# Positive Solutions of a Diffusive Predator-Prey System including Disease for Prey and Equipped with Dirichlet Boundary Condition

Authors: Xiaoqing Wen; Yue Chen; Hongwei Yin
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

We study a three-dimensional system of a diffusive predator-prey model including disease spread for prey and with Dirichlet boundary condition and Michaelis-Menten functional response. By semigroup method, we are able to achieve existence of a global solution of this system. Extinction of this system is established by spectral method. By using bifurcation theory and fixed point index theory, we obtain existence and nonexistence of inhomogeneous positive solutions of this system in steady state.


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## Body

## 1. Introduction

In recent decade, predator-prey systems are significant in mathematical biology and applications [1–3]. These systems are usually described by ordinary differential equation (ODEs). Besides, epidemic models have been deeply studied by differential equations in [4, 5], based on the pioneer work of the classical SIR model by Kermack and Mckendrick. However, epidemic disease can spread within a species except for interactions among species. For this case, some ecoepidemiological models were introduced in [6–8] to study how disease affects the dynamic of predator-prey systems. Since we live in the real world, spatial factor plays an important role in populations dynamic. Moreover, space could induce some significant phenomena different from the corresponding ODE systems. At present, a lot of literature has showed that predator-prey systems with spatial diffusion can produce a variety of pattern formation by Turing bifurcation [9–11] and references wherein. In this paper, we are interested in the following model having Holling-type II functional response with disease in the prey:(1a)ut-Δu=ua-u-v-βuv,x,t∈Ω×R+,vt-Δv=βuv-bv-cwvmw+v,x,t∈Ω×R+,wt-Δw=-dw+ewvmw+v,x,t∈Ω×R+.There are two species in the model: prey(u+v) and predator (w). In (1a), u is susceptible prey, and v denotes infected prey. Only susceptible prey u is capable of reproducing and its reproducing rate is a. However, the infected prey does possess the capability of reproducing and still contributes to population growth of susceptible prey u according to the logistic growth law. The disease is transmitted only within prey in the form of βuv, and β∈(0,1) is called the transmission coefficient. The disease is not genetically inherited. The infected populations do not recover or become immune. Infected individuals are less active and can be caught more easily by predator resulting in increasing survival of susceptible prey. Catching rate of predators is a ratio-dependent Michaelis-Menten functional response function; that is, cwv/mw+v. Infected prey has a death rate b>0 and d>0 is a death rate of predators. Prey and predators are inhabited in a bounded region Ω⊆RN with the smooth boundary ∂Ω, where N is a spatial dimension number. Δ is the Laplace operator and represents diffusion of individuals from a high density region to low density one. Zhang et al. [12] have studied (1a) under the homogeneous Neumann boundary condition and obtain permanence and stability.However, few author, as far as we know, investigated mathematical property of (1a) equipped with Dirichlet boundary condition, such as existence of global time-dependent positive solution and inhomogeneous positive solutions in steady state. To this end, in this paper we make the first attempt to fill this gap and study existence of a global positive solution (1a) with homogeneous Dirichlet boundary conditions:(1b)ux,t=vx,t=wx,t=0,x,t∈∂Ω×R+,which imply that predator and prey die out on the boundary ∂Ω and initial values(1c)ux,0=u0x≥0,vx,0=v0x≥0,wx,0=w0x≥0,x∈Ω.Besides, we first attempt to investigate existence and nonexistence of inhomogeneous positive solutions of (1a), (1b), and (1c) in steady state as follows:(2)-Δu=ua-u-v-βuv,x∈Ω,-Δv=βuv-bv-cwvmw+v,x∈Ω,-Δw=-dw+ewvmw+v,x∈Ω,u=v=w=0,x∈∂Ω.The remainder of the paper is organized as follows: in Section2, some necessary notations and theoretical results are introduced; in Section 3, we examine existence of a global positive solution to (1a), (1b), and (1c); in Section 4 we discuss extinction of system (1a), (1b), and (1c); existence and nonexistence of inhomogeneous positive solutions are discussed by bifurcation theory and degree theory in Section 5.


## 2. Preliminary

In this section, some notations and basic well-known results are introduced. For anyq∈C(Ω¯), the linear eigenvalue problem [13, 14],(3)-Δu+qxu=λu,x∈Ω,u=0,x∈∂Ω,has an infinite sequence of eigenvalues. Let λ1(q) be the principal eigenvalue of (9) which is simple and its eigenfunction does not change sign in Ω. We denote λ1(0) by λ1 for simplicity and the eigenfunction corresponding to λ1 is denoted by φ1(x). q1(x)≤q2(x) with q1 being not identically equal to q2 implies λ1q1≤λ1(q2).Now, consider the nonlinear boundary value problem(4)-Δu=ua-u,x∈Ω,u=0,x∈∂Ω.If a>λ1 then (4) has a unique positive solution (cf. [15, 16]). We denote this unique positive solution by ua and ua<a.Next, the fixed point index theory, which is used later, is introduced. LetE be a real Banach space. W is called a wedge if W is a closed convex set and βW⊂W for all β≥0. For y∈W, we define (5)Wy=x∈E:∃r=rx>0,s.t.y+rx∈W,Sy=x∈W¯y:-x∈W¯y.We always assume that E=W-W¯. Let T:Wy→Wy be a compact linear operator on E. We say that T has property α on W¯y if there exists t∈(0,1) and w∈W¯y∖Sy such that (1-tT)w∈Sy. Let A:W→W be a compact operator with a fixed point y∈W and A is Fréchet differentiable at y. Let L=A′(y) be the Fréchet derivative of A at y. Then L maps W¯y into itself. We denote by indexW(A,y) the fixed point index of A at y relative to W.Proposition 1 (see [16–18]).
Assume thatI-L is invertible on W¯y. Then (i)
ifL has property α on W¯y, then indexW(A,y)=0;(ii)
ifL does not have property α on W¯y, then indexW(A,y)=(-1)σ, where σ is the sum of multiplicities of all eigenvalues of L which is greater than 1.For a linear operatorL, we denote by r(L) the spectral radius of L.Proposition 2 (see [16, 18]).
Letq(x)∈C(Ω¯) and M be a positive constant such that M-q(x)>0 on Ω¯. Let λ1(q) be the first eigenvalue of the problem(6)-Δu+qxu=λu,x∈Ω,u=0,x∈∂Ω.We have the following conclusions:(a)
λ1(q)<0⇒r[(M-Δ)-1(M-q(x))]>1,(b)
λ1(q)>0⇒r[(M-Δ)-1(M-q(x))]<1,(c)
λ1(q)=0⇒r[(M-Δ)-1(M-q(x))]=1.Now, we introduce the degree calculations, which was introduced by Dancer and Du in [19]. Assume that E1 and E2 are ordered Banach spaces with positive cones W1 and W2, respectively. Let E=E1⊕E2 and W=W1⊕W2. Then E is an ordered Banach space with positive cone W. Let D be an open set in W containing 0 and Fi:D¯→Wi be completely continuous operators, i=1,2. Denote by (u,v) a general element in W with u∈W1 and v∈W2. Let F:D¯→W be defined by (7)Fu,v=F1u,v,F2u,v.Define W2ϵ=v∈W2:vE2<ϵ.Theorem 3 (see [15]).
Suppose thatU⊂W1∩D is relatively open and bounded and (8)F1u,0≠uforu∈∂U;F2u,0=0foru∈U¯.Suppose that F2:D→W2 extends to a continuously differentiable mapping of a neighborhood of D into E2, W2-W2 is dense in E2, and Φ={u∈U:u=F1(u,0)}. Then the following are true: (i)
degW(I-F,U×W2(ε),0)=0 for ε>0 small, if, for any u∈Φ, the spectral radius r(F2′(u,0)|W2)>1 and 1 is not an eigenvalue of F2′(u,0)|W2 corresponding to a positive eigenvector;(ii)
degW(I-F,U×W2(ε),0)=degW(I-F1|W1,U,0) for ε>0 small, if, for any u∈Φ, the spectral radius r(F2′(u,0)|W2)<1.


## 3. Existence of a Global Positive Solution of (1a), (1b), and (1c)

For convenience, we introduce some notations. Forx,y∈R, we denote x∧y≔min{x,y}, x∨y≔max{x,y}, x+≔x∨0, and x-≔(-x)∨0 and extend these notations to real-valued functions. If (V,≥) is partially ordered vector space, we denote its positive cone by V+≔{v∈V:v≥0}. For p∈[1,+∞), p>N/2, let X={Lp(Ω)}3. Assume z=(u,v,w)∈X and its norm is defined as z=up+vp+wp.Hence, system (1a), (1b), and (1c) can be rewritten as an abstract differential equation:(9)dzdt=Az+Hz,t∈R+,z0=z0,t=0,where z=(u(x,t),v(x,t),w(x,t))T and z0=(u0,v0,w0)T,(10)A=Δ000Δ000Δwith (11)
D
A
=
u
,
v
,
w
∈
C
2
Ω
¯
3
,
u
∂
Ω
=
v
∂
Ω
=
w
∂
Ω
=
0
,
H
z
=
u
a
-
u
-
v
-
β
u
v
β
u
v
-
b
v
-
c
w
v
m
w
+
v
-
d
w
+
e
w
v
m
w
+
v
.Next, the existence of the local solution for (9) will be proved. First, we give out the following lemma.Lemma 4.
For everyz0∈X+, Cauchy problem (9) has a unique maximal local solution(12)z∈C0,Tmax;X∩C10,Tmax;X∩C0,Tmax;DAp,which satisfies the following Duhamel formula for t∈[0,Tmax):(13)zt=etApz0+∫0tet-sApHzsds,where Ap is a operator and the closure of A in X. Moreover, if Tmax<∞ then limsupTmax-z(t)=∞.Proof.
Since the operatorAp is the closure of A in X, it generates an analytic, condensed, strong continuous operator semigroup etAp. Moreover, we observe that H:D(Ap)→X is locally Lipschitz on a bounded set [20]. Via Theorem 7.2.1 in [21], we complete this proof.Lemma 5.
For initial-boundary problem (1a), (1b), and (1c), the components of its solution u(x,t), v(x,t), w(x,t) are nonnegative.Proof.
To proveu(x,t),v(x,t),w(x,t)≥0 for t∈[0,Tmax) and x∈Ω, we consider the following system:(14a)∂u′∂t=Δu′+u+′a-u+′-v+′-βu+′v+′,(14b)∂v′∂t=Δv′+βu+′v+′-bv+′-cw+′v+′mw+′+v+′,(14c)∂w′∂t=Δw′-dw+′+ew+′v+′mw+′+v+′with boundary and initial conditions (15)u′∂Ω=v′∂Ω=w′∂Ω=0,u′x,0=u0,v′x,0=v0,w′x,0=w0.After multiplying (14a), (14b), and (14c) by u-′, v-′, and w-′, respectively, and integrating by parts on the domain Ω, we obtain (16)-12ddt∫Ωu-′2dx-∫Ω∇u-′2dx=0,-12ddt∫Ωv-′2dx-∫Ω∇v-′2dx=0,-12ddt∫Ωw-′2dx-∫Ω∇w-′2dx=0.Hence, for t∈(0,Tmax), (17)u-′·,t22+v-′·,t22≤u-′·,022+v-′·,022=0.Consequently, (18)u′x,t≥0,v′x,t≥0,w′x,t≥0.Now, we can know that (u′(x,t),v′(x,t),w′(x,t)) is a solution of (1a), (1b), and (1c), and according to Lemma 4, we finally get that(19)ux,t=u′x,t≥0,vx,t=v′x,t≥0,wx,t=w′x,t,t∈0,Tmax.Next, we will prove the global existence of the positive solution for (1a), (1b), and (1c). Via Lemmas 4 and 5, we only need to show that the solution of (1a), (1b), and (1c) is uniformly bounded, that is, dissipation.Theorem 6.
Let(u(x,t),v(x,t),w(x,t)) be the solution of (1a), (1b), and (1c) with initial value (1c); then (20)0≤ux,t≤c1,0≤vx,t≤c2,0≤wx,t≤c3for (x,t)∈Ω×[0,Tmax), where ci, i=1,2,3 are defined by (21)c1≔max⁡a,u0∞,c2≔max⁡βa+bc1b1+β,βu0∞1+β+v0∞,c3≔max⁡a+deβc1cd1+β+ec2c,ecβu0∞1+β+v0∞+w0∞.Proof.
In view of Lemma5, we only prove the upper bound of (1a), (1b), and (1c) for (x,t)∈Ω×[0,Tmax). From (1a), we have(22)ut≤Δu+ua-u,inΩ×R+,ux,t=0,on∂Ω×R+,ux,0=u0,inΩ.We get u(x,t)≤c1, (x,t)∈Ω×(0,Tmax) by maximum principle, where c1=max{a,u0∞}.
Letv1=βu/1+β+v. From (1b) we have (23)v1t-Δv1=β1+βua-u-bv-cwvmw+v≤aβu1+β-bv1-βu1+β≤βa+bc11+β-bv1.Hence, by maximum principle we have that (24)vx,t≤v1x,t≤max⁡βa+bc1b1+β,βu0∞1+β+v0∞≔c2,t∈0,T.By using the same method, we consider w1=e/cβu/1+β+v+w; then from (1c) we have that (25)w1t-Δw1=eβc1+βua-u-becv-dw≤aeβc1+βu-dw1-eβuc1+β-evc≤a+deβc1c1+β+dec2c-dw1.Thus, the maximum principle shows that (26)wx,t≤w1x,t≤max⁡a+deβc1cd1+β+ec2c,ecβu0∞1+β+v0∞+w0∞≔c3.Remark 7.
From Theorem6, we know that the upper bound of the solution of (1a), (1b), and (1c) is independent of the maximal time Tmax, which implies that the solution of (1a), (1b), and (1c) is a global solution.By Lemmas4 and 5 and Theorem 6, we can get the following theorem.Theorem 8.
System (1a), (1b), and (1c) has a unique, nonnegative, and bounded solution z(x,t)=(u(x,t),v(x,t),w(x,t)), such that (27)zx,t∈C0,∞,X∩C10,∞,X∩Cx,∞,DA.


## 4. Extinction of (1a), (1b), and (1c)

Theorem 9.
Let(u(x,t),v(x,t),w(x,t)) be a positive solution of (1a), (1b), and (1c), then we have the following: (i)
ifa≤λ1 and e<d then (u,v,w)→(0,0,0) as t→∞;(ii)
ifa<λ1 and -b<λ1(-βua) then (u,v,w)→(ua,0,0) as t→∞.Proof.
(i) First, we observe that solution(u,v,w) of (1a), (1b), and (1c) satisfies(28)ut-Δu≤ua-u,x,t∈Ω×R+,u=0,x,t∈∂Ω×R+.Let ε be a sufficiently small positive constant such that ε≤b/β. Since a≤λ1, we see that u(x,t)→0 uniformly as t→∞ by using comparison principle for elliptic problems. Then there exists Tε≥0 such that u(x,t)≤ε for all t≥Tε. Therefore, we have(29)vt-Δv≤vβε-b,x,t∈Ω×Tε,∞,v=0,x,t∈∂Ω×Tε,∞,which concludes that v(x,t)→0 uniformly as t→∞. Similarly, there exists Tε′>0 such that v(x,t)≤ε for all t>Tε′. Further, since ewv/mw+v is monotone increasing with respect to the variable v, we have(30)wt-Δw≤weεmw+ε-d≤we-d,x,t∈Ω×Tε′,∞,w=0,x,t∈∂Ω×Tε′,∞,which concludes that w(x,t)→0 as t→∞.
(ii) Letε be a sufficiently small positive constant such that (31)aε<λ1b-βuaβ,bε<a-λ1β+1.
Sincea>λ1, we get from (28) and comparison theorem that(32)limsupt→∞⁡ux,t≤ua.Then, there exists Tε>0 such that u(x,t)≤ua+ε for all t>Tε. Thus, we have(33)vt-Δv≤vβua+ε-b,x,t∈Ω×Tε,∞,v=0,x,t∈∂Ω×Tε,∞.Since ε satisfies (a) and -b<λ1(-βua), we conclude that v(x,t)→0 uniformly as t→∞. Similarly, we can see that w(x,t)→0 uniformly as t→∞.
Besides, there existsTε′≥0 such that v(x,t),w(x,t)<ε for all t>Tε′. Hence, we have(34)ut-Δu≥ua-u-β+1ε,x,t∈Ω×Tε′,∞,u=0,x,t∈∂Ω×Tε′,∞.Since ε satisfies (b), we get that(35)liminft→∞⁡ux,t≥ua-β+1ε.Thus, from (32), (35), and the arbitrariness of ε, we can conclude that u(x,t)→ua as t→∞. We finish this proof.


## 5. Existence and Nonexistence of Positive Solutions of (2)

To prove the existence of positive solution of (2) by using fixed point index theory, we need a priori estimate for nonnegative solutions of (1a), (1b), and (1c). So we first give the following theorem.Theorem 10.
Any nonnegative solution(u(x),v(x),w(x)) of (2) has a priori bounds:(36)ux≤a,vx≤c4,wx≤c5,where (37)c4≔βa+bab1+β,c5≔aa+deβcd1+β+ec4c.Proof.
From the first equation of (2), we have(38)-Δu≤ua-u,x∈Ω,u=0,x∈∂Ω,and we get u(x)≤a by maximum principle. Let v1=βu/1+β+v with v1∂Ω=0. From (1b), we have (39)-Δv1=β1+βua-u-bv-cwvmw+v≤aβu1+β-bv1-βu1+β≤βa+ba1+β-bv1.Thus, by maximum principle, we get (40)vx≤v1x≤βa+bab1+β=c4.We set w1=e/cβu/1+β+v+w with w1∂Ω=0, and then we have (41)-Δw1=eβc1+βua-u-becv-dw≤aeβc1+βu-dw1-eβuc1+β-evc≤aa+deβc1+β+dec4c-dw1.By using maximum principle again, we obtain (42)w≤w1≤aa+deβcd1+β+ec4c=c5.Therefore, we complete this proof.Now, we introduce some notations:E=C01(Ω¯)3, where C01(Ω)=ϕ∈C1Ω¯:ϕ=0,on∂Ω,W=K3, where K=ϕ∈C1Ω¯:ϕ≥0inΩandϕ=0on∂Ω,D=u,v,w∈W:u<a+1,v<c4+1,w<c5+1, according to Theorem 10.Taking (43)M=max⁡a2+ac4+βac4,bc4+cm,dc5,we can define a positive and compact operator F:E→E by(44)
F
u
,
v
,
w
=
-
Δ
+
M
-
1
u
a
-
u
-
v
-
β
u
v
+
M
u
β
u
v
-
b
v
-
c
w
v
m
w
+
v
+
M
v
-
d
w
+
e
w
v
m
w
+
v
+
M
w
.It is obvious that F having a positive fixed point in interior of D implies that (2) has a positive solution.Next, we give the degree ofI-F in D relative to K and the fixed point index of F at the trivial solution (0,0,0) of (1a), (1b), and (1c) relative to K.Lemma 11.
Assume thata>λ1, then one has(i)
degW1-F,D=1,(ii)
indexWF,0,0,0=0.Proof.
(i) It is easy to see thatF has no fixed point on ∂Ω, so degW(I-F,D) is well defined. For s∈[0,1], we define a positive and compact operator Fs:E→E by (45)
F
s
u
,
v
=
-
Δ
+
M
-
1
s
u
a
-
u
-
v
-
s
β
u
v
+
M
u
s
β
u
v
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s
b
v
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s
c
w
v
m
w
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v
+
M
v
-
d
s
w
+
s
e
w
v
m
w
+
v
+
M
w
,and then F1=F. For each s, a fixed point Fs is a solution of the following problem:(46)-Δu=sua-u-v-sβuv,x∈Ω,-Δv=sβuv-sbv-scwvmw+v,x∈Ω,-Δw=-dsw+sewvmw+v,x∈Ω,u=v=0,x∈∂Ω.Form Theorem 10Fs has no fixed point ∂Ω and degW(1-Fs,D) is well defined. Hence, (47)deg⁡I-F,D=degWI-F1,D=degWI-F0,D.Note that (46) has only the trivial solution (0,0,0) when s=0. Set (48)L=F0′0,0,0=-Δ+M-1M000M000M.Let L(ξ1,ξ2,ξ3)=(ξ1,ξ2,ξ3) for some (ξ1,ξ2,ξ3)∈W¯(0,0,0)=K×K×K. Obviously, (ξ1,ξ2,ξ3)=(0,0,0).
Hence,I-L has no nontrivial kernel in W¯(0,0,0). Since λ1>0, it is easy to see that r((-Δ+M)-1(M))<1 by Proposition 2. It implies that L has no property α. Therefore, by Proposition 1, we get the first result of Lemma 11.
(ii) It is obvious thatF(0,0,0)=(0,0,0). Let L=F′(0,0,0); then (49)L=-Δ+M-1a+M000-b+M000-d+M.Assume that L(ξ1,ξ2,ξ3)=(ξ1,ξ2,ξ3) for some (ξ1,ξ2,ξ3)∈W¯(0,0,0), and then(50)-Δξ1=aξ1,x∈Ω,ξ1=0,x∈∂Ω,(51)-Δξ2=-bξ2,x∈Ω,ξ2=0,x∈∂Ω,(52)-Δξ3=-dξ3,x∈Ω,ξ3=0,x∈∂Ω.Since b>0 and d>0, from (51) and (52), we know that ξ2=ξ3=0. If ξ1∈K with ξ1≢0, then a=λ1 from (50), which contradicts with the assumption. Thus (ξ1,ξ2,ξ3)=(0,0,0). So I-L has nontrivial kernel in W¯(0,0,0).
Fora>λ1, by Proposition 2, we see that r0=r((-Δ+M)-1(a+M))>1, and r0 is the principle eigenvalue of (-Δ+M)-1(a+M) with a corresponding eigenfunction ϕ0>0. Since S(0,0,0)={(0,0,0)}, we see that (ϕ0,(0,0,0))∈W¯(0,0,0)∖S(0,0,0). Set s∈1/r0∈(0,1), and then (I-sL)(ϕ0,(0,0,0))=(0,0,0)∈S(0,0,0). Thus, L has the property α. By Proposition 1 we have indexW(F,(0,0,0))=0.Now, we give out the index of the semitrivial solution(ua,0,0) of (2).Lemma 12.
Assume thata>λ1, and then one has(i)
if-b>λ1(-βua), then indexW(F,(ua,0,0))=0;(ii)
if-b<λ1(-βua), then indexW(F,(ua,0,0))=1.Proof.
LetL=F′(ua,0,0); then (53)L=-Δ+M-1a-2ua+M-1+βua00βua-b+M000-d+M.Let L(ξ1,ξ2,ξ3)=(ξ1,ξ2,ξ3) for some (ξ1,ξ2,ξ3)∈W¯(ua,0,0)=C01(Ω¯)×K×K. Then(54)-Δξ1+2ua-aξ1=-1+βuaξ2,x∈Ω,-Δξ2+b-βuaξ2=0,x∈Ω,-Δξ3+dξ3=0,x∈Ω,ξ1=ξ2=ξ3=0,x∈∂Ω.Since ξ2,ξ3∈K, -b≠λ1-βua and d>0. So we get that ξ2=ξ3=0. Hence, from the first equation of (54) we have(55)-Δξ1+2ua-aξ1=-1+βuaξ2,x∈Ω,ξ1=0,x∈∂Ω.If ξ1≠0, then λ1(2ua-a)=0 by Proposition 2. On the other hand, λ1(2ua-a)>λ1(ua-a)=0, and we get a contradiction. Hence, (ξ1,ξ2,ξ3)=(0,0,0). Then I-L has no nontrivial kernel in W¯(ua,0,0).
(i) Since-b>λ1(-βua), by Proposition 1, we have r0=r(-Δ+M)-1(βua-b+M)>1 which is an eigenvalue of (-Δ+M)-1(βua-b+M) with a corresponding eigenfunction ϕ0>0. Since S(ua,0,0)=C01(Ω¯)×{0}×{0}, we have 0,ϕ0,0∈W¯ua,0,0∖S(ua,0,0). Set s∈1/r0∈(0,1), and then(56)I-sL0ϕ00=0ϕ00-s-Δ+M-1-1+βuaϕ0βua-b+Mϕ00=s-Δ+M-11+βuaϕ000∈Sua,0,0.Hence, L has the property α on W¯(ua,0,0) resulting in indexW(F,(ua,0,0))=0 according to Proposition 1.
(ii) First, we prove thatL does not have the property α on W¯(ua,0,0).
Since-b<λ1(-βua), we have r((-Δ+M)-1(-βua))<1. On the contrary, we suppose that L has the property α on W¯(ua,0,0). Then there exist s∈(0,1) and (ξ1,ξ2,ξ3)∈W¯(ua,0,0) such that (I-sL)(ξ1,ξ2,ξ3)∈S(ua,0,0). So(57)-Δ+M-1βua-b+Mξ2=ξ2s.Then 1/s is an eigenvalue of the operator (-Δ+M)-1(βua-b+M), which contradicts r((-Δ+M)-1(βua-b+M))<1. So L does not have the property α on W¯(ua,0,0). By Proposition 2, we have indexW(F,(ua,0,0))=(-1)σ, where σ is the sum of the multiplicities of all eigenvalues of L. Next, we will prove that σ=0.
Assume that1/ρ>1 is an eigenvalue of L with corresponding eigenfunction (ξ1,ξ2,ξ3), and then L(ξ1,ξ2,ξ3)=1/ρ(ξ1,ξ2,ξ3); that is,(58)-Δξ1+Mξ1=ρa-2ua+Mξ1-1+βuaξ2,x∈Ω,-Δξ2+Mξ2=ρβua-b+Mξ2,x∈Ω,-Δξ3+Mξ3=ρ-d+Mξ3,x∈Ω,ξ1=ξ2=ξ3=0,x∈∂Ω.Since 0<ρ<1 and d>0, it follows from the third equation (58) that ξ3=0.
We suppose thatξ2≠0, and it follows from the second equation of (58) and Proposition 2 that (59)0=λ1M1-ρ-ρβua-b>λ1-βua+b,which contradicts -b<λ1(-βua), and so ξ2=0.
Ifξ1≠0, it follows from the first equation of (58) that (60)0=λ1M1-ρ-ρa-2ua>λ12ua-a>λ1ua-a=0.Thus, we get the contradiction and this implies that ξ1=0. Hence, (ξ1,ξ2,ξ3)=(0,0,0), which implies that L has no eigenvalue being greater than 1; that is, σ=0, and indexW(F,(ua,0,0))=1. We complete the proof.Next, we consider the other semitrivial solutions of (2). Let us consider the following three subsystems:(61)-Δu=ua-u-v-βuv,x∈Ω,-Δv=βuv-bv,x∈Ω,u=v=0,x∈∂Ω.(62)-Δu=ua-u,x∈Ω,-Δw=-dw,x∈Ω,u=w=0,x∈∂Ω.(63)-Δv=-bv-cwvmw+v,x∈Ω,-Δw=-dw+ewvmw+v,x∈Ω,v=w=0,x∈∂Ω.It is obvious that (63) has trivial solution (v,w)=(0,0) and that (62) only has a trivial solution (u,w)=(0,0) and a semitrivial solution (u,w)=(ua,0) if and only if a>λ1. For subsystem (61), a lot of papers [15, 22] and references wherein have discussed systems in this case, so we directly give out some results about (61).Theorem 13.
Equation (61) has a positive solution (u∗,v∗) with u∗<ua if and only if a>λ1 and -b>λ1(-βua). Moreover, if a-β-1>λ1 and -b>λ1(-βu(a-β-1)), then the positive solution (u∗,v∗) satisfies u(a-β-1)≤u∗ and v¯≤v∗, where v¯ is the unique positive solution of the following equations: (64)-Δv=vβua-β-1-b,x∈Ω,v=0,x∈∂Ω.Let{(u,v)} be the set of all positive solutions of (61), denoted by Φ, and Ψ={(u,v,0),(u,v)∈Φ}. It is easy to see that (61) has semitrivial solution (u∗,v∗,0)∈Ψ from Theorem 13. Next, we give the degree of (I-F) in Ψ relative to W.Lemma 14.
AssumeΨ≠∅, and then one has the following:(i)
if-d>λ1-e, for any (u∗,v∗,0)∈Ψ, then degW(I-F,Ψ)=0;(ii)
if-d<λ1-e, for any (u∗,v∗,0)∈Ψ, then degW(I-F,Ψ)=1.Proof.
Let(65)E1=C01Ω¯2,E2=C01Ω¯,W1=K2,W2=K.Define (66)F1u,v,w=-Δ+M-1ua-u-βuv+Muβuv-bv-cvwmw+v+Mv,F2u,v,w=-Δ+M-1evwmw+v-dw.Then F=(F1,F2) and we are in the framework to use Theorem 3. We choose a neighborhood U⊂W1∩D of Ψ∩W1 such that (ua,0)∉U¯.
Now,F1(u,v,0)=(u,v) with (u,v)∈U¯ if and only if (u,v,0)∈Ψ. By the results in [22], we have (67)degW1I-F1W1,U,0=1,ifa>λ1,-b>λ1-βua,-1,ifa<λ1,-b<λ1-βua,0,ifa-λ1·-b-λ1-βua<0.For any (u∗,v∗,0)∈Ψ, a direct calculation shows that (68)F2′u∗,v∗,0W2w=-Δ+M-1M-b+ew.Notice that, by our choice of M,M-b+e>0, so it follows from the maximum principle that F2′(u∗,v∗,0)|W2 is u0-positive in the sense of [23] with u0=(-Δ)-1. Hence, r(F2′(u∗,v∗,0)|W2)>0 and is the only eigenvalue corresponding to a positive eigenvector. By a simple calculation, we can easily show that r(F2′(u∗,v∗,0)|W2)>1 if and only if -d>λ1-e and r(F2′(u∗,v∗,0)|W2)<1 if and only if -d<λ1-e. Therefore, by Theorem 3 and the discussion above, we have (69)degWI-F,U×W2ε,0=0,if-d>λ1-e,degWI-F1W1,U,0,if-d<λ1-e.Again since the degree does not depend on the particular choice of U and ε, Ψ≠∅ means a>λ1 and -b>λ1-βua. Thus, we complete this proof.From the discussion above, we can obtain the existence of positive solutions of (2).Theorem 15.
Assumea>λ1. Equation (2) has at least one positive solution if (70)a>λ1+β+1,-b>λ1-βua-β-1,-d>λ1-e,and then (2) has at least one positive solution.Proof.
Sincea>λ1, we obtain degW(I-F,D)=1 and indexW(F,(0,0,0))=0 from Lemma 11. Hence, we have(71)indexWF,ua,0,0+degWI-F,D=1.Since -b>λ1(-βu(a-β-1))>λ1(-βua), we have indexW(F,(ua,0,0))=0 from Lemma 12. Moreover, Theorem 13 implies Ψ≠∅. Let (u∗,v∗,0)∈Ψ. Since u∗≥u(a-β) and v∗≥v¯ by Theorem 13, we have -d>λ1-e and thus degW(I-F,Ψ)=0 from Lemma 14. Therefore, indexW(F,(ua,0,0))+degW(I-F,Ψ)=0. So (71) holds. We complete this proof.Next, we give out the result of nonexistence of positive solutions.Theorem 16.
If any one of the following conditions holds, then (2) has no positive solution:(i)
a<λ1;(ii)
a>λ1 and -b≤λ1(-βua).Proof.
Assume that (2) has positive solution (u,v,w), and then u,v,w satisfy the following three equations, respectively:(72)-Δu=ua-u-v-βuv,x∈Ω,u=0,x∈∂Ω,(73)-Δv=βuv-bv-cwvmw+v,x∈Ω,v=0,x∈∂Ω,(74)-Δw=ewvmw+v-dw,x∈Ω,w=0,x∈∂Ω.(i) Since u>0 in Ω, we get from (72) that (75)a=λ1u+β+1v>λ1.By Proposition 2 and comparison property of principle eigenvalue, this contradicts a<λ1. Then (73) and (74) have v=0 and w=0.
(ii) Sincev>0 in Ω, we get from (73) that (76)-b=λ1cwmw+v-βu>λ1-βua,by Proposition 2 and comparison property of principle eigenvalue, we get a contradiction -b≤λ1(-βua).



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