# Fixed Point Theory and Positive Solutions for a Ratio-Dependent Elliptic System

Authors: Jingmei Liu; Aixia Qian
Journal: Journal of Function Spaces
 (2019)

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## Abstract

We consider a ratio-dependent predator-prey model under zero Dirichlet boundary condition. By using topological degree theory and fixed index theory, we study the necessary and sufficient conditions for the existence of positive solutions. Then we present the asymptotic behavior analysis of positive solutions, by bifurcation theory and energy estimates.


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## Body

## 1. Introduction and Preliminaries

There is growing results about elliptic system, which comes from biological and physiological evidence, such as [1–9]. A more suitable general predator-prey model should be based on the so-called ratio-dependent theory, which asserts that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. In recent years, such model has been studied extensively, and many important phenomena have been observed (see [4–6, 10–12] and references therein).In this paper, we consider the following semilinear elliptic system with ratio-dependent function response and Dirichlet boundary condition:(1)-△u=uλ-u-buvu+mv,inΩ-△v=v-k+uu+mv,inΩu=v=0,on∂Ω.Here Ω is a bounded domain in RN(N≥1) with smooth boundary ∂Ω. In the biological model, the two unknown functions u(x),v(x) represent the spatial distribution density of the prey and predator. λ is termed the birth rate of the prey, while k is the death rate of the predator and b is a positive constant sometimes referred to as the conversion rate. The Holling-Tanner interaction term buv/u+mv represents the rate at which the prey is consumed by the predator with m>0.We remark that problem (1) with homogeneous Neumann boundary condition was discussed in [6]. We will consider what is more general in this paper, where the parameters λ,b are positive and k∈R. What is more, since (1) comes from the biological module, our results and methods are different from those in [13–16] and references therein, which are also the Dirichlet boundary problems.We say that a solution(u,v) of (1) is positive solution if both u(x)>0,v(x)>0 for all x∈Ω and ∂u/∂n(x)<0,∂v/∂n(x)<0 for all x∈∂Ω; i.e., (u,v) is also a coexistence state of (1).Now, we give some notations, definitions and well-known facts which will be used in the sequel.For eachq∈C(Ω¯), let λ1(q) be the principal eigenvalue of(2)-△u+qxu=λu,x∈Ωu=0,x∈∂Ω.As is well known, the principal eigenvalue λ1(q) is given by the following variational characterization(3)λ1q=infϕ∈H01Ω,ϕL2Ω=1⁡∫Ω∇ϕ2+qxϕ2dxWe denote λ1(0) by λ1; some useful properties can be seen [6, Proposition A.1] or [17, Lemma 5.2].Forq∈C(Ω¯), let p be a sufficiently large constant such that p-q(x)>0 for all x∈Ω¯. Define a compact linear operator T:C(Ω¯)→C(Ω¯) by u=Tv=(-△+p)-1p-qxv, where u∈C(Ω¯) is the unique solution of the following problem:(4)-△u+pu=p-qxv,x∈Ωu=0,x∈∂Ω.Denote r(T) be the spectral radius of T. Then the relationship between λ1(q) and r(T) can be given as [2, Proposition 1].Theorem 1 (see [2]).
( 1 )   λ 1 ( q ) > 0 ⇔ r ( T ) < 1;
( 2 )   λ 1 ( q ) < 0 ⇔ r ( T ) > 1;
( 3 )   λ 1 ( q ) = 0 ⇔ r ( T ) = 1.
From Theorem1, we see that it is crucial to determine the eigenvalue λ1(q). The following theorem is established by [2] (see also Theorem 2.4 [1]).Theorem 2 (see [1, 2]).
Letq(x)∈L∞(Ω) and ϕ≥0,ϕ≢0 in Ω with ϕ=0 on ∂Ω. Then we have(a)
If0≢-△ϕ+q(x)ϕ≤0, then λ1(q)<0;(b)
If0≢-△ϕ+q(x)ϕ≥0, then λ1(q)>0;(c)
If-△ϕ+q(x)ϕ≡0, then λ1(q)=0.
Some concepts of cone, total wedge, topological degree and fixed point index in a cone can be seen in [18–21] and so on.LetE be a Banach space and W⊂E be a total wedge. Let A:W→W be a compact operator with a fixed point y∈W and let D be a relatively open subset of W such that A has no fixed point on the boundary of D. We denote by degW⁡(I-A,D) the degree of I-A in D relative to W and by indexW(A,y) the fixed point index of A at y relative to W.Theorem 3 (see [21, 22]).
Assume thatW is a total wedge, and let A:W→W be a compact operator with a fixed point y∈W and it is Frechlet differentiable at y. Let L=A′(y) be the Frechlet derivative of A at y. Then L maps W¯y into itself. Moreover, if I-L is invertible on W¯y, then the following results hold:(i)
IfL has property a on W¯y, then indexW(A,y)=0;(ii)
IfL does not has property a 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.


## 2. Existence of Positive Solutions of (1)

In the sequel, for simplicity of notation and more transparent analysis, we redefine(5)pu,v=uvu+mv,u,v≠0,00,u,v=0,0It is easy to see that, for u≠0,(6)pu′u,v=mv2u+mv2⇒pu′u,0=0,pv′u,v=u2u+mv2⇒pv′u,0=1,pu′′u,v=-2mv2u+mv3⇒pu′′u,0=0,pv′′u,v=-2mu2u+mv3⇒pv′′u,0=-2mu,puv′′u,v=pvu′′=2muvu+mv3⇒puv′′u,0=0.while (7)pu′0,0=pv′0,0=0.Under the definition of p(u,v), it is obvious that (1) has a trivial solution (0,0) and only one semitrivial solution (uλ,0) (if λ>λ1,k>-λ1), where uλ is the unique positive solution of(8)-△u=λu-u2,x∈Ωu=0,x∈∂Ω.Lemma 4.
If model (1) has a positive solution, then λ>λ1 and -λ1<k<1-λ1.Proof.
Assume(u,v) is a positive solution of (1), and then(9)-△u<λu,x∈Ω-△v<1-kv,x∈Ωu=v=0,x∈∂Ω.It follows from the property of the principal eigenvalue that λ>λ1 and 1-k>λ1, that is, k<1-λ1.
On the other hand, since(u,v) satisfies(10)-△v=v-k+uu+mv,x∈Ωv=0,x∈∂Ωone has(11)0=λ1k-uu+mv<λ1k=k+λ1.So k>-λ1.Remark 5.
Above lemma is about the necessary condition for (1) to have positive solutions. Next we show that λ>λ1 and -λ1<k<1-λ1 are also the sufficient conditions for the existence of a positive solution of (1). We will use fixed point index theory.Lemma 6.
Assumingk>-λ1, then any positive solution (u,v) of (1) has a priori bounds(12)ux≤λ,vx≤λλ+kb-△+k-11CΩ¯.Proof.
Since part 1 is a simple consequence of a standard comparison argument, we omit it. For part 2, by a direct calculation, we get that(13)-△u+bv=uλ-u-buvu+mv+buvu+mv-kbv=-kbv+uλ-u=-ku+bv+uλ+k-u.and hence(14)-△+ku+bv=uλ+k-u.As λ1+k>0, we have(15)v=1b-u+-△+k-1uλ+k-u.Therefore λ+k>0. Otherwise, if λ+k≤0, then (-△+k)(u+bv)<0, which implies u+bv<0, a contradiction. Consequently λ+k>0 is necessary for the existence of a positive solution; in such case(16)bv≤bv+u=-△+k-1uλ+k-u≤λλ+k-△+k-11,then(17)v≤λλ+kb-△+k-11CΩ¯.Remark 7.
Above result shows that the coexistence states of (1) have a priori bounds as soon as k varies in compact subinterval of (-λ1,1-λ1).
Now we introduce the following notations:(18)E=CΩ¯×CΩ¯,W=K×K,whereK=u∈CΩ¯:ux≥0inΩ¯,D=u,v∈W:u≤λ+1,v≤Q+1,where Q=λ(λ+k)/b(-△+k)-1(1)C(Ω¯)
From Lemma6, nonnegative solutions of (1) must be in D. Define a positive and compact operator A:E→E by(19)Au,v=-△+p-1uλ-u-buvu+mv+puvuu+mv-k+pvwhere p is s sufficiently large number such that p+λ-u-bv/u+mv>0 and p+u/u+mv-k>0 for u,v∈[0,λ]×[0,Q].Remark 8.
Note that (1) is equivalent to (u,v)=A(u,v) by elliptic regularity, and therefore in order to show the existence of positive solutions of (1), it suffices to prove A has nontrivial fixed points in D.We next compute the fixed point index of the trivial and semitrivial solution of (1). We have shown that (1) has a trivial solution (u,v)=(0,0) and a semitrivial solution (uλ,0) since λ>λ1 and k>-λ1. Moreover the following lemma holds.Lemma 9.
Assuming thatλ>λ1 and k>-λ1, then(i)
degW⁡(I-A,D)=1;(ii)
indexW(A,(0,0))=0;(iii)
indexW(A,(uλ,0))=0 if k<1-λ1; and indexW(A,(uλ,0))=1 if k>1-λ1.Proof.
(i) For eacht∈[0,1], we define a positive and compact operator At:E→E by(20)Atu,v=-△+p-1tuλ-u-buvu+mv+putv-k+uu+mv+pvThen A1=A,At has no fixed point on ∂D and At(D¯)⊂W. Thus degW⁡(I-At,D) is well defined for all t∈[0,1]. By the homotopy invariance of Laray-Schauder degree and (0,0) the only fixed point of A0 in D, we obtain(21)degW⁡I-A,D=degW⁡I-A0,D=indexWA0,0,0.Set(22)L=A0′0,0=-△+p-1p00pAssume that L(ξ1,ξ2)=(ξ1,ξ2) for some (ξ1,ξ2)∈W¯(0,0)=K×K. It is easy to see (ξ1,ξ2)=(0,0). Thus I-L is invertible on W¯(0,0). Since λ1>0, we see r(L)<1 by Theorem 1, this implies that L does not have property a on W¯(0,0). Thus by Theorem 3 (ii), indexW(A0,(0,0))=1.
(ii) Observe thatA(0,0)=(0,0). Let L=A′(0,0) and then(23)L=-△+p-1λ+p00-k+pAssuming that L(ξ1,ξ2)=(ξ1,ξ2) for some (ξ1,ξ2)∈W¯(0,0), then(24)-△ξ1=λξ1,inΩξ1=0,on∂Ω.-△ξ2=-kξ2,inΩξ2=0,on∂Ω.Since λ>λ1 and k>-λ1, we see that ξ1=ξ2=0. Thus I-L is invertible on W¯(0,0). By Theorem 1, we see rλ≔r((-△+p)-1(λ+p))>1 and rλ is the principle eigenvalue of the operator (-△+p)-1(λ+p) with a corresponding eigenfunction ϕλ>0 in Ω. Set tλ=1/rλ∈(0,1) and then (ϕλ,0)∉S0,0=0,0, but(25)I-tλLϕλ,0=0,0∈S0,0.This shows that L has property a, and thus indexW(A,(0,0))=0 by Theorem 3 (i).
(iii) The second part is a straightforward consequence of Lemma4 and (i)(ii). In fact, from Lemma 4, the nonnegative solutions of (1) are (0,0) and (uλ,0) if k>1-λ1, so by the properties of topological degree, we have(26)degW⁡I-A,D=indexWA,0,0+indexWA,uλ,0.Combining (i) and (ii) above and (26), we have indexW(A,(uλ,0))=1.
Next, give the proof of the first part. ObserveA(uλ,0)=(uλ,0). Let L=A′(uλ,0), and then(27)L=-△+p-1λ-2uλ+p-b0-k+p+1Assume that L(ξ,η)=(ξ,η) for some (ξ,η)∈W¯(uλ,0)=C(Ω¯)×K; i.e., (ξ,η) satisfies(28)-△ξ+2uλ-λξ=-bη,inΩ-△η-η=-kη,inΩξ=η=0,on∂Ω.Taking η∈K, it follows from the second equation of (28) and Theorem 2 that -k=-1+λ1, if η≠0, which contradicts k≠1-λ1. So η=0. Then we get from the first equation of (28)(29)-△ξ+2uλ-λξ=0,inΩξ=0,on∂Ω.If ξ≠0, then λ1(2uλ-λ)=0 by Theorem 2. On the other hand, λ1(2uλ-λ)>λ1(uλ-λ)=0, and we get a contradiction. Therefore (ξ,η)=(0,0) and I-L is invertible on W¯(uλ,0).
We claim thatL has property a on W¯(uλ,0). In fact, set(30)B=-△+p-1-k+1+pSince k<1-λ1, by Theorem 1 (ii), rk≔r(B)>1 is an eigenvalue of B with a corresponding eigenfunction ϕk>0. Since S(uλ,0)=C(Ω¯)×0, we see (0,ϕk)∈W¯(uλ,0)∖S(uλ,0). Set tk=rk-1∈(0,1), and then(31)I-tkL0ϕk=0ϕk-tk-△+p-1-bϕk-k+p+1ϕk=-△+p-1tkbϕk0∈Suλ,0.This proves that L has property a. Therefore indexW(A,(uλ,0))=0.Lemma 10.
Ifλ>λ1 and -λ1<k<1-λ1, then (1) has a positive solution.Proof.
By Lemma9, we have(32)degW⁡I-A,D-indexWA,0,0-indexWA,uλ,0=1.Hence (1) has at least a positive solution.From Lemmas4 and 10, we get the following.Lemma 11.
Model (1) admits a positive solution if and only if λ>λ1 and -λ1<k<1-λ1.


## 3. Structure of Solutions withk as a Bifurcation Parameter

In this section we shall regardk as a bifurcation parameter and suppose that all other constants are fixed. For all values of k, we have the branch of zero solutions of (1) S0=k,0,0:k∈R. Suppose λ>λ1 and k>-λ1, and then (1) also has the branch of semitrivial solutions S1=k,uλ,0:k∈R. We next give some results about the bifurcation from (uλ,0).Lemma 12.
(i) The trivial steady state(0,0) is locally asymptotically stable if λ<λ1 and k>λ1, while it is unstable if λ>λ1 or k<-λ1.
(ii) Assume thatλ>λ1. Then the semitrivial solution steady state (uλ,0) is locally asymptotically stable if k>1-λ1, while it is unstable if k<1-λ1.Proof.
From the linearization principle eigenvalue problem(33)-△ϕ-λϕ-2uϕ-bpu′u,vϕ-bpv′u,vψ=ηϕ,inΩ-△ψ--kψ+pu′u,vϕ+pv′u,vψ=ηψ,inΩϕ=ψ=0,on∂Ω.The stability of (uλ,0) is determined by the problem(34)-△ϕ-λϕ+2uλϕ+bψ=ηϕ,inΩ-△ψ+kψ-ψ=ηψ,inΩϕ=ψ=0,on∂Ω.Since (34) is not completely coupled, we only need to consider the following two eigenvalue problems:(35)-△ϕ-λϕ+2uλϕ=ηϕ,inΩϕ=0,on∂Ω.and(36)-△ψ+kψ-ψ=ηψ,inΩψ=0,on∂Ω.Then an eigenvalues of (34) must be an eigenvalues of (35) or (36) (see [11, 12] and references therein). Denote the principle eigenvalue of (35) and (36) by λ∗ and λ∗, respectively. Then(37)λ∗=λ12uλ-λ>λ1uλ-λ=0λ∗=λ1k-1=k-1+λ1.Combining above results, one can see that if k>1-λ1, then all eigenvalues of (34) are positive, and thus (uλ,0) is locally asymptotically stable. On the other hand, if k<1-λ1, then (34) has a negative eigenvalue, which implies the instability of (uλ,0).
Similarly, combining with the definition ofp(u,v), we can get (i).First we shall obtain a local result on bifurcation fromS1, as the results of [23].Lemma 13.
Letλ>λ1 be fixed. Then k=1-λ1 is a bifurcation value of (1) where positive solutions bifurcate from the line of semi-trivial solutions S1. The set of positive solutions to (1) near (1-λ1,uλ,0) is a smooth curve(38)Γ=ks,uλ-us,vs,s∈0,δsuch that k(0)=1-λ1, u(s)=sφ1(x)+os, and v(s)=sφ2(x)+o(|s|). Moreover(39)k′0=-m∫Ωφ23x/uλdx∫Ωφ22xdxand k=1-λ1 is the unique bifurcation value for which positive solutions bifurcate form S1.Proof.
We apply the similar proof of Lemma12 [4]. By changing the variables w=uλ-u, define(40)Gk,w,v=△w+λw-2uλw+w2+bpuλ-w,v△v+puλ-w,v-kvA simple calculation implies that(41)Gkk,w,v=0-vGkw,vk,w,vϕ,ψ=0-ψGw,vk,0,0ϕ,ψ=△ϕ+λϕ-2uλϕ+bψ△ψ+1-kψGw,vw,vk,0,0ϕ,ψ2=2ϕ2-2muλψ2-2muλψ2
By letting(w,v)=(0,0), we can find that only when k=1-λ1, does G(w,v)(k,0,0)[ϕ,ψ]=0 have a solution with ψ>0. Thus k1=1-λ1 is the only bifurcation point along S1, where positive solution of (1) bifurcates. At (k,w,v)=(k1,0,0), it is easy to verify that the kernel(42)NGw,vk1,0,0=spanφ1,φ2,where (φ1,φ2)≠(0,0) satisfies(43)△ϕ+λϕ-2uλϕ+bψ=0,inΩ△ψ+1-k1ψ=0,inΩϕ=ψ=0,on∂Ω.Since k1=1-λ1, we can choose φ2>0, and then(44)φ1=-△+2uλ-λ-1bφ2>0.The range of the operator is given by(45)RGw,vk1,0,0=f,g∈Y1×Y2:∫Ωgxφ2xdx=0which is of codimension one and(46)Gkw,vk1,0,0φ1,φ2=0,-φ2∉RGw,vk1,0,0,since ∫Ωφ22(x)dx>0. Thus we can conclude that the set of positive solutions to (1) near (k1,uλ,0) is Γ. Moreover(47)k′0=-Gw,vw,vk1,0,0φ1,φ22,l2Gkw,vk1,0,0φ1,φ2,l=-m∫Ωφ23x/uλdx∫Ωφ22xdx<0,where l is the linear functional defined by [f,g],l=∫Ωg(x)φ2(x)dx. Therefore the bifurcation at (k1,uλ,0) is always subcritical.
Suppose that there is a sequencekn,un,vn of coexistence states of (1) with limn→∞⁡(kn,un,vn)=(k,uλ,0) for some k∈R. Then from the v-equation of (1), we find for every n≥1(48)-△vnvnCΩ¯=-knvnvnCΩ¯+vnvnCΩ¯unun+mvnor equivalently(49)vnvnCΩ¯=-△-1-kvnvnCΩ¯+vnvnCΩ¯unun+mvn+k-kn-△-1vnvnCΩ¯By the compactness of (-△)-1, it is easy to see that along some subsequence, relabeled by n, we have(50)limn→∞⁡vnvnCΩ¯=ϕ>0for some ϕ∈C01(Ω¯) with ϕC(Ω¯)=1. Thus passing the limit n→∞ in (49), we find that(51)ϕ=-△-1-k+1ϕ⇔-△ϕ=-k+1ϕ.Therefore k=1-λ1, which concludes the proof.Remark 14.
Above results together with Lemma4 show that the bifurcation of nonnegative solutions from S1 at (k1,uλ,0) must be to the left, while Lemma 4 also shows that the branch of nontrivial solutions cannot extend too far to the left.
We now investigate the global nature of the above curve of nontrivial nonnegative solution in thek-(u,v) plane, i.e., in R×C(Ω¯)×C(Ω¯), to show that hypotheses of Th3.2 [24] are satisfied.
WritingU=uλ-u and V=v, it is easy to check that (U,V) is a nonnegative solution of (1) if and only if 0≤U≤uλ,V≥0 and (U,V) satisfies(52)-△U=λU-2uλU+U2+buλ-UVuλ-U+mV-△V=-kV+uλ-UVuλ-U+mV.Rewrite (52) as(53)UV=-△-1Uλ-2uλ+U+buλ-UVuλ-U+mV-kV+uλ-UVuλ-U+mVDefine T:R×C1(Ω¯)×C1(Ω¯)→C1(Ω¯)×C1(Ω¯) by(54)Tk,u,v=-△-1λ-2uλb0-k+1uv+-△-1u2+buλ-uvuλ-u+mv-bv-v+uλ-uvuλ-u+mv=Kku,v+Rk,u,vThen K(k) is a linear compact operator and the Frechlet derivative R(u,v)(k,0,0)=0. Let H=I-T and then H(k,u,v)=0 with 0≤u≤uλ and v≥0 if and only if (k,uλ-u,v) is a nonnegative solution of (1).
We must calculate the indexi(T(k,·),0) when k is close to k0≔1-λ1. This index is equal to (-1)β, where β is the sum of the algebraic multiplicities of eigenvalue of K(k)>1.
Suppose thatμ>0 is an eigenvalue of K(k). Then there exists a nonzero function v such that(55)-μ△v-v=-kvinΩ,v=0on∂Ω.i.e., -k is an eigenvalue of (55). Conversely, if μ≥1 and -k is an eigenvalue of (55) with corresponding eigenfunction v, then (u,v) is an eigenfunction of K(k) corresponding to the eigenvalue μ, where u is the unique solution of(56)-μ△u-λu+2uλu=bvinΩ,u=0on∂Ω.Since all eigenvalues of -△-λ+2uλ are positive and μ≥1, it follows that -△-λ+2uλ is invertible. The eigenvalues of (55) form an increasing sequence r1(μ)<r2(μ)≤r3(μ)≤⋯ and μ→ri(μ) is a continuous increasing function. Thus μ≥1 is an eigenvalue of K(k) if and only if k=ri(μ) for some μ. Clearly ri(1)=λi(-1).
(i) Supposek>k0⇔-k<-k0 and μ≥1 is an eigenvalue of K(k). Then -k is an eigenvalue of (55) and -k> the least eigenvalue of -△-1, i.e., -k>λ1(-1)=λ1-1. But -k<-k0=λ1-1, a contradiction. Hence K(k) has no eigenvalues >1 and so i(T(k,·),0)=1.
(ii) Now suppose-r2(1)<k<k0, i.e., r2(1)>-k>-k0=r1(1). Since μ→r1(μ) is increasing with limμ→∞⁡r1(μ)=∞, there exists a unique μ>1 (μ1 say) such that -k=r1(μ1). Since -k<r2(1), it follows that -k<ri(μ) for i=2,3,⋯ and μ≥1. Thus μ1 is the only eigenvalue of K(k) which is greater than 1. We now to show that μ1 is a simple eigenvalue of K(k). The discussion above shows that N(K-μ1I)=spanϕ,ψ, where ψ is the principal eigenfunction corresponding to the eigenvalue -k of(57)-μ1△v-v=-kvinΩ,v=0on∂Ω.and ϕ=(-μ△-λ+2uλ)-1ψ. Thus dim⁡N(K(k)-μ1I)=1. Suppose that (ϕ,ψ)∈R(K(k)-μ1I). Then there exists v such that(58)-μ1△v+kv-v=-△ψ=μ1-1-kψ+ψonΩ.Multiplying by ψ and integrating over Ω shows that ψ=0, which is impossible. Hence R(K(k)-μ1I)∩N(K(k)-μ1I)=0 and so μ1 is a simple eigenvalue of K(k). Thus i(T(k,·),0)=-1 whenever k<k0.
Therefore Theorem 3.2 [24] can be applied to T.
Let(59)P1=u∈CΩ¯:ux>0forx∈Ωand∂u∂nx<0forx∈∂ΩP=k,u,v:k∈Randu,v∈P1By arguments similar to those used in Section 4 of [24], it can be proved that there exists a continuum C in R×C(Ω¯)×C(Ω¯) emanating from (k0,uλ,0) such that(i)
if(k,u,v)∈C, then (uλ-u,v)=T(k,uλ-u,v);(ii)
if(k,u,v)∈C and u,v≥0, then (k,u,v) is a solution of (1).(iii)
close to the bifurcation point(k0,uλ,0), C consists of the points (k,u,v) on the curve given by Lemma 13.
ClearlyC⊂P in a neighbourhood of the bifurcation point (k0,uλ,0).Theorem 15.
Assumeλ≠λ1+b/m(i)
If(k,u,v)∈C-k0,uλ,0, then u,v∈P1, i.e., C-{(k0,uλ,0)}⊂P;(ii)
C is unbounded in R×C(Ω¯)×C(Ω¯).Proof.
(i) Suppose thatC contains a point (k,u,v)≠(k0,uλ,0) which lies outside of P. Then there exists a point (k^,u^,v^)∈C-k0,uλ,0∩∂P which is the limit of a sequence of points {(kn,un,vn)} in C∩P. As (k^,u^,v^)∈∂P, either u^∈∂P1 or v^∈∂P1.
(1) Supposev^∈∂P1. Then v^≥0 for x∈Ω and either v^(x)=0 for some x∈Ω or ∂v/∂n(x)=0 for some x∈∂Ω. It follows that(60)-△v^+M+k-v^u^+mv^v^=μv^≥0inΩwhere M is a constant chosen sufficient large so that the term in the square bracket is positive for all x∈Ω. It follows from the strong maximum principle that v^≡0. A similar argument shows that if u^∈∂P1, then u^≡0. Thus u^≡0 or v^≡0.
(2) Supposeu^≡0 and v^≡0. Then (k^,u^,v^)=(k^,0,0), and so (k^,u^,v^) lies on the trivial branch of solutions S0. The only trivial nonnegative solutions which are close to S0 lie on the semitrivial S1, and so there cannot exist a sequence in C∩P converging to (k^,u^,v^). Therefore it is impossible that both u^ and v^ are identically zero.
(3) Supposeun→u^≡0, then vn↛0 and we find from the first equation of (1)(61)-△ununCΩ¯=ununCΩ¯λ-un-bununCΩ¯vnun+mvnThen by Lp- theory and bootstrapping arguments, ∃ϕ>0 such that
lim n → ∞ ⁡ u n / u n C ( Ω ¯ ) = ϕ and(62)-△ϕ=λ-bmϕ,ϕ∂Ω=0.which is impossible, since λ-b/m≠λ1.
(4) Suppose thatv^≡0. Then u^≠0,(k^,u^,v^)∈S1 and there bifurcate from (k^,u^,v^) nontrivial, nonnegative solutions. While we have shown that k0 is the unique bifurcation value for which positive solutions bifurcate from S1, then k^=k0. That is impossible.
Therefore, if(k,u,v)∈C-(k0,uλ,0), then (k,u,v)∈P.
(ii)C must satisfy one of the three alternatives discussed before Theorem 15. Because of (i) above, C contains no pair of points of the form (k,uλ-u,v) and (k,uλ+u,-v) and C cannot join up with another bifurcation point of the form (k,uλ,0) on S1. Hence C joins (k,uλ,0) to ∞; i.e., C is unbounded.
Based on above preparations, we have the bifurcation results as follows.Theorem 16.
Assume(63)k>-λ,λ>λ1andλ≠λ1+km.Then there exists an unbounded component C+=C-k0,uλ,0 of the set of positive solutions of (1) such that (k0,uλ,0)∈C+¯ and (k,uλ,0)∉C+¯ if k≠k0. i.e., PkC+=(-λ1,1-λ1), where Pk stands for the projection operator into the k- component of the term.


## 4. Asymptotic Behavior Analysis of Positive Solutions of (1)

In this section, we give the sketch of asymptotic behavior for positive solution as the parameterk→-λ1.Suppose (63) and let (u,v) be a coexistence of (1). Then, since u≤uλ≤λ, there exists a constant M such that uC(Ω¯)≤M. Hence owing to Lemma 6, there exists a constant C(k) such that(64)vCΩ¯≤1b-△+k-1λλ+kCΩ¯≔Ck.Note that(65)limk→-λ1⁡Ck=∞.Theorem 17.
Supposeλ-b/m>λ1. Let {kn} be a sequence such that(66)-λ1<kn<1-λ1,limn→∞⁡kn=-λ1.For each n≥1, let (kn,un,vn) be a coexistence of (1); then(67)limn→∞⁡vn=∞uniformlyinanycompactsubsetofΩand(68)limn→∞⁡un=u0uniformlyonΩ¯,where u0 is the unique positive solution of(69)-△u=λ-bmu-u2,inΩu=0,on∂ΩProof.
By the global bifurcation theory or Theorem15, we get vnC(Ω¯)→∞. From the second equation of (1)(70)-△vnvnCΩ¯=-knvnvnCΩ¯+unun+mvnvnvnCΩ¯,inΩvn∂Ω=0.Since 0<unvn/un+mvn≤un/m≤λ/m in Ω¯, by using Lp- regularity theory and Schauder bootstrapping technique (see [17] Chapter 5), ∃ϕ>0 in Ω such that vn/vnC(Ω¯)→ϕ. In fact, ϕ is the principal eigenfunction corresponding to λ1. This implies that for any compact subset Ω0 of Ω, vn→∞ uniformly in Ω0.
On the other hand, from the first equation of (1)(71)-△un=unλ-un-bunun/vnCΩ¯+mvn/vnCΩ¯vnvnCΩ¯,inΩun∂Ω=0.Similar to arguments above, ∃u0>0 in Ω, such that un→u0 on Ω¯ uniformly and u0 satisfies(72)-△u0=u0λ-bm-u02,inΩu0∂Ω=0.



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