On the symbiotic Lotka–Volterra model with diffusion and transport effects

ON THE SYMBIOTIC LOTKA-VOLTERRA MODEL WITH DIFFUSION AND TRANSPORT EFFECTS M. Delgado1, J. L´ opez-G´ omez2and A. Su´ arez1 1Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla C. Tarfia s/n. 41012-Sevilla, Spain 2Departamento de Matem´atica Aplicada Universidad Complutense 28040-MADRID, Spain Abstract. In this work we analyze the existence, stability and multiplicity of coexistence states for a symbiotic Lotka-Volterra model with general diffusivities and transport effects. Global bifurcation theory, blowing up arguments for a priori bounds, singular perturbation results, singularity theory and fixed point index in cones are among the techniques used to get our results and to explain the drastic change of behavior exhibited by the dynamics of the model between the cases of weak and strong mutualism between the species. Our methodology works out to treat much more general classes of symbiotic models. AMS Subject Classification: 35K57, 35B25, 35B32, 35B45, 35B50. Key words and phrases: Blowing up for a priori bounds in systems. Local and global bifurcation theory. Singularity theory. Fixed point index in cones. Singular perturbations. 1. Introduction. In this paper we analyze the existence, multiplicity and stability of coexistence states for the following problem L1u=λu −a(x)u2+b(x)uv L2v=µv −d(x)v2+c(x)uv in Ω ,(1.1a) u=v= 0 on ∂Ω,(1.1b) Typeset by A M S-T EX Typeset by A M S-T EX 1 2 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ where Ω is a bounded domain of RNwith boundary ∂Ω of class C2regularity, Lk, k= 1 ,2 are two second order uniformly elliptic operators of the form Lk=− N X i,j=1 aijk(x)∂i∂j+ N X j=1 bjk(x)∂j+ck(x)k= 1 ,2,(1.2) with aijk ∈C(Ω) , bjk , ck∈L∞(Ω) , i , j ∈ {1, ..., N}, k ∈ {1,2},(1.3) and a,b,c,d∈C(Ω) satisfy a(x)>0, d(x)>0, for each x∈Ω, and b≥0, c≥0 in Ω, b6= 0, c6= 0; λ,µ∈Rwill be regarded as bifurcation parameters. Under these assumptions, (1.1) provides us with a model for symbiotic species, where Ω is the inhabiting region, u(x) and v(x) are the densities of each of the species, a(x) and d(x) describe the limiting effects of crowding in each population, b(x) and c(x) are the interaction rates between the species, the operators Lk−ck(x), k= 1,2, measure the diffusivities and the external transport effects of the species, and λ−c1(x), µ−c2(x) are the growth rates of the species, positive on favorable regions and negative on unfavorable ones. In this model we are assuming that Ω is fully surrounded by inhospitable areas, because both population densities are subject to homogeneous Dirichlet boundary conditions. In this work our attention will be focused into the problem of analyzing the existence, stability and multiplicity of the non-negative solution couples (u, v) of (1.1). Due to the structure of (1.1) and thanks to the strong maximum principle, if (u, v) is a solution of (1.1) with u6= 0 (resp. v6= 0), then u(resp. v) is strongly positive in the sense of Section 2. Therefore, (1.1) admits three types of non-negative component-wise solution couples. Namely, the trivial one, (0,0); those with one component positive and the other zero, (u, 0) or (0, v), referred as the semi-trivial positive solutions, and those with both components positive, the coexistence states. The symbiotic model has attracted much less attention in the literature than its competing and predator-prey counterparts, due basically to the absence of a priori bounds for the coexistence states in high spatial dimensions (N≥6) under strong mutualism (bc −ad large). This lack of a priori bounds was observed originally in [16], where it was shown that the positive solutions of the parabolic problem associated with (1.1) may blow up in finite time when L1=L2=−∆ and bc > ad, and in [21], where it was shown that if in addition λ=µ, then the coexistence states of (1.1) are given by the positive solutions of −∆w=λw +w2in Ω , w|∂Ω= 0 ,(1.4) and that thanks to the results of [13], (1.4) possesses uniform a priori bounds in any compact subinterval of λif, and only if, 2 <N+2 N−2, i.e. if N≤5. The absence of a priori bounds for the coexistence states of (1.1) makes very involved the problem of finding out global sufficient conditions for the existence of a coexistence SYMBIOTIC SPECIES 3 state, since most of the technical tools available to attack this kind of problems involve either degree theory, i.e. global bifurcation theory, or monotonicity techniques, where the existence of a priori bounds is needed. Nevertheless, although most of the attention has been focused into the very special case when L1=L2=−∆ and a,b,c,dare constants, in recent years some substantial progress has been carried out into the analysis of these problems. The study of symbiotic species actually started in [19], where it was constructed monotonic sequences which approximate the solutions of (1.1). In [17] the method of sub and supersolutions for systems, coming from [28], was used to show that if λ > σ1and µ>σ1, then (1.1) possesses a coexistence state if, and only if, bc < ad, where σ1is the principal eigenvalue of −∆ in Ω under homogeneous Dirichlet boundary conditions. This result was generalized in [20] to cover some more general classes of symbiotic kinetics. The first global result about the existence of coexistence states for the symbiotic model was found in [25] by using global bifurcation theory, where it was shown that if any of the semi-trivial positive solutions is linearly unstable, then the model possesses a coexistence state provided bc < ad; global in the sense that if some of the semi-trivial states is stable, then there are choices of the several parameters involved in the setting of (1.1) for which the model does not admit a coexistence state (cf. Section 11 here in for further details). Almost simultaneously, in [35] was found the same result included in [25], but this time using the method of sub and supersolutions. More recently, allowing the coefficients of the model to vary, the technique of decoupling was shown to work out to get the same result as in [25] and [35], [5]. In [21] and [23] fixed point index in cones and global bifurcation theory were shown to work out to get the corresponding results for wider classes of models. Although the global results of [21] work out to show that a global continuum of coexistence states emanates from each of the surfaces of semi-trivial positive solutions along their curves of change of stability in the space of the parameters (λ, µ), the first global result in the case bc > ad was found in [27], where it was shown that if N≤5 and some of the semi-trivial positive solutions is linearly stable, then the model possesses a coexistence state. We point out that this result was obtained for the special case when L1=L2=−∆ and all coefficients are constant. In [27], the blowing up argument of [13] was adapted to show the existence of a priori bounds in case N≤5 and then the fixed point index in cones was used to complete the proof. In this work we extend and complete all the previous features, obtaining in addition some optimal non-existence and multiplicity results for all ranges of the parameters in the general setting of (1.1), and in addition we analyze the bifurcation equations of (1.1) at (λ, µ) = (σΩ 1[L1], σΩ 1[L2]). Hereafter, given an elliptic operator L,σΩ 1[L] will stand for the principal eigenvalue of Lin Ω under homogeneous Dirichlet boundary conditions. Our analysis of the bifurcation equations at (σΩ 1[L1], σΩ 1[L2]) explains the drastic change of behavior of the global continuum of coexistence states as some of the interactions between the species, bor c, grows acrossing the critical value given by Theorem 10.1 in Section 10. Namely, the global manifold of coexistence states linking the two surfaces of semi-trivial positive solutions turns backwards in the parameter space (λ, µ) changing 4 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ its relative position with respect to each the surfaces of semi-trivial solutions, as the amplitude of b, or c, grows. To state our main results, we have to introduce some of notation. Given λ > σΩ 1[L1] (resp. µ > σΩ 1[L2]), (θλ,0) (resp. (0, θµ)) will stand for the unique semi-trivial solution of (1.1) of the form (u, 0), u > 0 (resp. (0, v), v > 0). Moreover, for any f∈L∞(Ω) we denote fL:= ess inf Ωf , fM:= ess sup Ω f . Among our main results we list the following ones: •If bMcM< aLdLand any of the semitrivial positive solutions is linearly unstable, then (1.1) possesses a coexistence state. If in addition λ > σΩ 1[L1] and µ > σΩ 1[L2], then there exists I0>0 such that if min {bM, cM}< I0, then the coexistence state is unique and exponentially asymptotically stable. •If bMcM< aLdLand for (λ, µ) = (λ0, µ0) some of the semitrivial positive solutions is linearly stable and (1.1) possesses a coexistence state, then it possesses a coexistence state for each (λ, µ) satisfying λ≥λ0,µ≥µ0, and at least two coexistence states if λ > λ0,µ > µ0and some of the semi-trivial positive solutions is linearly stable. •If bMcM< aLdL, then for each λ∈R, there exists µext(λ)∈Rsuch that (1.1) does not admit a coexistence state if µ≤µext(λ). Similarly, for each µ∈R, there exists λext(µ)∈Rsuch that (1.1) does not admit a coexistence state if λ≤λext(µ). •If L1=L2,N≤5, bLcL−aMdM>max {aMbM−aLbL, dMcM−dLcL},(1.5) and some of the semitrivial positive solutions is linearly stable, then (1.1) possesses a coexistence state. •Assume that L1=L2,N≤5, (1.5), and that there exists (λ, µ)=(λ0, µ0) for which (1.1) possesses a coexistence state being any of the semi-trivial states linearly unstable. Then, (1.1) possesses a coexistence state for each (λ, µ) satisfying λ≤λ0and µ≤µ0, and at least two coexistence states if λ < λ0,µ < µ0and any of the semi-trivial states is linearly unstable. •Assume L1=L2,N≤5 and (1.5). Then, for each λ∈Rthere exists µext(λ)∈R such that (1.1) does not admit a coexistence state if µ≥µext(λ). Similarly, for each µ∈R, there exists λext(µ)∈Rsuch that (1.1) does not admit a coexistence state if λ≥λext(µ). We now describe the distribution and contains of this paper. In Section 2 we give an extension of Theorem 2.5 in [22] to cover our general setting here in, and then use it to infer some basic monotonicity properties of principal eigenvalues. Most of these results come from Section 2 of [3]. SYMBIOTIC SPECIES 5 In Section 3 we study the single boundary value problem L1u=λ u −a(x)u2in Ω , u|∂Ω= 0 .(1.6) A particular attention is paid to the behavior of its unique positive solution as λ↑ ∞, showing that lim λ↑∞ θ[L1,λ,a] λ=a−1(1.7) uniformly on any compact subset of Ω, where θ[L1,λ,a]stands for the unique positive solution of (1.6). This result extends the corresponding singular perturbation result in Section 3 of [12] to our general setting here in, and it is the basic technical tool to get our non-existence results in Section 7. In Section 4 we characterize the attractive character of each of the semitrivial positive solutions in terms of several parameters involved in the setting of (1.1) through by the principal eigenvalues of some related second order elliptic operators. Then, we analyze the shape of the curves of change of stability in the space of the parameters (λ, µ). Section 5 is devoted to the abstract results concerning the existence of global continua of coexistence states emanating from the surfaces of semitrivial positive solutions along their respective curves of change of stability. The analysis throughout this work shows that these results are optimal, reducing the problem of finding out coexistence states for (1.1) to the problem of finding out a priori bounds for the component-wise positive solutions of (1.1). The methodology adopted in this section comes from the abstract theory developed in [21] for general systems with two species. In Section 6 we analyze the existence of coexistence states for the case of small interaction coefficients. How small should they are is measured by condition bMcM< cLdL.(1.8) Precisely, we will find out some non-existence results and then we will use the theory of Section 5 to show that (1.1) possesses a coexistence state if any of the semitrivial positive solutions is linearly unstable. The analysis of Section 11 for the case of constant coefficients will show the optimality of our results. In Section 7 we analyze the existence of coexistence states for the case of large interaction coefficients. How large should they are is measured by condition (1.5). Notice that if any coefficient is assumed to be constant, then (1.5) becomes into bc > ad . (1.9) By technical reasons for most of the results in this section we need assuming that L1=L2, assumption needed in all previous references. We begin the section giving a necessary condition for the existence of coexistence states which is totally new even for the simplest symbiotic models where L1=L2=−∆ and any coefficient is constant. Namely, if (0, θ[L1,µ,d]) (resp. (θ[L1,λ,a])) is linearly unstable, then (1.1) does not admit a 6 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ coexistence state if µ(resp. λ) is sufficiently large (cf. Theorem 7.1 here in). This nonexistence result is based upon (1.7), finding out the behavior of an eventual sequence of coexistence states for µ, or λ, large. Then, we adapt the blowing up argument of [13] to show that uniform a priori bounds for the coexistence states of (1.1) are available if N≤5. We should point out that our blowing up argument differs substantially from the corresponding argument of [27] and that we need a general Liouville type result much sharper than the corresponding result in [27]. These additional difficulties coming from the fact that in this work we are dealing with a general elliptic operator and with spatially varying coefficients. We refer to Section 7 for further details. Bringing together the non-existence results and the a priori bounds, it follows from the global results in Section 5 that if any of the semitrivial positive solutions is linearly stable, then (1.1) possesses a coexistence state. In Section 8 we use the abstract theory of [2] to show that the method of sub and supersolutions is valid for (1.1). Then, we use it to analyze the structure of the set of λ’s and µ’s for which (1.1) possesses a coexistence state and to get our multiplicity results, those already stated in the list above. In Section 9 we obtain simple readily computable conditions in terms of the several coefficients involved in the setting of (1.1) ensuring that (1.1) has a unique stable coexistence state, and then consider the parabolic problem associated with (1.1) to show that there is a dense subset of the set of initial data such that any solution starting there in converges to the coexistence state as time grows to infinity. In Section 10, considering (λ, µ) as the main bifurcation parameters we describe the possible local bifurcation diagrams near the co-dimension two singularity (λ, µ) = (σΩ 1[L1], σΩ 1[L2]) . For this, we apply the general results of [10] where one of the authors developed a singularity theory to deal with this type of two parameter bifurcation problems. Finally, in Section 11 we restrict ourselves to the original Lotka-Volterra symbiotic model with diffusion, L1=L2=−∆ and a,b,c,dconstants, for which we can give some sharper existence and non-existence results and can go further in the analysis of the bifurcation equation around the co-dimension two bifurcation point, obtaining in addition some global results about the nature of the local bifurcations to coexistence states from the surfaces of semitrivial positive solutions along their curves of change of stability. As a result from this analysis we can explain the drastic change of behavior of the global manifold of coexistence that links the two surfaces of semitrivial positive solutions along their curves of change of stability as bc acrosses the critical value ad passing from values where bc < ad to values where bc > ad. 2. The maximum principle. Main properties of the principal eigenvalues. In this section we give an extension of Theorem 2.5 in [22] to cover our setting here and then we infer some basic properties of principal eigenvalues which will be used throughout SYMBIOTIC SPECIES 7 this paper. We will consider a uniformly elliptic operator of the form L=− N X i,j=1 aij(x)∂i∂j+ N X j=1 bj(x)∂j+e(x),(2.1) with aij ∈C(Ω) , bj, e ∈L∞(Ω) , i , j ∈ {1, ..., N},(2.2) and use the natural product order on Lp(Ω) ×Lp(∂Ω). Recall that p > N implies W2,p(Ω) ⊂C2−N p−ε(Ω) with compact imbedding for all ε > 0 and that each u∈ W2,p(Ω) is a.e. twice classically differentiable in Ω (e.g. Theorem VIII.1 of [33]). Suppose that p>N. Then u∈W2,p(Ω) is said to be strongly positive if u(x)>0 for x∈Ω and ∂nu(x)<0 for all x∈∂Ω with u(x) = 0, where nis the outward unit normal on ∂Ω. The operator Lis said to satisfy the strong maximum principle in Ω if p > N,u∈W2,p(Ω), and (Lu, u)>(0,0) imply that uis strongly positive. Consider the eigenvalue problem Lu=σu in Ω , u = 0 on ∂Ω,(2.3) in W2,p(Ω) and let Lpdenote the closure of the operator L|W2,p(Ω)∩W1,p 0(Ω) in Lp(Ω). Then, (2.3) can be reformulated as the eigenvalue equation Lpu=σu in Lp(Ω) .(2.4) It is an easy consequence of standard regularity theory that the spectrum and the eigenspaces of Lpare independent of p > N. Moreover, from the strong maximum principle and the generalization of the Krein Rutman Theorem of [32] together with Theorem 3 in [29], the following result holds (cf. Section 2 of [3]). Theorem 2.1. There exists a least eigenvalue of (2.4), denoted by σΩ 1[L]and called principal eigenvalue of Lin Ω. This eigenvalue is simple and possesses a unique eigenfunction, up to multiplicative constants, which can be taken positive, the so called principal eigenfunction of Lin Ω. Moreover, the principal eigenfunction is strongly positive and σΩ 1[L]is the only eigenvalue of (2.4) possessing a positive eigenfunction. Furthermore, any other eigenvalue σof (2.4) satisfies Re σ > σΩ 1[L] and (Lp+ν)−1∈ L(Lp(Ω)) is positive, compact and irreducible for ν > −σΩ 1[L]. If p > N a function u∈W2,p(Ω) is said to be a positive supersolution of Lin Ω if u≥0 and (Lu, u)≥(0,0). If in addition (Lu, u)>(0,0), then it is said that uis a positive strict supersolution. Similarly, a function u∈W2,p(Ω) is said to be a positive subsolution of Lin Ω if u≥0 and (Lu, u)≤(0,0). If in addition (Lu, u)<(0,0), then it is said that uis a positive strict subsolution. ¿From the strong maximum principle it is easily seen that any positive strict supersolution is strongly positive. Moreover, the following characterization of the strong maximum principle holds (cf. Theorem 2.5 in [22] and Theorem 2.4 in [3]). 8 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Theorem 2.2. The following assertions are equivalent: (i) σΩ 1[L]>0; (ii) Lpossesses a positive strict supersolution in Ω; (iii) Lsatisfies the strong maximum principle in Ω. From this characterization we can readily get the following properties of σΩ 1[L] which will be used throughout this work. For selfadjoint operators, these properties are easily obtained from the variational characterization of the principal eigenvalue. Theorem 2.3. (i) Monotonicity with respect to the potential: Let V1,V2∈L∞(Ω) such that V1≤V2and V1< V2on a set of positive measure. Then, σΩ 1[L+V1]< σΩ 1[L+V2].(2.5) (ii) Continuity with respect to the potential: If Vn∈L∞(Ω),n≥1is a sequence of potentials such that lim n→∞ kVn−Vk∞,Ω= 0 , then lim n→∞ σΩ 1[L+Vn] = σΩ 1[L+V]. (iii) If Ω1is a proper subdomain of Ωwith ∂Ω1of class C2, then σΩ1 1[L]> σΩ 1[L].(2.6) Proof. (i) Let ϕ1be the principal eigenfunction associated with σΩ 1[L+V1]. Then, (L+V2)ϕ1=σΩ 1[L+V1]ϕ1+ (V2−V1)ϕ1> σΩ 1[L+V1]ϕ1 on a set of positive measure, and hence ϕ1is a positive strict supersolution of L+V2− σΩ 1[L+V1]. Thus, thanks to Theorem 2.2, we find that σΩ 1[L+V2−σΩ 1[L+V1]] >0. This relation implies (2.5). (ii) For any ε > 0 there exists N0∈Nsuch that V−ε≤Vn≤V+ε∀n≥N0. Thus, by Part (i) we find that σΩ 1[L+V]−ε≤σΩ 1[L+Vn]≤σΩ 1[L+V] + ε . This completes the proof. (iii) Let ϕdenote the principal eigenfunction associated with σΩ 1[L]. Then, (L−σΩ 1[L])ϕ= 0 in Ω1and ϕ > 0 on ∂Ω1. Thus, ϕis a positive strict supersolution of L−σΩ 1[L] in Ω1 and hence, it follows from Theorem 2.2 that σΩ1 1[L−σΩ 1[L]] >0. This relation implies (2.6). ¤ SYMBIOTIC SPECIES 9 3. The logistic equation. The semi-trivial positive solutions of (1.1) are given by the positive solutions of a semilinear elliptic boundary value problem of the form Lw=γw −f(x)w2in Ω , w= 0 on ∂Ω,(3.1) where Lis a second order uniformly elliptic operator of the form (2.1) with coefficients satisfying (2.2), γ∈R, and f∈C(Ω) satisfies f(x)>0 for each x∈Ω. If p > N and w∈W2,p(Ω) ∩W1,p 0(Ω) is a positive solution of (3.1), then (L+fw)w=γw and thanks to Theorem 2.1 we have that γ=σΩ 1[L+fw] (3.2) and that wis strongly positive. Therefore, w(x)>0 for each x∈Ω and ∂nw(x)<0 for each x∈∂Ω. The following result characterizes the existence of positive solutions for (3.1). Theorem 3.1. If p>N, then the problem (3.1) possesses a positive solution in W2,p(Ω) ∩W1,p 0(Ω) if, and only if, γ > σΩ 1[L]. Moreover, it is unique if it exists. Let θ[L,γ,f]denote it. Then, lim γ↓σΩ 1[L]θ[L,γ,f]= 0 (3.3) uniformly in Ω. Condition (3.3) says that the positive solutions bifurcate from the trivial state w= 0 at the critical value of the parameter γ=σΩ 1[L]. This result is well known under some additional regularity conditions on the several coefficients involved in the model setting, e.g. see [14]. By the sake of completeness we shall give a short self-contained proof of it. Proof of Theorem 3.1. Let wbe a positive solution of (3.1). Then, thanks to Theorem 2.1, we have (3.2) and hence Theorem 2.3(i) implies γ=σΩ 1[L+fw]> σΩ 1[L]. Therefore, γ > σΩ 1[L] is necessary for the existence of a positive solution. Assume γ > σΩ 1[L]. It is easily seen that large positive constants provide us with supersolutions of (3.1) and that if ϕ > 0 stands for the principal eigenfunction associated with σΩ 1[L], then εϕ provide us with arbitrarily small positive subsolutions if ε > 0 is sufficiently small. Therefore, (3.1) possesses at least a positive solution for each γ > σΩ 1[L]. We point out that the method of sub and supersolutions works out thanks to the validity of the strong maximum principle. 16 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ is an eigenvalue to a positive eigenfunction, say ψ, of the second equation of (4.5). Since τ1<0, (4.7) implies σΩ 1[L1+ 2aθ[L1,λ,a]−λ−τ1]>0, and therefore, thanks to the strong maximum principle, the first equation of (4.5) with τ=τ1possesses a unique solution. Namely, u= (L1+ 2aθ[L1,λ,a]−λ−τ1)−1(bθ[L1,λ,a]ψ). Therefore, under condition (4.2) τ1<0 is an eigenvalue of (4.5) and hence the state (θ[L1,λ,a],0) is linearly unstable. Finally if we assume (4.3), it is easily seen that τ1= 0 is an eigenvalue of (4.5) and that any other eigenvalue has positive real part. Therefore, under condition (4.3) the state (θ[L1,λ,a],0) is linearly neutrally stable. The results concerning with the other semi-trivial state follow by symmetry interexchanging L1,λ,aand bby L2,µ,dand c, respectively. ¤ By Proposition 4.1 we shall refer to the curve (4.3) in the (λ, µ)-plane as the curve of change of stability of the semi-trivial positive solution (θ[L1,λ,a],0). Similarly, the curve (4.4) will be refereed as the curve of change of stability of (0, θ[L2,µ,d]). The following result provides us with the global behavior of these curves. Proposition 4.2. The mapping F(λ)defined by F(λ) := σΩ 1[L2−c(x)θ[L1,λ,a]], λ > σΩ 1[L1],(4.8) is continuous strictly decreasing and satisfies lim λ↓σΩ 1[L1]F(λ) = σΩ 1[L2],lim λ↑∞ F(λ) = −∞.(4.9) Similarly, the mapping G(µ)defined by G(µ) := σΩ 1[L1−b(x)θ[L2,µ,d]], µ > σΩ 1[L2],(4.10) is continuous strictly decreasing and satisfies lim µ↓σΩ 1[L2]G(µ) = σΩ 1[L1],lim µ↑∞ G(µ) = −∞.(4.11) Proof. The continuity and monotonicity of F(λ) can be easily obtained from Theorem 3.1, Corollary 3.3 and Theorem 2.3(ii). The first relation of (4.9) follows from (3.3) and Theorem 2.3(ii). We now show the second relation of (4.9). Since c∈C(Ω), c≥0, c6= 0, there exists a ball Bwith B⊂Ω such that cL:= min B c > 0. SYMBIOTIC SPECIES 17 On the other hand, by Theorem 3.4 lim λ↑∞ θ[L1,λ,a] λ=a−1uniformly in B , and hence, there exists λ0such that for λ > λ0 θ[L1,λ,a]>λ 2 maxBain B . Therefore, Theorem 2.3 implies F(λ)< σB 1[L2−c(x)θ[L1,λ,a]]< σB 1[L2]−cL 2 maxBaλ for each λ > λ0. This completes the proof. The same argument shows the corresponding properties of G(µ). ¤ By Proposition 4.2 the curves of change of stability of the semi-trivial positive solutions meet at (σΩ 1[L1], σΩ 1[L2]). The next result provides us with the tangents of these curves and their concavity or convexity character at this co-dimension two singularity. Lemma 4.3. Let ϕj, ϕ∗ jbe the principal eigenfunctions associated with Ljand L∗ j, respectively, j= 1,2, where ∗stands for the adjoint and ZΩ ϕ2 j= 1 ,ZΩ ϕjϕ∗ j= 1 , j = 1 ,2. Then, θ[L1,λ,a]= (λ−σΩ 1[L1])m−1 a,1ϕ1+ (λ−σΩ 1[L1])2m−2 a,1U1+O((λ−σΩ 1[L1])3), θ[L2,µ,d]= (µ−σΩ 1[L2])m−1 d,1ϕ2+ (µ−σΩ 1[L2])2m−2 d,1U2+O((µ−σΩ 1[L2])3),(4.12) σΩ 1[L2−c(x)θ[L1,λ,a]] = σΩ 1[L2]−mc,a(λ−σΩ 1[L1]) −Mc,a(λ−σΩ 1[L1])2 +O((λ−σΩ 1[L1])3), σΩ 1[L1−b(x)θ[L2,µ,d]] = σΩ 1[L1]−mb,d(µ−σΩ 1[L2]) −Mb,d(µ−σΩ 1[L2])2 +O((µ−σΩ 1[L2])3), (4.13) as λ↓σΩ 1[L1]and µ↓σΩ 1[L2], where ma,1:= ZΩ aϕ2 1ϕ∗ 1>0, md,1:= ZΩ dϕ2 2ϕ∗ 2>0, mc,a := m−1 a,1ZΩ cϕ1ϕ2ϕ∗ 2, mb,d := m−1 d,1ZΩ bϕ2ϕ1ϕ∗ 1, 18 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Mc,a := ZΩ c(x)(m−1 a,1ψ2ϕ1+m−2 a,1U1ϕ2)ϕ∗ 2−mc,a ZΩ ψ2ϕ∗ 2. Mb,d := ZΩ b(x)(m−1 d,1ψ1ϕ2+m−2 d,1U2ϕ1)ϕ∗ 1−mb,d ZΩ ψ1ϕ∗ 1, and we have denoted by βi,i= 1,2, and ψi,i= 1,2, the unique solutions of the following linear problems in Ωunder homogeneous Dirichlet boundary conditions (L1−σΩ 1[L1])β1=ma,1ϕ1−a(x)ϕ2 1,ZΩ β1ϕ1= 0 , (L2−σΩ 1[L2])β2=md,1ϕ2−d(x)ϕ2 2,ZΩ β2ϕ2= 0 , (L1−σΩ 1[L1])ψ1= (−mb,d +m−1 d,1b(x)ϕ2)ϕ1,ZΩ ψ1ϕ1= 0 , (L2−σΩ 1[L2])ψ2= (−mc,a +m−1 a,1c(x)ϕ1)ϕ2,ZΩ ψ2ϕ2= 0 , U1:= β1−ma,2 ma,1·ϕ1, U2:= β2−md,2 md,1·ϕ2, where ma,2:= 2 ZΩ aβ1ϕ1ϕ∗ 1−ma,1ZΩ β1ϕ∗ 1, md,2:= 2 ZΩ dβ2ϕ2ϕ∗ 2−md,1ZΩ β2ϕ∗ 2. Proof. The relations (4.12) follow from the main theorem of [7] applied to (3.1) with (L, γ, f)=(L1, λ, a) and (L, γ, f)=(L2, µ, d). Assume (L, γ, f)=(L1, λ, a). For λ≃σΩ 1[L1], the semi-trivial branch (λ, θ[L1,λ,a]) may be parametrized by two analytic functions λ(s) = σΩ 1[L1] + ∞ X j=1 λjsj, θ[L1,λ,a](s) = sϕ1+ ∞ X j=1 ujsj+1 , s ≃0, where ZΩ ujϕ1= 0 , j ≥1.(4.14) Substituting these expansions into (3.1) and identifying the terms of order two and three in syields (L1−σΩ 1[L1])u1=λ1ϕ1−a(x)ϕ2 1in Ω , u1|∂Ω= 0 ,(4.15a) (L1−σΩ 1[L1])u2=λ1u1+λ2ϕ1−2a(x)ϕ1u1in Ω , u2|∂Ω= 0 ,(4.15b) respectively. From (4.14) and the Fredholm alternative applied to (4.15) it is easily seen that λ1=ma,1, u1=β1, λ2=ma,2. SYMBIOTIC SPECIES 19 To obtain the first relation of (4.12), it suffices calculating sas a function of λfrom λ(s). Doing so, we obtain that s(λ) = m−1 a,1(λ−σΩ 1[L1]) −ma,2 m3 a,1 (λ−σΩ 1[L1])2+O((λ−σΩ 1[L1])3). Indeed, substituting this expansion into the expansion of θ[L1,λ,a](s), the first relation of (4.12) get shown. By standard perturbation results (cf. [18]), the principal eigenvalues in the left hand sides of (4.13) vary analytically with λand µ. Thus, there exist Kj∈R,j= 1 ,2, such that σΩ 1[L2−c(x)θ[L1,λ,a]] = σΩ 1[L2] + K1(λ−σΩ 1[L1]) +K2(λ−σΩ 1[L1])2+O((λ−σΩ 1[L1])3).(4.16) Moreover, if Ψ(λ)>0 stands for the principal eigenfunction of σΩ 1[L2−c(x)θ[L1,λ,a]], i.e. (L2Ψ(λ)−c(x)θ[L1,λ,a]Ψ(λ) =σΩ 1[L2−cθ[L1,λ,a]]Ψ(λ) in Ω Ψ(λ) =0 on ∂Ω , (4.17) normalized so that ZΩ Ψ(λ)2= 1 ,ZΩ (Ψ(λ)−ϕ2)ϕ2= 0 ,(4.18) then Ψ(λ) admits a unique expansion of the form Ψ(λ) = Ψ0+ (λ−σΩ 1[L1])Ψ1+ (λ−σΩ 1[L1])2Ψ2+O((λ−σΩ 1[L1])3).(4.19) Using (4.18) gives Ψ0=ϕ2,ZΩ Ψjϕ2= 0 , j ≥1.(4.20) Now, substituting (4.16), (4.19) into (4.17), using (4.12), (4.20) and identifying the terms with the same order in λ−σΩ 1[L1], we find that (L2−σΩ 1[L2])Ψ1= (K1+m−1 a,1c(x)ϕ1)ϕ2,(4.21) (L2−σΩ 1[L2])Ψ2=c(x)(m−1 a,1ϕ1Ψ1+m−2 a,1U1ϕ2) + K1Ψ1+K2ϕ2.(4.22) Applying Fredholm’s alternative to (4.21) yields K1=−m−1 a,1ZΩ c(x)ϕ1ϕ2ϕ∗ 2=−mc,a ,Ψ1=ψ2. Now, substituting these values into (4.22) and applying Fredholm’s alternative gives K2=−ZΩ c(x)(m−1 a,1ϕ1ψ2+m−2 a,1U1ϕ2)ϕ∗ 2+mc,a ZΩ ψ2ϕ∗ 2=−Mc,a . 20 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ By symmetry, θ[L2,µ,d]and σΩ 1[L1−b(x)θ[L2,µ,d]] have the expansions given in the statement. The proof is completed. ¤ By (4.13), the tangents to the curves of change of stability of the semi-trivial positive solutions (4.3) and (4.4) at the singularity (σΩ 1[L1], σΩ 1[L2]) are given, respectively, by the stright lines µ=σΩ 1[L2]−mc,a(λ−σΩ 1[L1]) , λ =σΩ 1[L1]−mb,d(µ−σΩ 1[L2]) .(4.23) Close to the singularity (σΩ 1[L1], σΩ 1[L2]) the convexity or concavity of these curves is given by the sign of Mc,a and Md,b, respectively. Although in general the problem of ascertaining the sign of these quantities might be very difficult to handle with, as they depend upon some unknown solutions of certain homogeneous Dirichlet boundary value problems, there are some special cases where these signs can be easily found out, as the following result shows. Lemma 4.4. If L1=L2is a selfadjoint operator and the coefficients aand care constants, then Mc,a >0.(4.24) By symmetry, if band dare constant, then Mb,d >0. Therefore, if a,b,cand dare constant, then the curves of change of stability are concave in a neighborhood of (σΩ 1[L1], σΩ 1[L2]). Proof. Since L1=L2is a selfadjoint operator, we have that ϕ1=ϕ2=ϕ∗ 1=ϕ∗ 2. Hence, ZΩ ψ2ϕ∗ 2=ZΩ ψ2ϕ2= 0 and Mc,a := cm−1 a,1ZΩ ψ2ϕ2 1+cm−2 a,1ZΩ U1ϕ2 1.(4.25) Moreover, ma,1=aZΩ ϕ3 1, ma,2= 2aZΩ β1ϕ2 1, U1=β1−2RΩβ1ϕ2 1 RΩϕ3 1 ϕ1,(4.26) and by the uniqueness of the solution of the corresponding boundary value problem in the orthogonal complement of ϕ1, we find that ψ2=−c a2RΩϕ3 1 β1.(4.27) SYMBIOTIC SPECIES 21 Thus, substituting (4.26) and (4.27) into (4.25) gives Mc,a =−ca−2(ZΩ ϕ3 1)−2(1 + c/a)ZΩ β1ϕ2 1.(4.28) To complete the proof of (4.24), it remains to show that ZΩ β1ϕ2 1<0.(4.29) Indeed, from the β1-equation it is easily seen that ZΩ β1(L1−σΩ 1[L1])β1=−aZΩ β1ϕ2 1,(4.30) since RΩβ1ϕ1= 0. Moreover, β1changes of sign in Ω, and hence the variational characterization of σΩ 1[L1] implies that ZΩ β1(L1−σΩ 1[L1])β1>0. Therefore, (4.30) implies (4.29). This completes the proof. ¤ In Figure 1 we have represented the curves of change of stability of the semi-trivial positive solutions in the case when a,b,cand dare constant and L1=L2is selfadjoint. λ µ µ= λ= σ σΩ Ω 1 1 F G [L [L1 1 (λ) (µ) ] ] Figure 1: The curves of change of stability. 22 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ 5. The existence of unbounded continua of coexistence states. Although with less regularity on the several coefficients involved into our setting the abstract theory of [21] applies to (1.1) if the solutions of (1.1) are regarded as fixed points of a compact operator on (C1 0(Ω))2. This observation provides us with the following result, where the notations introduced in the previous sections will be kept. Theorem 5.1. Fix λ > σΩ 1[L1]and regard to µ∈Ras the bifurcation parameter. Then, the point (µ, u, v) = (σΩ 1[L2−cθ[L1,λ,a]], θ[L1,λ,a],0) is the only bifurcation point to coexistence states from the semi-trivial state (θ[L1,λ,a],0). Moreover, the maximal component (closed and connected) of coexistence states emanating from (θ[L1,λ,a],0) at µ=F(λ), say C+ (µ,u,0) ⊂R×C1 0(Ω) ×C1 0(Ω), is unbounded. Now, fix µ < σΩ 1[L2]and regard to λ∈Ras the bifurcation parameter. By Proposition 4.2 there exists a unique λµ> σΩ 1[L1]such that µ=F(λµ). Then, the point (λ, u, v) = (λµ, θ[L1,λµ,a],0) is the only bifurcation point to coexistence states from the curve (θ[L1,λ,a],0). Moreover, the maximal component (closed and connected) of coexistence states emanating from (θ[L1,λ,a],0) at λ=λµ, say C+ (λ,u,0) ⊂R×C1 0(Ω) ×C1 0(Ω), is unbounded. Similarly, if we fix µ > σΩ 1[L2]and regard to λ∈Ras the bifurcation parameter, then the point (λ, u, v) = (σΩ 1[L1−bθ[L2,µ,d]],0, θ[L2,µ,d]) is the only bifurcation point to coexistence states from the semi-trivial state (0, θ[L2,µ,d]) and the maximal component (closed and connected) of coexistence states emanating from (0, θ[L2,µ,d])at λ=G(µ), say C+ (λ,0,v)⊂R×C1 0(Ω) ×C1 0(Ω), is unbounded. Finally, fix λ < σΩ 1[L1]and regard to µ∈Ras the bifurcation parameter. By Proposition 4.2 there exists a unique µλ> σΩ 1[L2]such that λ=G(µλ). In this case, the point (µ, u, v) = (µλ,0, θ[L2,µλ,d]) is the only bifurcation point to coexistence states from the curve (0, θ[L2,µ,d])and the maximal component (closed and connected) of coexistence states emanating from (0, θ[L2,µ,d]) at µ=µλ, say C+ (µ,0,v)⊂R×C1 0(Ω) ×C1 0(Ω), is unbounded. Proof. The local bifurcations are obtained as an application of the main theorem of [7] using rather standard arguments. It remains to show that each of the continua of coexistence states emanating from the semi-trivial states are unbounded in the phase space. We shall show this for the continuum C+ (µ,u,0). The argument can be easily adapted to cover the remaining cases. By Theorem 4.1 in [21] the continuum C+ (µ,u,0) satisfies some of the following alternatives: Either SYMBIOTIC SPECIES 23 (i) C+ (µ,u,0) is unbounded in R×C1 0(Ω) ×C1 0(Ω); or (ii) there exists µ∞∈Rsuch that λ=σΩ 1[L1−bθ[L2,µ∞,d]] (5.1) and (µ∞,0, θ[L2,µ∞,d])∈closure C+ (µ,u,0) ; or (iii) there exists a positive solution ˆ θ[L1,λ,a]6=θ[L1,λ,a]of L1u=λu −au2in Ω , u|∂Ω= 0 ,(5.2) such that (σΩ 1[L1−bˆ θ[L1,λ,a]], θ[L1,λ,a],0) ∈closure C+ (µ,u,0) ; or (iv) λ=σΩ 1[L1] and (σΩ 1[L2],0,0) ∈closure C+ (µ,u,0) . Since we are assuming that λ > σΩ 1[L1], alternative (iv) is not possible. Moreover, by Theorem 3.1 θ[L1,λ,a]is the unique positive solution of (5.2) and hence, alternative (iii) is not possible either. Notice that (5.1) is not possible either, since σΩ 1[L1−bθ[L2,µ∞,d]]≤σΩ 1[L1]. Therefore, alternative (i) must occur. This completes the proof. ¤ 6. Coexistence regions for small interaction coefficients. As an easy consequence from Corollary 3.3 we obtain the following result. Lemma 6.1. Assume that bMcM< aLdL,(6.1) and that (1.1) possesses a coexistence state, say (u, v). Then,        λ >(c1)L cMbM aLdL +σΩ 1[L1]µ1−cMbM aLdL¶−bM dL (µ−(c2)L), µ >(c2)L cMbM aLdL +σΩ 1[L2]µ1−cMbM aLdL¶−cM aL (λ−(c1)L), (6.2) and        uM≤(λ−(c1)L)dL+ (µ−(c2)L)bM aLdL−bMcM , vM≤(µ−(c2)L)aL+ (λ−(c1)L)cM aLdL−bMcM . (6.3) Proof. From (1.1) it is easily seen that u=θ[L1,λ+bv,a], v =θ[L2,µ+cu,d]. 24 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Moreover, by Lemma 3.2 and Corollary 3.3 we have θ[L1,λ+bv,a]≤θ[L1,λ+bMvM,aL]≤λ+bMvM−(c1)L aL . Thus, uM≤λ+bMvM−(c1)L aL .(6.4a) Similarly, vM≤µ+cMuM−(c2)L dL .(6.4b) ¿From (6.4), relations (6.3) follow readily. Moreover, the second relation of (6.3) implies λ+bMvM≤λaLdL+bMaL(µ−(c2)L)−cMbM(c1)L aLdL−bMcM , and therefore, since θ[L1,λ+bMvM,aL]≥u > 0, we find from Theorem 3.1 that λaLdL+bMaL(µ−(c2)L)−cMbM(c1)L aLdL−bMcM > σΩ 1[L1].(6.5a) Similarly, µaLdL+cMdL(λ−(c1)L)−cMbM(c2)L aLdL−bMcM > σΩ 1[L2].(6.5b) Relations (6.2) follow readily from (6.5). This completes the proof. ¤ Note that if λand µsatisfy (6.2), then the following relations hold      λ >(c1)L−bM dL (µ−(c2)L), µ >(c2)L−cM aL (λ−(c1)L), (6.6) and therefore, the right hand sides of (6.3) are positive. Indeed, it is easily seen from Theorems 2.2, 2.3 that σΩ 1[L1] = σΩ 1[L1−c1+c1]> σΩ 1[L1−c1]+(c1)L>(c1)L.(6.7) Thus, we find from (6.1) and (6.7) that cMbM aLdL (c1)L+σΩ 1[L1](1 −bMcM aLdL )>(c1)L, SYMBIOTIC SPECIES 25 and hence, (c1)L cMbM aLdL +σΩ 1[L1]µ1−cMbM aLdL¶−bM dL (µ−(c2)L)>(c1)L−bM dL (µ−(c2)L). Similarly, (c2)L cMbM aLdL +σΩ 1[L2]µ1−cMbM aLdL¶−cM aL (λ−(c1)L)>(c2)L−cM aL (λ−(c1)L). This shows the claim above. Under assumption (6.1), (6.2) provides us with a simple readily computable necessary condition for the existence of a coexistence state. Moreover, (6.3) shows that we have a priori bounds in L∞(Ω) for the coexistence states of (1.1) uniformly on compact subsets of the parameter space (λ, µ). By the Lp-estimates of Agmon, Douglis and Nirenberg we have uniform a priori bounds in W2,p(Ω) for all p∈[2,∞). Notice that the boundary of the non-existence region given by (6.2) consists of the stright lines λ= (c1)L cMbM aLdL +σΩ 1[L1](1 −cMbM aLdL )−bM dL (µ−(c2)L), µ= (c2)L cMbM aLdL +σΩ 1[L2](1 −cMbM aLdL )−cM aL (λ−(c1)L). In Figure 2 we have represented these lines together with the curves of change of stability of semi-trivial positive solutions. µ λ µ= λ= F G (λ) (µ) Figure 2: Estimating the coexistence region. 32 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ and since this limit is positive and bounded away from zero, the existence of α > 0 satisfying (7.11) is easily obtained from (7.23). This completes the proof. ¤ Proof of Theorem 7.1. (i) Assume (7.1), (7.2) and pick µ≥λ > σΩ 1[L]. If (1.1) possesses a coexistence state, say (u, v), then we find from the first equation of (1.1) that λ=σΩ 1[L+au −bv]≤σΩ 1[L+aMu−bLv].(7.24) Moreover, thanks to Lemma 7.2(i), we find from (7.2) that u≤bM+dM cL+aL v≤bL aM v . Thus, aMu−bLv≤0, and (7.24) gives λ≤σΩ 1[L], which is impossible. Therefore, (1.1) can not admit a coexistence state. This completes the proof of Part (i). Part (ii) follows by symmetry, interexchanging the roles of λ,aand bby µ,dand c, respectively. We now prove (iii). Assume (7.1), (7.4) and fix λ < σΩ 1[L]. We argue by contradiction assuming that there exists a sequence of coexistence states of (1.1), say (µn, un, vn), n≥1, such that µn>max{µ0(λ),0},n≥1, and limn↑∞ µn=∞. Without loss of generality we can assume that µn≥λfor each n≥1. Let Ω1⊂Ω an arbitrary subdomain of Ω with Ω1⊂Ω. By lemma 7.3(i), there exists α=α(Ω1)>0 such that for each n≥1vn µn≥αin Ω1. Moreover, by Lemma 7.2(i), we have that for each n≥1 un µn≤bM+dM cL+aL vn µn . Thus, by (7.4) there exists ε > 0 such that for each n≥1 un µn≤bL aM vn µn−εin Ω1. Hence, aMun−bLvn≤ −εaMµnin Ω1∀n≥1.(7.25) On the other hand, we find from the first equation of (1.1) that λ=σΩ 1[L+aun−bvn]≤σΩ1 1[L+aMun−bLvn] and therefore, (7.25) gives λ≤σΩ1 1[L]−εaMµn↓ −∞ as n→ ∞. This contradiction shows that (1.1) does not admit a coexistence state for µlarge and completes the proof of this part. Part (iv) follows by symmetry. ¤ SYMBIOTIC SPECIES 33 7.2. A priori bounds for N≤5.The following result provides us with uniform a priori bounds in L∞for the coexistence states of (1.1). Theorem 7.4. Under condition (7.1), if N≤5,bLcL> aMdMand for some α > 0 max {|λ|,|µ|} ≤ α , then there exists a constant C=C(α, Ω, a, b, c, d)such that kukL∞(Ω) ≤C , kvkL∞(Ω) ≤C , for any coexistence state (u, v)of (1.1). This result is optimal in the sense that if N > 5, then there are choices of the several coefficients and of Ω for which the uniform a priori bounds are lost (cf. the final comments in Section 5 of [21] and Theorem 1.4 of [27]). For instance, if a,b,c,dare constants and λ=µ, then for any coexistence state (u, v) of (1.1) it is easily seen that (L−λ+au +dv)((b+d)v−(c+a)u) = 0 and hence, v=c+a b+du , (7.26) since σΩ 1[L−λ+au +dv]>0. Therefore, (u, v) is a coexistence state of (1.1) if, and only if, (7.26) holds and uis a positive solution of Lu=λu +bc −ad b+du2in Ω , u|∂Ω= 0 .(7.27) If bc < ad, then the coefficient of u2in (7.27) is negative and hence the positive solutions of (7.27) possesses uniform a priori bounds on compact subintervals of λ. On the contrary, when bc > ad the coefficient of u2in (7.27) is positive and therefore (7.27) is a superlinear problem. In this case it is well known that a priori bounds are available if 2<N+2 N−2(cf. [13]), i.e. if N≤5, while in the case when N≥6 the a priori bounds are in general lost and the structure of the set of positive solutions can change drastically as either the geometry of Ω changes or the spatial dimension Nincreases. Being the higher dimensional case outside the scope of this work we send to the interested reader in further details to [4] and [8]. In the special case when L=−∆ and a,b,cand dare constants Theorem 7.4 is given by Lemma 4.3 of [27], but the proof of [27] can not be adapted to cover our current situation here, as it will become clear later. The main difficulty coming from the fact that now the coefficients are not constant. To prove Theorem 7.4 we will argue by contradiction using the blowing up argument introduced in [13] for the case of one single equation. It should be noted that our blowing up argument is somewhat different from the corresponding argument used in [27]. 34 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Proof of Theorem 7.4. We shall prove the result in case λ≥µ. By symmetry, the result is also true when µ≥λ. If the conclusion of Theorem 7.4 is false, then there exists a sequence of coexistence states (λk, µk, uk, vk), k≥1, with −α≤µk≤λk≤α, such that lim sup k→∞ (kukkL∞(Ω) +kvkkL∞(Ω)) = ∞.(7.28) We claim that lim sup k→∞ kukkL∞(Ω) = lim sup k→∞ kvkkL∞(Ω) =∞.(7.29) Indeed, if {kvkkL∞(Ω)}k≥1is bounded by some positive constant β, then we find from the first equation of (1.1) that Luk≤(α+bMβ)uk−au2 k and therefore, it follows from Lemma 3.2 and Corollary 3.3, that {kukkL∞(Ω)}k≥1is also bounded. By (7.28) this is impossible. Similarly, if {kukkL∞(Ω)}k≥1is bounded, then {kvkkL∞(Ω)}k≥1is also bounded. Therefore, (7.29) is satisfied. By chosing a subsequence, if necessary, we can assume that lim k→∞ kukkL∞(Ω) =∞,lim k→∞(λk, µk) = (λ∞, µ∞),(7.30) for some (λ∞, µ∞)∈R2satisfying −α≤µ∞≤λ∞≤α. Note that thanks to Lemma 7.2(ii) we have that vk≤cM+aM bL+dL uk∀k≥1.(7.31) For each k≥1, pick xk∈Ω such that Mk:= uk(xk) = kukkL∞(Ω) .(7.32) Since Ω is bounded, without loss of generality we can assume that lim k→∞ xk=x∞∈Ω.(7.33) Now, we consider two different situations, accordingly with whether x∞∈Ω or x∞∈ ∂Ω. Assume that x∞∈Ω. Then, δ:= d(x∞, ∂Ω)/2>0. Moreover, setting ρk:= M−1/2 k, k ≥1, SYMBIOTIC SPECIES 35 we have limk→∞ ρk= 0, since thanks to (7.30) and (7.32) limk→∞ Mk=∞. Now, it is easily seen that the change of variables y:= x−xk ρk ,(zk, wk) := ρ2 k(uk, vk), k ≥1,(7.34) transforms the system of (1.1) into Akzk=ρ2 kλkzk−a(xk+ρky)z2 k+b(xk+ρky)zkwk, Akwk=ρ2 kµkwk−d(xk+ρky)w2 k+c(xk+ρky)zkwk,(7.35) where Ak=− N X i,j=1 aij(xk+ρky)∂i∂j+ρk N X j=1 bj(xk+ρky)∂j+ρ2 ke(xk+ρky),(7.36) provided xk+ρky∈Ω. By definition of δ, for ksufficiently large, |x−xk| ≤ δimplies x=xk+ρky∈Ω. Hence, |y| ≤ δ ρkimplies x=xk+ρky∈Ω and so (7.35) holds. Since limk→∞ δ ρk=∞, given R > 0 arbitrary BR⊂Bδ/ρkfor ksufficiently large, where for any τ > 0Bτstands for the ball of radius τcentered at the origin. Now, from the definition of ρkwe have that zk=ρ2 kuk=uk Mk and hence, kzkkL∞(BR)= 1 , zk(0) = 1 ,∀k≥1.(7.37) Moreover, thanks to (7.31) and (7.37), we find that kwkkL∞(BR)≤cM+aM bL+dL∀k≥1.(7.38) Now the same compactness argument of the proof of Theorem 1.1 in [13] shows that given any p > N and passing to a suitable subsequence, again relabeled by k, there exists (z, w)≥(0,0) in W2,p(BR)∩C1,ν (BR), 0 < ν < 1, such that lim k→∞(zk, wk) = (z, w) in (W2,p(BR)∩C1,ν(BR))2. By H¨older continuity z(0) = 1. Moreover, passing to the limit as k→ ∞ in (7.35) gives − N X i,j=1 aij(x∞)∂i∂jz=−a(x∞)z2+b(x∞)zw , − N X i,j=1 aij(x∞)∂i∂jw=−d(x∞)w2+c(x∞)zw , (7.39) 36 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ in BR, for any R > 0. By a standard diagonal sequence argument it is easily seen that z,w∈W2,p loc (RN) and that (7.39) holds true in the whole of RN. Moreover, standard elliptic regularity theory implies that z,w∈C2(RN). Furthermore, by a linear change of coordinates (cf. [13] pg. 890), (7.39) can be reduced to −∆z=−a(x∞)z2+b(x∞)zw −∆w=−d(x∞)w2+c(x∞)zw in RN.(7.40) ¿From (7.40), it is easily seen that (−∆ + a(x∞)z+d(x∞)w)(w−c(x∞) + a(x∞) b(x∞) + d(x∞)z) = 0 . Since (z, w)≥(0,0) and z(0) = 1, the potential V:= a(x∞)z+d(x∞)w satisfies V≥0 and V6= 0. Therefore, due to the following lemma, whose proof we postpone up to conclude the proof of Theorem 7.4, we find that w=c(x∞) + a(x∞) b(x∞) + d(x∞)z . (7.41) Lemma 7.5. Assume that either D=RNor D=RN +, where RN +={x∈RN:xN≥0}. If V∈L∞(D)∩Cν(D),V≥0,V6= 0, then θ= 0 is the only bounded solution of (−∆ + V)θ= 0 in D . (7.42) Substituting (7.41) into the first equation of (7.40) and rearranging terms gives −∆z=b(x∞)c(x∞)−a(x∞)d(x∞) b(x∞) + d(x∞)z2in RN.(7.43) Since bLcL> aMdM,b(x∞)c(x∞)> a(x∞)d(x∞) and hence, thanks to Theorem 1.1 of [13], z= 0 is the unique non-negative solution of (7.43), because N≤5. This is a contradiction with z(0) = 1. Therefore, x∞∈∂Ω. Now, the same argument as in Case 2 of the proof of Theorem 1.1 in [13] shows that the problem −∆z=−a(x∞)z2+b(x∞)zw −∆w=−d(x∞)w2+c(x∞)zw in RN +.(7.44) SYMBIOTIC SPECIES 37 possesses a non-negative solution couple (z, w) with z(0) = 1. The same argument as above shows that this is impossible. This contradiction shows the existence of uniform a priori bounds and completes the proof of the theorem. ¤ We now prove Lemma 7.5, which is a Liouville type result interesting in its own right. In the proof we use the concepts and results in Chapter 4 of [31]. Proof of Lemma 7.5. Thanks to Theorem 3.3(iii) in page 148 of [31], the Schr¨odinger operator ∆ −Vis subcritical on D, i.e. it possesses a Green function G(x, y) on D. Therefore, thanks to Theorem 3.8(i) in page 151 of [31] for each non-negative p∈Cν 0(D), p6= 0, there exists positive solutions u∈C2,ν(D) of (−∆ + V)u=p . (7.45) Moreover, (7.45) possesses a minimal solution u0, given by u0(x) = ZD G(x, y)p(y)dy , and any other solution of (7.45) must be given by u=u0+θ , for some some positive solution θof (7.42). The minimality of u0shows that θ= 0 is the unique solution of (7.42). This completes the proof. ¤ Remark 7.6. (a) Although (7.31) implies w≤cM+aM bL+dLz, this does not necessarily entails w≤c(x∞) + a(x∞) b(x∞) + d(x∞)z(7.46) and hence, Lemma 4.5 of [27] can not be applied to show that (z, w) = (0,0) is the unique solution of (7.40). In fact, our corresponding Liouville type result is substantially sharper than Lemma 4.5 of [27], as we do not need assuming (7.46) to infer z=w= 0. (b) By the Lpestimates of Agmon, Douglis & Nirenberg and Morrey’s Theorem, Theorem 7.4 provides us with a uniform a priori bounds in C1 0(Ω) ×C1 0(Ω) for the coexistence states of (1.1) on any compact subset of the (λ, µ)-plane. 7.3. On the existence of coexistence states in case N≤5.As an immediate consequence, from Theorem 5.1, Theorem 7.1 and Theorem 7.4 we obtain the following result. Theorem 7.7. (i) If N≤5,(7.4) and λ < σΩ 1[L−bθ[L,µ,d]],(7.47) are satisfied, then (1.1) possesses a coexistence state. 38 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ (ii) If N≤5,(7.5) and µ < σΩ 1[L−cθ[L,λ,a]],(7.48) are satisfied, then (1.1) possesses a coexistence state. (iii) If N≤5and either (7.4) or (7.5) is satisfied, then (1.1) possesses a coexistence state provided λ < σΩ 1[L], µ < σΩ 1[L].(7.49) Proof. We first show Part (i). Fix λ < σΩ 1[L] and consider µas the main bifurcation parameter. By Theorem 7.1 (iii) there exists µ=µ(λ) such that λ > σΩ 1[L−bθ[L,µ(λ),d]] and (1.1) does not admit a coexistence state for µ > µ(λ). Moreover, by Theorem 5.1 the continuum C+ (µ,0,v)of coexistence states emanating from (0, θ[L,µ,d]) at µλis unbounded, where µλis the unique value of µ>σΩ 1[L] for which λ=σΩ 1[L− bθ[L,µ,d]]. Furthermore, (7.4) implies bLcL> aMdMand hence, we conclude from Theorem 7.4 that (1.1) possesses a coexistence state for each µ < µλ. This completes the proof of Part (i). Part (ii) follows by symmetry and Part (iii) is an easy consequence from Parts (i), (ii). ¤ In practice, the verification of conditions (7.47) and (7.48) is far from easy, as each of them involves the evaluation of the principal eigenvalue of a second order elliptic operator whose associated potential is given through by a positive solution of a semilinear elliptic boundary value problem. The next results provide us with some easily computable sufficient conditions in terms of the several coefficients involved in the setting of (1.1) so that (7.47), or (7.48), holds. Our analysis extends to the case of general second order elliptic operators the estimates of Theorem 2.3 (c) in [26], found for the special case of operators in divergence form. Lemma 7.8. Assume that Lis a differential operator of the form (2.1) whose coefficients satisfy (2.2). For γ > σΩ 1[L], let θ[L,γ,f]denote the positive solution of (3.1). Then, there exists a positive constant K=K(L, f, Ω) ≥max ½kϕk∞ mf,1 ,1 fL¾(7.50) such that kθ[L,γ,f]k∞≤K(γ−σΩ 1[L]) ∀γ≥σΩ 1[L], where ϕis the principal eigenfunction associated with L, normalized so that ZΩ ϕ2= 1 , and mf,1is the constant defined in the statement of Lemma 4.3. Proof. Thanks to Lemma 4.3 dθ[L,γ,f] dγ c{γ=σΩ 1[L]}=ϕ mf,1 , SYMBIOTIC SPECIES 39 and hence, there exist δ > 0 and a constant C > 0 such that kθ[L,γ,f]k∞≤C(γ−σΩ 1[L]) (7.51) for each γ∈[σΩ 1[L], σΩ 1[L] + δ]. On the other hand, it follows from Corollary 3.3 that θ[L,γ,f]≤γ−eL fL . Thus, there exists a constant ˆ C > 0 such that kθ[L,γ,f]k∞ γ−σΩ 1[L]≤γ−eL γ−σΩ 1[L]·1 fL≤ˆ C1 fL for each γ≥σΩ 1[L] + δ. This completes the proof. ¤ Theorem 7.9. Assume that Lis a differential operator of the form (2.1) whose coefficients satisfy (2.2), and let K1:= K(L, a, Ω),K2:= K(L, d, Ω) denote the two constants whose existence was shown by Lemma 7.8. Then, the following assertions are true: (i) If N≤5,(7.4) and λ < σΩ 1[L], λ < min{σΩ 1[L]−bMK2(µ−σΩ 1[L]) , σΩ 1[L]−bM dL (µ−eL)} are satisfied, then (1.1) possesses a coexistence state. (ii) If N≤5,(7.5) and µ < σΩ 1[L], µ < min{σΩ 1[L]−cMK1(λ−σΩ 1[L]) , σΩ 1[L]−cM aL (λ−eL)} are satisfied, then (1.1) possesses a coexistence state. Proof. By Lemma 7.8, we have that kθ[L,λ,a]k∞≤K1(λ−σΩ 1[L]) ,kθ[L,µ,d]k∞≤K2(µ−σΩ 1[L]) . Thus, it follows from Theorem 2.3 that σΩ 1[L−bθ[L,µ,d]]≥σΩ 1[L−bMkθ[L,µ,d]k∞]≥σΩ 1[L−bMK2(µ−σΩ 1[L])] =σΩ 1[L]−bMK2(µ−σΩ 1[L]) . Similarly, σΩ 1[L−cθ[L,λ,a]]≥σΩ 1[L]−cMK1(λ−σΩ 1[L]) . On the other hand, Corollary 3.3 implies θ[L,λ,a]≤λ−eL aL , θ[L,µ,d]≤µ−eL dL , and the same argument as above shows that σΩ 1[L−cθ[L,λ,a]]≥σΩ 1[L]−cM aL (λ−eL), σΩ 1[L−bθ[L,µ,d]]≥σΩ 1[L]−bM dL (µ−eL). Theorem 7.7 completes the proof. ¤ 40 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ 8. The maximum principle. Multiplicity results. In this section we use the abstract theory of [2] to show that the method of sub and supersolutions is valid for (1.1). Then, we use it to analyze the structure of the set of λ’s (or µ’s) for which (1.1) possesses a coexistence state and to get some multiplicity results of coexistence states. The basic technical tool to prove these results is the strong maximum principle for linear cooperative systems. The validity of the strong maximum principle is guaranteed if, for instance, we assume that b(x)>0, c(x)>0,∀x∈Ω.(8.1) So, for the rest of this section we shall assume that this condition is satisfied. 8.1. The strong maximum principle for cooperative systems. If (u0, v0) is a coexistence state of (1.1), then its linearized stability is given by the eigenvalues of the linearization of (1.1) at (u0, v0), i.e. by the τ’s for which the following problem has some solution (u, v)∈W2,p 0(Ω) ×W2,p 0(Ω), (u, v)6= (0,0), p > N, µL10 0L2¶µu v¶=Aµu v¶+τµu v¶,(8.2) where A=µλ−2au0+bv0bu0 cv0µ−2dv0+cu0¶.(8.3) Note that thanks to (8.1) the off-diagonal entries of this matrix are positive and so the coupling matrix Ais of cooperative type. More generally, we consider the linear cooperative eigenvalue problem (8.2) with (u, v)∈W2,p 0(Ω) ×W2,p 0(Ω) for some p>N and A=µα(x)β(x) γ(x)ρ(x)¶,(8.4) where α,β,γ,ρ∈C(Ω) and the off-diagonal entries, βand γ, are positive almost everywhere in Ω. In the sequel we set L:= µL10 0L2¶−A(8.5) and suppose that p>N. Now, to state the maximum principle we need some of notation. Given (u, v)∈Lp(Ω) ×Lp(Ω), it is said that (u, v)≥0 if u≥0 and v≥0. If in addition u6= 0 or v6= 0, then it is said that (u, v)>0. A couple (u, v)∈ W2,p 0(Ω) ×W2,p 0(Ω) is said to be strongly positive if u(x)>0, v(x)>0 for all x∈Ω and ∂nu(x)<0, ∂nv(x)<0 for all x∈∂Ω, where nis the outward unit normal at x. Definition 8.1. The operator Ldefined by (8.5) is said to satisfy the strong maximum principle in Ωif x:= (u, v)∈W2,p 0(Ω) ×W2,p 0(Ω) and Lx > 0imply that xis strongly positive. SYMBIOTIC SPECIES 41 Definition 8.2. A function x:= (u, v)∈W2,p(Ω) ×W2,p(Ω) is said to be a supersolution of Lin Ωif x|∂Ω≥0and Lx≥0. If in addition Lx > 0, or x|∂Ω>0, then it is said that xis a strict supersolution. Now, using Theorems 2.1, 2.2 of Section 2, the proof of Theorem 2.1 in [24] can be easily adapted to cover our general setting providing us with the following general versions of Theorems 2.1, 2.2 of Section 2. Theorem 8.3. There exists a least eigenvalue of (8.2), denoted by σΩ 1[L]and called principal eigenvalue of Lin Ω. This eigenvalue is simple and possesses a unique eigenfunction, up to multiplicative constants, which can be taken positive, the so called principal eigenfunction of Lin Ω. Moreover, the principal eigenfunction is strongly positive and σΩ 1[L]is the only eigenvalue of (8.2) possessing a positive eigenfunction. Furthermore, any other eigenvalue σof (8.2) satisfies Re σ > σΩ 1[L] and (L+ν)−1∈ L(Lp(Ω)×Lp(Ω)) is positive, compact and irreducible for ν > −σΩ 1[L]. Theorem 8.4. The following assertions are equivalent: (i) σΩ 1[L]>0; (ii) Lpossesses a positive strict supersolution in W2,p(Ω) ×W2,p(Ω); (iii) Lsatisfies the strong maximum principle. Moreover, the following generalized maximum principle holds. Theorem 8.5. If Lsatisfies the strong maximum principle, then any strict supersolution x:= (u, v)∈W2,p(Ω) ×W2,p(Ω) of Lis positive in Ω. In fact, u(x)>0and v(x)>0for all x∈Ω. It will simply said that Lsatisfies the generalized maximum principle in Ω. Proof. It is based upon Theorem Anof [34]. Thanks to Theorem 8.4, σΩ 1[L]>0. Let h > 0 denote the principal eigenfunction associated with σΩ 1[L]>0. We have that Lh > 0 in Ω. Therefore, thanks to Theorem Anof [34], some of the following options occurs: Either (i) x > 0 in Ω, or (ii) x= 0 in Ω, or (iii) x=αh for some α < 0. Since, we are assuming that xis a strict supersolution, the options (ii) and (iii) are excluded. Therefore, x > 0 in Ω. Corollary 2 of [34] completes the proof. ¤ Thanks to these results, for any operator Lof the type (8.5) there exists ωsuch that L+νsatisfies the the generalized maximum principle for all ν > ω. Therefore, the proof of Theorem 9.4 of [2] carries over mutatis mutandis to our present situation, showing that the method of sub and supersolutions works out for the nonlinear model (1.1). To state our result we need to introduce the concept of sub and supersolution. Definition 8.6. A positive function x= (u, v)∈W2,p(Ω) ×W2,p(Ω) is said to be a subsolution of (1.1) if L1u≤λu −a(x)u2+b(x)u v L2v≤µv −d(x)v2+c(x)u v in Ω, 48 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Lemma 8.13. (i) Let (µ, u, v) = (µ0, u0, v0)be a coexistence state of (1.1) such that σΩ 1[Lµ0]>0,(8.16) where Lµ0is the operator defined by (8.5) with A(x)given by (8.3). Then, there exists ε > 0and a differentiable mapping (u, v):(µ0−ε, µ0+ε)→P2such that (u(µ0), v(µ0)) = (u0, v0)and (µ, u(µ), v(µ)) is a coexistence state of (1.1) for each µ∈(µ0−ε, µ0+ε). Moreover, the mapping µ→(u(µ), v(µ)) is strictly increasing and there exists a neighborhood Qof (µ0, u0, v0)in R×(Ce(Ω))2such that if (µ, u, v)∈ Q is a solution of (1.1), then (u, v) = (u(µ), v(µ)). (ii) Assume σΩ 1[Lµ0] = 0, instead of (8.15), and let Φdenote the principal eigenfunction associated with σΩ 1[Lµ0]. Then, there exists ε > 0and a differentiable mapping (µ, u, v) : (−ε, ε)→R×P2such that (µ(0), u(0), v(0)) = (µ0, u0, v0)and for each s∈(−ε, ε) (µ(s), u(s), v(s)) is a coexistence state of (1.1). Moreover, µ(s) = µ0+ ˆµ(s),(u(s), v(s)) = (u0, v0) + sΦ + (ˆu(s),ˆv(s)) ,(8.17) where ˆµ(s) = 0(s),ˆu(s) = o(s)and ˆv(s) = o(s)as s→0, and there exists a neighborhood Qof (µ0, u0, v0)in R×(Ce(Ω))2such that if (µ, u, v)∈ Q is a solution of (1.1), then (µ, u, v) = (µ(s), u(s), v(s)) for some s∈(−ε, ε). Furthermore, sgn µ0(s) = sgn σΩ 1[Ls],(8.18) where Ls=µL10 0L2¶−µλ−2au(s) + bv(s)bu(s) cv(s)µ(s)−2dv(s) + cu(s)¶. If σΩ 1[Lµ]>0, then the Leray-Schauder formula implies that the local index i(Kµ,(uµ, vµ)) = 1 and therefore, thanks to Lemma 8.11, (1.1) must have a further coexistence state. Therefore, in this case the proof is completed. Now, assume that σΩ 1[Lµ] = 0 and let (µ(s), u(s), v(s)) denote the curve of coexistence states through by (µ, uµ, vµ), for s= 0, whose existence is guaranteed by Lemma 8.13. Since Φ >0, (u(s), v(s)) is strictly increasing and hence, if µ(s) = µfor some s6= 0, then (1.1) possesses two coexistence states. Namely, (uµ, vµ) and (u(s), v(s)). Thus, without loss of generality we can assume that µ(s)6=µ∀0<|s|< ε . (8.19) SYMBIOTIC SPECIES 49 We claim that µ(s)< µ ∀s∈(−ε, 0) .(8.20) Indeed, if there exists s1<0 such that µ1:= µ(s1)≥µ, then (u(s1), v(s1)) <(u(0), v(0)) = (uµ, vµ)≤(uµ1, vµ1),(8.21) since (u(s), v(s)) is increasing in sand the minimal solution is non-decreasing in µ. Here, (uµ1, vµ1) stands for the minimal coexistence state of (1.1) for µ=µ1. Relation (8.21) contradicts the minimality of (uµ1, vµ1). Thus, (8.20) get shown. Moreover, by (8.19), either µ(s)< µ for all s∈(0, ε), or µ(s)> µ for all s∈(0, ε), so we can distinguish two cases: Case a: Assume that µ(s)< µ for all s∈(0, ε). Then, since µ < µ∗and (1.1) possesses a coexistence state for each value of the parameter in [µ, µ∗], there exists a sequence of coexistence states (µn, un, vn), n≥1, such that limn→∞ µn=µand µn> µ for all n≥1. By the existence of uniform a priori bounds, without loss of generality we can assume that lim n→∞(un, vn) = (u0, v0), for some non-negative solution (u0, v0) of (1.1). Since λ < σΩ 1[L1] and µ > µλ, with a similar argument as in the proof of Theorem 8.8, it is easily seen that (µ, u0, v0) is a coexistence state. Moreover, by the uniqueness obtained as an application of Lemma 8.13(ii), (µn, un, vn)6∈ Q for each n≥1 and hence, (µ, u0, v0)6∈ Q. In particular, (µ, u0, v0)6= (µ, uµ, vµ) and therefore, (1.1) possesses at least two coexistence states. Case b: Now, assume that µ(s)> µ ∀s∈(0, ε).(8.22) Then, thanks to Lemma 8.13(ii), (µ, uµ, vµ) is an isolated solution of (1.1) and so i(Kµ,(uµ, vµ)) is well defined. By Lemma 8.11, to complete the proof of Theorem 8.10, it suffices to show that i(Kµ,(uµ, vµ)) = 1 .(8.23) By (8.22) there exists s1∈(0, ε) for which µ0(s1)>0. By (8.18), σΩ 1[Ls1]>0 and therefore, we find from Theorem 8.3 and the linearized stability principle that (u(s1), v(s1)) is exponentially asymptotically stable. Thus, Leray-Schauder’s formula implies i(Kµ(s1),(u(s1), v(s1))) = 1 .(8.24) Since (µ(s1), u(s1), v(s1)) is non-degenerate and s→(u(s), v(s)) is increasing there exists δ > 0 such that if ρ1:= k(u(s1), v(s1))ke−δ , ρ2:= k(uµ, vµ)ke−δ , then (1.1) does not admit a coexistence state in [µ(s1), µ(s1) + δ]×∂(Pρ1\Pρ2). 50 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Moreover, by the uniqueness of Lemma 8.13(ii), δ > 0 can be chosen so that (1.1) does not have a coexistence state in Pρ1\Pρ2for µ=µ(s1) + δeither. Thus, the homotopy invariance implies i(Kµ(s1), Pρ1\Pρ2) = 0 .(8.25) Now, for δ > 0 sufficiently small set ρ:= k(u(s1), v(s1)ke+δ . By (8.24), (8.25), we find that i(Kµ(s1), Pρ\Pρ2) = 1 . Moreover, by the monotonicity of (u(s), v(s)) and the uniqueness given by Lemma 8.13(ii), (1.1) does not admit a coexistence state on [µ, µ(s1)] ×∂(Pρ\Pρ2). This implies (8.23) and completes the proof of the theorem. ¤ Similarly, for the case of small interaction coefficients we have the following result. Theorem 8.14. Assume (6.1). Then following assertions are true: (i) Assume µ>σΩ 1[L2]and Λ=[λ∗,∞)with λ∗< σΩ 1[L1−b(x)θ[L2,µ,d]]. Then, (1.1) possesses at least two coexistence states for each λ∈(λ∗, σΩ 1[L1−b(x)θ[L2,µ,d]]). (ii) Assume λ > σΩ 1[L1]and M= [µ∗∞)with µ∗< σΩ 1[L2−c(x)θ[L1,λ,a]]. Then, (1.1) possesses at least two coexistence states for each µ∈(µ∗, σΩ 1[L2−c(x)θ[L1,λ,a]]). Proof. Being the proof rather similar to the proof of Theorem 8.10, we are only to sketch it. By symmetry, it suffices to show Part (ii). Let (µ∗, u∗, v∗) be a coexistence state of (1.1). Then, it is easily seen that for each µ∈(µ∗, σΩ 1[L2−c(x)θ[L1,λ,a]]) x= (u∗, v∗), x = (K1, K2), is an ordered sub-supersolution pair of (1.1) provided K1and K2are sufficiently large positive constants. Moreover, thanks to Lemma 6.2, if K1and K2are sufficiently large, then any coexistence state of (1.1) lies in the order interval [0, x]. Therefore, (1.1) possesses a maximal coexistence state within the interval [x, x], denoted by (uµ, vµ). Thanks to Proposition 7.8 of [2], (uµ, vµ) is weakly stable and so σΩ 1[Lµ]≥0 where Lµ is the operator defined by (8.5) with A(x) given by (8.3) and (u0, v0) = (uµ, vµ). If σΩ 1[Lµ]>0 the same argument of the proof of Theorem 8.10 completes the proof of Theorem 8.11. If σΩ 1[Lµ] = 0 arguing as in the proof of Theorem 8.10 we find that µ(s)> µ ∀s∈(0, ε), SYMBIOTIC SPECIES 51 and two different situations may arise: Case a. If µ(s)> µ for s∈(−ε, 0), then the same argument of the proof of Theorem 8.10 applies to complete the proof of this one. Case b. If µ(s)< µ for s∈(−ε, 0), then there exists s1<0 such that µ0(s1)>0 and hence, i(Kµ(s1),(u(s1), v(s1))) = 1 . Now, setting ρ1:= k(uµ, vµ)ke+δ, ρ2:= k(u(s1), v(s1))ke+δ, ρ := k(u(s1), v(s1))ke−δ. yields i(Kµ(s1), Pρ1\Pρ2)=0, i(Kµ(s1), Pρ1\Pρ) = 1 , i(Kµ, Pρ1\Pρ) = 1 , and therefore, i(Kµ,(uµ, vµ)) = 1 . This completes the proof. ¤ 9. On the uniqueness of the coexistence state. In this section we give a uniqueness result in the case of small interaction coefficients. When the interaction coefficients are large we already know that (1.1) exhibits a superlinear character and so its number of coexistence states might vary drastically when the geometry of the support domain Ω changes, [8]. Our main uniqueness result is the following. Theorem 9.1. Assume that (6.1),(6.8) and (8.1) are satisfied and that for any coexistence state (u0, v0)of (1.1) µu0 v0¶Mµv0 u0¶M <³a b´Lµd c¶L .(9.1) Then, (1.1) possesses a unique coexistence coexistence. Moreover, it is exponentially asymptotically stable. After the proof of this theorem we shall use Theorem 8.7 to get some upper estimates of the left hand side of (9.1), giving rise to very simple easily computable sufficient conditions, in terms of the several coefficients involved in the model setting, for the uniqueness of the coexistence state. Proof. Under conditions (6.1) and (6.8) we have uniform a priori bounds for the nonnegative solutions of (1.1) and hence the fixed point index in cones can be used as in Section 8.3. By Proposition 4.1 the semi-trivial positive solutions (θ[L1,λ,a],0) and (0, θ[L2,µ,d]) are linearly unstable, if they exist, and a rather standard index computation shows that each of them has local index zero (cf. [23] for details). Moreover, the state (0,0) has index zero and the global index equals one. Therefore, by the principle 52 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ of linearized stability, it suffices to show that under condition (9.1) any coexistence state is linearly asymptotically stable, since by Leray-Schauder formula any linearly asymptotically stable solution has local index one. Let (u0, v0) be a coexistence state of (1.1). Then, the spectrum of the linearization of (1.1) at (u0, v0) is given by the τ’s for which the following problem has some solution (u, v)∈W2,p 0(Ω) ×W2,p 0(Ω), (u, v)6= (0,0), p > N, (L1+ 2au0−bv0−λ)u=bu0v+τu , (L2+ 2dv0−cu0−µ)v=cv0u+τv . (9.2) By Theorem 8.3 if we are able to show that there exist u > 0 and v > 0 such that (L1+ 2au0−bv0−λ)u > bu0v , (L2+ 2dv0−cu0−µ)v > cv0u , (9.3) then the principal eigenvalue of (9.2) will be positive and therefore, the linearized stability of (u0, v0) will follow from Theorem 8.3. Taking (u, v) = (αu0, βv0), where α > 0 and β > 0 have to be found, (9.3) becomes into αau0> βbv0, βdv0> αcu0.(9.4) Now, due to (9.1), it is rather clear that there exist α > 0 and β > 0 satisfying (9.4). This completes the proof. ¤ The following result provides us with a sufficient condition for (9.1) to be hold. Proposition 9.2. Assume L1=L2,b(x)>0and c(x)>0for each x∈Ω, σΩ 1[L1]>0, bMcM< aLdL, λ > σΩ 1[L1], µ > σΩ 1[L1],(9.5) and aMdM 16aLdL(aLdL−bMcM)2·(dLλ2+bMµ2)(aLµ2+cMλ2) (λ−σΩ 1[L1])(µ−σΩ 1[L1]) ·Ãsup Ω ψ ϕ!2 <1 bMcM ,(9.6) where ϕ > 0is the principal eigenfunction associated with σΩ 1[L1], normalized so that kϕkL∞(Ω) = 1 and ψ > 0is the unique solution of L1ψ= 1 in Ω, ψ|∂Ω= 0 . Then (1.1) has exactly one coexistence state. Proof. We claim that for each t > 1 the couple (ut, vt) defined by ut:= t(dLλ2+bMµ2) 4(aLdL−bMcM)ψ , vt:= t(aLµ2+cMλ2) 4(aLdL−bMcM)ψ , SYMBIOTIC SPECIES 53 is a strict supersolution of (1.1). To prove this it suffices to show that 1≥ψ·[λ−t(a(x)K1−b(x)K2)ψ], 1≥ψ·[µ−t(d(x)K2−c(x)K1)ψ],(9.7) where K1=dLλ2+bMµ2 4(aLdL−bMcM), K2=aLµ2+cMλ2 4(aLdL−bMcM). Since sup ξ≥0 (A−Bξ)ξ=A2 4B, we find that for each t≥1, ψ·[λ−t(a(x)K1−b(x)K2)ψ]≤λ2 4t(a(x)K1−b(x)K2)≤λ2 4(aLK1−bMK2). Similarly, ψ·[µ−t(d(x)K2−c(x)K1)ψ]≤µ2 4(aLK2−bMK1). Thus, the following conditions imply (9.7) λ2= 4(aLK1−bMK2), µ2= 4(aLK2−bMK1). Since these conditions are satisfied by the choice of K1and K2itself, the claim above get shown. Now, we need the following generalized version of the sweeping maximum principle of [28], whose proof is postponed up to the end of the proof of Proposition 9.2. Lemma 9.3. Let x= (u, v)∈W2,p 0(Ω) ×W2,p 0(Ω),p > N, be a solution of the problem L1u=f(x, u, v) L2v=g(x, u, v)in Ω, u=v= 0 on ∂Ω, where fand gare two continuous functions in xand of class C1in (u, v),fincreasing in v, and gincreasing in u. For each t∈(t0, t1], let xt= (ut, vt)∈W2,p 0(Ω) ×W2,p 0(Ω) be a strict supersolution of this problem. Assume that xtis continuous and strictly increasing in t, that xt1−xis strongly positive, and that ∂nxtis continuous in t, where nstands for the outward unit normal to Ω. Then, x≤xt0. 54 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Thanks to Lemma 9.3, we find that u0≤dLλ2+bMµ2 4(aLdL−bMcM)ψ , v0≤aLµ2+cMλ2 4(aLdL−bMcM)ψ , (9.8) for any coexistence state (u0, v0) of (1.1). Similarly, it follows from Lemma 3.2 that u0≥θ[L1,λ,a]≥λ−σΩ 1[L1] aM ϕ , v0≥θ[L1,µ,d]≥µ−σΩ 1[L1] dM ϕ . (9.9) Finally, using (9.8) and (9.9), it is easily seen that (9.6) implies (9.1). Theorem 9.1 completes the proof. ¤ Proof of Lemma 9.3. Let t∗denote the infimum of the set of t∈(t0, t1) for which x−xt is strongly positive. We claim that t∗=t0. On the contrary, assume that t∗> t0. By our assumptions it is rather clear that there exists K > 0 such that each of the mappings u→f(·, u, v) + Ku , v →g(·, u, v) + Kv , is increasing and K > −min{σΩ 1[L1], σΩ 1[L2]}. Since xt∗is a strict supersolution of the problem, some of its components, say ut∗, satisfies (L1+K)(ut∗−u)> f(·, ut∗, vt∗) + Kut∗−f(·, u, v)−Ku > 0. Thus, the strong maximum principle implies that ut∗−uis strongly positive. This contradicts the minimality of t∗and completes the proof. ¤ Note that, thanks to the strong maximum principle, ϕand ψare strongly positive and hence, supΩ ψ ϕis well defined. The estimates given by the following result will be used to find out another sufficient condition for (9.1). Lemma 9.4. Assume L1=L2,b(x)>0and c(x)>0for each x∈Ω, and bMcM< aLdL, λ ≥µ > σΩ 1[L1]. Then, for any coexistence state (u, v)of (1.1) the following estimates hold M1θ[L1,µ,d]≤u≤N1θ[L1,λ,a],(9.10) M2θ[L1,µ,d]≤v≤N2θ[L1,λ,a],(9.11) where N1=aM(dL+bM) aLdL−cMbM , N2=aM(aL+cM) aLdL−cMbM , M1= max ½dL(bL+dM) aMdM−cLbL ,(bL+dL)[dM(aM+cM)−cLdL] aM[dM(aM+cM)−cL(bL+dL)]¾, SYMBIOTIC SPECIES 55 M2= max ½dL(cL+aM) aMdM−cLbL ,dM(aM+cM) dM(aM+cM)−cL(bL+dL)¾. Proof. Since N1aL−bMN2=aM, N2dL−N1cM=aM, for each t≥1 we have that t(N1aL−N2bM)−aM≥0, t(N2dL−N1cM)−aM≥0.(9.12) Now, thanks to (9.12) it is easily seen that for each t > 1 the couple (ut, vt) defined by (ut, vt) := t(N1θ[L1,λ,a], N2θ[L1,λ,a]) is a strict supersolution of (1.1). Therefore, thanks to Lemma 9.3, the upper estimates in (9.10) and (9.11) get shown. Now, in order to prove the validity of the lower estimates in (9.10), (9.11) we will adapt a device coming from [17]. A reiterative application of Lemma 3.2 shows that αnθ[L1,µ,d]≤u , βnθ[L1,µ,d]≤v , (9.13) for each n≥1, where αn=dL+bLβn−1 aM , βn=dL+cLαn−1 dM , α0=dL/aM, β0= 1 . Thus, passing to the limit as n→ ∞ yields αθ[L1,µ,d]≤u , βθ[L1,µ,d]≤v , where α=dL(bL+dM) aMdM−bLcL , β =dL(cL+aM) aMdM−bLcL . This provides us with half of the lower estimates in (9.10), (9.11). Now, it follows from Lemma 7.2 (ii) that v/K ≥θ[L1,µ,d], where K=dM(cM+aM) dM(aM+cM)−cL(bL+dL). Thus, L1=λu −a(x)u2+b(x)uv ≥µu −a(x)u2+bLKθ[L1,µ,d]u and hence, uis a supersolution of L1w= (µ+bLKθ[L1,µ,d])w−a(x)w2in Ω , w= 0 on ∂Ω.(9.14) 56 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ Therefore, Theorem 3.1 implies θ[L1−bLKθ[L1,µ,d],µ,a]≤u . Finally, a further application of Lemma 3.2 shows that Bθ[L1,µ,d]≤θ[L1−bLKθ[L1,µ,d],µ,a], where B=(bL+dL)[dM(aM+cM)−cLdL] aM[dM(aM+cM)−cL(bL+dL)] . This completes the proof. Note that Kand Bare positive constants. ¤ Now, as an immediate consequence from Theorem 9.1 and Lemma 9.4 we obtain the following result. Corollary 9.5. Assume L1=L2,b(x)>0and c(x)>0for each x∈Ω, bMcM< aLdL, λ ≥µ > σΩ 1[L1], and N1 M2·N2 M1Ãsup Ω θ[L1,λ,a] θ[L1,µ,d]!2 <aLdL bMcM .(9.15) Then, (1.1) possesses a unique coexistence state. Note that since θ[L1,λ,a]and θ[L1,µ,d]are strongly positive, supΩ θ[L1,λ,a] θ[L1,µ,d]is well defined. Remark 9.6. (i) If a,b,cand dare assumed to be constant, then M1=d(b+d) ad −cb , M2=d(c+a) ad −cb , although in case a=b=c= 1 there are choices of d(x) for which some of these relations fails. (ii) If a,b,cand dare constant, then (9.15) becomes into the condition found in Theorem 3.3 of [17]. (iii) As a consequence from Proposition 9.2 and Corollary 9.5, it follows that if one of the interaction coefficients (bor c) is small, then (1.1) possesses a unique coexistence state. For some special classes of domains and differential operators, how small should be bor cto have uniqueness can be estimated in terms of the several coefficients of the model. For instance, if Ω = (0, π), L1=L2=−d2 dx2and a=d= 1, then σΩ 1[L1] = 1, ϕ(x) = sin(x), ψ(x) = x(π−x)/2, supΩ ψ ϕ=π/2 and the estimate (9.15) becomes into R(λ, µ) := sup Ω θ[L1,λ,1] θ[L1,µ,1] <1 √bc .(9.16) SYMBIOTIC SPECIES 57 Some explicit estimates of R(λ, µ) were found in [17] and [1]. Namely, in [17] it was shown that R2(λ, µ)≤λ3 (µ−1)2.(9.17) Therefore, thanks to Corollary 9.5, (1.1) possesses a unique coexistence state provided bc < (µ−1)2 λ3.(9.18) (iv) In many cases Proposition 9.2 is sharper than Corollary 9.5. Indeed, in the previous example (9.6) becomes into bc < 64 π2·(λ−1)(µ−1)(1 −bc)2 (λ2+bµ2)(µ2+cλ2).(9.19) Thus, if λ= 2, µ= 1.5 and c= 1, (9.18) becomes into b < 1/32 ≃0.031, while (9.19) becomes into b < b0with b0≃0.099. Therefore, in this case (9.19) is sharper than (9.18). Under the assumptions of Theorem 9.1, the problem of the global attractivity of the coexistence state with respect to the cone of positive functions in both components is very difficult to handle with. This is in strong contrast with the competing species counterpart of (1.1), where due to the compressivity of the model (cf [14]) the uniqueness of a stable coexistence state implies its global attractivity as a result from the abstract theory of [9]. Nevertheless, the presence of uniform a priori bounds in the context of Theorem 9.1 allows us to apply the following result of [15] to the parabolic system associated with (1.1). Theorem 9.7. Assume that Tis a strongly positive monotone continuous dynamical system on Xwhere the cone Khas non-empty interior and Xis separable. Moreover, assume that O(x)(the positive semi-orbit of x) is compact for each x∈X. Then, there exists a dense subset Aof Xsuch that if x∈A, then ω(x)(the ω-limit of x), is contained in the set of stationary points. Using this result we obtain the following one. Theorem 9.8. Assume that bMcM< aLdL,λ>σΩ 1[L1],µ>σΩ 1[L2],b(x)>0, c(x)>0, for each x∈Ω, and that (1.1) possesses a unique coexistence state, say (uc, vc). Consider the following parabolic reaction diffusion problem ∂tu+L1u=λu −au2+buv , ∂tv+L2v=µv −dv2+cuv , in Ω×(0,∞), u|∂Ω=v|∂Ω= 0 , t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈Ω, 64 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ We should point out that all the previous information is of local nature, i.e. it provides us with the bifurcation directions to coexistence states from the semi-trivial states for values of the parameters close to the co-dimension two singularity (σΩ 1[L1], σΩ 1[L2]). Being the problem of finding out global information about the nature of these local bifurcations very difficult to handle with in our general setting, in the next section we will restrict ourselves to the consider the very special case when L1=L2=−∆ and all the coefficients are constant. In particular, it will be shown that there are ranges of the parameters for which there is a change of the bifurcation direction to coexistence states provided ad−bc > 0 is sufficiently small. This will provide us with some sufficient conditions so that the model exhibits at least two coexistence states accordingly to the multiplicity results of Section 8. If A= 1, then Theorem 10.1 can not be applied and the complexity of the bifurcation diagrams increase. In this case, Theorem 5.1 (ii) of [11] gives the following result. Theorem 10.3. Assume A= 1, and set ε=sign a0, c =−a2 3a2c0|a0|−1, where a0=−1 2 a4 a3 b2+b3−1 2 a2 a3 b5+a2 a4 b6+1 2 a2a4 a3 (d1+d3)−1 2a2(d2+d4), c0=a4 a2 b1−1 2 a3 a2 b2+b4−1 2 a3 a4 b5−1 2a3(d1+d3) + 1 2 a2 3 a4 (d2−d4). Then, if a0c0((c0)2−(a0)2)6= 0,fis K-equivalent to Ãr(λ−r+s−εs2) s(µ+r−s−cr2)!. Moreover, the universal unfolding of fis given by µr(λ−r+ (1 + β)s−εs2) s(µ+r−s−cr2)¶(10.8) and cis a modal parameter. Here, β≃0is an unfolding parameter. From (10.8), the bifurcation directions to coexistence states can be very easily found out. In our present situation, the signs of the prqs−qrpsdepend on the parameter t, as shown by the following identities (prqs−qrps)(Λ1,0(t)) = −β+2(1+β)ct+... , (prqs−qrps)(Λ0,1(t)) = −β+2εt+... . Notice that since Λ1,0(0) = Λ0,1(0) = 0, when tgrows Λ1,0(t) and Λ0,1(t) separate from (σΩ 1[L1], σΩ 1[L2]). SYMBIOTIC SPECIES 65 The list bellow provides us with all the bifurcation directions as sgrows from zero. Without lost of generality, we can assume that ε= 1. 1. Bifurcation directions along Λ0,1 1.1If β > 0, then for values of the parameters sufficiently close to (σΩ 1[L1], σΩ 1[L2]) the bifurcation to coexistence states is subcritical, up to some value of the parameter where it becomes into supercritical. 1.2If β < 0, then the bifurcation is always supercritical. 2. Bifurcation directions along Λ1,0 2.1If c > 0 and β > 0, then the situation described in case 1.1 occurs. 2.2If c > 0 and β < 0, then the bifurcation direction is supercritical. 2.3c < 0 and β > 0, then the bifurcation direction is subcritical. 2.4If c < 0 and β < 0, then for values of the parameters sufficiently close to (σΩ 1[L1], σΩ 1[L2]) the bifurcation is supercritical, while after some critical value becomes subcritical. We should point out that, due to the symmetry of the problem, if L1=L2=−∆ and a0=c0= 0, then fis much more degenerate than (10.8). To treat these degenerate situations we refer to the Appendix of [10]. 11. The special case L1=L2=−∆with constant coefficients. Throughout this section we assume that L1=L2=−∆ and that a,b,cand dare constant. After a change of variables we can assume that a=d= 1 . In the sequel we use the notation σ1[q] := σΩ 1[−∆ + q], σ1:= σ1[0] , θγ:= θ[−∆,γ,1] , and extend the definition of θγtaking θγ:= 0 for γ≤σ1. As an immediate consequence from the results in the previous sections we obtain the following global theorem, which is a substantial improvement of all the previous results in the references. Theorem 11.1. (i) Assume bc < 1. Then, the following assertions are true: (i.1) If any of the semi-trivial positive solutions is linearly unstable, then (1.1) possesses a coexistence state. If in addition λ > σ1,µ > σ1, then there exists I0>0such that if either b < I0or c < I0, then the coexistence state is unique and exponentially asymptotically stable. (i.2) If for (λ, µ)=(λ0, µ0)some of the semi-trivial positive solutions is linearly stable and (1.1) possesses a coexistence state, then it possesses a coexistence state for each (λ, µ)satisfying λ≥λ0,µ≥µ0, and at least two coexistence states if λ > λ0,µ > µ0 and some of the semi-trivial positive solutions is linearly stable. 66 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ (i.3) For each λ∈R, there exists µext(λ)∈Rsuch that (1.1) does not admit a coexistence state if µ≤µext(λ). Similarly, for each µ∈R, there exists λext(µ)∈Rsuch that (1.1) does not admit a coexistence state if λ≤λext(µ). Moreover, thanks to Lemma 6.2, µext(λ)≥(1 −bc)σ1−cλ , λext(µ)≥(1 −bc)σ1−bµ . (11.1) (ii) Assume bc > 1. Then, the following assertions are true: (ii.1) If N≤5and some of the semi-trivial positive solutions is linearly stable, then (1.1) possesses a coexistence state. (ii.2) If N≤5and there exists (λ, µ) = (λ0, µ0)for which (1.1) possesses a coexistence state being any of the semi-trivial states linearly unstable, then (1.1) possesses a coexistence state for each (λ, µ)satisfying λ≤λ0and µ≤µ0, and at least two coexistence states if λ < λ0and µ < µ0and any of the semi-trivial states is linearly unstable. (ii.3) For each λ∈R, there exists µext(λ)∈Rsuch that (1.1) does not admit a coexistence state if µ≥µext(λ). Similarly, for each µ∈R, there exists λext(µ)∈Rsuch that (1.1) does not admit a coexistence state if λ≥λext(µ). The first goal of this section is finding out sharper estimates than (11.1) for the values of λext(µ) and µext(λ) in the case bc < 1. Our main result in this direction reads as follows: Theorem 11.2. Assume bc < 1and λ > σ1, λ ≥µ > σ1[−c1 + b 1−bcθλ].(11.2) Then, u≤1 + b 1−bcθλ, v ≤θ[−∆−c1+b 1−bc θλ,µ,1] ,(11.3) for any coexistence state (u, v)of (1.1). Therefore, if λ > σ1and µ≤max{σ1[−c1 + b 1−bcθλ], σ1(1 −bc)−cλ }(11.4) then (1.1) does not admit a coexistence state. By symmetry, the same result holds if µ > σ1and λ≤max{σ1[−b1 + c 1−bcθµ], σ1(1 −bc)−bµ }. Proof. Thanks to Lemma 7.2, (1+b)v≤(1+c)uand hence, we find from the u-equation of the system that −∆u≤λu −1−bc 1 + bu2. SYMBIOTIC SPECIES 67 Thus, Lemma 3.2 implies the first upper estimate of (11.3). Substituting this estimate into the v-equation of the system gives (−∆−c1 + b 1−bc θλ)v≤µv −v2, and Lemma 3.2 completes the proof of (11.3). The remaining assertions follow readily from Theorem 3.1 and Theorem 11.1 (i.3). ¤ Remark 11.3. The curve defined by the right hand side of (11.4) meets (σ1, σ1) at the value λ=σ1, since limλ↓σ1θλ= 0 and hence, lim λ↓σ1 max{σ1[−c1 + b 1−bcθλ], σ1(1 −bc)−cλ }= lim λ↓σ1 σ1[−c1 + b 1−bcθλ] = σ1, thanks to the continuous dependence of the principal eigenvalue with respect to the potential. Therefore, the estimate of the extinction region given by (11.4) is optimal for values of λ≃σ1. Moreover, (11.4) is also optimal for values of λvarying on compact subintervals of [σ1,∞) provided bis sufficiently small, as the following result shows. Theorem 11.4. Assume bc < 1,λ > σ1and µ < σ1[−cθλ]. Then, there exists b0= b(λ)>0such that (1.1) does not admit a coexistence state if b∈[0, b0]. Moreover, b(λ) varies continously with λ. Proof. The function h(b) := −c1 + b 1−bc , is decreasing and it satisfies h(0) = −c , lim b↑c−1h(b) = −∞. Thus, there exists a unique b0=b(λ)>0 such that µ=σ1[−c1 + b0 1−b0cθλ]< σ1[−cθλ]. Therefore, for b∈[0, b0] we have that µ≤σ1[−c1 + b 1−bc θλ]≤σ1[−cθλ] and Theorem 11.2 completes the proof. ¤ Remark 11.5. Thanks to the estimate (4.10) in the proof of Theorem 4.1 in [25], we find that σ1[−c1 + b 1−bcθλ]≤σ1−c1 + b 1−bc (λ−σ1) 68 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ and therefore, the following estimate for µext(λ) is obtained µext(λ)≥(σ1−c1+b 1−bc (λ−σ1) if λ≤σ1b(2−bc)+1 b(c+1) , σ1(1 −bc)−cλ if λ > σ1b(2−bc)+1 b(c+1) . This estimate provides us with some very readily computable sufficent condition in terms of the several coefficients involved in the model setting for the extinction of the species v. In Figure 4 we have represented the curve of change of stability of (θλ,0) together with the boundary of the extinction region given by the estimate (11.4); for values of (λ, µ) in the bright grey region the model possesses a coexistence state, while for the values of (λ, µ) in the darker region the species vis driven to extinction by u. λ µ µ= (σ F 1 (λ) ,σ 1) Figure 4: The coexistence and extinction regions. In the next result we complete the local analysis of Section 10 by giving some sufficient conditions for completely ascertaining the bifurcation directions to coexistence states in the case bc > 1. SYMBIOTIC SPECIES 69 Theorem 11.6. Assume bc > 1,bc ≥2 + cand fix λ>σ1. Then the bifurcation direction to coexistence states from (µ, u, v)=(σ1[−cθλ], θλ,0) is subcritical. By symmetry, if bc > 1,bc ≥2 + band we fix µ>σ1, then the bifurcation direction from (λ, u, v) = (σ1[−bθµ],0, θµ)is subcritical. Proof. Let (µ(s), u(s), v(s)) denote the local curve of coexistence states emanating from (θλ,0) at µ=σ1[−cθλ]. The main theorem of [7] guarantees that µ(s) is real analytic in sand hence it possesses an expansion of the form µ(s) = σ1[−cθλ] + sµ1(λ) + O(s2),as s→0, for some µ1(λ)∈R. A rather standard calculation shows that (cf. [6] and [10] for details) µ1(λ) = (2 + c)−1[(2 + c−bc)ZΩ ϕ3 λ−bc(λ−σ1[−cθλ]) ZΩ ϕ2 λR(λ)ϕλ],(11.5) where R(λ) := (−∆+2θλ−λ)−1and ϕλ>0 is the principal eigenfunction associated with σ1[−cθλ] normalized so that kϕλk2= 1. This completes the proof. ¤ Modulo the change of band cby −band −c, respectively, the formula (5.2) of [10] provides us with the sign of µ1(λ) for λ≃σ1. Lemma 11.7. (i) If λis sufficiently close to σ1, then sign µ1(λ) = sign (1 −bc). (ii) Similarly, for µ≃σ1, sign λ1(µ) = sign (1 −bc), where λ1(µ) = dλ ds |s=0. Here, λ(s)stands for the λ-component of the curve of coexistence states emanating from (λ, u, v) = (σ1[−bθµ],0, θµ), whose existence is guaranteed by Theorem 5.1. We now show how change the bifurcation directions to coexistence states along the semi-trivial branches as bc grows from the critical value 1, so completing the results of Section 10. For this we will use the local bifurcation analysis already done in Section 10. Interexchanging the roles of band cin [10] by −bby −chere, we obtain the bifurcation equation λr −rp(r, s, λ, µ, b, c) = 0 , µs −sq(r, s, λ, µ, b, c) = 0 ,(11.6) where q(r, s, λ, µ, b, c) = p(s, r, µ, λ, c, b) and p(r, s, λ, µ, b, c) = M(r−bs) + N[2r2−b(3 −c)rs −b(1 −b)s2] +K{5r3−b(c2−4c+ 10)r2s−3b[(1 −b)(1 −c)−b]rs2 −b(b2−2b+ 2)s3} +L[2λr2−b(3λ−cµ)rs −b(µ−bλ)s2] +O(4,(r, s, λ, µ)) , 70 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ where M,N,K,Lare the constants defined by (3.6) in [10]. We should point out that if bc = 1, then the constants a0and c0of the statement of Theorem 10.2 equal zero, and so Theorem 10.2 does not cover this case. This is why to analyze the change of criticality of the local bifurcations from the semi-trivial branches third order terms are needed. Our main result in this direction is the following, where the notations introduced in Section 10 are kept. Theorem 11.8. If bc −1>0is sufficiently small, then there exists a unique change of criticality in a neighborhood of the origin along each of the curves Mλand Mµ. Proof. After some strightforward manipulations, we find that λ1(t) = Mt + 2Nt2+ (5K+ 2LM)t3+O(t4),(11.7a) µ1(t) = −Mct −Nc(1 −c)t2−(Kc(c2−2c+ 2) + LMc(1 + c2))t3+O(t4).(11.7b) Thus, setting Jac1(t) = (prqs−psqr)(t, 0, λ1(t), µ1(t)) , and substituting (11.7) in it gives Jac1(t) = εM2+ε(4 −3c)NMt + [2(c+ 1)2(KM −N2) + εFc]t2+O(t3), where ε:= 1 −bc , Fc=M2L(3c2+ 4) + KM(4c2−7c+ 13) + N2(2c2−8c+ 2) . Making the change of variables ε=−τ2, s =s0τ , and setting Jac1(τ, s0) := Jac1(−τ2, s0τ) τ2, P := 2(c+ 1)2(KM −N2), it is easily seen that Jac1(τ, s0) = −M2−(4 −3c)NMs0τ+Ps2 0−τ2s2 0Fc+O(s3 0τ), We already know that P > 0 (cf. [10], pg. 109). Moreover, we have that Jac1(0,M √P) = 0 , Ds0Jac1(0,M √P) = 2√PM 6= 0 . Thus, thanks to the implicit function theorem, there exists a unique function s0such that for each τ≃0 s0(0) = M(P)−1/2,Jac1(τ, s0(τ)) = 0 . Henceforth, Jac1(−τ2, s0(τ)τ) = 0 . Therefore, there exists a unique t(ε)>0 such that Jac1(t(ε)) = 0 . By symmetry, the remaining assertions get shown. This completes the proof. ¤ SYMBIOTIC SPECIES 71 Some further discussion. We now summarize the information given by the results in the last two sections. For this, it is convenient regarding band cas the main parameters of the model. More precisely, we will fix c > 0 and vary b. Thanks to Lemma 10.2, if b < c−1, then the bifurcation directions to coexistence states are supercritical. Thanks to Theorem 11.8, there exists ε0=ε0(c)>0 such that if c−1<b<(1 + ε0)c−1then the bifurcation directions are subcritical for (λ, µ) close to (σ1, σ1), in fact this holds in a √bc −1-neighborhood of (σ1, σ1), while they become supercritical outside this neighborhood, within another slightly larger neighborhood of (σ1, σ1). Now, since the curves bc = 2+band bc = 2+cin the statement of Theorem 11.6 meet at (b, c) = (2,2), changing their relative positions as cacrosses 2, two different cases must be considered. If c < 2, then we find from Theorem 11.6 that (1 + ε0)c−1<1 + 2 c, since for bc ≥c+ 2 all bifurcation directions from (θλ,0) became subcritical. If c < 1, then our results do not provide us with any further global information about the bifurcation directions along (0, θµ), while in case 1 <c<2 it follows from Theorem 11.6 that if bincreases up to acrossing some critical value, necessarily less than 2 c−1, then all bifurcations to coexistence states from (0, θµ) will change to subcritical either. In case c > 2 these global changes in the nature of the bifurcations occur in the converse order. Now, any bifurcation direction from (θλ,0) is subcritical if b > 2 c+1 and moreover all bifurcation directions from any of the semi-trivial states are subcritical if b > 1 + 2 c. c b bc=1 bc=2+b bc=2+c G L G λ Gµ Figure 5: Varying band c. In Figure 5 we have summarized all the previous information. The first quadrant is divided into four regions. The bright grey region stands for bc < 1, where we only have local information; the black region, which is a thin streep above bc > 1, where we know 72 M. DELGADO, J. L´ OPEZ-G´ OMEZ AND A. SU´ AREZ that the local change of criticality occurs; the regions Gλand Gµ, in between bc = 2 + c and bc = 2+b, where we know that the bifurcation direction from one of the semi-trivial branches, respectively (θλ,0) and (0, θµ), is always subcritical but no global information about the nature of the bifurcation along the remaining semi-trivial branch is available; in the region G, thanks to Theorem 11.6 all bifurcation directions are subcritical, and finally the region L, where only local information is supplied by our analysis. By the continuous dependence of the bifurcation directions with respect to (λ, µ, b, c), if we move away from Ltowards Gλ∪Gµ(or the region G), any point of change of criticality on any of the semi-trivial branches should vary along this branch up to either meet with another point of change of criticality or grow up to infinity. In the first case, both points of change of criticality shrink at the meeting value and then dismiss. 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