# Qualitative Analysis of a Three-Species Reaction-Diffusion Model with Modified Leslie-Gower Scheme

Authors: Xiaoni Wang; Gaihui Guo; Jian Li; Mengmeng Du
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory.


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## Body

## 1. Introduction

Three-species reaction-diffusion models with Holling-type II functional response have been a familiar subject for the analysis. Taking more practical factors into consideration, a model with a modified Leslie-Gower scheme is worthy to explore. Leslie-Gower’s scheme indicates that the carrying capacity of the predator is proportional to the population size of the prey. The existing works [1–3] are all about models with this scheme. As a matter of fact, predators prefer to prey on other prey in the event of a shortage of favorite prey, so the research of the modified Leslie-Gower model springs up. Aziz-Alaoui and Okiye [4] focused on a two-dimensional continuous time dynamical system modeling a predator-prey food chain and gave the main result of the boundedness of solutions, the existence of an attracting set, and the global stability of the coexisting interior equilibrium, which was based on a modified version of the Leslie-Gower scheme and Holling-type II scheme. Singh and Gakkhar [5] investigated the stabilization problem of the modified Leslie-Gower type prey-predator model with the Holling-type II functional response. The analysis of models with a modified Leslie-Gower scheme can be also found in [6–10].Nonconstant positive steady states have received increasing attention in recent years, see [11–18] and references therein. Ko and Ryu [19] showed that the predator-prey model with Leslie-Gower functional response had no nonconstant positive solution in homogeneous environment, but the system with a general functional response might have at least one nonconstant positive steady state under some conditions. Zhang and Zhao [20] analyzed a diffusive predator-prey model with toxins under the homogeneous Neumann boundary condition, including the existence and nonexistence of nonconstant positive steady states of this model by considering the effect of large diffusivity. Shen and Wei [21] considered a reaction-diffusion mussel-algae model with state-dependent mussel mortality which involved a positive feedback scheme. Wang and his partners [22] considered a tumor-immune model with diffusion and nonlinear functional response and investigated the effect of diffusion on the existence of nonconstant positive steady states and the steady-state bifurcations. Hu and Li [23] were concerned about a strongly coupled diffusive predator-prey system with a modified Leslie-Gower scheme and established the existence of nonconstant positive steady states. Qiu and Guo [24] analyzed a stationary Leslie-Gower model with diffusion and advection.Motivated by the mentioned above, we consider a three-species reaction-diffusion model with a modified Leslie-Gower and Holling-type II scheme under the homogeneous Neumann boundary condition as follows:(1)∂u∂t=uA−A1u−B1v−C1w1+D1u,t>0,∂v∂t=vB−A2v−B2u,t>0,∂w∂t=wC−C2w1+D2u,t>0,where u and v represent the density of two competitors, respectively, while w stands for the density of the predator who preys on u. A, B, and C are all positive as the intrinsic growth rates, A1 and A2 regard as influencing factors within diverse populations themselves while B1 and B2 are influencing factors between different populations. All of them are nonnegative. C1w/1+D1u and C2w/1+D2u are the modified Leslie-Gower scheme, and C1, C2, D1, and D2 are positive. Applying the following scaling to (1), as well as assuming C1D2/D1C2=1 for simplicity of calculation:
(2)m1=D1C1u,m2=A2D1A1C1v,m3=D1A1C1w,s=A1C1D1t,still using u,v,w,t replace m1,m2,m3,s, the following ODE system can be logically obtained:
(3)∂u∂t=ua−u−α1v−wβ1+u,t>0,∂v∂t=vb−v−α2u,t>0,∂w∂t=wc−wβ2+u,t>0,u0≥0,v0≥0,w0≥0,where a=AD1/A1C1,b=BD1/A1C1,c=CD1/A1C1,α1=B1/A2,α2=B2/A1,β1=1/C1,β2=1/C2.It is clear that0,0,0,a,0,0,0,b,0,0,0,cβ2, and 0,b,cβ2 are nonnegative constant solutions of system (3). a−α1b/1−α1α2,b−α2a/1−α1α2,0 is a semitrivial solution when it satisfies a−α1bb−α2a>0. When aβ1>cβ2, u̇,0,ẇ is a semitrivial solution where
(4)u̇=a−β1−c+β1−a+c2−4cβ2+4aβ12,(5)ẇ=cβ2+u0.System (3) yields that
(6)α1α2−1u2+a−β1−α1b+α1α2β1−cu+β1a−α1β1b−β2c=0.If the following alternative conditions hold:(7)iα1α2>1anda<bα2,(8)iiα1b+cβ2β1<a<bα2,there exists the unique positive equilibrium u∗,v∗,w∗ as
(9)u∗=−a+β1+α1b−α1α2β1+c+Δ2α1α2−1,v∗=b−α2u∗,w∗=cβ2+u∗,where
(10)Δ=a−β1−α1b+α1α2β1−c2−4α1α2−1β1a−α1β1b−β2c.Taking the diffusion into account, the corresponding PDE system can be written as(11)∂u∂t−d1Δu=ua−u−α1v−wβ1+u,t>0,x∈Ω,∂v∂t−d2Δv=vb−v−α2u,t>0,x∈Ω,∂w∂t−d3Δw=wc−wβ2+u,t>0,x∈Ω,∂u∂n=∂v∂n=∂w∂n=0,t>0,x∈∂Ω,ux,0=u0x,vx,0=v0x,wx,0=w0x,x∈Ω¯,whereΩ⊂RN is a smooth bounded domain, n is the outward unit normal vector on ∂Ω, Δ is the Laplace operator, and diffusion coefficients are d1,d2,d3>0.The rest of this paper is arranged as follows. In Section2, the stability of constant solutions for the ODE system is discussed. In Section 3, the stability of constant solutions for the PDE system is studied. In Section 4, we focus on the priori estimates of positive steady states. In the last two sections, we have a discussion about the nonexistence and existence of nonconstant positive steady states under different conditions.


## 2. Stability of Constant Solutions for the ODE System

In this section, we discuss the stability of constant solutions with the condition of their existence for the ODE system.Theorem 1.
For the ODE system (3), let Γ=a,b,c,α1,α2,β1,β2 and 1/β1+u∗≜B.
(i)
0,0,0,a,0,0,0,b,0,0,0,cβ2 and a−α1b/1−α1α2,b−α2a/1−α1α2,0 are all unconditionally unstable(ii)
IfΓ satisfies a<b/α2, then u̇,0,ẇ is unstable; if b<α2a−β1−c holds, u̇,0,ẇ is local asymptotically stable(iii)
IfΓ satisfies a>α1b+cβ2/β1, then 0,b,cβ2 is unstable; if a<α1b+cβ2/β1 holds, 0,b,cβ2 is local asymptotically stable(iv)
IfΓ and B satisfy 2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c<0, then u∗,v∗,w∗ is unstable; if 2u∗−a−c≥0 and c−α1α2u∗≥0 holds, u∗,v∗,w∗ is local asymptotically stableProof.
The Jacobian matrix of the ODE system (3) is
(12)J=a−2u−α1v−β1wβ1+u2−α1u−uβ1+u−α2vb−α2u−2v0w2β2+u20c−2wβ2+u.
Obviously, we can obtain(13)J=a000b000cat 0,0,0 and its corresponding characteristic polynomial is
(14)φλ=λ−aλ−bλ−c=0,
so its eigenvalues areλ1=a>0, λ2=b>0, and λ3=c>0. Therefore, 0,0,0 is unstable to system (3).By the same manner, we know thata,0,0,0,b,0,0,0,cβ2, and a−α1b/1−α1α2,b−α2a/1−α1α2,0 are all unstable to ODE system (3).The Jacobian matrix of the ODE system atu̇,0,ẇ is
(15)J=−u̇+u̇a−u̇β1+u̇−α1u̇−u̇β1+u̇0b−α2u̇0c20−c.The characteristic polynomial is(16)λ−b−α2u̇λ+u̇−u̇a−u̇β1+u̇λ+c+c2u̇β1+u̇=0.When the eigenvalue satisfiesλ1=b−α2u̇>0, it deduces that a<b/α2, so we can see that u̇,0,ẇ is unstable to ODE system (3). When λ1=b−α2u̇<0, we consider that
(17)λ2+u̇−u̇a−u̇β1+u̇+cλ+cu̇−cu̇a−u̇β1+u̇+c2u̇β1+u̇=0.Letp1=u̇−u̇a−u̇/β1+u̇+c,p2=cu̇−cu̇a−u̇/β1+u̇+c2u̇/β1+u̇and take value for u̇,ẇ as (4) and (5), we know that p1=u̇2+aβ1−cβ2+cβ1/β1+u̇, p2=cu̇2u̇+β1+c−a/β1+u̇>0. With the existence condition aβ1>cβ2, p1>0 and p2>0 hold, such that equation (17) has two solutions with negative real parts.Because ofaβ1>cβ2,
(18)λ1=b−α2u̇=b−α2a−β1−c+β1−a+c2−4cβ2+4aβ12<b−α2a−β1−c+β1−a+c22holds, then λ1<0 if b−α2a−β1−c<0. So we can conclude that when b<α2a−β1−c, u̇,0,ẇ is local asymptotically stable to ODE system (3).The Jacobian matrix of the ODE system at0,b,cβ2 is
(19)J=a−α1b−cβ2β100−α2b−b0c20−c.The characteristic polynomial is(20)λ−a+α1b+cβ2β1λ+bλ+c=0.The corresponding eigenvalues are λ1=a−α1b−cβ2/β1,λ2=−b<0,λ3=−c<0. If a>α1b+cβ2/β1, 0,b,cβ2 is unstable. Otherwise, a<α1b+cβ2/β1, 0,b,cβ2 is local asymptotically stable to ODE system (3).The Jacobian matrix of the ODE system atu∗,v∗,w∗ is
(21)J=a11a12a13a21a22a23a31a32a33=a−2u∗−α1b−α2u∗−β1cB2β2+u∗−α1u∗−u∗B−α2b−α2u∗−b+α2u∗0c20−c.The corresponding characteristic polynomial isλ3+A1λ2+A2λ+A3=0, where
(22)A1=2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c,A2=a−2u∗−α1b−α2u∗−β1cB2β2+u∗α2u∗−b−c+c2u∗B+α1α2u∗−cα2u∗−b,A3=cα2u∗−ba−2u∗−α1b−2α2u∗−β1cB2β2+u∗−cu∗B.WhenΓ satisfies 2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c<0, then A1<0, u∗,v∗,w∗ is unstable applying the Hurwitz criterion [25]. When 2u∗−a−c≥0,c−α1α2u∗≥0, we can find A1>0,A2>0,A3>0,A1A2−A3>0. So u∗,v∗,w∗ is local asymptotically stable to ODE system (3).The proof is complete.


## 3. Stability of Constant Solutions for the PDE System

In this section, the stability of the constant solutions with the condition of their existence for the PDE system is discussed.Let0=μ0<μ1<μ2<μ3<⋯ as the eigenvalues of the operator −Δ over Ω under the homogeneous Neumann boundary condition and Eμi be the corresponding eigenspace while φij∣j=1,2,⋯,dimEμi is a set of the orthogonal basis of Eμi, X=U∈C1Ω¯×C1Ω¯×C1Ω¯∣∂nU=0,x∈∂Ω, and Xij=cφij∣c∈R3. Then, X=⊕i=o∞⊕j=1dimEμiXij.Theorem 2.
For the PDE system (11), let Γ=a,b,c,α1,α2,β1,β2 and 1/β1+u∗≜B.
(i)
0,0,0,a,0,0,0,b,0,0,0,cβ2 and a−α1b/1−α1α2,b−α2a/1−α1α2,0 are all unconditionally unstable(ii)
IfΓ satisfies a<b/α2, then u̇,0,ẇ is unstable; if b<α2a−β1−c and d1/d3>a+β2/β1 holds, u̇,0,ẇ is uniformly asymptotically stable(iii)
IfΓ satisfies a>α1b+cβ2/β1, then 0,b,cβ2 is unstable; if a<α1b+cβ2/β1 holds, 0,b,cβ2 is uniformly asymptotically stable(iv)
IfΓ and B satisfy 2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c<0, then u∗,v∗,w∗ is unstable; if 2u∗−a−c≥0 and c−α1α2u∗≥0 holds, u∗,v∗,w∗ is uniformly asymptotically stableProof.
The linearization of (11) at the positive constant solution U∗ can be expressed by Ut=DΔ+GUU∗U where U=u,v,wT,U∗=u∗,v∗,w∗T,D=diagd1,d2,d3 and GUU∗ is the Jacobian matrix at U∗. For each i≥0, ⊕j=1dimEμiXij is invariant under the operator DΔ+GUU∗. And λ is an eigenvalue of DΔ+GUU∗ on ⊕j=1dimEμiXij if and only if λ is an eigenvalue of the matrix −μiD+GUU∗.
The Jacobian matrix of PDE system (11) is
(23)J=a−2u−α1v−β1wβ1+u2−d1μi−α1u−uβ1+u−α2vb−α2u−2v−d2μi0w2β1+u20c−2wβ2+u−d3μi.According to the Theorem1, 0,0,0,a,0,0,0,b,0,0,0,cβ2,a−α1b/1−α1α2,b−α2a/1−α1α2,0 are all unstable to ODE system (3). Hence, there exist the eigenvalue with positive real parts in the PDE system. It means that 0,0,0,a,0,0,0,b,0,0,0,cβ2,a−α1b/1−α1α2,b−α2a/1−α1α2,0 are all unstable to PDE system (11).The Jacobian matrix of the PDE system atu̇,0,ẇ is
(24)J=−u̇+u̇a−u̇β1+u̇−d1μi−α1u̇−u̇β1+u̇0b−α2u̇−d2μi0c20−c−d3μi.The characteristic polynomial is
(25)λ−b−α2u̇−d2μiλ+u̇−u̇a−u̇β1+u̇+d1μiλ+c+d3μi+c2u̇β1+u̇=0.When the eigenvalue satisfiesλ1=b−α2u̇>0, it deduces that a<b/α2, there exists an eigenvalue with positive real part, and u̇,0,ẇ is unstable to PDE system (11).It is clear that eigenvalueλ1μi=b−α2u̇−d2μi<0 as λ1=b−α2u̇<0. Then, we discuss the following equation emphatically:
(26)λ2+u̇−u̇a−u̇β1+u̇+d1μi+c+d3μiλ+u̇−u̇a−u̇β1+u̇+d1μic+d3μi+c2u̇β1+u̇=0.Let(27)p3=u̇−u̇a−u̇β1+u̇+d1μi+c+d3μi=d1+d3μi+p1,p4=u̇−u̇a−u̇β1+u̇+d1μic+d3μi+c2u̇β1+u̇=d1d3μi2+cd1+u̇−u̇a−u̇β1+u̇d3μi+p2.It shows thatp3>0 on account of p1>0. When cd1+u̇−u̇a−u̇/β1+u̇d3>0, we know p4>0 holds. So the eigenvalues all have negative real parts.The Jacobian matrix of PDE system (11) at 0,b,cβ2 can be written as
(28)a−α1b−cβ2β1−d1μi00−α2b−b−d2μi0c20−c−d3μi.The characteristic polynomial is(29)λ−a+α1b+cβ2β1+d1μiλ+b+d2μiλ+c+d3μi=0.The corresponding eigenvalues are λ1μi=a−α1b−cβ2/β1−d1μi,λ2μi=−b−d2μi<0 and λ3μi=−c−d3μi<0. If a>α1b+cβ2/β1, there exists an eigenvalue with positive real part; 0,b,cβ2 is unstable to PDE system (11). On the contrary, if a<α1b+cβ2/β1, the eigenvalues all have negative real parts.The Jacobian matrix of the PDE system atu∗,v∗,w∗ is
(30)GUU∗=a−2u∗−α1b−α2u∗−β1cB2β2+u∗−d1μi−α1u∗−u∗B−α2b−α2u∗−b+α2u∗−d2μi0c20−c−d3μi.Its characteristic polynomial isλ3+A1μiλ2+A2μiλ+A3μi=0, where
(31)A1μi=d1+d2+d3μi+A1,A2μi=d1d2+d1d3+d2d3μi2−d1a22+d2a11+d1a33+d3a11+d2a33+d3a22μi+A2,A3μi=d1d2d3μi3−d1d2a33+d1d3a22+d2d3a11μi2+d3a11a22+d2a11a33+d1a22a33−d3a12a21−d2a13a31μi+A3.When2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c<0, there exists an eigenvalue with positive real part; u∗,v∗,w∗ is unstable to PDE system (11).WhenA1>0 and d1,d2,d3>0, A1μi>0 holds. Similarly, A2μi>0 since A2>0 and d1,d2,d3>0. If 2u∗−a−c≥0,c−α1α2u∗≥0, we have d3a11a22+d2a11a33+d1a22a33−d3a12a21−d2a13a31>0 and d1d2a33+d1d3a22+d2d3a11<0. As a result of A3>0 and d1,d2,d3>0, A3μi>0 can be obtained. What is more, A1A2−A3>0 leads to A1μiA2μi−A3μi>0. Thus, the eigenvalues all have negative real parts.In the following, we shall prove that there exists a positive constantκ when the corresponding eigenvalues all have negative real parts, such that
(32)Reλ1μi,Reλ2μi,Reλ3μi<−κ,foralli≥1.Letλ=μiζ, then
(33)ψiλ=μi3ζ3+A1μiμi2ζ2+A2μiμiζ+A3μi≜ψ~iζ.Sinceμi→∞ as i→∞, it follows that
(34)limi→∞ψ~iζμi3=ζ3+d2+d2+d3ζ2+d1d2+d1d3+d2d3ζ+d1d2d3≜ψ¯ζ.Applying the Hurwitz criterion, the three rootsζ1,ζ2,ζ3 of ψ¯ζ=0 all have negative real parts. Thus, there exists a positive constant κ′ such that Reζ1,Reζ1,Reζ1≤−κ′. By continuity, there exists i0 such that the three roots ζi1,ζi2,ζi3 of ψ¯ζ=0 satisfy
(35)Reζi1,Reζi2,Reζi3≤−κ′2,foralli≥i0.Hence,Reλ1μi,Reλ2μi,Reλ3μi≤−μiκ′/2≤−κ′/2foralli≥i0.Let−κ′′=max1≤i≤i0Reλ1μi,Reλ2μi,Reλ3μi,κ=minκ′,κ′′. Then, for i≥1,
(36)Reλ1μi,Reλ2μi,Reλ3μi<−κ.Therefore, the constant solutions are uniformly asymptotically stable when the corresponding eigenvalues all have negative real parts.The proof is complete.


## 4. A Priori Estimates of Positive Steady States

The corresponding steady-state problem of system (11) is
(37)−d1Δu=ua−u−α1v−wβ1+u,x∈Ω,−d2Δv=vb−v−α2u,x∈Ω,−d3Δw=wc−wβ2+u,x∈Ω,∂u∂n=∂v∂n=∂w∂n=0,x∈∂Ω.Two lemmas are listed here for the preliminary.Lemma 3.
(Harnack inequality [26]).
Letωx∈C2Ω∩C1Ω¯ be a positive solution to Δω+cxω=0,x∈Ω, where cx∈CΩ¯, satisfying the homogeneous Neumann boundary condition. Then, there exists a positive constant C∗=C∗N,Ω,c∞ such that
(38)maxΩ¯ω≤C∗minΩ¯ω.Lemma 4.
(maximum principle [27]).
Suppose thatg∈CΩ×R1 and bj∈CΩ¯,j=1,2,⋯,N.(i)
ifωx∈C2Ω∩C1Ω¯ satisfies(39)Δω+∑j=1Nbjxωxj+gx,ωx≥0,x∈Ω,∂ω∂ν≤0,x∈∂Ω,and ωx0=maxΩ¯ωx, then gx0,ωx0≥0.(ii)
ifωx∈C2Ω∩C1Ω¯ satisfies(40)Δω+∑j=1Nbjxωxj+gx,ωx≤0,x∈Ω,∂ω∂ν≥0,x∈∂Ω,and ωx0=minΩ¯ωx, then gx0,ωx0≤0.The results of upper and lower bounds can be stated as follows.Theorem 5.
(upper bounds).
Assuming thatu,v,w is a positive solution of system (37), we get
(41)maxΩ¯u≤a,(42)maxΩ¯v≤b,(43)maxΩ¯w≤cβ2+a.Proof.
Sinceua−u−α1v−w/β1+u≤ua−u and vb−v−α2u≤vb−v, such that maxΩ¯u≤a,maxΩ¯v≤b according to Lemma 4. Because of maxΩ¯u≤a, it is evident that maxΩ¯w≤cβ2+a.
The proof is complete.Theorem 6.
(lower bounds).
FixΓ and d¯1,d¯2,d¯3 as positive constants. Assume that
(44)d1,d2,d3∈d¯1,∞×d¯2,∞×d¯3,∞,then there exists a positive constant C¯=C¯Γ,Ω,N,d¯1,d¯2,d¯3 who can make every positive solution u,v,w of system (37) satisfy
(45)minΩ¯ux>C¯,(46)minΩ¯vx>C¯,(47)minΩ¯wx>C¯.Proof.
Let(48)c1x=d1−1a−u−α1v−wβ1+u,c2x=d2−1b−v−α2u,c3x=d3−1c−wβ2+u.
In view of (41), (42), and (43), a positive constant C¯=C¯Ω,N,D¯,Γ can be easily found, such that
(49)c1x∞,c2x∞,c3x∞≤C¯,where d1,d2,d3>D¯. Thus, u, v, and w satisfy that
(50)Δu+c1xu=0,x∈Ω,∂u∂n=0,x∈∂Ω,Δv+c2xv=0,x∈Ω,∂v∂n=0,x∈∂Ω,Δw+c3xw=0,x∈Ω,∂w∂n=0,x∈∂Ω.
According to the Harnack inequality in Lemma3, there must be a positive constant C∗=C∗Ω,N,D¯,Γ, such that
(51)maxΩ¯u≤C∗minΩ¯u,maxΩ¯v≤C∗minΩ¯v,maxΩ¯w≤C∗minΩ¯w.
Suppose that (45), (46), and (47) hold of no account.
There must be a sequenced1i,d2i,d3ii=1∞ with d1i,d2i,d3i∈d¯1,∞×d¯2,∞×d¯3,∞, such that the corresponding positive solutions ui,vi,wi of system (37) reach the qualification
(52)maxΩ¯ui→0,ormaxΩ¯vi→0ormaxΩ¯wi→0i→∞.
Then, we applyui,vi,wi to the system of (37) and integrate by parts, so we obtain that
(53)∫Ωuia−ui−α1vi−wiβ1+uidx=0,∫Ωvib−vi−α2uidx=0,∫Ωwic−wiβ2+uidx=0.
There exists a subsequence ofd1i,d2i,d3ii=1∞ according to the Lp-regularity theory and Sobolev embedding theorem, but we still use d1i,d2i,d3ii=1∞ to represent for convenience. So there must be u∗,v∗,w∗ and d1¯,d2¯,d3¯ as the limiting of ui,vi,wi and d1i,d2i,d3i when i→∞. They can be written as follows:
(54)ui,vi,wi→u∗,v∗,w∗∈C2Ω×C2Ω×C2Ω,d1i,d2i,d3i→d1¯,d2¯,d3¯∈d¯1,∞×d¯2,∞×d¯3,∞.
Leti→∞, we get that
(55)∫Ωu∗a−u∗−α1v∗−w∗β1+u∗dx=0,∫Ωv∗b−v∗−α2u∗dx=0,∫Ωw∗c−w∗β2+u∗dx=0.
We now discuss the following three cases.Case 1.
u∗≡0. Since vi→v∗ as i→∞, b−vi−α2ui>0,x∈Ω¯ holds for every i≫1, so that
(56)∫Ωvib−vi−α2uidx>0,which contradicts with (55).Case 2.
v∗≡0,u∗≠0. Since ui→u∗ as i→∞, a−ui−α1vi−wi/β1+ui>0,x∈Ω¯ holds for every i≫1, so that
(57)∫Ωuia−ui−α1vi−wiβ1+uidx>0,which contradicts with (55).Case 3.
w∗≡0,u∗≠0,v∗≠0. Since wi→w∗ as i→∞, c−wi/β2+ui>0,x∈Ω¯ holds for every i≫1, so that
(58)∫Ωwic−wiβ2+uidx>0,which contradicts with (55).The proof is complete.


## 5. Nonexistence of Nonconstant Positive Steady States

We prove the nonexistence of nonconstant positive steady states of system (37) in this section.Theorem 7.
Letμ1 is the smallest positive eigenvalue of operator −Δ over Ω under the homogeneous Neumann boundary conditions and fixed positive constants d2∗,d3∗ satisfy μ1d2∗>b and μ1d3∗>c+1, then there exists a positive constant D1=D1Γ,d2∗,d3∗ such that when d1>D1, d2≥d2∗ and d3≥d3∗, system (37) has no nonconstant positive steady states.Proof.
Assume thatu,v,w is the positive solution of (37). For any ϕ∈L1Ω, let ϕ¯=1/∣Ω∣∫Ωϕdx. The differential equation (37) multiplies u−u¯,v−v¯,w−w¯ and integrates by parts over Ω to get
(59)∫Ωd1∇u2dx=∫Ωua-u-α1v-wβ1+u-u-a-u--α1v--w-β1+u-u--u-dx=∫Ωau-u-2-u+u-u-u-2-α1uv-u-v-u-u--uwβ1+u-u-u--uw-β1+uu-u-β1+uβ1+u-dx,(60)∫Ωd2∇v2dx=∫Ωvb-v-α2u-v-b-v--α2u-⋅v-v-dx=∫Ωbv-v-2-α2uv-u-v-v-v--v+v-v-v-2dx,(61)∫Ωd3∇w2dx=∫Ω∫Ωwc-wβ2+u-w-c-w-β2+u-⋅w-w-dx=∫Ωcw-w-2-w2β2+u-w-w--w-2β2+uw-w-β2+uβ2+u-dx.
Combine (59), (60), and (61), we have
(62)∫Ωd1∇u2+d2∇v2+d3∇w2dx≤∫Ωau-u-2+bv-v-2+cw-w-2+α1u-+α2v-u-u-v-v-+1+w-2β22u-u-w-w-dx≤∫Ωa+α1a+α2b2ε1+1+c2β2+a22ε2β22⋅u-u-2dx+∫Ωb+ε1α1a+α2b2v-v-2dx+∫Ωc+1+ε2c2β2+a22β22w-w-2dx,where ε1,ε2 are the arbitrary small positive constants arising from Young inequality. Meanwhile, applying the Poincaré inequality μ1∫Ωf−f¯2dx≤∫Ω∇f2dx, we gain that
(63)μ1∫Ωd1u−u¯2+d2v−v¯2+d3w−w¯2dx≤∫Ωa+1+C1∗ε1,ε2u−u¯2dx+∫Ωb+ε1α1a+α2b2v−v¯2dx+∫Ωc+1+ε2c2β2+a22β22w−w¯2dx
for some positive constantsC1∗ε1,ε2. Choose ε1,ε2>0 very small such that
(64)μ1d2∗≥b+ε1α1a+α2b2,(65)μ1d3∗≥c+1+ε2c2β2+a22β22.
Hence, (65) implies that v=v¯=constant, w=w¯=constant, and u=u¯=constant if d1>D1≜μ1−1a+1+C1∗ε1,ε2.
The proof is complete.


## 6. Existence of Nonconstant Positive Steady States

In this part, we discuss the existence of nonconstant positive solutions of (37) by using the degree theorem.Fix theΓ,d1,d3 still as positive number and define X+=U∈X∣U>0,x∈Ω¯,i=1,2,3,Bl=U∈X∣l−1<u,v,w<l,x∈Ω¯,l>0. Then, (37) can be noted as
(66)−DΔU=GU,x∈Ω,∂U∂n=0,x∈∂Ω.SoU is a positive solution to (37) if and only if
(67)FU≜U−I−Δ−1D−1GU+U=0,U∈X+,where I−Δ−1 is the inverse of I−Δ in X under the homogeneous Neumann boundary condition. And if FU≠0 on ∂B, the Leray-Schauder degree degF·,0,B can be well defined. Besides, we note that
(68)DUFU∗≜I−I−Δ−1D−1GUU∗+I.The index ofFU at U∗ can be either 1 or -1 if DUFU∗ is invertible, which is defined as indexF·,U∗=−1r, where r is the total number of eigenvalues with negative real parts of DUFU∗.Letλ be an eigenvalue of DUFU∗ on Xij for each integer i≥1 and each integer 1≤j≤dimEμi, if and only if it is an eigenvalue of the matrix
(69)I−11+μiD−1GUU∗+I=11+μiμiI−D−1GUU∗.Hence,DUFU∗ is invertible if and only if, for all i≥1,i∈Z, the matrix I−1/1+μiD−1GUU∗+I is nonsingular. Let
(70)Hμ=HU∗;μ=detμI−D−1GUU∗=1d1d2d3detμD−GUU∗.We can know that ifHμi≠0, the number of negative eigenvalues of DUFU∗ on Xij is odd if and only if Hμi<0 for every 1≤j≤dimEμi. According to this, we can form the following result.Proposition 8.
Assume that the matrixμiI−D−1GUU∗ is nonsingular for all i≥1, then
(71)indexF·,U∗=−1σ,
whereσ=∑i≥1,Hμi<0dimEμi.For calculating the sign ofHμi, we firstly consider the index of F·,U∗. The calculation shows that
(72)detμD−GUU∗=Φ3d2μ3+Φ2d2μ2+Φ1d2μ−detGUU∗≜Φd2;μ,with
(73)Φ1d2=d3a11a22+d2a11a33+d1a22a33−d3a12a21−d2a13a31,Φ2d2=−d2d3a11+d1d3a22+d1d2a33,Φ3d2=d1d2d3,where aij are shown as (21).Consider the dependence ofΦ on d2. Let μ¯1d2,μ¯2d2, and μ¯3d2 be the three roots of Φd2;μ=0, so that μ¯1d2μ¯2d2μ¯3d2=detGUU∗/Φ3d2. The computation leads to detGUU∗<0. Therefore, one of μ¯1d2,μ¯2d2,μ¯3d2 is real and negative, and the product of the other two is positive.Considering the following limits:(74)limd2→∞Φ3d2d2=d1d3,limd2→∞Φ2d2d2=−d1a33+d3a11,limd2→∞Φ1d2d2=a11a33−a13a31,limd2→∞Φd2d2=μd1d3μ2−d1a33+d3a11μ+a11a33−a13a31.We establish the following result.Proposition 9.
Assume the parameters satisfy (7) or (8) and satisfy 2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c<0. If a11a33−a13a31<0, there is a positive constant D2, such that when d2≥D2, the three roots μ¯1d2,μ¯2d2,μ¯3d2 of Φd2;μ=0 are all real and satisfy
(75)limd2→∞μ¯1d2=d1a33+d3a11−d1a33+d3a112+4d1d3a13a31−a11a332d1d3<0,limd2→∞μ¯2d2=0,limd2→∞μ¯3d2=d1a33+d3a11+d1a33+d3a112+4d1d3a13a31−a11a332d1d3≜μ~>0,(76)−∞<μ¯1d2<0<μ¯2d2<μ¯3d2,(77)Φd2;μ<0,μ∈−∞,μ¯1d2∪μ¯2d2,μ¯3d2,(78)Φd2;μ>0,μ∈μ¯1d2,μ¯2d2∪μ¯3d2,+∞.
Now, we prove the existence of nonconstant positive solutions of (37) when d2 is sufficiently large.Theorem 10.
Let the parametersd1,d3 are fixed, Γ satisfies (7) or (8), and satisfies 2u∗−a+α1+1b−α2u∗+β1cB2β2+u∗+c<0. If a11a33−a13a31<0, μ~∈μn,μn+1 for some n≥1, and the sum σn=∑i=1ndimEμi is odd. Then, D2 must be as a positive constant such that (37) has one nonconstant positive solution at least if d2≥D2.Proof.
There exists a positive constantD2 by Proposition 9, such that for d2≥D2, (76), (77), and (78) hold and
(79)0=μ0<μ¯2d2<μ1,μ¯3d2∈μn,μn+1.
We will testify that for anyd2≥D2, system (37) has at least one nonconstant positive solution and the proof is proved by contradiction. Assume on the contrary that the statement is not true for some d¯2≥D2. Afterwards, we fix d2=d¯2, d1∗=C1∗/μ1i, d2∗=C2∗/μ1i, d3∗=C3∗/μ1i, and
(80)d^1≥maxd1∗,d1,d^2≥d2∗,d^3≥maxd3∗,d3.
As fort∈0,1, make Dt=diagd1t,d2t,d3t with dit=tdi+1−td^i,i=1,2,3 and think about the problem
(81)−DtΔU=GU,x∈Ω,∂U∂n=0,x∈∂Ω.
U is a nonconstant positive solution of (37) if and only if it is a positive solution of (81) when t=1. Obviously for any 0≤t≤1, U∗ is the unique constant positive solution of (81). U is a positive solution of (81) if and only if
(82)Ft;U≜U−I−Δ−1D−1tGU+U=0,U∈X+.
It is evident thatF1;U=FU. F0;U=0 has been shown in Theorem 7, which has only positive solution U∗ in X+. After computing, we get that
(83)DUFt;U∗=I−I−Δ−1D−1tGUU∗+I.
Specifically,(84)DUF0;U∗=I−I−Δ−1D∧−1GUU∗+I,DUF1;U∗=I−I−Δ−1D−1GUU∗+I=DUFU∗,where D^=diagd^1,d^2,d^3. From (70) and (72), we know that
(85)Hμ=1d1d2d3Φd2;μ.
In view of (76) - (79), and (85), it follows that
(86)Hμ0=H0>0,Hμi<0,1≤i≤n,Hμi>0,i≥n+1.
Thus,0 is not an eigenvalue of the matrix μiI−D−1GUU∗ for any i≥1, and
(87)∑i≥0,Hμi<0dimEμi=∑i=1ndimEμi,which is odd. Because of Proposition 8, it can be true that
(88)indexF1;·,U∗=−1r=−1σn=−1.
The same method is available to indexF0;·,U∗=−10=1.
According to Theorems5 and 6, we can find a positive constant C, such that the positive solutions of (81) can meet the demand C−1<u,v,w<C for all ∀0≤t≤1. So, Ft;U≠0 on ∂BC. By using the homotopy invariance of the topological degree, it is clear that
(89)degF1;·,0,BC=degF0;·,0,BC.
Moreover, by our assumption, both equationsF1;·=0 and F0;·=0 have only the positive solution U∗ in BC, so
(90)degF0;·,0,BC=indexF0;·,U∗=1,degF1;·,0,BC=indexF1;·,U∗=−1,
which is contradictory with (89).
The proof is complete.



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*Source: 6650783-2021-05-04.xml*