Then, the following statements for system (6) hold.(1)
us is locally asymptotically stable if and only if for every i∈ℕ, all the eigenvalues of the linearization matrix ℒ have negative real part.(2)
us is unstable if and only if there exists an i∈ℕ, such that the linearization matrix ℒ has at least one eigenvalue with positive real part.
By applying the Routh–Hurwitz criterion or Corollary 2.2 in [23], we have the following stability and instability results.Corollary 1.
(1)
E1 is locally asymptotically stable if s<r and b<gm, here m=mins,r−s/r,rs2/r−s.(2)
E2 is locally asymptotically stable if g<b<1/a, 1−ab/1−ag<s<eb−g/1−ab, and ab−1ag−1d3+ab−1g−bα32+g−bag−1α31>0.(3)
E3 is locally asymptotically stable if ag<1, eα13<α31, and g+u¯2w¯α33+d3+u¯v¯α32>su¯α21+2v¯α22+w¯α23+d2.(4)
u˜ is locally asymptotically stable if s<δ≔r1−ab−eg+abr−r+eg2+4erg−b, b<g,ag<1, and e/rα23≤α21≤α13≤α31/e≤α21α33/eα23.(5)
Ifag<1 and s>f, then there exists a positive constant α˜13 such that u˜ is unstable when α13>α˜13.(6)
Ifag>1, then there exists a positive constant d˜3 such that u˜ is unstable when d3>d˜3.Proof.
We only prove the case of (4) and (5), other cases can be treated similarly. With the help of the Maple application, we can evaluate thatγi>0,i=1,2,3 and γ1γ2−γ3>0 due to b<g,ag<1, s<δ≔r1−ab−eg+abr−r+eg2+4erg−b and e/rα23<α21<α13<α31/e<α21α33/eα23. Therefore, Routh–Hurwitz criterion shows that u˜ is locally asymptotically stable.
On the contrary, the constant equilibrium is unstable if and only if there exists ani∈ℕ such that γ3<0 or γ1γ2<γ3. Thanks to calculate γ1γ2−γ3 is complicated, we now only discuss the case of γ3<0.
In fact,γ3=detμiP−J=β4μi3+β5μi2+β6μi+β7≔Cα13;μi, where β4>0. In order to study whether cross-diffusion has unstable effect, we assume that u˜ is stable in the corresponding self-diffusion system, i.e., ag<1, which implies that β7=−j33v˜1−ag+j13j22j31>0. Let μ˜1,μ˜2,andμ˜3 be the three roots of Cα13;μi with Reμ˜1≤Reμ˜2≤Reμ˜3. Then, μ˜1μ˜2μ˜3=−β4/β1<0, at least one of μ˜1,μ˜2,andμ˜3 is real and negative, and the product of the other two is positive.
Now, we consider the following limits:(40)C1≔limα13⟶∞β1α13=2w˜3α23α33+2u˜α21α33+4v˜α22α33+2d3α33+d3α23w˜2+2v˜α32u˜α21+v˜α22+d2d3+v˜α32+d3u˜α21+2v˜α22,C2≔limα13⟶∞β2α13=rw˜−eu˜u˜w˜α21+2v˜w˜α22+w˜2α23+2u˜v˜+gv˜−bu˜w˜α32+2w˜2g+uα33,C3≔limα13⟶∞β3α13=s−2eu˜g+u˜.
Note that(41)limα13⟶∞Cα13;νiα13=μiC1μi2+C2μi+C3,and C1>0, C3<0 since s<2eu˜, i.e. s>r1−ab−eg+abr−r+eg2+4erg−b. A continuity argument yields that μ˜1 is real and negative. Moreover, as μ˜2μ˜3>0, μ˜2 and μ˜3 are real and positive and(42)limα13⟶∞μ˜1=−C2−C2−4C1C32C1<0,limα13⟶∞μ˜2=0,limα13⟶∞μ˜3=−C2+C2−4C1C32C1>0.
Thus, there exists a positive constantα˜13 such that, for all α13>α˜13, the three roots μ˜1,μ˜2,andμ˜3 of Cα13;μi are all real and satisfy(43)−∞<μ˜1<0<μ˜2<μ˜3,α3=Cα13;μi<0,whenμi∈−∞,μ˜1∪μ˜2,μ˜3,α3=Cα13;μi>0,whenμi∈μ˜1,μ˜2∪μ˜3,+∞.
Therefore, the characteristic equationϕ=0 has at least one positive eigenvalue when μi∈μ˜2,μ˜3.Remark 2.
Remark1 and conclusion (5) of Corollary 1 indicate that if ag<1 and s>f, the large cross-diffusion rate α13 has destabilizing effect for positive equilibrium point u˜,v˜,w˜. This means that under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs if the cross-diffusion rate is sufficiently large.
To demonstrate the results of stability and instability foru˜, we give the following examples.Example 1.
Assume that the spatial domainΩ=0,10π.(1)
Let(44)α11=α12=α21=α22=α23=α32=1,α13=1.2,α31=3,α33=4,a=0.5,b=1,g=1.5,s=3,e=2,r=4,d1=3,d2=3,d3=2.Then, the equilibrium pointu˜,v˜,w˜=0.725,0.326,0.388, γ1=5.809>0, γ2=10.163>0, γ3=5.082>0, and γ1γ2−γ3=53.959>0. From Corollary 1 (4), u˜ is locally asymptotically stable.(2)
Let(45)α11=α12=α21=α22=α23=α31=α32=α33=1,d1=3,d2=3,d3=2,a=0.5,b=1.3,g=1.5,s=3,e=5,r=4,and chooseα13=300. Then, the equilibrium point u˜,v˜,w˜=0.453,0.302,0.184, ag=0.75<1, s=3>f=1.195, and γ3=−0.663<0. Therefore, u˜ is unstable thanks to Corollary 1 (5).(3)
Letαij=0.1,i,j=1,2,3, a=2.8,b=1,g=3,s=3,e=3,r=4.5,d1=3,andd2=3, and choose d3=200. Then, the equilibrium point u˜,v˜,w˜=0.7,0.189,0.2, ag=8.4>1 and γ3=−1.643<0. Thus, u˜ is unstable due to Corollary 1 (6).


## 3. The Uniform Boundedness and the Global Existence of Solutions for (6)

In this section, the uniform boundedness and the global existence of solutions for (6) are investigated when the space dimension is one. For simplicity, let Ω=0,1, denote ⋅k,p=⋅Wpk0,1,⋅p=⋅Lp0,1,andQT=Ω×0,T. From the consequence of a series of important papers [24–26] by Amann, we have, if u0,v0,w0∈Wp1Ω with p>1, then (6) has a unique nonnegative solution u,v,w∈C0,T,Wp1Ω∩C∞0,T,C∞Ω, where T≤+∞ is the maximal existence time for the local solution. If the solution u,v,w satisfies the estimates(46)supu⋅,twp1Ω,v⋅,twp1Ω,w⋅,twp1Ω:0<t<T<∞,then T=+∞. Moreover, if u0,v0,w0∈Wp20,1, then u,v,w∈C0,∞,Wp20,1.In order to establish the timing uniformW21−estimate of the solution for system (6), the following corollaries to the Gagliardo–Nirenberg-type inequality play important roles.Lemma 2.
There exists a universal constantC such that(47)u2≤Cux21/3u12/3+u1,∀u∈W210,1,(48)u4≤Cux21/2u11/2+u1,∀u∈W210,1,(49)u5/2≤Cux22/5u13/5+u1,∀u∈W210,1,(50)ux2≤Cuxx23/5u12/5+u1,∀u∈W220,1.Theorem 5.
Letu0,v0,w0∈W220,1 and u,v,w be the unique nonnegative solution for (6) in the maximal existence interval 0,T. Assume that(H1)
ab<1,b<g.(H2)
8α11α21α31>α21α132+α122α31,8α12α22α32>α32α212+α232α12,8α13α23α33>α23α312+α322α13.Then, there existt0>0 and positive constants M,M′ which depend only on di,αiji,j=1,2,3,a,b,g,s,r,e, such that(51)supu⋅,t1,2,v⋅,t1,2,w⋅,t1,2:t∈t0,T≤M′,(52)maxux,t,vx,t,wx,t:x,t∈0,1×t0,T≤M,and T=∞. Moreover, if di≥1i=1,2,3 and d2/d1,d3/d1∈d¯,d¯, where d¯ and d¯ are positive constants, then M′andM depend on d¯andd¯ but do not on d1,d2,andd3.In this section, we always denote thatC is Sobolev embedding constant or other kind of universal constant, Aj,Bj,andCj are some positive constants which depend only on αiji,j=1,2,3,a,b,g,s,r,e, and Kj are positive constants depending on di and αiji,j=1,2,3,a,b,g,s,r,e. When d1,d2,d3≥1, d1/d2,d3/d2∈d¯,d¯, Kj depend on d¯andd¯ but do not on d1,d2,andd3.Proof.
By the maximum principle, one can obtain that the solutions of (6) with nonnegative initial values are always nonnegative. We will give the W21-estimates of the solution u,v,w for (6) next.
Firstly, we establishL1-estimates of the solution u,v,w for (6). Taking integrations of the first three equations in (6) over the domain [0, 1], respectively, and then combining the three integration equalities linearly, we have(53)ddt∫01u+ρv+1ewdx≤C1−l∫01u+ρv+1ewdx,where C1=s2/erΩ,ρ=1/2maxa,1/g+1/b and l=min1−ρb,ρg−1/ρ,s. It follows from (H1) that there exists a positive constant τ0 such that(54)∫01udx,∫01vdx,∫01wdx≤M0,t>τ0,where M0=C1/lmax1,1/ρ,e. Moreover, there exists a positive constant M0′ which depends on a,b,g,s,r,e and the L1−norm of u0,v0, and w0, such that(55)∫01udx,∫01vdx,∫01wdx≤M0′,t≥0.
Secondly, we will obtainL2-estimates of u,v,w. We multiply the first three equations in (6) by u,v,andw, respectively, and integrate over [0, 1] to have(56)12ddt∫01u2dx≤−d1∫01ux2dx−∫01p11ux2dx−∫01p12uxvxdx−∫01p13uxwxdx+∫01uv+au2v+u2wdx,12ddt∫01v2dx≤−d2∫01vx2dx−∫01p22vx2dx−∫01p21uxvxdx−∫01p23vxwxdx+b∫01uvdx,12ddt∫01w2dx≤−d3∫01wx2dx−∫01p33wx2dx−∫01p31uxwxdx−∫01p32vxwxdx+s∫01w2dx,where pij,i,j=1,2,3, are given by (35) for u,v,w. Let d∗=mind1,d2,d3. Then, we obtain that(57)12ddt∫01u2+v2+w2dx≤−d∗∫01ux2+vx2+wx2dx−∫01qux,vx,wxdx+b+1∫01uvdx+a∫01u2vdx+s∫01w2dx,where(58)qux,vx,wx=p11ux2+p22vx2+p33wx2+p12+p21uxvx+p23+p32vxwx+p13+p31uxwx,is a positive semidefinite quadratic form of ux,vx,andwx if H2 holds. So, (H2) implies that(59)12ddt∫01u2+v2+w2dx≤−d∗∫01ux2+vx2+wx2dx+b+1∫01uvdx+a∫01u2vdx+s∫01w2dx.
Now, we proceed in the following two cases:(1)
t≥τ0. Notice by (48) that u44≤Cux22u12+u14≤CM02ux22+M02. Then, by the Young inequality,(60)a∫01u2vdx≤a∫01u42εdx+a∫01ε2v2dx=a2CM022d∗∫01v2dx+d∗2∫01ux2dx+d∗M022.Substituting (60) into (59), we have(61)12ddt∫01u2+v2+w2dx≤−d∗2∫01ux2+vx2+wx2dx+C2∫01u2+v2+w2dx+d∗M022.Inequality (47) implies that u26≤Cux22u14+u16≤CM04ux22+M02. Therefore,(62)−∫01ux2+vx2+wx2dx≤3M02−19CM02∫01u2+v2+w2dx3,which together with (59) infers that(63)12ddt∫01u2+v2+w2dx≤−C3d∗∫01u2+v2+w2dx3+C2∫01u2+v2+w2dx+2d∗M02.This means that there exist positive constantsτ1≥τ0 and M1 depending on a,b,g,s,r,e,dii=1,2,3 such that(64)∫01u2dx,∫01v2dx,∫01w2dx≤M1,t≥τ1.Whend∗≥1, M1 is independent of d∗ since the zero point of the right-hand side in (63) can be estimated by positive constants independent of d∗.(2)
t≥0. Replacing M0 with M0′ and repeating estimates (60)–(63), one can obtain a new inequality which is similar to (64). The coefficients of this new inequality depend not only on dii=1,2,3,a,b,g,s,r,e but also on initial functions u0,v0,w0. Then, there exists a positive constant M1′ depending on a,b,g,s,r,e,dii=1,2,3 and the L2-norm of u0,v0,andw0 such that(65)∫01u2dx,∫01v2dx,∫01w2dx≤M1′,t≥0.
Finally,L2−estimates of ux,vx, and wx will be obtained. We introduce the following scaling:(66)u˜=ud2,v˜=vd2,w˜=wd2,t˜=d1t.
Denotingξ=d2/d1andη=d3/d1 and using u,v,w,andt instead of u˜,v˜,w˜,andt˜, respectively, then system (6) reduces to(67)ut=Pxx+fu,v,w,0<x<1,t>0,vt=Qxx+ku,v,w,0<x<1,t>0,wt=Rxx+hu,v,w,0<x<1,t>0,uxx,t=vxx,t=wxx,t=0,x=0,1,t>0,ux,0=u0x,vx,0=v0x,wx,0=w0x,0<x<1,where P=u+α11ξu2+α12ξuv+α13ξuw,Q=ξv+α21ξuv+α22ξv2+α23ξvw,R=ηw+α31ξuw+α32ξvw+α33ξw2, and fu,v,w=−d1−1u+d1−1v+aξuv+ξuw,ku,v,w=bd1−1u−gd1−1v−ξuv,hu,v,w=wsd1−1−rξw−eξu.
We still divide the subsequent discuss into two cases:(1)
t≥τ1∗=d1τ1 (namely, t≥τ1 in the original scale). It is clear that(68)∫01udx,∫01vdx,∫01wdx≤M0d2−1,∫01u2dx,∫01v2dx,∫01w2dx≤M1d2−2,P1,Q1,R1≤DK1d2−1.Multiplying the first three equations in (67) by Pt,Qt,andRt and integrating them over the domain [0, 1], respectively, then adding up the three integration equalities, we have(69)12y¯′t=−∫01ut2dx−ξ∫01vt2dx−η∫01wt2dx−ξ∫01qut,vt,wtdx+∫011+p11ξutf+p12ξvtf+p13ξwtfdx+ξ∫01p21utk+1+p22vtk+p23wtkdx+∫01p31ξuth+p32ξvth+η+p33ξwthdx,wherey¯=∫01Px2+Qx2+Rx2dx, q is given by (58). It is not hard to verify by (H2) that there exists a positive constant C4 depending only on αiji,j=1,2,3 such that(70)qut,vt,wt≥C4u+v+wut2+vt2+wt2.Therefore,(71)12y¯′t≤−∫01ut2dx−ξ∫01vt2dx−η∫01wt2dx−C4ξ∫01u+v+wut2+vt2+wt2dx+∫011+p11ξutfdx+∫01ξ1+p22vtkdx+∫01η+p33ξwthdx+∫01p12ξvtfdx+∫01p13ξwtfdx+∫01p21ξutkdx+∫01p23ξwtkdx+∫01p31ξuthdx+∫01p32ξvthdx.By the Hӧlder inequality, one can obtain the following estimates:(72)∫01u4dx≤M11/3d1−2/3∫01u5dx2/3,∫01u2v2dx≤M11/3d1−2/3∫01u5dx1/3∫01v5dx1/3,∫01u3dx≤M12/3d1−4/3∫01u5dx1/3,∫01uv2dx≤M12/3d1−4/3∫01v5dx1/3,∫01u3wdx≤M11/3d2−2/3∫01u5dx1/2∫01w5dx1/6,∫01uvwdx≤M12/3d2−4/3∫01u5dx1/6∫01v5dx1/6,∫01u2vwdx≤M11/3d2−2/3∫01u5dx1/3∫01v5dx1/6∫01w5dx1/6.Applying the Young inequality, the Hӧlder inequality, the Gagliardo–Nirenberg inequality, and estimate (68), one can obtain the following estimates for the terms on the right-hand side of (71):(73)−∫01ut2dx≤−12∫01Pxx2dx+∫01f2dx,−ξ∫01vt2dx≤−ξ2∫01Qxx2dx+ξ∫01k2dx,−η∫01wt2dx≤−η2∫01Rxx2dx+η∫01h2dx,∫01f2dx≤2M1d1−2d2−2+a2ξ2M11/3d2−2/3∫01u5dx1/3∫01v5dx1/3+ξ2M11/3d2−2/3∫01u5dx1/3∫01w5dx1/3+2aξd1−1d2−4/3M12/3∫01v5dx1/3+2ξd1−1d2−2/3M12/3∫01u5dx1/6∫01v5dx1/6+2aξ2d2−2/3M11/3∫01u5dx1/3∫01v5dx1/6∫01w5dx1/6,ξ∫01k2dx≤ξd1−2d2−2M1b2+g2+ξ3d2−2/3M11/3∫01u5dx1/3∫01v5dx1/3+ξ2gd1−1d2−4/3M12/3∫01v5dx1/3,η∫01h2dx≤ηs2d1−2d2−2M1+ηd2ξ2d2−2/3M11/3∫01u5dx1/3ηr2ξ2d2−2/3M11/3+ηr2ξ2d2−2/3M11/3∫01u5dx2/3+ηdrξ2d2−2/3M11/3∫01u5dx1/6∫01w5dx1/2.Thus,(74)−∫01ut2dx−ξ∫01vt2dx−η∫01wt2dx≤−12∫01Pxx2dx−ξ2∫01Qxx2dx−η2∫01Rxx2dx+C51+ξ+ηd1−2d2−2M1+C6d2−4/3M12/3ξ1+η∫01u5+v5dx1/3+C7d2−2/3M11/3ξ21+η∫01u5+v5+w5dx2/3,(75)∫01utfdx≤d1−1∫01uutdx+d1−1∫01utvdx+aξ∫01uutvdx+ξ∫01uutwdx≤d1−112ϵ∫01udx+ϵ2∫01uut2dx+d1−112ϵ∫01vdx+ϵ2∫01vut2dx+aξ12ϵ∫01uv2dx+ϵ2∫01uut2dx+ξ12ϵ∫01uw2dx+ϵ2∫01uut2dx≤d1−1d2−1ϵM0+a2ϵM12/3d2−4/3ξ∫01v5dx1/3+ξ2ϵM12/3d2−4/3∫01w5dx1/3+ϵd1−12∫01vut2dx+ϵ2d1−1+aξ+ξ∫01uut2dx,2α11ξ∫01uutfdx≤α11ϵM12/3d1−1d22/3∫01u5dx1/3+∫01v5dx1/3+3α11a+15ϵξ2∫01u5dx+2α11a5ϵξ2∫01v5dx+2α115ϵξ2∫01w5dx+α11ξϵ2d1−1+aξ+ξ∫01uut2dx,2α12ξ∫01vutfdx≤α12ϵM12/3d1−1d2−4/3∫01v5dx1/3+α12a+15ϵξ2∫01u5dx+2α125ϵξ2∫01w5dx+α123a+410ϵξ2∫01v5dx,α13ξ∫01wutfdx≤α13ϵM12/3d1−1d2−4/3∫01w5dx1/3+α13a+1/25ϵξ2∫01u5dx+2aα135ϵξ2∫01v5dx+2α13a+15ϵξ2∫01w5dx+α13d1−12ϵξ∫01vut2dx+ϵ2α13ξd1−1+α13ξ2+1∫01uut2dx,ξ∫01vtkdx≤b+g2ϵM0d1−1d2−1ξ+ξ22ϵM12/3d2−4/3∫01u5dx1/3+bd1−12ϵξ∫01uvt2dx+ϵ2ξgd1−1+ξ∫01vvt2dx,α21ξ∫01uvtkdx≤α21b+g2ϵM12/3d1−1d2−4/3∫01u5dx1/3+2α215ϵξ2∫01u5dx+α2110ϵξ2∫01v5dx+α21bd1−12ϵξ∫01uvt2dx+α21ϵξ2gd1−1+ξ∫01vvt2dx,2α22ξ∫01vvtkdx≤α22b+gϵM12/3d1−1d2−4/3∫01u5dx1/3+α225ϵξ2∫01u5dx+4α225ϵξ2∫01v5dx+α22b+gd1−1ϵξ∫01vvt2dx+α22ϵξ2∫01uvt2dx,α23ξ∫01wvtkdx≤α23b+g2ϵM12/3d1−1d2−4/3∫01w5dx−4/3+α23ϵξ2bd1−1+ξ∫01uvt2dx+α235ϵξ2∫01u5dx+2∫01v5dx+2∫01w5dx+α23gd1−12ϵξ∫01vvt2dx,η∫01wthdx≤sϵd1−1d2−1M0+r2ϵM12/3d2−4/3ηξ∫01w5dx1/3+e2ϵM12/3d2−4/3ηξ∫01w5dx1/3ηϵ2sd1−1+rξ+eξ∫01wwt2dx,α31ξ∫01uwthdx≤α31s2ϵM12/3d1−1d2−4/3ξ∫01u5dx1/3+α31r+2e5ϵξ2∫01u5dx1/3+α313r+e10ϵξ2∫01w5dx1/3+α31ξϵ2sd1−1+rξ+eξ∫01wwt2dx,α32ξ∫01vwthdx≤α32s2ϵM12/3d1−1d2−4/3ξ∫01v5dx1/3+2α32e5ϵξ2∫01u5dx+α32r+2e5ϵξ2∫01v5dx+α32e+3r/25ϵξ2∫01w5dx+α32ϵξ2sd1−1+rξ+eξ∫01wwt2dx,2α33ξ∫01wwthdx≤α33sϵM12/3d1−1d2−4/3ξ∫01w5dx1/3+2α33e5ϵξ2+α335r+3e5ϵξ2∫01w5dx+α33ϵξsd1−1+rξ+eξ∫01wwt2dx,α12ξ∫01uvtfdx≤α12ϵM12/3d1−1d2−4/3∫01u5dx1/3+α125ϵ2a+32ξ2∫01u5dx+aα1210ϵξ2∫01v5dx+α125ϵξ2∫01w5dx+α12ϵξ2d1−1+ξ∫01uvt2dx+α12ϵξ2d1−1+aξ∫01vvt2dx,α12ξ∫01uwtfdx≤α12ξ2ϵM12/3d1−1d2−4/3∫01u5dx1/3+∫01v5dx1/3+3α12a+110ϵξ2∫01u5dx+aα125ϵξ2∫01v5dx+α125ϵξ2∫01w5dx+α12ξϵd1−1+aξ2+ξ2∫01uwt2dx,α21ξ∫01vutkdx≤α21ξ2ϵM12/3d1−1d2−4/3b∫01u5dx1/3+g∫01v5dx1/3+α215ϵξ2∫01u5dx+3α2110ϵξ2∫01v5dx+α21ϵξ2bd1−1+gd1−1+ξ∫01vut2dx,α23ξ∫01vwtkdx≤α23ξ2ϵM12/3d1−1d2−4/3b∫01u5dx1/3+g∫01v5dx1/3+α235ϵξ2∫01u5dx+3α2310ϵξ2∫01v5dx+α23ϵξ2bd1−1+gd1−1+ξ∫01vwt2dx,α31ξ∫01wuthdx≤α31sξ2ϵM12/3d1−1d2−4/3∫01w5dx1/3+α31e5ϵξ2∫01u5dx+α31ξ22ϵr+35e∫01w5dx+α31ϵξ2sd1−1+rξ+eξ∫01wut2dx,α32ξ∫01wvthdx≤α32sξ2ϵM12/3d1−1d2−4/3∫01w5dx1/3+α32e5ϵξ2∫01u5dx+α32ξ22ϵr+35e∫01w5dx+α32ϵξ2sd1−1+rξ+eξ∫01wvt2dx.Therefore,(76)∫011+p11ξutfdx+ξ∫011+p22vtkdx+∫01η+p33ξwthdx+ξ∫01p12vtfdx+ξ∫01p13wtfdx+ξ∫01p21utkdx+ξ∫01p23wtkdx+ξ∫01p31uthdx+ξ∫01p32vthdx≤λϵξ∫01u+v+wut2+vt2+wt2dx+C8ϵM0d1−1d2−21+ξ+η+C9ϵM12/3d2−4/3ξ1+d1−1+ξ+η∫01u5+v5+w5dx1/3+C10ϵξ2∫01u5+v5+w5dx,whereλ is a positive constant. Choose a small enough positive number ε=εdi,αij,i,j=1,2,3,a,b,g,s,r,e such that λε<C4. Combining (74) and (76), one can obtain(77)12y¯′t≤−12∫01Pxx2dx−ξ2∫01Qxx2dx−η2∫01Rxx2dx+B1K2d1−1d2−1+B2K3d1−4/3z1/3+B3K4d1−2/3z2/3+B4K5z,wherez=∫01u5+v5+w5dx,K2=1+ξ+ηM0+M1d1−1d2−1,K3=M12/3ξ1+d1−1+ξ+η,K4=M11/2ξ21+η,andK5=ξ2. Clearly, P≥α11ξu2,Q≥α22ξv2,andR≥α33ξw2. It follows from (49) that P5/25/2≤CPx2P13/2+P15/2, and(78)z≤B5ξ−5/2∫01P5/2+Q5/2+R5/2dx≤B6ξ−5/2K13/2d1−3/2y¯1/2+B6ξ−5/2K15/2d1−5/2,z1/3≤B7ξ−5/6K11/2d1−1/2y¯1/6+B7ξ−5/6K15/6d1−5/6,z2/3≤B8ξ−5/3K1d1−1y¯1/3+B8ξ−5/3K15/3d1−5/3.Moreover, it follows by (50) that Px210/3≤CPxx22P14/3+P110/3≤B9K14/3d1−4/3Pxx22+K12d1−2. So,(79)−ξ2∫01Pxx2dx−12∫01Qxx2dx−η2∫01Rxx2dx≤−B10min1,ξ,ηK1−4/3d14/3y¯5/3+1+ξ+ηK12d1−2.Substituting the above inequality into (80) and then multiplying it by d22, we have(80)12y′t≤−A1min1,ξ,ηK1−4/3d24/3y5/3+A2ξ−5/6K11/2K3d2−1/6y1/6+A3ξ−5/3K1K4d2−1/3y1/3+A4ξ−5/2K13/2K5d2−1/2y1/2+A5K121+ξ+η+K2ξ+K15/6K3ξ−5/6d2−1/6+K15/3K4ξ−5/3d2−1/2+K15/2K5ξ−5/2d2−1/2,wherey=∫01d2Px2+d2Qx2+d2Rx2dx. Iinequality (77) implies that there exist τ2˜>0 and a positive constant M2˜ depending only on a,b,g,s,r,e,d1,d2,d3,αiji,j=1,2,3 such that(81)∫01d2Px2dx,∫01d2Qx2dx,∫01d2Rx2dx≤M2˜,t≥τ2˜.In the case thatd1,d2,d3≥1,d2/d1,d3/d1∈d¯,d¯, the coefficients of inequality (83) can be estimated by some constants depending on d¯andd¯ but doing not on d1,d2,andd3. So, M2˜ depends on αiji,j=1,2,3,a,b,g,r,s,e,d¯,d¯ and is irrelevant to d1,d2,andd3 when d1,d2,d3≥1, d2/d1,d3/d1∈d¯,d¯. Since(82)uxvxwx=J−1PxQxRx,J=PuPvPwQuQvQwRuRvRw,bydetJ>ξη, we have(83)d2ux+d2vx+d2wx≤Ld2Px+d2Qx+d2Rx,0<x<1,t>0,whereL is a constant depending only on ξ,η,αiji,j=1,2,3. After scaling back and contacting estimates (84) and (85), there exist positive constants τ2 and M2 depending on αiji,j=1,2,3,dii=1,2,3,a,b,g,r,s,e such that(84)∫01ux2dx,∫01vx2dx,∫01wx2dx≤M2,t≥τ2.Whend1,d2,d3≥1, d2/d1,d3/d1∈d¯,d¯, M2 depends on d¯andd¯ but do not on d1,d2,d3≥1.(2)
t≥0. Modifying the dependency of the coefficients in inequalities (68)–(84), namely, replacing M0andM1 with M0′andM1′, there exists a positive constant M2′ depending on αiji,j=1,2,3,dii=1,2,3,a,b,g,r,s,e and the W21−norm of u0,v0,w0 such that(85)∫01ux2dx,∫01vx2dx,∫01wx2dx≤M2′,t≥0.
Furthermore, whendi≥1i=1,2,3,ξ,η∈d¯,d¯, M2′ depends on d¯,andd¯ but do not on dii=1,2,3.
Summarizing estimates (54), (64), and (84) and the Sobolev embedding theorem, there exist positive constants MandM′ depending only on αiji,j=1,2,3,dii=1,2,3,a,b,g,r,s,e such that (51) and (54) hold. In particular, MandM′ depend only on αiji,j=1,2,3,a,b,g,r,s,e,d¯,d¯ but do not on dii=1,2,3 when di≥1i=1,2,3,ξ,η∈d¯,d¯.
Similarly, there exists a positive constantM′′ depending on αiji,j=1,2,3,dii=1,2,3,a,b,g,r,s,e and the initial functions u0,v0,andw0 such that(86)u⋅,t1,2,v⋅,t1,2,w⋅,t1,2≤M′′,t≥0.
Furthermore, in the case thatdi≥1i=1,2,3,ξ,η∈d¯,d¯, M′′ depends only on d¯andd¯ but do not on dii=1,2,3. Thus, T=+∞. This completes the proof of Theorem 5.


## 4. Asymptotic Behavior

In this section we will discuss the global asymptotic behavior of the positive equilibrium pointu˜ for (6).Theorem 6.
Assume that all conditions in Theorem5 are satisfied. Furthermore, if ag<1 and(87)4λρu˜v˜w˜d1d2d3>u˜M2λα23v˜+ρα32w˜2d1+2α11M+α12M+α13M+λv˜M2α13u˜+ρα31w˜2d2+α21M+2α22M+α23M+ρw˜M2α12u˜+λα21v˜2d3+α31M+α32M+2α33M,where λ=2−agu˜+g/g2,ρ=1/e, M is given by (58), then the positive equilibrium point u˜=u˜,v˜,w˜ for (6) is globally asymptotically stable.Proof.
Letu,v,w be a solution of (6) with u0,v0,w0≥≡0. From the strong maximum principle for parabolic equations, we can prove that u,v,w>0 for any t>0.
Define(88)Vu,v,w=∫01u−u˜−u˜lnuu˜+λv−v˜−v˜lnvv˜+ρw−w˜−w˜lnww˜dx,where λ=2−agu˜+g/g2andρ=1/e. Obviously, Vu,v,w is nonnegative and Vu,v,w=0 if and only if u=u˜,v=v˜,andw=w˜. The time derivative of Vu,v,w for system (6) satisfies that(89)dVu,v,wdt=−∫01u˜u2d1+p11ux2+λv˜v2d2+p22vx2+ρw˜w2d3+p33wx2+α12u˜u+λα21v˜vuxvx+α13u˜u+ρα31w˜wuxwx+λα23v˜v+ρα32w˜wvxwxdx−∫01vu˜uu−u˜2+bλuv˜vv−v˜2+ρrw−w˜2−gλ+1u˜+au−u˜v−v˜+1−ρeu−u˜w−w˜dx.
It is easy to know that the second integrand in the above equality is positive definite by the choices ofλ and ρ, and the first integrand in the above equality is positive semidefinite if(90)4λρu˜v˜w˜d1+p11d2+p22d3+p33+α12u˜v+λα21v˜uα13u˜w+ρα31w˜uλα23v˜w+ρα32w˜v>ρw˜α12u˜v+λα21v˜u2d3+p33+λv˜α13u˜w+ρα31w˜u2d2+p22+u˜λα23v˜w+ρα32w˜v2d1+p11.
By the maximum-norm estimate in Theorem5, condition (87) implies that (90) holds. Therefore, if the all conditions in Theorem 6 hold, there exists a positive constant δ such that(91)dHu,v,wdt≤−δ∫01u−u˜2+v−v˜2+w−w˜2dx,dHu,v,wdt<0,u,v,w≠E3.
Using integration by parts, the Hölder inequality and (52), one can easily verify that d/dt∫01u−u˜2+v−v˜2+w−w˜2dx is bounded from above. Then, from Lemma 1 and (91), we have(92)u⋅,t−u˜2⟶0,v⋅,t−v˜2⟶0,w⋅,t−w˜2⟶0,t⟶∞.
It follows from the Gagliardo–Nirenberg inequalityu⋅,t∞≤Cu1,21/2u21/2 that(93)u⋅,t−u˜∞⟶0,v⋅,t−v˜∞⟶0,w⋅,t−w˜∞⟶0,t⟶∞.
Namely,u,v,w converges uniformly to u˜,v˜,w˜. By the fact that Vu,v,w is decreasing for t≥0, it is obvious that u˜,v˜,w˜ is globally asymptotically stable. So, the proof of Theorem 6 is completed.Example 2.
The following parameters satisfy all the conditions of Theorem6:(94)α11=2,α12=1,α13=10,α21=32,α22=12,α23=1,α31=10,α32=2,α33=3,a=12,b=12,g=1,s=3,e=1,r=5,d1,d2,d3≫1.Remark 3.
In model (6), if coefficients of reaction functions satisfy the conditions of Theorem 6, the diffusion matrix is positive definite and the self-diffusion coefficients are large enough; then, the model has no nonconstant positive steady state.



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