# Positive Solutions for a General  Gause-Type Predator-Prey Model with  Monotonic Functional Response

Authors: Guohong Zhang; Xiaoli Wang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study a general Gause-type predator-prey model with
monotonic functional response under Dirichlet boundary condition. Necessary and
sufficient conditions for the existence and nonexistence of positive solutions for this
system are obtained by means of the fixed point index theory. In addition, the local
and global bifurcations from a semitrivial state are also investigated on the basis of
bifurcation theory. The results indicate diffusion, and functional response does help
to create stationary pattern.


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## Body

## 1. Introduction

In this paper, we are interested in the following semilinear elliptic system with monotonic functional response under Dirichlet boundary condition:(1.1)-d1Δu=ug(u)-p(u)vinΩ,-d2Δv=-cv+m(x)p(u)vinΩ,u=v=0on⁡∂Ω,
where Ω is a bounded domain in RN(N≥1 is an integer) with a smooth boundary ∂Ω. The two functions u and v represent the densities of the prey and predator, respectively. The positive constants d1 and d2 are the diffusion coefficients of the corresponding species, c is the death rate of the predator, and m(x), which is assumed to be space dependent, represents the conversion rate of the prey to predators. The function g(u) denotes the growth rate of the prey species in the absence of predator. Throughout this paper, we impose the following hypotheses on the function g(u).(ℍ1)
g∈C1([0,∞)),g(0)>0,-g̃<gu(u)<0 for all u≥0 with a positive constant g̃; there exists a unique positive constant K such that g(K)=0.Obviously, the classical Logistic growth rateg(u)=r(1-(u/K)) satisfies (ℍ1). The function p(u) denotes the functional response of predators to prey. According to different biology backgrounds, the functional response p(u) may have several forms and many important results on the dynamics of predator-prey systems with different functional response have been obtained (see [1–20] and references therein). In many predator-prey interactions, the functional responses satisfies the following hypotheses.(ℍ2)
p∈C2([0,∞)),p(0)=0,0<pu(u)<p̂ for all u>0 with a positive constant p̂.It is easy to see that Holling-typeI, Holling-type II, Holling-type III, and Ivelev functional response satisfy hypothesis (ℍ2).In this work, we aim to understand the influence of diffusion and functional response on pattern formation, that is, the positive solutions of (1.1).  Throughout this paper, a solution (u,v) of (1.1) is called a positive solution if u(x)>0,v(x)>0 for all x∈Ω and (∂w/∂νx),(∂v/∂νx)<0 for all x∈∂Ω, where ∂νx stand for the outward unit norm to Ω at x. As a consequence, the results indicate the stationary pattern arises when the diffusion coefficient enter into certain regions. In other words, we show that diffusion does help to create stationary pattern and diffusion and functional response can become determining factors in the formation pattern. Furthermore, we also investigate the properties of the nonconstant positive solution by using local bifurcation theory introduced by  Crandall and Rabinowitz in [21] and global bifurcation theory introduced by López-Gómez and Molina-Meyer in [22]. We remark that problem (1.1) with Neumann boundary conditions was discussed in [5] recently. We point out that our results about the existence and nonexistence of positive solutions are different from [5] (see Corollary 3.8 and Remark 3.9).The rest of this paper is organized as follows. In Section2, some necessary preliminaries are introduced. In Section 3, we will give a priori upper bounds for positive solutions and investigate the existence and nonexistence of positive solutions of (1.1). In Section 4, the local bifurcations about parameter c are investigated. Finally, the results about global bifurcations are obtained in Section 5.


## 2. Some Preliminaries

In order to give the main results and complete the corresponding proofs, we need to introduce some necessary notations and theorems as the following.For eachh∈Cα(Ω)(0<α<1), let λ1(h) denote the principle eigenvalue of the following eigenvalue problem:(2.1)-d1Δu+h(x)u=λuinΩ,u=0on∂Ω.
Let λ1*(h) denote the principle eigenvalue of the following eigenvalue problem:(2.2)-d2Δu+h(x)u=λuinΩ,u=0on∂Ω,
and denote λ1(0),λ1*(0) by λ1,λ1* for simplicity. It is easy to know that λ1(h),λ1*(h) is strictly increasing (see [23, 24]).In order to calculate the indexes at the trivial and semitrivial states by means of the fixed point index theory, we also need to introduce the following theorem.Theorem 2.1 (see [9, 13]).
Assumeh∈Cα(Ω)(0<α<1) and M is a sufficiently large number such that M>h(x) for all x∈Ω¯. Define a positive and compact operator 𝕃=(-d1Δ+M)-1(M-h(x)). Denote the spectral radius of 𝕃 by r(𝕃).(i)
λ1(h)>0 if and only if r(𝕃)<1;(ii)
λ1(h)<0 if and only if r(𝕃)>1;(iii)
λ1(h)=0 if and only if r(𝕃)=1.It is easy to see that the corresponding conclusions in Theorem2.1 are also correct if the positive and compact operator 𝕃=(-d1Δ+M)-1(M-h(x)) is replaced by 𝕃=(-d2Δ+M)-1(M-h(x)).From Theorem2.1, we see that it is crucial to know the sign of the eigenvalue λ1,k(h) to determine the spectral radius of 𝕃. The following theorem give some sufficient conditions to determine the sign of the eigenvalue λ1,k(h).Theorem 2.2 (see [7, 9, 10, 23, 24]).
Leth(x)∈L∞(Ω) and φ≥0, φ≢0 in Ω with φ=0 on ∂Ω. Then one has(i)
if0≢-Δφ+h(x)φ≤0, then λ1(h(x))<0;(ii)
if0≢-Δφ+h(x)φ≥0, then λ1(h(x))>0;(iii)
if-Δφ+h(x)φ≡0, then λ1(h(x))=0.Consider the following equation:(2.3)-d1Δφ=φg(φ)inΩ,φ=0on⁡∂Ω,
where Ω is a bounded domain in RN(N≥1 is an integer) with a smooth boundary ∂Ω.Theorem 2.3 (see [7, 23, 24]).
Assume that the functiong(φ):Ω¯→R satisfies the following hypotheses:(i)
g(φ)∈C1(Ω) and gφ(φ)<0 for all φ≥0;(ii)
g(φ)≤0 for φ≥C, where C is a positive constant.Then, (2.3) has a unique positive solution if λ1(-g(0))<0.LetΘ(g(φ)) be the unique  positive solution of (2.3) when the unique positive solution exists. Denote Θ(g(0)) by Θ for simplicity.Remark 2.4.
It is easy to see that if the functiong(φ) satisfies the hypothesis (ℍ1), then it must satisfies the conditions (i) and (ii) in Theorem 2.3. We also point out that the condition λ1(-g(0))<0 holds if and only if g(0)>λ1. Therefore, if the function g(φ) satisfies the hypothesis (ℍ1) and g(0)>λ1, then (2.3) has a unique positive solution.
Now, we introduce the fixed point index theory which plays an important role in finding the sufficient conditions for the existence of positive solutions of model (1.1).
Let𝔼 be a real Banach space and let 𝕎⊂𝔼 be the natural positive cone of 𝔼. 𝕎⊂𝔼 is a closed convex set. 𝕎 is called a total wedge if τ𝕎⊂𝕎 and 𝕎-𝕎=𝔼. For y∈𝕎, define 𝕎y={x∈𝔼:y+γ∈𝕎forsomeγ>0} and Sy={x∈𝕎¯y:-x∈𝕎¯y}. Then, 𝕎¯y is a wedge containing 𝕎, y, -y, while Sy is a closed subset of 𝔼 containing y. Let T be a compact linear operator on 𝔼 which satisfies T(𝕎¯y)⊂𝕎¯y. We say that T has property α on 𝕎¯y if there is a t∈(0,1) and an ω∈𝕎¯y∖Sy such that (I-tT)ω∈Sy. Let 𝒜:𝕎→𝕎 be a compact operator with a fixed point y∈𝕎 and 𝒜,  a 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 deg𝕎(I-𝒜,𝔻) the degree of I-𝒜 in 𝔻 relative to 𝕎, index𝕎(𝒜,y) the fixed point index of 𝒜 at y relative to 𝕎. Then, the following theorem can be obtained.Theorem 2.5 (see [5, 11, 13]).
Assume thatI-𝕃 is invertible on 𝕎y. (i)
If𝕃 have property α on 𝕎y, then index𝕎(𝒜,y)=0;(ii)
If𝕃 does not have property α on 𝕎y, then index𝕎(𝒜,y)=(-1)σ, where σ is the sum of algebraic multiplicities of the eigenvalues of 𝕃 which are greater than 1.Finally, we introduce a result about global bifurcation, which was introduced by  López-Gómez and Molina-Meyer in [22] and we state here for convenience.LetU be an ordered Banach space whose positive cone P is normal and has nonempty interior, and consider the nonlinear abstract equation:(2.4)F(λ,u)=L(λ)u+R(λ,u),
where (ℍ𝔏)
𝔏(λ):=IU-𝔑(λ)∈ℒ(U),λ∈ℝ, is a compact and continuous operator pencil with a discrete set of singular values, denoted by 𝔊.(ℍℜ)
ℜ∈C(ℝ×U;U) is compact on bounded sets and
(2.5)lim⁡u→0R(c,u)‖u‖C(Ω¯)=0
uniformly on compact intervals of ℝ.(ℍℙ)
The solutions of (2.4) satisfy the strong maximum principle in the sense that
(2.6)(c,u)∈R×(P∖{0}),F(c,u)=0⟹u∈IntP,
where IntP stands for the interior of the cone P.Define the parity mappingC:𝔊↦{-1,0,1} by(2.7)C(σ):=12lim⁡ɛ↦0[Ind(0,N(σ+ɛ))-Ind(0,N(σ-ɛ))],σ∈G.
Then, thanks to [25, Theorem  6.2.1],  2.4 possesses a component emanating from (λ,0) at λ0 if C(λ0)∈{-1,1}. Such a component will be subsequently denoted by ℭλ0. Then, the following abstract result hold.Theorem 2.6.
Suppose thatλ0∈𝔊 satisfies C(λ0)≠0,
(2.8)N[L(λ0)]=span⁡[φ0],φ0∈P∖{0},
and 𝔑(λ0) is strongly positive in the sense that
(2.9)N(λ0)(P∖{0})⊂IntP.
Then, there exists a subcomponent ℭλ0P of ℭλ0 in ℝ×IntP such that (λ0,0)∈ℭ¯λ0P.
Moreover, ifλ0 is the unique singular value for which 1 is an eigenvalue of 𝔑(λ) to a positive eigenvector, then ℭλ0P must be unbounded in ℝ×U.Remark 2.7.
When we are working in a product-ordered Banach space, the conditions (2.6) and (2.9) can be modified as
(2.10)(c,u,v)∈[R×(P∖{0})×(P∖{0})],F(c,u,v)=0⟹(u,v)∈IntP×IntP,N(c)([P∖{0}]×[P∖{0}])⊂IntP×IntP.
For the technical details, one can refer to [25, Theorem  7.2.2] and [26, Proposition  2.2]. To avoid a repetition, we omitted it herein.


## 3. Existence and Nonexistence of Stationary Pattern

At first,  we introduce the following lemma which gives the necessary condition for (1.1) to have positive solutions.Lemma 3.1.
If problem (1.1) has a positive solution, then g(0)>λ1 and -λ1*<c<-λ1*(-mp(Θ)).Proof.
Assume(u,v) is a positive solution of (1.1). Then, it is obvious  that g(0)>λ1 and u<Θ by maximum principle. Because (u,v) satisfies
(3.1)-d2Δv=-cv+mp(u)vinΩ,v=0on⁡∂Ω,
we have
(3.2)0=λ1*(c-mp(u))>λ1*(c-mp(Θ))=c+λ1*(-mp(Θ)),0=λ1*(c-mp(u))<λ1*(c)=c+λ1*.
So, -λ1*<c<-λ1*(-mp(Θ)).In the rest of this section, we shall prove that the necessary conditions in Lemma3.1 are also sufficient conditions by means of the fixed point index theory. So, we need to obtain a priori bound for the positive solutions of (1.1).Theorem 3.2.
Assumec>-λ1* and (u,v) is a positive solution of (1.1). Then, one has
(3.3)u≤g(0),v≤g(0)(cd1md2+mg(0))‖(-Δ+cd2)-1‖C(Ω¯).Proof.
It is obvious thatu(x)≤g(0) by the maximum principle. From (1.1), we can find that
(3.4)-Δ(d1mu+d2v)=-cv+mp(u)v+mug(u)-mp(u)v=-cv+mug(u)=-cd2(d1mu+d2v)+u(cd1d2m+mg(u))
and hence
(3.5)(-Δ+cd2)(d1mu+d2v)=u(cd1d2m+mg(u)).
Therefore,
(3.6)v≤1d2(d2v+d1mu)≤(-Δ+cd2)-1ud2(cd1d2m+mg(u))≤g(0)d2(cd1md2+mg(0))(-Δ+cd2)-1(1)≤g(0)d2(cd1md2+mg(0))‖(-Δ+cd2)-1‖C(Ω¯).Now, we introduce the following notations:(3.7)E={C01(Ω¯)×C01(Ω¯)},N={φ∈C01(Ω¯):φ≥0inΩ¯},W=N×N,D={(u,v)∈W:u≤(K+1),v≤(R+1)},whereR=(g(0)/d2)((cd1m)/d2+mg(0))∥(-Δ+(c/d2))-1∥C(Ω¯). Take q sufficiently large with q>max{g(0)+pu(0)R,-c+p(0)} such that ug(u)-p(u)v+qu and -cv+p(u)v+qv are, respectively, monotone increasing with respect to u and v for all (u,v)∈[0,K]×[0,R].Define a positive and compact operatorℜ:𝔼→𝔼 by
(3.8)R(u,v)=((-d1Δ+q)-1[ug(u)-p(u)v+qu](-d2Δ+q)-1[-cv+p(u)v+qv]).Remark 3.3.
(i) By the maximum principle, it is easy to see that v≡0 if u≡0 in Ω in system (1.1). On the other hand, if v≡0, then we have -d1Δu=ug(u) in Ω and u=0 on ∂Ω. From the assumption (ℍ1), we see that (Θ,0) is the only semitrivial solution of (1.1) if g(0)>λ1. Moreover, (1.1) does not have any other constant solution except the trivial solution (0,0).
(ii) Observe that (1.1) is equivalent to (u,v)=ℜ(u,v). Then, it is sufficient to prove that ℜ has a nonconstant positive fixed point in 𝔻 to show that (1.1) has a positive solution.
(iii) From the Remarks (i) and (ii), we can see that it is necessary to calculate the fixed point index of ℜ at (0,0) and (Θ,0). By Kronecker's existence theorem [23], we also need to calculate the topological degree of ℜ in 𝔻 to prove that the necessary conditions in  Lemma 3.1 are also sufficient.
At first, we shall calculate the topological degree of the operatorℜ in 𝔻 and the fixed point index of the operator at (0,0), that is, deg𝕎(I-ℜ,𝔻) and  index𝕎(ℜ,(0,0)). It is easy to see that ℜ has no fixed point on ∂𝔻. Then, the deg𝕎(I-ℜ,𝔻) is well defined.
Forμ∈[0,1], we define a positive and compact operator ℜμ:𝔼→𝔼 by
(3.9)Rμ(u,v)=((-d1Δ+q)-1[μ(ug(u)-p(u)v)+qu](-d2Δ+q)-1[μ(-cv+p(u)v)+qv]).
Observe that
(3.10)R′(0,0)=((-d1Δ+q)-1(g(0)+q)00(-d2Δ+q)-1(-c+q))
and S(0,0)=(0,0), 𝕎¯(0,0)=ℕ×ℕ; we can obtain the following lemma and we omit the proofs because the calculations are standard.Lemma 3.4.
Assume thatg(0)>λ1 and c>-λ1*. Then, one has(i)
deg𝕎(I-ℜ,𝔻)=1,(ii)
index𝕎(ℜ,(0,0))=0.Now,  we need to calculate the fixed point index of the operatorℜ at (Θ,0), that is, index𝕎(ℜ,(Θ,0)).Lemma 3.5.
Assume thatg(0)>λ1 and c>-λ1*, Then, one has (i)
if-c>λ1*(-mp(Θ)), then index𝕎(ℜ,(Θ,0))=0;(ii)
if-c<λ1*(-mp(Θ)), then index𝕎(ℜ,(Θ,0))=1.Proof.
(i) Observeℜ(Θ,0)=(Θ,0). Let 𝕃=ℜ′(Θ,0). Then,
(3.11)L=R′(Θ,0)=((-d1Δ+q)-1(g(Θ)+Θgu(Θ)+q)(-d1Δ+q)-1(-p(Θ))0(-d2Δ+q)-1(-c+mp(Θ)+q)).
Assume 𝕃(ξ1,ξ2)=(ξ1,ξ2) for some (ξ1,ξ2)∈𝕎¯(Θ,0)=C01(Ω¯)×ℕ. Then,
(3.12)-d1Δξ1-(g(Θ)+Θgu(Θ))ξ1=-p(Θ)ξ2inΩ,-d2Δξ2-mp(Θ)ξ2=-cξ2inΩ,ξ1=ξ2=0on⁡∂Ω.
Taking account of ξ2∈ℕ, if ξ2≠0, then we can see from the second equation of  (3.12) that -c=λ1*(-mp(Θ)). This contradicts -c≠λ1*(-mp(Θ)). So, ξ2=0. Then, we can get from the first equation of  (3.12) that
(3.13)-d1Δξ1-(g(Θ)+Θgu(Θ))ξ1=0inΩ,ξ1=0on⁡∂Ω.
If ξ1≠0, then λ1(-g(Θ)-Θgu(Θ))=0. On the other hand, λ1(-g(Θ)-gu(Θ)Θ)>λ1(-g(Θ))=0, which is a contradiction. Therefore, (ξ1,ξ2)=(0,0) and I-𝕃 is invertible on 𝕎¯(Θ,0).
We claim that𝕃 has property α on 𝕎¯(Θ,0). In fact, set
(3.14)Ψ=(-d2Δ+q)-1(-c+mp(Θ)+q).
Since -c>λ1*(-mp(Θ)), we can see that r(Ψ)>1 is an eigenvalue of Ψ with a corresponding eigenfunction Φc>0 by Theorem 2.1. Because S(Θ,0)=C01(Ω¯)×{0}, we know that (0,Φc)∈𝕎¯(Θ,0)∖S(Θ,0). Then, we have
(3.15)(I-rc-1L)(0ϕc)=(0ϕc)-rc-1(-(-d1Δ+q)-1p(Θ)ϕc(-d2Δ+q)-1(-c+mp(Θ)+q)ϕc)=(-(-d1Δ+q)-1rc-1p(Θ)ϕc0)∈S(Θ,0).
This establishes our claim. Hence, index𝕎(ℜ,(Θ,0))=1.
(i)
From Remark3.3, we know that the unique nonnegative solutions of (1.1) are (0,0) and (Θ,0) if -c<λ1*(-mp(Θ)). Thus, we have
(3.16)deg⁡W(I-R,D)=indexW(R,(0,0))+indexW(R,(Θ,0)).
From Lemma 3.4, we know that deg𝕎(I-ℜ,𝔻)=1 and index𝕎(ℜ,(0,0))=0. Therefore, we have index𝕎(ℜ,(Θ,0))=1.Now, we can prove thatg(0)>λ1 and -λ1*<c<-λ1*(-mp(Θ)) are also the sufficient conditions for model (1.1) to have a positive solution.Lemma 3.6.
Ifg(0)>λ1 and -λ1*<c<-λ1*(-mp(Θ)), then model (1.1) has at least one positive solution.Proof.
Ifg(0)>λ1 and -λ1*<c<-λ1*(-mp(Θ)), by Lemmas 3.4 and 3.5, then we have
(3.17)deg⁡W(I-R,D)-indexW(R,(0,0))-indexW(R,(Θ,0))=1.
Hence, model (1.1) has at least one positive solution by Kronecker's existence theorem [8].From Lemmas3.1 and 3.6, we can get the following theorem.Theorem 3.7.
Problem (1.1) has at least one positive solution if and only if g(0)>λ1 and -λ1*<c<-λ1*(-mp(Θ)).
Letλ0 denote the principle eigenvalue of the following eigenvalue problem:
(3.18)-Δu=λuinΩ,u=0on⁡∂Ω.
Then, it is easy to see that the condition g(0)>λ1 is equivalent to d1<g(0)/λ0 and the condition -λ1*<c<-λ1*(-mp(Θ)) is equivalent to (λ1*(-mp(Θ))+mp(Θ))/λ0<d2<(λ1*+mp(Θ))/λ0. Therefore, one can get the following corollary from Theorem 3.7.Corollary 3.8.
Problem (1.1) has no positive solution if one of the following conditions hold:(i)
d1>g(0)/λ0;(ii)
d2>(λ1*+mp(Θ))/λ0;(iii)
d2<(λ1*(-mp(Θ))+mp(Θ))/λ0.Remark 3.9.
From Corollary3.8, we can see that if the prey diffuses so rapidly that d1>g(0)/λ0, then no positive solution exists. On the other hand, if the predator diffuses so rapidly that d2>(λ1*+mp(Θ))/λ0 or diffuses so slowly that d2<(λ1*(-mp(Θ))+mp(Θ))/λ0, then we can also observe the same phenomena. These results are different from the corresponding results in paper [5]. In paper [5], if the predator diffuse so rapidly that d2>D̃(d1), where D̃(d1) is a constant, then the corresponding model has at least one positive solution (see [5],  Theorem  3.8). How to explain these differences? The key point, we think, lies in the boundary conditions. Different from the reflecting boundary conditions, that is, Neumann boundary condition in [5], the prey and the predator in our model both face lethal boundary conditions, that is, Dirichlet conditions in our model. Therefore, the more rapidly the prey or the predator diffuses, the more possibly they encounter the lethal boundary and then the more possibly they cannot coexist.


## 4. Local Bifurcation

In this subsection,  we will employ the local bifurcation theory [21] to investigate the positive solution branches of (1.1) which bifurcate from the semitrivial solution (Θ,0) if g(0)>λ1. We choose c as the bifurcation parameter and denote by Γu={(c,Θ,0):c∈R}  the semitrivial solution set with the parameter c. The next proposition gives the local bifurcation branch of positive solution of (1.1).Theorem 4.1.
Assume thatg(0)>λ1. A branch of positive solutions of (1.1) bifurcates from Γu if and only if c=-λ1*(-mp(Θ)). More precisely, there exists a positive number δ such that when 0<s<δ, the local bifurcation positive solutions (c(s),u(s),v(s)) from (-λ1*(-mp(Θ)),Θ,0) have the following form:
(4.1)c(s)=-λ1*(-mp(Θ))+c1s+O(s2),u(s)=Θ+sϕ*+O(s2),v(s)=sψ*+O(s2),
where ψ*=(-d1Δ-g(Θ)-gu(ξ)Θ)-1ϕ* with ξ between Θ and u and ϕ* is the positive eigenfunction corresponding to c=-λ1*(-mp(Θ)) of the following eigenvalue problem with ∫Ωϕ2=1:
(4.2)-d2Δϕ-mp(Θ)ϕ=-cϕinΩ,ϕ=0on⁡∂Ω.
Furthermore, the bifurcation is subcritical, that is, c′(0)<0.Proof.
Let us introduce the change of variablew=Θ-u, which shifts the semitrivial solution (Θ,0) to (0,0).
Introduce an operatorΦ:ℝ×C02+α×C02+α↦ℝ×C0α×C0α as the following:
(4.3)Φ(c,w,v)=(d1Δ+w(g(Θ)+gu(ξ)Θ-gu(ξ)w)+p(Θ-w)vd2Δ+(-cv+mp(Θ-w)v)),
where ξ is between Θ and u. We will seek for the degenerate point of the linearized operator Φ(w,v)(c,0,0). By a simple calculation, we have
(4.4)Φ(w,v)(c,0,0)(ϕψ)=(d1Δϕ+(g(Θ)+gu(ξ)Θ)ϕ+p(Θ)ψd2Δψ+(-c+mp(Θ))ψ).
When c=c*=-λ1*(-mp(Θ)), it is easy to show that KerΦ(w,v)(c*,0,0)=Span{(ϕ*,ψ*)}, where ψ*=(-d1Δ-g(Θ)-gu(ξ)Θ)-1ϕ*>0 in Ω.
If(ϕ̃,ψ̃)∈ Range Φ(u,v)(c*,0,0), then there exist (ϕ,ψ)∈C02+α×C02+α such that
(4.5)d1Δϕ+(g(Θ)+gu(ξ)Θ)ϕ+p(Θ)ψ=ϕ̃inΩ,d2Δψ+(-c*+mp(Θ))ψ=ψ̃inΩ,ϕ=ψ=0on⁡∂Ω.
By the Fredholm alternative theorem, it is easy to see that the first equation of (4.5) is solvable if and only if ∫Ωψ̃ψ*dx=0.
For such a solution, the first equation enables us to obtainϕ=(-d1Δ-g(Θ)-gu(ξ)Θ)-1(p(Θ)ψ-ϕ̃). Therefore, we know that codim RangeΦ(u,v)(c*,0,0)=1. In order to use the local bifurcation theorem [21] at the degenerate point, we need to verify that Range Φ(w,v)c(c*,0,0)(ϕ*,ψ*)∉ RangeΦ(w,v)(c*,0,0). Here, it can be calculated that
(4.6)Φ(w,v)c(c*,0,0)(ϕ*ψ*)=(0-ψ*).
Suppose for contradiction that Φ(w,v)c(c*,0,0)(ϕ*,ψ*)∈ RangeΦw,v(c*,0,0). By (4.4) and (4.6), there exist (ϕ,ψ)∈𝔼 such that
(4.7)d1Δϕ+(g(Θ)+gu(ξ)Θ)ϕ+p(Θ)ψ=0inΩ,d2Δψ+(-c*+mp(Θ))ψ=-ψ*inΩ,ϕ=ψ=0on⁡∂Ω.
Then, multiplying the second equation of (4.7) by ψ* and integrating the resulting expression, we obtain that ∫Ω(ψ*)2dx=0, which obviously yields a contradiction. Consequently, we can apply the local bifurcation theorem to Φ at (c,0,0). Furthermore, by virtue of the Krein-Rutman theorem, we know that the possibility of other bifurcation points except c=c* is excluded.
In order to investigate the bifurcation direction from(-λ1*(-mp(Θ)),Θ,0), substituting (c(s),u(s),v(s)) into the second equation of (1.1) and differentiating it with respect to s, setting s=0, we have
(4.8)-d2Δvss(0)=2c1ϕ+λ1*(-mp(Θ))vss(0)-2mpw(Θ)ϕψ+mp(Θ)vss(0).
Multiplying (4.8) by ϕ and applying divergence theorem, we obtain
(4.9)-∫Ω2c1ϕ2dx=∫Ω(d2Δϕ+λ1*(-mp(Θ))ϕ+mp(Θ)ϕ)vss(0)dx-∫Ω2mpu(Θ)ϕ2ψdx.
By (4.2), the terms including vss(0) in (4.10) can be dropped out. Then, we can get
(4.10)c1=-∫Ωmpu(Θ)ϕ2ψdx∫Ωϕ2dx=-∫Ωmpu(Θ)ϕ2ψdx.
According to hypothesis (ℍ2), we have pu(Θ)>0 and c1<0. Then, we know that the bifurcation direction from (-λ1*(-mp(Θ)),Θ,0) is subcritical.Remark 4.2.
According to the theory of Rabinowitz [27], we can see that there is a continuum Cc* of the set of non-trivial solutions of (1.1) with (c*,0,0)∈C¯c* under the conditions of Theorem 4.1 and the continuum 𝒞c* consists of two subcontinua: 𝒞c*+, filled in by coexistence states, and 𝒞c*-, filled in by component-wise negative solution pairs in a neighborhood of (-λ1*(-mp(Θ)),Θ,0). However, this does not necessarily implies that the subcontinuum 𝒞c*+ satisfies the global alternative of Rabinowitz [27] by the reasons already explained by Dancer [12] and López-Gómez and  molina-meyer [22]. Instead, the existence of a global subcontinuum 𝒞c*+ of the set of positive solutions with (-λ1*(-mp(Θ)),Θ,0)∈𝒞c*+ follows by slightly adapting [22, Theorem  1.1]. Therefore, in the following subsection, we shall study the global bifurcation from (Θ,0) by using the global bifurcation theory of [22].


## 5. Global Bifurcation

In this subsection,  basing on the results in Theorem4.1, we can obtain the following results about global bifurcation from (Θ,0) by using the global bifurcation theory introduced by López-Gómez, Molina-Meyer in [22].Theorem 5.1.
Assume thatg(0)>λ1. Then, if one chooses c as the main continuation parameter of (1.1), there exists an unbounded component 𝒞c*+⊂R×𝔼 of the set of positive solutions of (1.1) such that
(5.1)(c,u,v)=(-λ1*(-mp(Θ)),Θ,0)∈Cc*+,PcCc*+=(-λ1*,-λ1*(-mp(Θ))),
where Pc stands for the projection operator into the c-component of the tern. Moreover, 𝒞c*+ must bifurcate from infinity at c=-λ1*.Proof.
Letw=Θ-u. Then, (1.1) is equivalent to the following problem:
(5.2)-d1Δw=w(g(Θ)+gu(ξ)Θ-gu(ξ)w)+p(Θ-w)vinΩ,-d2Δv=-cv+mp(Θ-w)vinΩ,w=v=0on⁡∂Ω,
where ξ is between Θ and u. Introduce an operator 𝔉:ℝ×𝔼↦𝔼 as the following:
(5.3)F(c,w,v)=(wv)-((-d1Δ)-1[w(g(Θ)+gu(ξ)Θ-gu(ξ)w)+p(Θ-w)v]-(d2Δ)-1[-cv+mp(Θ-w)v])
for every c∈ℝ and (w,v)∈𝔼. Obviously, 𝔉(c,0,0)=0 for all c∈ℝ and by elliptic regularity 𝔉(c,w,v)=0⇔(w,v) is a classic solution of (5.2).
Subsequently, for every(w,v)∈𝔼, we consider
(5.4)L(c)(wv)=(wv)-((-d1Δ)-1(g(Θ)+Θgu(ξ))(-d1Δ)-1p(Θ)0(-d2Δ)-1(-c+mp(Θ)))(wv)R(c,w,v)=(-(-d1Δ)-1[-gu(ξ)w2+p(Θ-w)v-p(Θ)v]-(-d2Δ)-1[mp(Θ-w)v-p(Θ)v]).
It is easy to see that ℜ(c,0,0)=0 and D(w,v)ℜ(c,0,0)=(0,0). Then, we have
(5.5)L(c)=D(w,v)F(c,0,0),F(c,w,v)=L(c)(wv)+R(c,w,v).
Define an operator
(5.6)N(c)(wv)=((-d1Δ)-1(g(Θ)+Θgu(ξ))(-d1Δ)-1p(Θ)0(-d2Δ)-1(-c+mp(Θ)))(wv).
By the Ascoli-Arzelá theorem and the classical Schauder estimates, we know that (5.6) is a compact linear operator. Owing to 𝔏(c)=I-𝔑(c), we can see that 𝔏(c) is Fredholm of index zero.
In order to complete the proof of Theorem5.1, we shall use [22, Theorem  1.1]. So, it is necessary to check the assumptions in Theorem  2.6.
Proof of(ℍ𝔏).
Since𝔏(c) is Fredholm of index zero, we know that c*∈ℭ if and only if c* is an eigenvalue of 𝔏(c), that is, if dim N[𝔏(c*)]≥1. Note that dim N[𝔏(c)]≥1 if and only if there exists (w,v)∈𝔼∖{(0,0)} such that
(5.7)-d1Δw=(g(Θ)+Θgu(ξ))w-p(Θ)vinΩ,-d2Δv=mp(Θ)v-cvinΩ,w=v=0on⁡∂Ω.
If v=0, then
(5.8)-d1Δw=(g(Θ)+Θgu(ξ))winΩ,w=0on⁡∂Ω
and hence w=0 (if w≠0, then we have 0=λ1(-g(Θ)-gu(ξ))>λ1(-g(Θ))=0, a contradiction). Then, we must have v≠0. So, dim N[𝔏(c*)]≥1 if and only if -c is an eigenvalue of -d2Δ-mp(Θ) in Ω. Consequently, the set of singular values of 𝔏(c) is indeed discrete and hence the assumption (ℍ𝔏) is fulfilled.Proof of(ℍℜ).
From the definition of the operatorℜ, it is easy to see that the assumption follows directly by a simple calculation.Proof of(ℍℙ).
It is easy to see that𝔼 can be regarded as an ordered Banach Space with respect to the order induced by the product cone ℙ. Using the the strong maximum principle, we can show that (c,w,v)∈[ℝ×(P∖{0})×(P∖{0})]∩Γ-1(0) imply that w,v>0 for all x∈Ω and ∂w/∂νx,∂v/∂νx<0. The assumption (ℍℙ) is fulfilled.Now, we can prove Theorem 5.1 according to the general framework of [22]. Firstly, note that C(σ)≠0 if and only if Ind(0,𝔑(λ)) changes as λ crosses σ; we can see that C(c*)≠0 from Theorem 4.1. Considering the operator 𝔑(c) defined by (5.6), it is not difficult to check that c=-λ1*(-mp(Θ)) is the unique value of c for which 1 is an eigenvalue of 𝔑(c) to a positive eigenfunction and
(5.9)N[L(-λ1*(mp(Θ)))]=N[I-N(-λ1*(mp(Θ)))]=span⁡[(ϕ,ψ)],
where (ϕ,ψ) are the corresponding eigenfunctions defined in Theorem 4.1. At last,  for g(0)>λ1 and -λ1*<c<-λ1*(-mp(Θ)), we can see that
(5.10)N(c)([P∖{0}]×[P∖{0}])⊂IntP=IntP×IntP.
Following from [22, Theorem  1.1],  we know that there exists an unbounded component 𝒞c*+⊂ℝ×𝔼 of the set of positive solutions of (1.1) such that (c,u,v)=(-λ1*(-mp(Θ)),Θ,0)∈𝒞c*+ and Pc𝒞c*+=(-λ1*,-λ1*(-mp(Θ))) due to Theorem 3.7.To complete the proof of Theorem 5.1, we suppose that -λ1*<c<-λ1*(-mp(Θ)) and let (u,v) be a positive solution of (1.1). Then, by Theorem 3.2, we have u(x)≤g(0) for all x∈Ω¯ and
(5.11)‖v‖C(Ω¯)≤g(0)(cd1md2+mg(0))‖(-Δ+cd2)-1‖C(Ω¯).
Note that
(5.12)lim⁡c→-λ1*g(0)(cd1md2+mg(0))‖(-Δ+cd2)-1‖C(Ω¯)=∞.
Therefore, we know that 𝒞c*+ must bifurcate from infinity at c=-λ1*. The proof of Theorem 5.1 is completed.



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*Source: 547060-2011-08-10.xml*