# Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species

Authors: Guohong Zhang; Xiaoli Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show
that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly
unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.


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## Body

## 1. Introduction

The study of the dynamic relationship between predator and prey has long been one of the most important themes in population dynamics because of its universal existence in nature and many different phenomena have been observed (see [1–13] and references therein). At the same time, since species need to interact with the environment, they are always subject to diseases in the natural world. So it is necessary and interesting to combine demographic as well as epidemic aspects in the standard classical population models. This type of systems is now known as ecoepidemic model.In fact, the importance of disease influence on the dynamics of plant as well as animal populations has been recognized and several such studies are reviewed in a number of recent publications. However, most of the previous researches on ecoepidemic models assume that the distribution of the predators and prey is homogeneous, which leads to the ODE system (see [14–23] and references therein). As we know, both predators and prey have the natural tendency to diffuse to areas of smaller population concentration. At the same time, some prey species always congregate and form a huge group to protect themselves from the attack of infected predator. So it is important to take into account the inhomogeneous distribution of the predators and prey within a fixed bounded domain Ω and consider the effect of diffusion and cross-diffusion.In order to construct the corresponding reaction-diffusion type model, we first propose the following assumptions, which are proper in biological background.(H1)
The disease spreads among the predator species only by contact and the disease incidence follows the simple law of mass action.(H2)
In the absence of predators, the prey populationu1 grows logistically with the intrinsic growth rate r>0 and carrying capacity r/B, in which B measures intraspecific competition of the prey.(H3)
The sound predator populationu2 has no other food sources, and μ>0 represents natural mortality. The infected predator population u3 cannot recover and their total death rate d>0 encompasses natural and disease-related mortality. The conversion factor of a consumed prey into a sound or infected predator is 0<e<1.(H4)
The sound and infected predators hunt the prey with different searching efficiencies, denoted, respectively, bym and pm, with 0<p<1. This is due to the fact that sound predators are more efficient to catch the prey than the infected ones, weakened by the infection.(H5)
Both predators and prey have the natural tendency to diffuse to areas of smaller population concentration and the natural dispersive forces of movements of the prey, sound predators, and infected predators ared1, d2, and d3, respectively.(H6)
The prey species congregate and form a huge group to protect themselves from the attack of infected predator.With the above assumptions, our model takes the following form, in which all parameters are assumed to be positive:(1)∂u1∂t-d1Δu1=ru1-Bu12-mu1u2-pmu1u3inΩ×(0,∞),∂u2∂t-d2Δu2=emu1u2-au2u3-μu2inΩ×(0,∞),∂u3∂t-d3Δ(u3+d4u1u3)=au2u3+empu1u3-δu3inΩ×(0,∞),∂νu1=∂νu2=∂νu3=0on⁡∂Ω×(0,∞),(u1(0,x),u2(0,x),u3(0,x))≥(0,0,0)on⁡Ω,
where Ω is a bounded domain in RN  (N≥1 is an integer) with a smooth boundary ∂Ω and ν is the outward unit rector on ∂Ω. The homogeneous Neumann boundary condition indicates that there is zero population flux across the boundary. In the diffusion terms, the constant di  (i=1,2,3), which is usually termed self-diffusion coefficient, represents the natural dispersive force of movement of an individual. The constant d3d4 could be referred to as cross-diffusion pressure, which describes a mutual interference between individuals.In fact, it is easy to see that the infected predatoru3 diffuses with flux:
(2)J=-∇(d3u3+d3d4u1u3)=-d3d4u3∇u1-(d3+d3d4u1)∇u3.
As d3d4u3<0, the part -d3d4u3∇u1 of the flux is directed toward the decreasing population density of the prey u1, which means that the prey species congregate and form a huge group to protect themselves from the attack of infected predator. We remark that this kind of nonlinear diffusion was first introduced by Shigesada et al. [24] and has been used in different type of population models [25–28]. We also point out that the corresponding ODE system of (1) with delay has been studied by [29], and they mainly investigate the stability and bifurcations related to the two most important equilibria of the ecoepidemic system, namely, the endemic equilibrium and the disease-free one.Since the first example of stationary patterns in a predator-prey system arising solely from the effect of cross-diffusion is introduced by Pang and Wang [30], recently, more attention has been given to investigate the effect of cross-diffusion in reaction-diffusion systems; see, for example, [31–36] and references therein. Here we point out that, to our knowledge, there is little work about ecoepidemic models with diffusion and cross-diffusion was discussed.In our work here, one of the main purposes is to study the existence of positive stationary solutions of (1) by using degree theory, which are the positive solutions of
(3)-d1Δu1=ru1-Bu12-mu1u2-pmu1u3inΩ,-d2Δu2=emu1u2-au2u3-μu2inΩ,-d3Δ(u3+d4u1u3)=au2u3+empu1u3-δu3inΩ,∂νu1=∂νu2=∂νu3=0on⁡∂Ω.
Hence we are interested in nonconstant positive solutions of (3), which correspond to coexistence states of prey and predators. For convenience, we denote Λ=(r,B,m,p,e,a,μ,δ). By a direct computation, we can show that (3) has a semitrivial constant steady state u0*=(u01*,u02*,u03*)=(μ/em,(erm-μB)/em2,0) if erm>μB and has a positive constant steady state u*=(u1*,u2*,u3*), where
(4)u1*=ra-m(δ-μp)aB,u2*=δ-empu*a,u3*=emu*-μa,
provided that
(5)ar>m(δ-μp),δemp>ra-m(δ-μp)aB>μem.
Here we remark that the semitrivial constant steady state u0* and the positive constant steady state u* are also called disease-free equilibrium and endemic equilibrium, respectively, in endemic models.The rest of this paper is organized as follows. In Section2, we will investigate the stability of disease-free equilibrium u0* and the endemic equilibrium u* and show that the cross-diffusion destabilizes a uniform equilibrium which is stable for the kinetic and self-diffusion reaction systems. In Section 3, a priori upper bounds and lower bounds for the nonconstant positive solutions of (3) are given. In Section 4, we study nonexistence of nonconstant positive solutions of model (3) when considering only the self-diffusion. Finally, in Section 5, we investigate the existence of the nonconstant positive solutions of (3) by using the Leray-Schauder degree theory, which explains why shrub ecosystem generates patterns.


## 2. Stability Analysis of the Constant Solutionsu0* and u*

In order to study the stability of the constant steady statesu0* and u* of (1), we first set up the following notation.Notation 1.
Consider the following.(i)
0=μ0<μ1<μ2<⋯ are the eigenvalues of -Δ in Ω under homogeneous Neumann boundary condition.(ii)
S(μi) is the set of eigenfunctions corresponding to μi.(iii)
Xij:=cφij:c∈R3, where φij are orthonormal basis of S(μi) for j=1,…,dim⁡[S(μi)].(iv)
X:={(u1,u2,u3)∈[C1(Ω¯)]3:∂u1/∂ν=∂u2/∂ν=∂u3/∂ν=0 on ∂Ω}, and so X=⊕i=1∞⊕j=1dim⁡[S(μi)]Xij.Now, we first consider system (1) without cross-diffusion and introduce the following system:
(6)∂u1∂t-d1Δu1=ru1-Bu12-mu1u2-pmu1u3inΩ×(0,∞),∂u2∂t-d2Δu2=emu1u2-au2u3-μu2inΩ×(0,∞),∂u3∂t-d3Δu3=au2u3+empu1u3-δu3inΩ×(0,∞),∂νu1=∂νu2=∂νu3=0on⁡∂Ω×(0,∞),(u1(0,x),u2(0,x),u3(0,x))≥(0,0,0)on⁡Ω.
Obviously, system (6) has the same disease-free equilibrium u0* and endemic equilibrium u* with system (1). From (6), we can get the following theorem.Theorem 1.
(i) Iferm>μB and (erm+μB)/em2<δ/a, the disease-free equilibrium u0* of system (6) is locally asymptotically stable.
(ii) Assume that (5) holds. The endemic equilibrium u* of system (6) is locally asymptotically stable.Proof.
(i) For simplicity, throughout this paper, we denote(7)G(u)=(G1(u)G2(u)G3(u))=(ru1-Bu12-mu1u2-pmu1u3emu1u2-au2u3+μu2au2u3+empu1u3-δu3).
By a direct calculation, we obtain
(8)Gu(u0*)=(-Bu1*-mu01*-pmu01*emu02*0-au02*00au02*-δ).
The linearization of (6) at u0* can be expressed by
(9)ut=(DΔ+Gu(u0*))u,
where
(10)D=(d1000d2000d3).
According to Notation 1, Xi is invariant under the operator DΔ+Gu(u0*), and λ is an eigenvalue of this operator on Xi if and only if it is an eigenvalue of the matrix -μiD+Gu(u0*).
A direct calculation shows that the characteristic polynomial of-μiD+Gu(u0*) can be given by
(11)ϕ0i(λ)=(λ+δ-au02*+d3μi)×[λ2+(Bu01*+d1μi+d2μi)λ+em2u01*u02*].
It follows from (11) that, if (erm+μB)/em2<δ/a, the corresponding eigenvalues have negative real parts for all i≥1, so we know that u0* is locally asymptotically stable.
(ii) SinceG(u*)=0, it follows from (7) that
(12)Gu(u*)=(-Bu1*-mu1*-pmu1*emu2*0-au2*empu3*au3*0).
The linearization of (6) at u* can be expressed by
(13)ut=(DΔ+Gu(u*))u,
where the matrix D is defined in (10). Direct calculation shows that the characteristic polynomial of -μiD+Gu(u*) is given by
(14)ϕi(λ)=λ3+c1λ2+c2λ+c3,
where
(15)c1=c1(μi)=(d1+d2+d3)μi+Bu1*,c2=c2(μi)=(d1d2+d1d3+d2d3)μi2+(d2+d3)Bu1*μi+(em2u1*u2*+em2p2u1*u3*+a2u2*u3*),c3=c3(μi)=d1d2d3μi3+d2d3Bu1*μi2+(d1a2u2*u3*+d2em2p2u1*u3*+d3em2u1*u2*)μi+Ba2u1*u2*u3*.
It is easy to see that c1, c2, and c3 are positive.
Notice that(16)c1c2-c3=(d2+d3)(d12+d1d2+d1d3+d2d3)μi3+Bu1*(d2+d3)(2d1+d2+d3)μi2+[(Bu1*)2(d2+d3)+d1(em2u1*u2*+em2p2u1*u3*)+d2(em2u1*u2*+a2u2*u3*)+d3(em2p2u1*u3*+a2u2*u3*)]μi+Bu1*(em2u1*u2*+em2p2u1*u3*)>0.
Then by the Routh-Hurwitz criterion, we know that, for each i≥1, all the three roots λi,1, λi,2, and λi,3 of characteristic equation ϕi(λ)=0 have negative real parts. Now we can prove that there exists a positive constant δ such that
(17)Re{λi1},Re{λi1},Re{λi1}≤-δ,i≥1.
In fact, let λ=μiξ; then we have
(18)ϕi(λ)=μi3ξ3+c1μi2ξ2+c2μiξ+c3≜ϕ~i(ξ).
Note that μi↦∞ as i↦∞. It follows that
(19)lim⁡i↦∞ϕ~i(ξ)μi3=ξ3+(d1+d2+d3)ξ2+(d1d2+d1d3+d2d3)ξ+d1d2d3≜ϕ~(ξ).
Using the Routh-Hurwitz criterion again, we can see that all the three roots ξ1, ξ2, and ξ3 of equation ϕ~(ξ)=0 have negative real parts. Thus, there exists a positive constant such that
(20)Re{ξ1},Re{ξ2},Re{ξ3}≤-δ¯.
By continuity, we know that there existsi0∈N such that the three roots ξi1, ξi2, and ξi3 of ϕ~i(ξ)=0 satisfy
(21)Re{ξi1},Re{ξi2},Re{ξi3}≤-δ¯2,i≥i0,
which implies that
(22)Re{λi1},Re{λi2},Re{λi3}≤-μiδ¯2≤-δ¯2,i≥i0.
Let
(23)-δ~=max⁡1≤i≤i0{Re{λi1},Re{λi2},Re{λi3}};
then δ~>0, and (17) holds for δ=min⁡{δ~,δ¯/2}. Thus the proof is completed by Theorem 5.1.1 of Henry [37].Remark 2.
From Theorem1, we can see that if only the free diffusion is introduced to the corresponding ODE system of (1), the uniform positive stationary solution is also locally stable, which means that only self-diffusion cannot induce Turing instability.We now consider the effect of the cross-diffusion and introduce the following theorem, which give the necessary conditions for the existence of nonconstant positive solution of system (3).Theorem 3.
Consider the following.(i)
Iferm>μB and (erm+μB)/em2<δ/a, the disease-free equilibrium u0* of system (1) is locally asymptotically stable.(ii)
Assume that (5) holds and d4>0 in (1). Suppose that b2<0 and b22-4b1b3>0, where bi is given in (31). If μ2*∈(μi,μi+1) and μ3*∈(μj,μj+1) for some j>i>0, where μ2* and μ3* are defined in (34), there exists a positive constant d3* such that the uniform stationary solution u* of (1) is unstable when d3≥d3*.Proof.
(i) For simplicity, we denote thatΦ(u)=(d1u1,d2u2,d3(u3+d4u4))T. Then the linearized system of system (1) at u0* is
(24)ut=(Φu(u0*)Δ+Gu(u0*))u,
where
(25)Φu(u0*)=(d1000d2000d3+d3d4u01*).
By some calculations, the characteristic polynomial of -μiΦu(u0*)+Gu(u0*) can be given by
(26)ψ0i(λ)=(λ+δ-au02*+d3μi+d3d4u01*μi)×(λ2+(Bu01*+d1μi+d2μi)λ+em2u01*u02*).
It is easy to see that, if (erm+μB)/em2<δ/a, all the corresponding eigenvalues of ϕ0i(λ)=0 have negative real parts for all i≥1, which implies that u0* is locally asymptotically stable.
(ii) The linearized system of system (1) at u* is
(27)ut=(Φu(u*)Δ+Gu(u*))u,
where
(28)Φu(u*)=(d1000d20d3d4u3*0d3+d3d4u1*).
By some calculations, the characteristic polynomial of -μiΦu(u*)+Gu(u*) can be given by
(29)Ψi(λ)=λ3+c¯1λ2+c¯2λ+c¯3,
where
(30)c¯1=(d1+d2+d3+d3d4u1*)μi+Bu1*,c¯2=[d1d2+d1d3+d2d3+(d1d3d4+d2d3d4)u1*]μi2+[B(d2+d3+d3d4)(u1*)2-d3d4pmu1*u2*]μi+em2u1*u2*+em2p2u1*u3*+a2u2*u3*,c¯3=d1d2d3(1+d4u1*)μi3+[Bu1*(d2d3+d2d3d4u1*)-d2d3d4pmu1*u3*]μi2+[(d1a2u2*u3*)+d2em2p2u1*u3*+(d3+d3d4u1*)em2u1*u2*+d3d4amu1*u2*u3*]μi+Ba2u1*u2*u3*≜Q3μi3+Q2μi2+Q1μi+Q0.
Let λi,1, λi,2, and λi,3 be the three roots of Ψi(λ)=0; then λi,1λi,2λi,3=-c¯3. In order to have at least one Reλi,j>0  (j=1,2,3), it is sufficient to prove that c¯3<0.
In the following we will find out the conditions such thatc¯3<0. Let Q~(μ)=Q3μ3+Q2μ2+Q1μ+Q0 and let μ~1, μ~2, and μ~3 be the three roots of Q~(μ)=0 with Re(μ~1)≤Re(μ~2)≤Re(μ~3). Notice that Q0>0 and Q3>0. Then μ~1μ~2μ~3=-Q0/Q3<0. Thus, one of the three roots μ~1, μ~2, and μ~3 is real and negative, and the product of the other two is positive.
Consider the following limits:(31)lim⁡d3↦∞Q0d3=0,lim⁡d3↦∞Q1d3=(1+d4u1*)em2u1*u2*+d4amu1*u2*u3*≜b1,lim⁡d3↦∞Q2d3=Bu1*(d2+d2d4u1*)-d2d4pmu1*u3*≜b2,lim⁡d3↦∞Q3d3=d1d2(1+d4u1*)≜b3.
It is easy to see that b1>0 and b3>0.
Note that(32)lim⁡d3↦∞Q~(μ)d3=b3μ3+b2μ2+b1μ=μ(b3μ2+b2μ+b1).
It follows that equation b3μ2+b2μ+b1=0 has two strictly positive solutions when the following conditions hold:
(33)b2<0,b22-4b1b3>0.
By a continuity argument, we know that, when d3 is large enough, μ~1 is real and negative, and μ~2 and μ~3 are real and positive as μ~2μ~3>0. Furthermore, we have
(34)lim⁡d3↦∞μ~1=0,lim⁡d3↦∞μ~2=-b2-b22-4b1b32b3≜μ2*>0,lim⁡d3↦∞μ~3=-b2+b22-4b1b32b3≜μ3*>0.
So there exists a positive number d3* such that, when d3>d3*, the following hold:
(35)-∞<μ~1<0<μ~2<μ~3;Q~(μ)<0ifμ∈(-∞,μ~1)∪(μ~2,μ~3);Q~(μ)>0ifμ∈(μ~1,μ~2)∪(μ~3,∞).
Since μ2*∈(μi,μi+1) and μ3*∈(μj,μj+1) for some j>i>0, we have Q~(μk)<0 when i+1<k<j+1. Thus we know that c¯3<0, and the proof is completed.


## 3. A Priori Estimates to the Positive Solution of (3)

In this section, we will give a priori estimates to the positive solution of (3). Let us first introduce two lemmas and we remark that the first lemma is due to Lou and Ni [38].Lemma 4 (maximum principle).
Suppose thatg∈C(Ω¯×R).(i)
Assume thatw∈C2(Ω)∩C1(Ω¯)and satisfies(36)Δw(x)+g(x,w(x))≥0inΩ,∂νw≤0on∂Ω.Ifw(x0)=max⁡Ω¯w(x), theng(x0,w(x0))≥0.(ii)
Assume thatw∈C2(Ω)∩C1(Ω¯)and satisfies(37)Δw(x)+g(x,w(x))≤0inΩ,∂νw≥0on∂Ω.Ifw(x0)=min⁡Ω¯w(x), theng(x0,w(x0))≤0.Next, we state the second lemma which is due to Lin et al. [39].Lemma 5 (Harnack inequality).
Assume thatc(x)∈C(Ω¯). Let w∈C2(Ω)∩C1(Ω¯) and satisfy
(38)Δw(x)+g(x,w(x))=0inΩ,∂νw=0on∂Ω.
Then there exists a positive constant C, depending only on ∥c(x)∥C(Ω¯) such that
(39)max⁡Ω¯w(x)≤Cmin⁡Ω¯w(x).Our results are the following theorems.Theorem 6 (upper bounds).
Any positive solutionu(x)=(u1(x),u2(x),u3(x))T of (3) satisfies
(40)max⁡Ω¯u1(x)≤M¯1,max⁡Ω¯u2(x)≤M¯2,max⁡Ω¯u3(x)≤M¯3,
where
(41)M¯1=rB,M¯2=erd1Bd2+er2μB,M¯3=erd3B[d1+e+d2r4μ+d3r4δ(1+d4rB)].Proof.
Letx0∈Ω¯ such that u1(x0)=max⁡Ω¯u1(x). Then by Lemma 4, it is clear that
(42)ru1(x0)-Bu12(x0)-mu1(x0)u2(x0)-pmu1(x0)u3(x0)≥0,
and so
(43)u1(x0)=max⁡Ω¯u1(x)≤rB≜M¯1.
Define y(x)=ed1u1(x)+d2u2(x); then y(x) satisfies
(44)-Δy=eru1-eBu12-epmu1u3-au2u3-μu2inΩ,∂νy=0on⁡∂Ω.
Let x1∈Ω¯ such that y(x1)=max⁡Ω¯y(x). Then, by Lemma 4, we can get
(45)eru1(x1)-eBu12(x1)-epmu1(x1)u3(x1)-au2(x1)u3(x1)-μu2(x1)≥0,
which implies
(46)u2(x1)≤1μ(eru1(x1)-eBu12(x1))≤er24μB.
So, by the definition of y(x), we have
(47)max⁡⁡Ω¯u2(x)≤1d2max⁡⁡Ω¯y(x)=1d2y(x1)=ed1d2u1(x1)+u2(x1)≤erd1Bd2+er24μB≜M¯2.
Letω(x)=u3(x)+d4u1(x)u3(x); then u3(x)=ω(x)/(1+d4u1(x)). Define z(x)=ed1u1(x)+d2u2(x)+d3ω(x); then z(x) satisfies
(48)-Δz=eru1(x)-eBu12(x)-μu2(x)-δω(x)1+d4u1(x)inΩ,∂νz=0on⁡∂Ω.
Let x2∈Ω¯ such that z(x2)=max⁡Ω¯z(x). Then, by using Lemma 4 again, we can obtain
(49)eru1(x2)-eBu12(x2)-μu2(x2)-δω(x2)1+d4u1(x2)≥0,
which implies
(50)ω(x2)≤1δ(1+d4u1(x2))(eru1(x2)-eBu12(x2))≤er24δB(1+d4rB).
It follows that
(51)max⁡Ω¯u3(x)≤max⁡Ω¯ω(x)≤1d3max⁡Ω¯z(x)=1d3z(x2)=1d3(ed1u1(x2)+d2u2(x2)+d3ω(x2))≤erd3B[d1+e+d2r4μ+d3r4δ(1+d4rB)]≜M¯3.
Then we obtain the three upper bounds in (40).Theorem 7.
There exist three positive constantsC1 (depending on r/d1, Ω), C2 (depending on emr/d2B,Ω), and C3 (depending on Λ, di, Ω) such that any positive solution u(x)=(u1(x),u2(x),u3(x)) of (3) satisfies
(52)max⁡Ω¯u1(x)min⁡Ω¯u1(x)≤C1,max⁡Ω¯u2(x)min⁡Ω¯u2(x)≤C2,max⁡Ω¯u3(x)min⁡Ω¯u3(x)≤C3,(i=1,2,3).Proof.
It is easy to see thatui(x)  (i=1,2) satisfies
(53)Δui(x)+ci(x)diui(x)=0inΩ,∂νui(x)=0on⁡∂Ω,
where c1(x)=r-Bu1-mu1u2-pmu1u3 and c2(x)=emu1-au3-μ. By (40), we know that
(54)∥c1(x)d1∥C(Ω¯)≤rd1,∥c2(x)d2∥C(Ω¯)≤emrd2B.
So by Lemma 5, we know that the first two inequalities of (52) hold. Define φ(x)=d3u3(x)+d3d4u1(x)u3(x); we have
(55)Δφ(x)+c3(x)φ(x)=0inΩ,∂νφ(x)=0on⁡∂Ω,
where c3(x)=(empu1+au2-δ)/d3(1+d4u1). By (40), we know that
(56)∥c3(x)∥C(Ω¯)≤emprd3B+erd1Bd2d3+er24μBd3.
Then Lemma 5 yields max⁡Ω¯φ(x)/min⁡Ω¯φ(x)≤C3* for some positive constant C3*(Λ,d1,d2,d3,Ω), and
(57)max⁡Ω¯u3min⁡Ω¯u3≤max⁡Ω¯φmax⁡Ω¯(1+d4u1)min⁡Ω¯φmin⁡Ω¯(1+d4u1)≤C3*max⁡Ω¯u1min⁡Ω¯u1≤C3*C1≜C3.
The proof is completed.Theorem 8 (lower bounds).
LetΛ, D1, D2, D3, and D4 be fixed positive constants. Assume that
(58)min⁡{empB(r-mM¯2),empB(r-pmM¯3)}>δ,
where M¯2 and M¯2 are given in (47) and (51). Then there exists a positive constant C_=C_(Λ,D1,D2,D3,D4) such that, when (d1,d2,d3)∈[D1,∞)×[D2,∞)×[D3,∞) and d4∈[0,D4], any positive solution u(x)=(u1(x),u2(x),u3(x)) of (3) satisfies
(59)min⁡Ω¯ui(x)≥C_,i=1,2,3.Proof.
Suppose that (59) fails. Then there exist sequences {d1,i,d2,i,d3,i,d4,i}i=1∞ with (d1,i,d2,i,d3,i)∈[D1,∞)×[D2,∞)×[D3,∞) and d4,i∈[0,D4] such that the corresponding positive solutions (u1,i,u2,i,u3,i) of (3) satisfy
(60)max⁡⁡Ω¯u1i⟼0,ormax⁡Ω¯u2i⟼0,ormax⁡⁡Ω¯u3i⟼0.
By a direct application of the maximum principle to the first equation of (3), we can obtain u1i≤r/B. Integrating by parts, we obtain that
(61)∫Ω‍(ru1i-Bu1i2-mu1iu2i-pmu1iu3i)dx=0,∫Ω‍(emu1u2i-au2iu3i-μu2i)dx=0,∫Ω‍(au2iu3i+empu1iu3i-δu3i)dx=0,
for i=1,2,…. By the standard regularity theorem for the elliptic equations, we know that there exists a subsequence of {u1,i,u2,i,u3,i}i=1∞, which we will still denote by {u1,i,u2,i,u3,i}i=1∞, and three nonnegative functions u1,u2,u3∈C2(Ω¯) such that
(62)(u1,i,u2,i,u3,i)⟼(u1,u2,u3)in[C2(Ω¯)]3asi⟼∞.
By (60), we know that
(63)u1≡0,oru2≡0,oru3≡0.
Furthermore, we assume that (d1,i,d2,i,d3,i,d4,i)↦(d_1,d_2,d_3,d_4)∈[D1,∞)×[D2,∞)×[D3,∞)×[0,D4]. Let i↦∞ in (61); we obtain
(64)∫Ω‍(ru1-Bu12-mu1u2-pmu1u3)dx=0,∫Ω‍(emu1u2-au2u3-μu2)dx=0,∫Ω‍(au2u3+empu1u3-δu3)dx=0.
Now, we consider the following three cases, respectively.
Case
1 (
u
1
≡
0
). Note that u1i↦u1, as i↦∞. Then we know that
(65)emu1i-au3i-μ<0on⁡Ω¯,∀i≫1.
Integrating the differential equation for u2i over Ω by parts, we have
(66)0=-d2i∫∂Ω‍∂νu2idx=-d2i∫Ω‍Δu2idx=∫Ω‍u2i(emu1i-au3i-μ)dx<0,∀i≫1,
which is a contradiction.
Case
2 (
u
2
≡
0
,
u
1
≠
0
on
Ω
¯
). By using Hopf boundary lemma, we know u1>0 on Ω¯. Then u1 and u3 satisfy the following equation:
(67)-d_1Δu1=u1(r-Bu1-pmu3)inΩ,∂νu1=0on⁡∂Ω.
Let u1(x0)=minΩ¯u1(x). It follows from Lemma 4 and (67) that
(68)r-Bu1(x0)-pmu3(x0)≤0;
that is,
(69)u1(x0)≥rB-pmBu3(x0)≥rB-pmM¯3B.
By using the assumption (emp/B)(r-pmM¯3)>δ, we know that
(70)empu1i-δ>0on⁡Ω¯,∀i≫1.
Integrating the differential equation for u3i over Ω by parts, we have
(71)0=-d3i∫∂Ω‍[(1+d4u1i)∂νu3i+d4u3i∂νu1i]dx=-d3i∫Ω‍Δ(u3i+d4u1iu3i)dx=∫Ω‍u3i(empu1i+au2i-δ)dx>0,∀i≫1,
which is a contradiction.
Case
3 (
u
3
≡
0
,
u
1
≠
0
, and
u
2
≠
0
on
Ω
¯
). By using Hopf boundary lemma, we know u1>0 and u2>0 on Ω¯.
Thenu1 and u2 satisfy the following equation:
(72)-d_1Δu1=u1(r-Bu1-mu2)inΩ,∂νu1=0on⁡∂Ω.
Let u1(x1)=minΩ¯u1(x). It follows from Lemma 4 and (72) that
(73)r-Bu1(x1)-mu2(x1)≤0;
that is,
(74)u1(x1)≥rB-mBu2(x1)≥rB-mM¯2B.
By using the assumption (emp/B)(r-mM¯2)>δ, we know that
(75)empu1i-δ>0on⁡Ω¯,∀i≫1.
Integrating the differential equation for u3i over Ω by parts, we have
(76)0=-d3i∫∂Ω‍[(1+d4u1i)∂νu3i+d4u3i∂νu1i]dx=-d3i∫Ω‍Δ(u3i+d4u1iu3i)dx=∫Ω‍u3i(empu1i+au2i-δ)dx>0,∀i≫1,
which is a contradiction. The proof is completed.


## 4. Nonexistence of Nonconstant Positive Solution of System (3) without Cross-Diffusion

In order to discuss the effect of cross-diffusion on the existence of nonconstant positive solution of system (3), we first give a nonexistence result when the cross-diffusion term is absent, which shows that the cross-diffusion coefficients do play important roles. The mathematical technique to be employed here is the energy method.Theorem 9.
Suppose thatd4=0 and d1≥r/μ2≜D1*, where μ2 is given in Notation 1. There exist positive constants D2* and D3*, depending on Λ, ɛ, Ω such that (3) has no nonconstant positive solution provided that d2≥D2* and d3≥D3*. Furthermore, one has
(77)ui≡u¯i,whereu¯i=1|ui|∫Ω‍uidx,i=1,2,3.Proof.
Assume thatu=(u1,u2,u3) is a positive solution of (3) with d4=0. Multiplying the ith equation of (3) by ui-u¯i and integrating the results over Ω by parts, we have
(78)d1∫Ω‍|∇u1|2dx=∫Ω‍(u1-u¯1)×(u¯1u¯3ru1-Bu12-mu1u2-pmu1u3-ru¯1+Bu¯12+mu¯1u¯2Bu12-mu1u2-pmu1u3+pmu¯1u¯3)dx=∫Ω‍(u1-u¯1)2×(r-Bu1-Bu¯1-mu2-pmu3)dx-∫Ω‍mu¯1(u1-u¯1)(u2-u¯2)dx-∫Ω‍pmu¯1(u1-u¯1)(u3-u¯3)dx;d2∫Ω‍|∇u2|2dx=∫Ω‍(u2-u¯2)×(emu1u2-au2u3-μu2-emu¯1u¯2+au¯2u¯3+μu¯2)dx=∫Ω‍(u2-u¯2)2(emu1-au3-μ)dx+∫Ω‍emu¯2(u1-u¯1)(u2-u¯2)dx-∫Ω‍au¯2(u2-u¯2)(u3-u¯3)dx;d3∫Ω‍|∇u3|2dx=∫Ω‍(u3-u¯3)×(empu1u3+au2u3-δu3-empu¯1u¯3-au¯2u¯3+δu¯2)dx=∫Ω‍(u3-u¯3)2(empu1-au2-δ)dx+∫Ω‍empu¯3(u1-u¯1)(u3-u¯3)dx+∫Ω‍au¯2(u2-u¯2)(u3-u¯3)dx.
Then it follows from (78) that
(79)d1∫Ω‍|∇u1|2dx+d2∫Ω‍|∇u2|2dx+d3∫Ω‍|∇u3|2dx=∫Ω‍(u1-u¯1)2×(r-Bu1-Bu¯1-mu2-pmu3)dx+∫Ω‍(u2-u¯2)2(emu1-au3-μ)dx+∫Ω‍(u3-u¯3)2(empu1-au2-δ)dx+∫Ω‍(emu¯2-mu¯1)(u1-u¯1)(u2-u¯2)dx+∫Ω‍(empu¯3-pmu¯1)(u1-u¯1)(u3-u¯3)dx+∫Ω‍(au¯3-au¯2)(u2-u¯2)(u3-u¯3)dx.
By Cauchy inequality with ɛ, we can get from (79) that
(80)d1∫Ω‍|∇u1|2dx+d2∫Ω‍|∇u2|2dx+d3∫Ω‍|∇u3|2dx≤∫Ω‍(u1-u¯1)2×(r-Bu1-Bu¯1-mu2-pmu3+2ɛ)dx+∫Ω‍(u2-u¯2)2×(emu1-au3-μ+(emu¯2-mu¯1)24ɛ+(au¯3-au¯2)24ɛ)dx+∫Ω‍(u3-u¯3)2×(empu1-au2-δ+(empu¯3-mpu¯1)24ɛ+(au¯3-au¯2)24ɛ)dx.
On the other hand, applying Poincaré inequality, we know that
(81)d1∫Ω‍|∇u1|2dx+d2∫Ω‍|∇u2|2dx+d3∫Ω‍|∇u3|2dx≥∫Ω‍d1μ2(u1-u¯1)2+∫Ω‍d2μ2(u2-u¯2)2+∫Ω‍d3μ2(u3-u¯3)2.
Then, by assumption, we can choose a sufficiently small positive constant ε0 such that
(82)d1μ2>r-Bu1-Bu¯1-mu2-pmu3+2ε0.
So by taking
(83)D2*>1μ2((emu¯2-mu¯1)24ε0emu1-au3-μ+(emu¯2-mu¯1)24ε0+(au¯3-au¯2)24ε0),D3*>1μ2((au¯3-au¯2)24ε0empu1-au2-δ+(empu¯3-mpu¯1)24ε0+(au¯3-au¯2)24ε0),
we can conclude that, when d4=0, (3) has only the positive constant solution ui≡u¯i for i=1,2,3. The proof is completed.


## 5. Existence of Nonconstant Positive Solution of System (3)

From Theorem9 we know that, when the cross-diffusion d3d4u1u3 is absent, (3) has no nonconstant positive solution under some conditions. In the following, we will discuss the effect of cross-diffusion on the existence of nonconstant positive solution of system (3) for certain values of diffusion coefficient d3, while the other parameters are fixed.Our main findings are the following theorem, which shows that the presence of cross-diffusion creates nonhomogeneous solution.Theorem 10.
Letd1, d2, and d4 be fixed and satisfy (33) and (58), and let μ2* and μ3* be defined in (34). If μ2*∈(μi,μi+1) and μ3*∈(μj,μj+1) for some j>i≥1 and the sum ∑n=i+1j‍m(μn) is odd, then there exists a positive constant d3* such that, if d≥d3*, (3) admits at least one nonconstant positive solution.In order to prove the above theorem by using Leray-Schauder theory, we start with some preliminary results. Throughout this section, Notation1 and G(u) defined in Section 2 will be used again. Define the set
(84)X+={u=(u1,u1,u1)T∈X:ui>0on⁡Ω¯,i=1,2,3},B(C)={u=(u1,u1,u1)T,C_≤ui≤C¯on⁡Ω¯,i=1,2,3},
where C¯=max⁡{M¯1,M¯2,M¯3} and C_ is given in Theorem 8. Then we will look for nonconstant positive solutions of (3) in the set B(C). Let Φ(u)=[d1u1,d2u2,d3(u3+d4u1u3)]T. Then (3) can be written as
(85)-ΔΦ(u)=G(u)inΩ,∂νu=0on⁡∂Ω.
Noting that the determinant of Φu(u) is positive for all u∈X+, we know that Φu-1(u) exists and det⁡Φu-1(u) is positive. Then, u is a positive solution to (85) if and only if
(86)F(u)=u-(I-Δ)-1×{Φu-1(u)[G(u)+∇uΦuu(u)∇u+u]}=0inX+,
where (I-Δ)-1 is the inverse of I-Δ in X with the no-flux boundary condition. As F(·) is a compact perturbation of the identity operator, for any B=B(C), the Leray-Schauder degree deg (F(·),0,B) is well defined if F(u)≠0 on ∂B. Note that
(87)DuF(u*)=I-(I-Δ)-1{Φu-1(u*)[Gu(u*)+I]}.
If DuF(u*) is invertible, the index of F at u* is defined as index(F(·),u*)=(-1)γ, where γ is the multiplicity of negative eigenvalues of DuF(u*) [40, Theorem 2.8.1]. For the sake of convenience, we denote
(88)H(μ)=H(u*,μ)≜det⁡[μI-Φu-1(u*)Gu(u*)]=det⁡[Φu-1(u*)]det⁡[μΦu(u*)-Gu(u*)].
By arguments similar to those in [41], we can conclude that the following proposition holds.Proposition 11.
Suppose that, for alln≥1, the matrix μnI-Φu-1(u*)Gu(u*) is nonsingular. Then index(F(·),u*)=(-1)σ, where σ=∑n≥1,H(μn)<0dim⁡S(μn).From Proposition11, we can see that, in order to compute index(F(·),u*), it is necessary to consider carefully the sign of H(μi). Noting that det⁡Φu-1(u*) is positive, then we only need to consider the sign of det⁡[μΦu*(u*)-Gu(u*)]. In fact, the direct calculation gives that the value of c¯3, which is given in (30), is equal to det⁡[μΦu*(u*)-Gu(u*)]. To study the existence of the positive solution of (3) with respect to the cross-diffusion constant d3d4, we will concentrate on the dependence of H(μi) on d3, and let d1, d2, and d4 be fixed. Hence, from Theorem 3, we first introduce the following proposition.Proposition 12.
Assume thatd4>0 and that (33) holds. Then there exists a positive number d3* such that, for all d3≥d3*, all the three roots μ~1, μ~2, and μ~3 of det⁡[μΦ(u*)-Gu(u*)]=0 are real and satisfy
(89)-∞<μ~1<0<μ~2<μ~3;det⁡⁡[μΦu(u*)-Gu(u*)]<0,ifμ∈(-∞,μ~1)∪(μ~2,μ~3);det⁡⁡[μΦu(u*)-Gu(u*)]>0,ifμ∈(μ~1,μ~2)∪(μ~3,∞).Proof of Theorem10.
By Proposition12 and our assumptions, there exists a positive constant d3* such that, when d3≥d3*, (89) holds and
(90)μi<μ~2<μi+1,μj<μ~3<μj+1.
Now, we show that, for anyd3≥d3*, (3) has at least one nonconstant positive solution. The proof, which will be accomplished by a contradict argument, is based on the homotopy invariance of the topological degree.
Suppose, on the contrary, that the assertion is not true. Letd^i=Di*  (i=1,2,3,4), where Di* is defined in Theorem 9. For t∈[0,1], define
(91)Φ(t;u)=[d^1u1+t(d1-d^1)u1,d^2u2+t(d2-d^2)u2,d^3+t(d3-d^3)(u3+td4u1u3)]T.
Now we consider the following problem:
(92)-ΔΦ(t;u)=G(u)inΩ,0≤t≤1,∂νu=0on⁡∂Ω.
Then u is a positive solution of (3) if and only if it is a positive solution of (92) for t=1. For 0≤t≤1, it is obvious that u* is the unique positive constant solution of (92) and u is a positive solution of (92) if and only if
(93)F(t;u)=u-(I-Δ)-1×{Φu-1(t;u)[G(u)+∇uΦuu(t;u)∇u]+u}=0inX+.
Clearly, F(1;u)=F(u). Theorem 9 shows that the only positive solution of F(0;u) is u* in B(C). By a direct computation, we have
(94)DuF(t;u*)=I-(I-Δ)-1{Φu-1(t;u*)Gu(u*)+I}.
In particular,
(95)DuF(0;u*)=I-(I-Δ)-1{D-1Gu(u*)+I},DuF(1;u*)=I-(I-Δ)-1{Φu-1(u*)Gu(u*)+I},
where D=diag⁡(d^1,d^2,d^3). In view of Proposition 12 and (90), it follows that
(96)H(μ1)=H(0)>0,H(μn)<0,i+1≤n≤j,H(μn)>0,1<n≤iorn≥j+1.
Therefore, zero is not an eigenvalue of the matrix μnI-Φu-1(u*)Gu(u*) for all n≥1, and
(97)∑n≥1,H(μn)<0‍m(μn)=∑n=i+1j‍m(μn)=an  odd  number.
Then Proposition11 yields
(98)index(I-F(1;·),u*)=(-1)γ=-1.
Similarly we can get that
(99)index(I-F(0;·),u*)=(-1)0=1.
According to Theorems 9 and (40) and (59), there exist positive constants C¯ and C_ such that, for all 0≤t≤1, the positive solutions of (3) satisfy C_≤u1,u2,u3≤C¯. Therefore, F(t;u)≠0 on ∂B(C) for all 0≤t≤1. By the homotopy invariance of the topological degree, we have
(100)deg⁡⁡(I-F(1;·),0,B(C))=deg⁡⁡(I-F(0;·),0,B(C)).
On the other hand, under our assumptions, the only positive solution of both F(1;u)=0 and F(0;u)=0 in B(C) is u*, and hence, by (98) and (99),
(101)deg⁡⁡(I-F(0;·),0,B(C))=index(I-F(0;·),u*)=1,deg⁡(I-F(1;·),0,B(C))=index(I-F(1;·),u*)=-1,
which contradicts (100). The proof is completed.



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*Source: 167856-2014-06-03.xml*