Spatially heterogeneous Lotka–Volterra competition

Nonlinear Analysis 165 (2017) 33–79

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Spatially heterogeneous Lotka–Volterra competition✩ Sergio Fernández-Rincón, Julián López-Gómez* Institute of Interdisciplinary Mathematics (IMI) of Complutense University, 28040-Madrid, Spain

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Article history: Received 27 April 2017 Accepted 19 September 2017 Communicated by Enzo Mitidieri MSC 2010: 35K40 35J57 35K57 92D25 Keywords: Competition Coexistence Permanence Founder control competition Uniqueness Stability Multiplicity Dynamics Singular perturbation

abstract This paper studies the dynamics of the spatially-heterogeneous diffusive Lotka– Volterra competing species model. It focuses special attention in ascertaining the linear stability and multiplicity of the coexistence steady states. One of our main findings establishes that, as soon as any steady-state solution of the non-spatial model is linearly unstable somewhere in the inhabiting territory, Ω , any steady state of the spatial counterpart perturbing from it therein (as the diffusion rates, d1 , d2 , move away from 0) must be linearly unstable. From this general principle one can derive a number of rather astonishing consequences, as the multiplicity of the coexistence steady states when the non-spatial model exhibits founder control competition somewhere in Ω , say Ωbi , even if Ωbi is negligible empirically. Actually, this is the first available multiplicity result for small diffusion rates. Finally, based on a celebrated identity by M. Picone (1910), we are able to establish a new, rather striking, uniqueness result valid for general spatially heterogeneous models. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we consider the Lotka–Volterra competition reaction–diffusion heterogeneous system ⎧ ∂u ⎪ ⎪ + d1 L1 u = λ(x)u − a(x)u2 − b(x)uv ⎪ ⎪ ∂t ⎪ ⎪ in Ω × (0, +∞), ⎨ ∂v 2 + d2 L2 v = µ(x)v − d(x)v − c(x)uv (1.1) ∂t ⎪ ⎪ ⎪ ⎪ ⎪ B u = B2 v = 0 on ∂Ω × (0, +∞), ⎪ ⎩ 1 u(·, 0) = u0 > 0, v(·, 0) = v0 > 0 in Ω , ✩ Partially supported by the Ministry of Economy and Competitiveness of Spain under Research Grant MT2015-65899-P, by the Institute of Interdisciplinary Mathematics of Complutense University, and by the Ministry of Education and Culture of Spain under Fellowship Grant FPU15/04755. * Corresponding author. E-mail addresses: [email protected] (S. Fern´ andez-Rinc´ on), [email protected] (J. L´ opez-G´ omez).

https://doi.org/10.1016/j.na.2017.09.008 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

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as well as its associated elliptic counterpart ⎧ ⎨d1 L1 u = λ(x)u − a(x)u2 − b(x)uv d L v = µ(x)v − d(x)v 2 − c(x)uv ⎩ 2 2 B1 u = B2 v = 0

in Ω ,

(1.2)

on ∂Ω ,

whose solutions are the steady states of the evolutionary model (1.1). In this model, Ω is a bounded domain of RN with boundary, ∂Ω , of class C 2 , and Li , i = 1, 2, are two self-adjoint uniformly elliptic operators in Ω of the type Li = −div(Ai ∇·) + Ci ,

i = 1, 2,

(1.3)

2 ¯ ¯ with Ai ∈ Msym N (C (Ω )) and Ci ∈ C(Ω ). Given any Banach space, X, we are denoting by MN (X) the set of matrices of order N with entries in X. Naturally, Msym N (X) stands for the subset of MN (X) consisting of all the symmetric matrices. As far as concerns ∂Ω , it is throughout assumed to be a (N − 1)-dimensional manifold of class C 2 consisting, for each i ∈ {1, 2}, of finitely many connected components of class C 2

ΓDi,j ,

ΓRi,k ,

1 ≤ j ≤ niD ,

1 ≤ k ≤ niR ,

for some integers niD , niR ≥ 0. By the definition of component, they must be disjoint (see, e.g., J. Munkres [30]) and each of them must be, simultaneously, a relatively open and closed subset of ∂Ω , because ∂Ω is a compact manifold without boundary. Some, or several, of these components might be empty, of course. We denote by i

ΓDi

=

nD ⋃ j=1

i

ΓDi,j ,

ΓRi

=

nR ⋃

ΓRi,j ,

i = 1, 2,

j=1

the Dirichlet and Robin portions of ∂Ω = ΓDi ∪ ΓRi ,

i = 1, 2.

Associated with these decompositions of ∂Ω , there are two boundary operators Bi , i = 1, 2, defined by { Di h := h in ΓDi , Bi h = for every h ∈ W 2,p (Ω ), p > N, (1.4) Ri h := ⟨n, Ai ∇h⟩ + βi h in ΓRi , where βi ∈ C(∂Ω ) and n stands for the outward normal vector field along ∂Ω . Thus, for each i = 1, 2, ΓDi and ΓRi are the portions of the edges of the inhabiting territory, ∂Ω , where the corresponding species, u, or v, obeys a boundary condition of Dirichlet (D) or Robin (R) type, respectively. In particular, we may ¯ ) satisfy denote Bi = D when ΓDi = ∂Ω . In most of this paper, we are assuming that λ, µ, a, b, c, d ∈ C(Ω b(x) > 0 and c(x) > 0 for all x ∈ Ω ,

min a > 0, ¯ Ω

min d > 0, ¯ Ω

(1.5)

¯. though in Sections 2–4 the hypothesis on b and c can be relaxed to b, c ≥ 0 in Ω Throughout this paper, for any given function h ∈ C(Ω ), we shall denote h+ := max{h, 0}. It is said that ¯ ), it is said that h is h is positive, h > 0 or h ⪈ 0 (in Ω ), if h ≥ 0 with h ̸= 0. Also, for any given h ∈ C 1 (Ω strongly positive (in Ω ), h ≫ 0, if it satisfies h(x) > 0

for all x ∈ Ω

and

∂h (x) := ⟨n(x), ∇h(x)⟩ < 0 ∂n

for all x ∈ h−1 (0) ∩ ∂Ω .

Except for the general existence results of Chapter 7 of [25], most of the available literature on Lotka– Volterra competing species models dealt with the very special cases when either ΓR1 = ΓR2 = ∅, or

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the species are subject to non-flux boundary conditions, where ΓD1 = ΓD2 = ∅ and β1 = β2 = 0; in particular, [5,8,10,11,12,14–16,18,19,29] as well as most of the references therein. Consequently, as the results of this paper are valid for general boundary conditions of mixed type, our findings are substantially more general than all previous existing results. As in most of the applications to Ecology, Environmental Sciences, Biology and Medical Sciences, the diffusion rates of the species, measured by d1 and d2 , are very small in comparison with the relative size of the remaining coefficients involved in the setting of the model, our attention in this paper is mainly focused into the problem of characterizing the dynamics of (1.1) for sufficiently small d1 and d2 , which seems to be an important mathematical challenge, as we are dealing with a singular perturbation problem for a parabolic system in the presence of spatial heterogeneities. The reader should be aware that no previous singular perturbation result under mixed boundary conditions is available, even for the single diffusive logistic equation! Much like in [11,12,14], also under general mixed boundary conditions the dynamics of (1.1) for small diffusion rates is based on the nature of the dynamics of the associated non-spatial model { ′ u (t) = λ(x)u(t) − a(x)u2 (t) − b(x)u(t)v(t) t > 0, (1.6) v ′ (t) = µ(x)v(t) − d(x)v 2 (t) − c(x)u(t)v(t) where x ∈ Ω is regarded as a parameter, though our new results here provide with some new findings that are extremely important from the point of view of the applications, as it will become apparent soon. According to the nature of the dynamics of (1.6), the inhabiting territory, Ω , consists of the following regions: ¯ : λ(x), µ(x) ≤ 0}, Ωext := {x ∈ Ω ¯ : λ(x), µ(x) > 0, λ(x)d(x) > µ(x)b(x), µ(x)a(x) > λ(x)c(x)}, Ωper := {x ∈ Ω ¯ : λ(x), µ(x) > 0, λ(x)d(x) < µ(x)b(x), µ(x)a(x) < λ(x)c(x)}, Ωbi := {x ∈ Ω u ¯ : λ(x) > 0, λ(x)d(x) > µ(x)b(x), µ(x)a(x) < λ(x)c(x)}, Ωdo := {x ∈ Ω

(1.7)

v ¯ : µ(x) > 0, λ(x)d(x) < µ(x)b(x), µ(x)a(x) > λ(x)c(x)}, Ωdo := {x ∈ Ω u v ¯ \ (Ωext ∪ Ωper ∪ Ωbi ∪ Ωdo Ωjunk := Ω ∪ Ωdo ).

¯ where (0, 0) As already suggested by the names given to each of these zones, Ωext consists of the set of x ∈ Ω ¯ where the is a global attractor with respect to the positive solutions of (1.6); Ωper stands for the set of x ∈ Ω λ(x) µ(x) semi-trivial positive steady-state solutions, ( a(x) , 0) and (0, d(x) ), are linearly unstable – and so, the model ¯ where ( λ(x) , 0) and (0, µ(x) ) are linearly stable, where (1.6) is permanent –, Ωbi consists of the set of x ∈ Ω a(x) d(x) ¯ (1.6) exhibits a genuine founder control competition, and the portions Ω u and Ω v stand for the zones of Ω do

do

where one of the semi-trivial steady states is positive and linearly stable, while the other one is non-positive, or it is positive but linearly unstable. Should it be the case, the linearly stable positive semi-trivial solution is a global attractor for the component-wise positive solutions of (1.6). Finally, we are denoting by Ωjunk ¯ of the union of the previous regions. It is folklore that in Ωper the non-spatial model the supplement in Ω possesses a unique coexistence steady state which is a global attractor for the component-wise positive solutions of (1.6), whereas in Ωbi there is a unique coexistence state which is a saddle point, whose stable manifold, linking (0, 0) to the coexistence state, divides the first quadrant, u > 0, v > 0, in two regions, each of them being the attraction source of one of the semi-trivial positive solutions. The first goal of this paper is establishing a new principle in the theory of competing species models, which may be formulated as follows: Principle of Parabolic Instability (PPI): As soon as any steady-state solution of the non-spatial model (1.6) is linearly unstable somewhere in Ω , any steady state of the spatial model (1.1) perturbing

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uniformly from it therein – as (d1 , d2 ) moves away from (0, 0) – must be linearly unstable with respect to (1.1). Therefore, any localized instability of a non-spatial steady-state solution has a global effect on the dynamics of the spatial model (1.1) for sufficiently small d1 > 0 and d2 > 0. This result has some astonishing µ(x) consequences. For example, when Ωper ̸= ∅, then the non-spatial semi-trivial solutions ( λ(x) a(x) , 0) and (0, d(x) ) are linearly unstable for all x ∈ Ωper . Thus, according to the PPI, rigorously established by Theorem 6.2 and Proposition 6.4, each of the semi-trivial positive solutions of the spatial model, which perturb from those of (1.6) by Theorem 3.1, must be linearly unstable with respect to (1.1) for sufficiently small d1 and d2 . Therefore, (1.1) must be permanent, which provides us with Theorem 2.1 of [14] and Corollary 4.6 of [12]. The most intriguing feature of this fact is that Ωper might be Bε (x0 ), for some x0 ∈ Ω , ε being the inverse of ¯ \Ω ¯ per ! Under this special patch the universe radius measured in Angstroms, while simultaneously Ωbi = Ω configuration, the smaller ε the smaller the values of d1 and d2 are so that (1.1) can be permanent. u v u Naturally, the same conclusion holds as soon as Ωdo and Ωdo are non-empty. Indeed, if Ωdo ̸= ∅, then, for µ(x) u u every x ∈ Ωdo with µ(x) > 0, the semi-trivial positive solution (0, d(x) ) must be unstable for all x ∈ Ωdo . Thus, according to Proposition 6.4, the semi-trivial positive steady-state solution of (1.1) perturbing from v it, (0, v), must be linearly unstable. By symmetry, since Ωdo ̸= ∅, also the semi-trivial positive steady state of the form (u, 0) must be linearly unstable for sufficiently small diffusion rates. Therefore, as in the previous u v case when Ωper ̸= ∅, also when Ωdo and Ωdo are non-empty, the problem (1.1) is permanent for sufficiently small d1 and d2 . The striking fact that these permanence results do not depend on the sizes of the patches u v Ωper , Ωdo and Ωdo reveals the strength of our Principle of Parabolic Instability. Actually, this might explain why in most of empirical studies on competing species permanence is much more usual than expected in heterogeneous environments (see, e.g., [3,28], and the references there in). Going beyond it was shown in [28] how “Most of field experiments and paleontology data corroborate that in the presence of refuge areas, the species persist during long periods of time, even under drastic changes in competition patterns as a result of sudden environmental ‘disasters’, so confirming that in many circumstances the competitive exclusion principle is false.” The second goal of this paper consists in establishing, as an important consequence of the Principle of Parabolic Instability, the multiplicity of coexistence steady states for sufficiently small diffusion rates when Ωbi ̸= ∅ in the symmetric model where d1 = d2 , L1 = L2 , B1 = B2 , λ = µ, a = d and b = c. For the validity of this result we need to assume (1.1) to be permanent for small diffusion rates in order to guarantee the existence of a stable coexistence steady state. We already know that the permanence for sufficiently small d1 u v and d2 can be reached by simply imposing that Ωper ̸= ∅, or Ωdo ̸= ∅ and Ωdo ̸= ∅. Although we were unable to establish the multiplicity result in the general non-symmetric case, we do make the following conjecture: Conjecture: If (1.1) is permanent for sufficiently small diffusion rates and Ωbi ̸= ∅, then (1.1) possesses at least three coexistence steady-state solutions for sufficiently small d1 and d2 . Two among them linearly stable and perturbing from each of the semi-trivial steady states in Ωbi and another one linearly unstable perturbing from the coexistence steady state of the associated non-spatial model in Ωbi as d1 and d2 move away from 0. Besides this result should not depend on the size of Ωbi , the number of coexistence steady-state solutions for sufficiently small d1 and d2 might depend on the number of components of Ωbi . Our multiplicity result allows us to establish the optimality of a substantial extension of the main uniqueness theorem of Hutson et al. [19] (see Theorem 1.1 and Proposition 3.5 therein) that establishes the uniqueness of a coexistence steady-state solution of (1.1) for sufficiently small d1 and d2 under the ¯ . In such case, the unique coexistence steady state must be a global rather natural assumption that Ωper = Ω attractor with respect to the component-wise positive solutions of (1.1). The multiplicity result when Ωbi ¯ is optimal for the uniqueness in the following sense. If we replace b(x) is non-empty shows that Ωper = Ω

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and c(x) by ρb(x) and ρc(x), where ρ > 0 is regarded as a real parameter, then there are choices of b(x) ¯ holds for all ρ ∈ (0, 1), but it fails at the single point x0 ∈ Ω when and c(x) for which condition Ωper = Ω ¯ ρ = 1. So, Ωper = Ω \ {x0 } if ρ = 1. As there are examples of b(x) and c(x) with non-empty Ωbi for ρ > 1 ¯ sufficiently close to 1, such that Ωbi shrinks to x0 as ρ ↓ 1, our multiplicity theorem shows that if Ωper = Ω ¯ fails at a single point x0 ∈ Ω , then the problem (1.1) might exhibit a bifurcation to multiple coexistence steady-steady solutions. Besides our extension of the uniqueness theorem of Hutson et al. [19], collected in Theorems 8.1 and 8.2 of Section 8, is new in its greatest generality, because is valid for a general class of selfadjoint differential operators subject to general mixed boundary conditions, the proof given in this paper overcomes the highly sophisticated technicalities of the proof of Proposition 3.5 of [19] by means of a quasi-cooperative version for mixed boundary conditions of Theorem 2.1 of L´opez-G´omez and Molina-Meyer [27], Theorem 6.3 of L´ opez-G´ omez and Sabina [29] and Theorem 2.4 of Amann and L´opez-G´omez [2]. Indeed, our proof is a rather direct, very elegant, application of the theorem of characterization of the strong maximum principle through the construction of an appropriate supersolution. It should be noted that Proposition 3.5 of Hutson et al. [19] was found for Neumann boundary conditions. The third goal of this paper is establishing a general, rather astonishing, uniqueness result covering the ¯ . Naturally, according to the main multiplicity theorem of this general case when Ωper is a proper subset of Ω paper, in order to get uniqueness one should assume that Ωbi = ∅. The simplest way to get it is imposing that bc ⪇ ad

in Ω .

(1.8) 2

3

2

3

b c Under (1.8), our general uniqueness theorem establishes that, if either ac , or ab2 d , or cdb , or ad 2 , is a positive constant in Ω , then any coexistence state of (1.2) must be linearly stable and hence, unique, if it exists. Naturally, these conditions hold when a(x), b(x), c(x) and d(x) are positive constants such that a = d = 1 and bc < 1, as it was recently imposed by He and Ni [15]. Consequently, our result, Theorem 9.1, provides us with an extremely sharp and substantial extension of Theorem 3.4(iii) of [15], because it is valid for general spatially heterogeneous systems subject to mixed boundary conditions. As a byproduct of Theorem 9.1, under the previous assumptions, as soon as the model possesses two non-degenerate semi-trivial positive steady states, (1.1) exhibits three different types of behavior. Namely, either both semi-trivial positive solutions are linearly unstable, and then the problem has a unique coexistence steady state which is a global attractor with respect to the component-wise positive solutions of (1.1), or one of the semi-trivial positive solutions is linearly stable, while the other one is linearly unstable, and in such case the stable one must be a global attractor, much like in the non-spatial model. As far as concerns the restrictions imposed on the function coefficients a(x), b(x), c(x) and d(x) in Theorem 9.1, it should be noted that each of them involves three of these four coefficients: either a, b, c in 3 b2 c3 is constant in Ω , or a, b, d if we impose that ab2 d is constant, or a, c, d when ad the assumption that ac 2 is 2 c constant, or b, c, d if, instead, db is constant in Ω . Thus, in either cases we have complete freedom to chose, arbitrarily, three of the coefficients, while the fourth one is uniquely determined by the remaining ones, up to a positive multiplicative constant chosen to satisfy (1.8). Rather astonishingly, in this uniqueness theorem λ(x) and µ(x) can be chosen arbitrarily. The distribution of this paper is as follows. Sections 2 and 3 are devoted to the generalized diffusive logistic equation under general mixed boundary conditions of non-classical type, in the sense that the coefficient β in the boundary operator (1.4) can change of sign. Precisely, Section 2 analyzes the existence of positive solutions, as well as some important comparison properties, and Section 3 derives a singular perturbation result, completely new in its greatest generality, by using a sophisticated technical device going back to Lemma 2.1 and Section 4.6 of [26]. This result is substantially more general than Theorem 4.2 of K. Nakashima et al. [31], for as here we allow β to change sign and the boundary conditions to be of

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general mixed type. Some previous singular perturbation results had been already given in [14,23] and [9], but these papers were left outside the list of references of K. Nakashima et al. [31]. Section 4 uses the singular perturbation result of Section 3 to derive a general singular perturbation theorem for the elliptic system (1.2) as (d1 , d2 ) → (0, 0) from the monotone scheme introduced by Hutson et al. in [18], later refined by the authors in [11,12]. Our singular perturbation result is substantially sharper than the previous ones of [12,11,18,19] because it is valid for general mixed boundary conditions and general differential operators in divergence form. All the previous ones were given for the −∆ in both equations under Dirichlet or Neumann boundary conditions. Section 5 contains a version of the theorem of characterization of the maximum principle for quasi-cooperative two species systems, as well as some important monotonicity properties of the underlying principal eigenvalues. Our results are refinements of some previous findings of [29], based on [27]. Section 6 uses most of the previous results to prove the Principle of Parabolic Instability stated above. Section 7 establishes the multiplicity of coexistence steady-state solutions when Ωbi ̸= ∅ provided (1.1) is permanent for sufficiently small d1 and d2 . So, corroborating the validity of the Conjecture above. Section 8 proves the ¯ . Finally, Section 9 derives the general uniqueness result uniqueness result in the special case when Ωper = Ω ¯ when Ωper is a proper subset of Ω through Picone’s identity. 2. Existence of positive solutions and comparison properties for the generalized diffusive logistic equation This section collects a series of results of technical nature that are going to be used throughout this paper. Some of them are folklore, while others are completely new with the generality stated herein. ¯ ), we will denote by σ1 [νL + h; B, Ω ] the principal eigenvalue Subsequently, given ν > 0 and h ∈ C(Ω of the operator νL + h in Ω subject to the boundary condition Bu = 0 on ∂Ω , where L and B are of the same type as those defined in (1.3) and (1.4), respectively. According to the Rayleigh–Courant formula, the principal eigenvalue admits the following variational characterization: { ∫ } ∫ ∫ 2 2 σ1 [νL + h; B, Ω ] = min ν ⟨∇u, A∇u⟩ + ν β u + (h + νC) u : ∥u∥2 = 1 , (2.1) 1 (Ω) ¯ u∈CB

Ω

ΓR

Ω

¯ ) stands for the set of test functions u ∈ C 1 (Ω ¯ ) such that Bu = 0 on ∂Ω . Subsequently, for where, CB1 (Ω k,p every p ∈ (1, +∞) and k ≥ 1, WB (Ω ) stands for the set of functions u ∈ W k,p (Ω ) such that Bu = 0 on ∂Ω . Since the associated normalized eigenfunction, φ, lies in WB2,p (Ω ) for every p > N , we have that, in ¯ ). This follows easily from the classical Lp -theory for mixed bvp’s collected in Chapter particular, φ ∈ CB1 (Ω 5 of [26]. Indeed, according to Theorem 5.11 of [26], any weak eigenfunction, φ, of [νL + h; B, Ω ] in Lp (Ω ), for some p ≥ 2, must be in the Sobolev space W 2,p (Ω ). By an additional bootstrapping argument involving the Sobolev imbeddings (collected, e.g., in [26, Th. 4.2]), this actually implies that φ ∈ W 2,p (Ω ) for all p > N . Indeed, suppose N ≥ 3. Then, since φ ∈ W 1,2 (Ω ) and 2 < N , it follows from [26, Cor. 4.1] that 2,p1 φ ∈ Lp1 (Ω ) with p1 = N2N (Ω ). Suppose 3 ≤ N ≤ 5. Then, 2p1 > N −2 . Thus, by [26, Th. 5.11], φ ∈ W ¯ and hence, thanks to [26, Th. 4.2(ii)], φ ∈ C(Ω ). In particular, φ ∈ Lp (Ω ) for all p > N and [26, Th. 5.11] ends the proof. When N = 6, p1 = 3 and, hence, 2p1 = 6 = N . However, φ ∈ W 2,3 (Ω ) ⊂ W 2,2 (Ω ), and ′ thus φ ∈ Lp1 (Ω ), with p′1 = N2N −4 = 6, by [26, Th. 4.2(ii)]. This gives the desired result arguing as before, ′ because 2p1 = 12 > 6. Next, assume that N ≥ 7. Then, 2 < 2p1 < N and, hence, due to [26, Th. 5.11] and p1 [26, Th. 4.2(i)], φ ∈ W 2,p1 (Ω ) ⊂ Lp2 (Ω ) with p2 := NN−2p > p1 > 2. Reiterating this argument yields 1 Np

k−1 φ ∈ Lpk (Ω ) and hence φ ∈ W 2,pk (Ω ), for all k ≥ 1, where pk := N −2p , k ≥ 2, unless 2pk−1 ≥ N for some k−1 k > 2. Suppose the first option occurs. Then, since pk is increasing and bounded, it must converge to some p p > 2. Then, letting k → ∞, we find that p = NN−2p , which implies p = 0, a contradiction. So, φ ∈ W 2,pk0 (Ω ), being 2pk0 ≥ N for some k0 ≥ 2 and 2pk < N for all 1 ≤ k < k0 . Consequently, if 2pk0 > N , then, by ¯ ), whereas if 2pk = N arguing like in the case N = 6 (W 2,pk0 (Ω ) ⊂ W 2,pk0 −1 (Ω )) [26, Th. 4.2(ii)], φ ∈ C(Ω 0 ends the proof of the previous claim when N ≥ 3. The lower dimensional case when N = 1, 2 is easier.

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39

Since our analysis will be focused on the dynamics of (1.1) for small diffusion rates, we will mainly analyze the behavior of the principal eigenvalue for such range of diffusions. The next result extends substantially and sharpens all the previous existing counterparts because it is valid for general mixed bvp’s. In particular, it extends Lemma 3.1 of [14] and Lemma 2.1 of [18] obtained for the very special case when L = −∆ under Dirichlet and Neumann boundary conditions, respectively. Except for Part (b), which is new, the proof has been adapted from the proof of Lemma 2.1 of [18]. Theorem 2.1. The principal eigenvalue Σ (ν, h) := σ1 [νL + h; B, Ω ] satisfies the following properties ¯ ) → R is C 1 and strictly increasing, that is, if h1 , h2 ∈ C(Ω ¯ ) with (a) For every ν > 0 the map Σ (ν, ·) : C(Ω h1 ≤ h2 , h1 ̸= h2 , then Σ (ν, h1 ) < Σ (ν, h2 ). ¯ ) the map Σ (·, h) : (0, +∞) → R is analytic. Moreover, if C ≥ 0 in Ω and β ≥ 0 on (b) For every h ∈ C(Ω ∂Ω , then Σ (·, h) is non-decreasing. If, in addition, either h is not constant, or ΓD ̸= ∅, or C ̸= 0 in Ω , or β ̸= 0 on ∂Ω , then the map Σ (·, h) is strictly increasing with ∂Σ ∂ν > 0. ¯ ) the map Σ (·, h) extends continuously to ν = 0 by (c) For every h ∈ C(Ω Σ (0, h) := lim Σ (ν, h) = min h. ν→0

¯ Ω

Proof . Part (a) follows adapting the argument of Lemma 3.1 of [14]. So, we omit its details. Taking into account that the family of operators L(ν) := νL + h is analytic in ν ∈ (0, ∞), the regularity of Σ (·, h) in Part (b) is a direct consequence from Sections VII.1.3 and VII.2 of [20] reasoning as in Section 9.1 of [26]. Naturally, their respective positive eigenfunctions normalized in L2 (Ω ), ϕ(ν), also are analytic in ν > 0. Thus, differentiating { νLϕ(ν) + hϕ(ν) = Σ (ν, h)ϕ(ν) in Ω , Bϕ(ν) = 0 on ∂Ω , with respect to ν yields { ∂Σ Lϕ(ν) + νLϕ′ (ν) + hϕ′ (ν) = (ν, h)ϕ(ν) + Σ (ν, h)ϕ′ (ν) ∂ν Bϕ′ (ν) = 0

in Ω , on ∂Ω .

Multiplying the differential equation of ϕ′ (ν) by ϕ(ν) and integrating in Ω , it follows from the Divergence Theorem that ∫ ∫ ∂Σ (ν, h) = ϕ(ν)Lϕ(ν) + ϕ(ν)(νL + h − Σ (ν, h))ϕ′ (ν) ∂ν Ω Ω ∫ ∫ = ⟨∇ϕ(ν), A∇ϕ(ν)⟩ + Cϕ2 (ν) Ω∫ Ω ∫ ∫ ′ − ϕ(ν)⟨n, A∇ϕ(ν)⟩ + ν ϕ (ν)⟨n, A∇ϕ(ν)⟩ − ν ϕ(ν)⟨n, A∇ϕ′ (ν)⟩ ∂Ω ∂Ω ∂Ω ∫ ∫ ∫ ∫ ∫ ∫ 2 = ⟨∇ϕ(ν), A∇ϕ(ν)⟩ + Cϕ2 (ν) + βϕ2 (ν) ≥ αA |∇ϕ(ν)| + Cϕ2 (ν) + βϕ2 (ν), Ω

Ω

ΓR

Ω

Ω

ΓR

where αA > 0 is the ellipticity constant of L. If C ≥ 0 and β ≥ 0 then ∂Σ ∂ν ≥ 0. If, in addition, h is not constant, or ΓD ̸= ∅, or C ̸= 0, or β ̸= 0, then at least one of the terms on the right hand side of the previous inequality is positive and therefore, ∂Σ ∂ν > 0. Note that if β = C = 0 and h is constant, then we are dealing with a Neumann problem if ΓD = ∅. In such case, ϕ(ν) is constant and Σ (ν, h) = h, constant, for all ν > 0. On the contrary, if ΓD ̸= ∅, then ϕ(ν) cannot be constant, because ϕ(ν) = 0 on ΓD implies ϕ(ν) = 0 on Ω , ∫ 2 which is impossible. Therefore, in such case, αA Ω |∇ϕ(ν)| > 0 and hence, also ∂Σ ∂ν (ν, h) > 0. This ends the proof of Part (b).

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

40

¯ ), we find As for Part (c), thanks to the variational characterization (2.1), for every ν > 0 and h ∈ C(Ω that σ1 [νL + h; B, Ω ] ≥ νσ1 [L; B, Ω ] + min h Ω

(2.2)

for every ν > 0. In particular, given ε > 0, σ1 [νL + h; B, Ω ] ≥ min h − ε Ω

for all ν <

ε . |σ1 [L; B, Ω ]|

Subsequently, consider the open subset of Ω , Ωε := h−1

((

min h, min h + ¯ Ω

¯ Ω

ε 2

))

.

¯ . Let uε ∈ C ∞ (Ω ) be such that supp uε ⊂ Ωε It is non-empty because we are assuming h to be continuous in Ω 0 and ∥uε ∥2 = 1. Then, according to (2.1), we have that, for every ν > 0, ∫ ∫ ∫ 2 Σ (ν, h) ≤ ν ⟨∇uε , A∇uε ⟩ + ν βuε + (h + νC) u2ε Ω ΓR Ω ) ∫ (∫ ∫ C u2ε + hu2ε ⟨∇uε , A∇uε ⟩ + =ν Ωε Ωε Ωε (∫ ) ∫ ε ≤ν ⟨∇uε , A∇uε ⟩ + C u2ε + min h + . ¯ 2 Ω Ωε Ωε Therefore, ⏐∫ ⏐−1 ∫ ⏐ ε ⏐⏐ 2⏐ σ1 [νL + h; B, Ω ] = Σ (ν, h) ≤ min h + ε for all ν < ⏐ ⟨∇uε , A∇uε ⟩ + C uε ⏐ . Ω 2 Ωε Ωε This ends the proof.

□

The next result establishes the monotonicity of the principal eigenvalue with respect to the domain. It will be very useful later. It brings together in a compact way Propositions 3.1 and 3.2 of [4]. Lemma 2.2. Let Ω0 be a subdomain of class C 2 of Ω such that ∂Ω0 ∩ ∂Ω consists of finitely many components of ∂Ω , if it is non-empty. Let B0 be any boundary operator of the type { h on ∂Ω0 ∩ Ω , B0 h := ˜ for every h ∈ W 2,p (Ω ), p > N, Bh on ∂Ω0 ∩ ∂Ω , ˜ = h, or Bh ˜ = Bh. Then, for every ν > 0 and h ∈ C(Ω ¯ ), where, on each component of ∂Ω0 ∩ ∂Ω , either Bh σ1 [νL + h; B, Ω ] < σ1 [νL + h; B0 , Ω0 ]

if (B, Ω ) ̸= (B0 , Ω0 ).

Proof . Set Σ := σ1 [νL + h; B, Ω ],

Σ0 := σ1 [νL + h; B0 , Ω0 ],

and let ϕ ≫ 0 be a principal eigenfunction associated with Σ . Then, B0 ϕ = ϕ > 0 on ∂Ω0 ∩ Ω as well as in the components of ∂Ω0 ∩ ∂Ω where B0 is the Dirichlet operator and B is of Robin type. Note that, since (B, Ω ) ̸= (B0 , Ω0 ), either Ω0 is a proper subset of Ω and hence ∂Ω0 ∩ Ω ̸= ∅, or Ω = Ω0 and then, as B ̸= B0 , there should be a component of ∂Ω0 ∩ ∂Ω where B0 is the Dirichlet operator and B is of Robin type. Should

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

41

there exist some component of ∂Ω0 ∩ ∂Ω where B0 = B, we must have B0 ϕ = Bϕ = 0, by definition of ϕ. Thus, ϕ is a positive strict supersolution of { (νL + h − Σ )u = 0 in Ω0 , B0 u = 0 on ∂Ω0 . Therefore, owing to Theorem 7.10 of [26], σ1 [νL + h − Σ ; B0 , Ω0 ] > 0, which provides us with the desired estimate, Σ0 = σ1 [νL + h, B0 , Ω0 ] > Σ . □ The semilinear elliptic boundary value problem { νLu = γ(x)u − m(x)u2 Bu = 0

in Ω , on ∂Ω ,

(2.3)

¯ ) and min ¯ m > 0, will arise very often throughout this paper. The next result characterizes where γ, m ∈ C(Ω Ω the existence and uniqueness of positive solutions of (2.3) in terms of the linearized stability of u = 0 with respect to its parabolic counterpart. It is a byproduct of Theorem 3.5 of [13]. Theorem 2.3. The problem (2.3) has a positive solution u ∈ 0. Furthermore, it is unique if it exists.

⋂

p>N W

2,p

(Ω ) if and only if σ1 [νL−γ; B, Ω ] <

L,B,Ω L,B,Ω The maximal non-negative solution of (2.3) will be throughout denoted by θ{ν,γ,m} . Thus, θ{ν,γ,m} = 0 if L,B,Ω σ1 [νL − γ; B, Ω ] ≥ 0, while θ{ν,γ,m} ≫ 0 if σ1 [νL − γ; B, Ω ] < 0. If there is no ambiguity from the context, L,B,Ω we might omit one, or several, of the involved parameters in θ{ν,γ,m} . Actually, through most of this paper the positive solution will be simply denoted by θ{ν,γ,m} . By Theorems 2.1 and 2.3, the next result holds. Among other things, it establishes that maxΩ¯ γ > 0 is a sufficient condition for the existence of a positive solution of (2.3) for small ν > 0.

Corollary 2.4. The following properties hold: (a) If maxΩ¯ γ < 0, then a maximal ν0 ∈ (0, +∞] exists such that θ{ν,γ,m} = 0 for ν ∈ (0, ν0 ). (b) If maxΩ¯ γ > 0, then a maximal ν0 ∈ (0, +∞] exists such that θ{ν,γ,m} ≫ 0 for ν ∈ (0, ν0 ). (c) If C ≥ 0 in Ω and β ≥ 0 on ∂Ω , then ν0 ∈ [0, +∞] exists such that θ{ν,γ,m} ≫ 0 for every ν ∈ (0, ν0 ) and θ{ν,γ,m} = 0 for every ν ∈ [ν0 , +∞). Proof . Thanks to Theorem 2.1(b) and (c) the map Σ (·, −γ) : (0, +∞) → R is analytic and extends continuously to ν = 0 by Σ (0, −γ) = min(−γ) = − max γ. ¯ Ω

¯ Ω

Thus, if maxΩ¯ γ < 0 then σ1 [νL − γ; B, Ω ] > 0 for sufficiently small ν > 0, whereas if maxΩ¯ γ > 0, then σ1 [νL − γ; B, Ω ] < 0. So, Parts (a) and (b) follow from Theorem 2.3. Similarly, the last assertion can be obtained from Theorem 2.3 as a consequence of the monotonicity of Σ (·, −γ) guaranteed by Theorem 2.1(b). □ It should be noted that, in Corollary 2.4(a), we have that σ1 [νL − γ; B, Ω ] ≥ 0 for all ν ∈ (0, ν0 ) and σ1 [ν0 L − γ; B, Ω ] = 0. Thus, ν0 < +∞ if σ1 [νL − γ; B, Ω ] < 0 for some ν > 0. An example of this can be constructed very easily when γ < 0 is a constant. Indeed, since Σ (ν, −γ) = σ1 [νL − γ; B, Ω ] = νσ1 [L; B, Ω ] − γ,

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

42

it is apparent that Σ (ν, −γ) < 0 for all ν > γ/σ1 [L, B, Ω ] if σ1 [L; B, Ω ] < 0. This holds, e.g., for sufficiently negative C(x). On the contrary, under the general assumptions of Theorem 2.1(b), Σ (ν, −γ) ≥ Σ (0, −γ) = − max γ > 0 ¯ Ω

and therefore, ν0 = +∞. The next result enables us to compare solutions of different problems in various domains. Lemma 2.5. Let Ω0 be a subdomain of class C 2 of Ω such that ∂Ω0 ∩∂Ω consists of finitely many components of ∂Ω , if it is non-empty. Let B0 be any boundary operator of the type introduced in the statement of Lemma ¯ ) be such that m ≤ m0 and γ ≥ γ0 . Then, for every ν > 0, 2.2. Let γ0 , m0 ∈ C(Ω L,B0 ,Ω0 L,B,Ω ≥ θ{ν,γ θ{ν,γ,m} 0 ,m0 }

in Ω0 .

L,B,Ω Moreover, if (Ω0 , B0 , γ0 , m0 ) ̸= (Ω , B, γ, m), then this estimate is strict if and only if θ{ν,γ,m} > 0.

Proof . If (Ω0 , B0 , γ0 , m0 ) = (Ω , B, γ, m), then L,B0 ,Ω0 L,B,Ω θ{ν,γ,m} = θ{ν,γ . 0 ,m0 }

Assume that (Ω0 , B0 , γ0 , m0 ) ̸= (Ω , B, γ, m). By Theorem 2.1(a) and Lemma 2.2, σ1 [νL − γ; B, Ω ] ≤ σ1 [νL − γ0 ; B0 , Ω0 ]. Thus, by Theorem 2.3, L,B0 ,Ω0 L,B,Ω θ{ν,γ,m} = θ{ν,γ =0 0 ,m0 } L,B,Ω θ{ν,γ,m}

>

L,B0 ,Ω0 θ{ν,γ 0 ,m0 }

=0

if σ1 [νL − γ; B, Ω ] ≥ 0, if σ1 [νL − γ; B, Ω ] < 0 ≤ σ1 [νL − γ0 ; B0 , Ω0 ].

L,B0 ,Ω0 Suppose that σ1 [νL − γ0 ; B0 , Ω0 ] < 0, i.e., θ{ν,γ > 0. If (Ω0 , B0 ) ̸= (Ω , B), then 0 ,m0 } L,B,Ω L,B,Ω B0 θ{ν,γ,m} = θ{ν,γ,m} >0

on some component of ∂Ω0 . If (γ0 , m0 ) ̸= (γ, m), then ( )2 ( )2 L,B,Ω L,B,Ω L,B,Ω L,B,Ω L,B,Ω νLθ{ν,γ,m} = γθ{ν,γ,m} − m θ{ν,γ,m} > γ0 θ{ν,γ,m} − m0 θ{ν,γ,m}

in Ω0 .

L,B0 ,Ω0 L,B,Ω Therefore, θ{ν,γ,m} − θ{ν,γ is a strict supersolution of 0 ,m0 } {[ ( )] L,B0 ,Ω0 L,B,Ω νL − γ0 + m0 θ{ν,γ,m} + θ{ν,γ u = 0 in Ω0 , 0 ,m0 } B0 u = 0 on ∂Ω0 .

Consequently, since (

) L,B0 ,Ω0 L,B0 ,Ω0 νL − γ0 + m0 θ{ν,γ θ{ν,γ =0 0 ,m0 } 0 ,m0 }

L,B0 ,Ω0 in Ω0 and θ{ν,γ > 0, Theorem 2.1(a) implies that 0 ,m0 } [ ( ) ] [ ] L,B0 ,Ω0 L,B0 ,Ω0 L,B,Ω σ1 νL − γ0 + m0 θ{ν,γ,m} + θ{ν,γ ; B , Ω > σ νL − γ + m θ ; B , Ω = 0. 0 0 1 0 0 0 0 {ν,γ0 ,m0 } 0 ,m0 }

So, owing to Theorem 7.10 of [26], we can conclude that L,B0 ,Ω0 L,B,Ω θ{ν,γ,m} > θ{ν,γ 0 ,m0 }

The proof is complete.

□

in Ω0 .

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

43

3. The singular perturbation problem for the generalized diffusive logistic equation This section focuses attention in the problem of ascertaining the limiting profile of the maximal nonnegative solution of (2.3) as ν ↓ 0 in the general case when the coefficients of the model also depend on the diffusion, ν, in a controlled way. Subsequently, we are going to denote by ΓR+ the union of the set of components of ΓR where the function γ is everywhere positive. The main result of this section can be stated −1 ¯ : γ+ (x) = 0}. as follows. Note that γ+ (0) = {x ∈ Ω −1 Theorem 3.1. Let K be a compact subset of Ω ∪ ΓR+ ∪ γ+ (0). Then,

γ+ m

lim θ{ν,γ,m} = ν↓0

uniformly in K,

where θ{ν,γ,m} stands for the maximal non-negative solution of (2.3). This theorem is a substantial generalization of Theorem 3.5 of [14] and of Theorem 3.3 of [12], which were established for the very special case when L = −∆ and ΓR = ∅, as well as of Lemma 2.5 of [18], which was found for the very special case when L = −∆, ΓD = ∅ and β ≡ 0 on ∂Ω . Astonishingly, Theorem 3.1 seems to be the first singular perturbation result for semilinear elliptic equations under mixed boundary conditions. It is optimal from two different points of view. First, because the boundary conditions are completely general; in particular, substantially more general that the ones considered in K. Nakashima et al. [31]. Secondly, ¯ , as B might be of Dirichlet type because in general the convergence cannot be expected to be uniform on Ω on some, or several, of the components of ∂Ω , where the positive solution must develop boundary layers. The available techniques do not work out to deal with our more general setting. Actually, our proofs are based on some rather sophisticated technical devices developed from Lemma 2.1 and Theorem 1.9 of [26], though the overall proof relies on a clever use of the method of sub and supersolutions, like in the available, less general, results. Naturally, from a technical point of view, it is much more intricate constructing these sub and supersolutions under arbitrary mixed boundary conditions. The proof of Theorem 3.1 will follow after a series of technical results, the first one providing us with a ¯. global uniform estimate for the positive solutions on Ω Ω Lemma 3.2. For every ε > 0, there exists ν0 = ν0 (ε) > 0 such that θ{ν,γ,m} ≤ ν ∈ (0, ν0 ).

γ+ m

¯ for all + ε in Ω

¯ ) be such that Proof . Fix ε > 0 and let Ψ ∈ C 2 (Ω γ+ ε γ+ ¯, + ≤Ψ ≤ + ε in Ω m 2 m

and BΨ ≥ 0

on ∂Ω .

(3.1)

¯) The construction of Ψ can be accomplished as follows. According to Lemma 2.1 of [26], there exist ψ ∈ C 2 (Ω and a constant τ > 0 such that ⟨n, A∇ψ⟩ ≥ τ

and ψ = 0

on ∂Ω .

¯ ), the proof of Theorem 1.9 of [26] can be adapted to show that there Since Ω is of class C 2 and A ∈ C 2 (Ω exists ρ0 > 0 such that, for every ρ ∈ [0, ρ0 ), the map Qρ : ∂Ω → Ω defined by Qρ (y) := y − ρ

A(y)n(y) , ∥A(y)n(y)∥

y ∈ ∂Ω ,

establishes a global diffeomorphism between ∂Ω and Qρ (∂Ω ), which is a C 2 -surface, like ∂Ω . Moreover, Q0 = I∂Ω and Qρ1 (∂Ω ) ∩ Qρ2 (∂Ω ) = ∅

if 0 ≤ ρ1 < ρ2 < ρ0 .

44

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

Subsequently, we denote by U the set ⋃

U :=

Qρ (∂Ω ),

ρ∈[0,ρ0 )

¯ consisting of all x ∈ Ω at a conormal distance which provides us with an open neighborhood of ∂Ω in Ω (distance along the conormal direction) less than ρ0 from ∂Ω . Actually, the Qρ ’s are level surfaces of the conormal distance. Similarly, we consider the map Π : U → ∂Ω of class C 2 defined by Π (x) := Q−1 ρ (x)

if x ∈ Qρ (∂Ω ).

¯ ) be such that Let Φ ∈ C ∞ (Ω ε γ+ 5ε γ+ 7ε γ+ γ+ (x) + < (x) + ≤ Φ(x) ≤ (x) + < (x) + ε, m 2 m 8 m 8 m

¯. ∀x ∈ Ω

Then, we can consider Ψ : U → R given through Ψ (x) := Φ(Π (x)) − 1 + eM ψ(x) ,

x ∈ U.

Note that Ψ (x) = Φ(Π (x)) = Φ(x) for all x ∈ ∂Ω . In particular, by the construction of Φ, γ+ 5ε γ+ 7ε (x) + ≤ Ψ (x) ≤ (x) + m 8 m 8

for all x ∈ ∂Ω .

(3.2)

Moreover, since Π is constant along any conormal direction, for every x ∈ ∂Ω , BΨ (x) = ⟨n, A(x)∇Ψ (x)⟩ + β(x)Ψ (x) = M eM ψ(x) ⟨n, A(x)∇ψ(x)⟩ + β(x)Φ(x) = M ⟨n, A(x)∇ψ(x)⟩ + β(x)Φ(x), because ψ = 0 on ∂Ω and Π (x) = x for all x ∈ ∂Ω . Thus, by the choice of ψ, BΨ (x) ≥ M τ + β(x)Φ(x) ≥ 0 for all x ∈ ∂Ω , provided M is large enough. Thanks to (3.2), by continuity, there exists a neighborhood of ∂Ω , U˜ ⊂ U, such that 9ε γ+ 15ε γ+ (x) + ≤ Ψ (x) ≤ (x) + m 16 m 16

˜ for all x ∈ U.

¯ satisfying all the properties Finally, it is rather obvious that Ψ can be extended to a C 2 -function in Ω required in (3.1). Consequently, setting ( ε )2 min m ¯ Ω ν0 := ∈ (0, +∞], 2 maxΩ¯ |LΨ | we find that, for every ν < ν0 , (γ ) (γ ( ε )2 γ+ ε) ε γΨ − mΨ 2 = mΨ − Ψ ≤ mΨ − − ≤ − mΨ ≤ − min m ≤ νLΨ , ¯ m m m 2 2 2 Ω i.e., Ψ is a positive supersolution of (2.3). By a comparison argument involving the strong maximum principle, we can infer that, for every 0 < ν < ν0 , θ{ν,γ,m} (x) ≤ Ψ (x) ≤

γ+ (x) ¯. □ + ε for all x ∈ Ω m(x)

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

45

The following result provides us with Theorem 3.1 in the special case when ΓR = ∅. −1 Ω Lemma 3.3. Let K ⊂ Ω ∪ γ+ (0) be a compact subset. Then, limν↓0 θ{ν,γ,m} =

γ+ m

uniformly in K.

Proof . By Lemma 3.2 it suffices to establish the lower estimates. First, we will get them for constant coefficients, γ and m. Then, the lower estimates will be derived in the general case. When γ ≤ 0, the required convergence follows directly from Lemma 3.2. So, assume that γ > 0 is constant, ¯r (x0 ) ⊂ Ω . Let φ ≫ 0 fix ε ∈ (0, 2γ/m), and consider x0 ∈ K and a positive r with r < dist(K, ∂Ω ). Then, B be the principal eigenfunction associated to σ1 [L; D, Br (x0 )] normalized so that ∥φ∥∞ = 21 , and consider the function ϕ ∈ ∩p>N W 2,p (Br (x0 )) defined through { ¯r (x0 ) \ B ¯r/2 (x0 ), φ in B ϕ := ¯r/2 (x0 ), φˆ in B where φˆ is any sufficiently smooth function chosen so that ϕ > 0 in Br (x0 ), ∥ϕ∥∞ = 1 and ϕ(x0 ) = 1. Then, setting (γ ε) γ ε − ϕ≤ − Ψ := m 2 m 2 and taking into account that γ − mΨ ≥ γ − m

(γ m

−

ε) = mε/2, 2

we have that min (γ − mΨ ) ≥ mε/2 > 0.

¯ r (x0 ) B

Thus, since νLΨ ≤ γΨ − mΨ 2 if and only if ν Lϕ ϕ ≤ γ − mΨ , there exists νx0 ,ε > 0 such that, for every ν ∈ (0, νx0 ,ε ), the function Ψ provides us with a strict subsolution of the problem { νLu = γu − mu2 in Br (x0 ), Du := u = 0 on ∂Br (x0 ). Hence, by a comparison argument involving the Strong Maximum Principle, we find that, for every ν < νx0 ,ε , D,B (x )

r 0 Ψ (x) ≤ θ{ν,γ,m} (x)

for all x ∈ Br (x0 ).

γ − 2ε and Ψ is continuous, there exist rx0 ,ε ∈ (0, r) such that Ψ (x) > Moreover, since Ψ (x0 ) = m all x ∈ Brx0 ,ε (x0 ). Therefore, for every ν ∈ (0, νx0 ,ε ), from Lemma 2.5 we can conclude that D,B (x )

B,Ω r 0 θ{ν,γ,m} (x) ≥ θ{ν,γ,m} (x) ≥ Ψ (x) ≥

γ −ε m

γ m

− ε for

in Brx0 ,ε (x0 ).

Finally, since K is compact, there are x1 , . . . , xn ∈ K such that K ⊂ ∪ni=1 Brxi ,ε (xi ) and so, for every B,Ω γ ν < mini νxi ,ε , the estimate θ{ν,γ,m} ≥m − ε holds in K. In the general case when γ and m are non-constants, for any given ε > 0, we have that ( γ )−1 γ+ + B,Ω θ{ν,γ,m} (x) ≥ 0 ≥ (x) − ε provided x ∈ [0, ε] m m for all ν > 0. Thus, it suffices to establish the lower estimate in the compact subset ( γ )−1 + K0 := K ∩ [ε, +∞) ⊂ Ω . m

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

46

¯r (x0 ) ⊂ Ω ∩ (γ+ /m)−1 (ε/2, +∞). Shortening r > 0, if necessary, we Let x0 ∈ K0 and r > 0 be such that B can obtain that minB¯r (x0 ) γ maxB¯r (x0 ) m

≥

ε γ+ ε γ (x) − = (x) − m 2 m 2

for all x ∈ Br (x0 ),

(3.3)

¯r (x0 ). Therefore, by Lemma 2.5, we find that, for every ν > 0, because γ(x) > 0 for all x ∈ B D,B (x )

D,B (x )

B,Ω r 0 r 0 ≥ θ{ν,γ,m} ≥ θ{ν,min θ{ν,γ,m} ¯

¯ r (x ) m} Br (x0 ) γ,maxB 0

in Br (x0 ).

(3.4)

Thanks to the first part of the proof, dealing with the case of constant coefficients, there exists νx0 ,ε > 0 such that, for every ν ∈ (0, νx0 ,ε ), D,B (x )

r 0 θ{ν,min ¯

¯ r (x ) m} Br (x0 ) γ,maxB 0

minB¯r (x0 ) γ

≥

maxB¯r (x0 ) m

−

ε 2

−

ε γ+ ≥ − ε in Br/2 (x0 ). 2 m

in Br/2 (x0 ).

Hence, thanks to (3.3) and (3.4), D,B (x )

B,Ω r 0 θ{ν,γ,m} ≥ θ{ν,min ¯

¯ r (x ) m} Br (x0 ) γ,maxB 0

≥

minB¯r (x0 ) γ maxB¯r (x0 ) m

Finally, as K0 is compact, there are x1 , . . . , xn ∈ K0 such that K0 ⊂ ∪ni=1 Bri /2 (xi ), and therefore, B,Ω θ{ν,γ,m} ≥

γ+ −ε m

in K0

for all ν < ν0 = mini νxi ,ε , which ends the proof. □ We are ready to complete the proof of Theorem 3.1. Proof of Theorem 3.1. Throughout this proof, we are assuming that maxΩ¯ γ > 0 since, otherwise, the result holds from Lemma 3.3. Thus, by Corollary 2.4, θ{ν,γ,m} ≫ 0 for sufficiently small ν > 0. Note that, −1 since Lemma 3.3 provides us with the result on any compact subset of Ω ∪ γ+ (0), it suffices to prove the j + theorem on a neighborhood of ΓR . Let ΓR be, with j ∈ {1, . . . , nR }, an arbitrary component of ΓR+ , and consider ε > 0 and r > 0 for which the open neighborhood of ΓRj Ωrj := {x ∈ Ω : dist (x, ΓRj ) < r} satisfies ∂Ωrj \ ΓRj ⊂ Ω ,

ε < min ¯j Ω r

γ . m

Since ∂Ω is of class C 2 , for sufficiently small r > 0, Ωrj is an open subdomain of Ω with boundary, ∂Ωrj , of class C 2 , because the distance function to the boundary is of class C 2 (see [21] if necessary). Moreover, ∂Ωrj ¯ rj \ Ω j is a compact subset of Ω , by Lemma consists of two components, ∂Ωrj ∩ Ω and ΓRj . Since Krj := Ω r/2 3.3, there exists νr,ε > 0 such that θ{ν,γ,m} ≥

γ ε − m 2

in Krj for every ν < νr,ε .

j γ In particular, θ{ν,γ,m} (x) ≥ m (x) − 2ε for every x ∈ ∂Ωr/2 ∩ Ω and ν < νr,ε . ¯ j ) such that Subsequently, arguing as in the proof of Lemma 3.2, we can consider a function Ψ ∈ C 2 (Ω r/2

0<

γ γ 3ε (x) − ε ≤ Ψ (x) ≤ (x) − m m 4

j for every x ∈ Ωr/2

(3.5)

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

47

and ( ) γ ⟨n, A∇Ψ ⟩ < − max |β(x)| (x) ≤ −βΨ x∈∂Ω m

on ΓRj .

(3.6)

Note that, by construction, such a function Ψ should satisfy θ{ν,γ,m} − Ψ ≥

ε 4

j ∩ Ω. on ∂Ωr/2

¯ ) such that The existence of Ψ can be established as follows. Consider Φ ∈ C ∞ (Ω γ γ 15ε γ 13ε γ 3ε (x) − ε < (x) − ≤ Φ(x) ≤ (x) − < (x) − m m 16 m 16 m 4

¯. for every x ∈ Ω

j Arguing as in Lemma 3.2, it becomes apparent that there exist a neighborhood, U ⊂ Ωr/2 , of the component ΓRj and a map Π : U → ΓRj of class C 2 such that for every x ∈ U there exists ξ(x) > 0 such that

Π (x) = x + ξ(x)A(Π (x))n(Π (x)) ∈ ΓRj , i.e., Π is the projection parallel to the conormal direction. ¯ ) and τ > 0 such that ψ ≤ 0 in Ω , ψ = 0 on Now, According to Lemma 2.1 of [26], there exist ψ ∈ C 2 (Ω ∂Ω , and ⟨n(x), A(x)∇ψ(x)⟩ ≥ τ

for all x ∈ ∂Ω .

Then, for any M > 0, we can define Ψ (x) := Φ(Π (x)) − M ψ(x),

x ∈ U.

Since ψ = 0 on ∂Ω , the function Ψ satisfies that, for every x ∈ ΓRj , ) ( 15ε γ 13ε γ (x) − , (x) − . Ψ (x) = Φ(x) ∈ m 16 m 16

(3.7)

Moreover, since Φ ◦ Π is constant along the conormal direction, we find that, for every x ∈ ΓRj , ⟨n(x), A(x)∇Ψ (x)⟩ = −M ⟨n(x), A(x)∇ψ(x)⟩ ≤ −M τ. Therefore, by choosing M>

( ) γ 1 max |β(x)| (x) , τ x∈∂Ω m

the estimate (3.6) holds. Finally, taking into account (3.7), shortening U and extending the function Ψ to j Ωr/2 , it becomes apparent that (3.5) also holds. j According to (3.5), we already know that γΨ − mΨ 2 is separated away from 0 on Ωr/2 . Thus, setting minΩ¯ j (γΨ − mΨ 2 ) ν1 :=

r/2

maxΩ¯ j |LΨ |

,

r/2

we have that νLΨ ≤ γΨ − mΨ 2

for every ν < ν1 .

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S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

Consequently, denoting ν0 := min{νr,ε , ν1 }, we find that, for every ν < ν0 , the function θ{ν,γ,m} − Ψ satisfies ⎧ j ⎪ ⎨[νL − γ + m(θ{ν,γ,m} + Ψ )](θ{ν,γ,m} − Ψ ) ≥ 0 in Ωr/2 , B(θ{ν,γ,m} − Ψ ) > 0 on ΓRj , ⎪ ε j ⎩θ >0 on ∂Ωr/2 ∩ Ω. {ν,γ,m} − Ψ ≥ 4 Therefore, θ{ν,γ,m} − Ψ provides us with a strict supersolution of { j , [νL − γ + m(θ{ν,γ,m} + Ψ )]u = 0 in Ωr/2 j B0 u = 0 on ∂Ωr/2 , where B0 stands for the boundary operator { B B0 := D

on ΓRj , j on ∂Ωr/2 ∩ Ω.

As established at the very beginning of the proof, maxΩ¯ γ > 0 and hence, θ{ν,γ,m} ≫ 0 for sufficiently small ν > 0. Thus, thanks to the monotony properties of the principal eigenvalues established by Theorem 2.1 and Lemma 2.2, we have that, for sufficiently small ν > 0, j j σ1 [νL − γ + m(θ{ν,γ,m} + Ψ ); B0 , Ωr/2 ] > σ1 [νL − γ + mθ{ν,γ,m} ; B0 , Ωr/2 ]

> σ1 [νL − γ + mθ{ν,γ,m} ; B, Ω ] = 0. Hence, owing to Theorem 7.10 of [26], we can conclude that θ{ν,γ,m} > Ψ ≥

γ+ −ε m

j in x ∈ Ωr/2

for all ν < ν0 . The proof is complete. □ The next results provide us with some variants of Theorem 3.1 for ‘floating’ coefficients, depending on ν. The second one, is a key ingredient in the proof of the singular perturbation result for the system. ¯ ), J ⊂ (0, +∞)2 with (0, 0) ∈ J, ¯ and families {γ(ν,δ) }(ν,δ)∈J , Theorem 3.4. Consider γ, m ∈ C(Ω ¯ ) such that {m(ν,δ) }(ν,δ)∈J ⊂ C(Ω lim (ν,δ)→(0,0)

γ(ν,δ) = γ

and

lim (ν,δ)→(0,0)

m(ν,δ) = m

¯. uniformly in Ω

(3.8)

Then, lim (ν,δ)→(0,0)

θ{ν,γ(ν,δ) ,m(ν,δ) } =

γ+ m

uniformly on compact subsets of Ω ∪ ΓR+ ∪ (γ+ )−1 (0), where ΓR+ is the subset of ΓR defined at the beginning of this section. Proof . Let ε > 0 be such that ε < minΩ¯ m. Then, by (3.8), νε > 0 exists such that γ − ε ≤ γ(ν,δ) ≤ γ + ε

and

0 < m − ε ≤ m(ν,δ) ≤ m + ε in Ω

for all ν, δ < νε , (ν, δ) ∈ J. Thanks to Lemma 2.5, for such range of ν and δ we have that θ{ν,γ−ε,m+ε} ≤ θ{ν,γ(ν,δ) ,m(ν,δ) } ≤ θ{ν,γ+ε,m−ε}

in Ω .

(3.9)

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

49

−1 Let K be a compact subset of Ω ∪ ΓR+ ∪ γ+ (0). By Theorem 3.1, letting (ν, δ) ∈ J approximate (0, 0) in the restriction of the estimate (3.9) to K yields

(γ + ε)+ (γ − ε)+ ≤ lim θ{ν,γ(ν,δ) ,m(ν,δ) } ≤ lim θ{ν,γ(ν,δ) ,m(ν,δ) } = m+ε m−ε (ν,δ)→(0,0) (ν,δ)→(0,0) uniformly in K. Letting ε → 0 ends the proof.

□

¯ to a uniform Essentially, the next result sharpens Theorem 3.4 by relaxing the uniform convergence in Ω −1 + convergence on compact subsets of Ω ∪ ΓR ∪ γ+ (0). ¯ ), J ⊂ (0, +∞)2 with (0, 0) ∈ J, ¯ and O ⊂ Ω ∪ Γ + ∪ γ −1 (0), an open Theorem 3.5. Consider γ, m ∈ C(Ω + R + ¯ ∩ Γ = ∅, or O ¯ ∩ Γ + consists of components subset, with respect to the induced topology, such that either O R R ¯ ) be such that of ΓR+ , each one contained in either O or RN \ O. Let {γ(ν,δ) }(ν,δ)∈J , {m(ν,δ) }(ν,δ)∈J ⊂ C(Ω lim (ν,δ)→(0,0)

γ(ν,δ) = γ

and

lim (ν,δ)→(0,0)

m(ν,δ) = m

(3.10)

uniformly on compact subsets of O. Assume that there exists k > 0 and M > 0 such that m(ν,δ) (x) ≥ k,

γ(ν,δ) (x) ≤M m(ν,δ) (x)

¯. ∀ (ν, δ) ∈ J, x ∈ Ω

(3.11)

Then, lim (ν,δ)→(0,0)

θ{ν,γ(ν,δ) ,m(ν,δ) } =

γ+ m

uniformly on compact subsets of O. Moreover, for every ε > 0 there exists νε > 0 such that, for every (ν, δ) ∈ J with ν, δ < νε , θ{ν,γ(ν,δ) ,m(ν,δ) } ≤ M + ε in

Ω.

The assumption (3.11) was unnecessary in the statement of Theorem 3.4 as it is a direct consequence of (3.8). Proof . To prove the convergence, we will obtain first the upper limit. Fix ε > 0 and consider a compact subset, K, of O. Subsequently, for each r > 0, we will denote by ¯ : dist(x, K) ≤ r} Kr := {x ∈ Ω the compact r-neighborhood of K. By construction, for sufficiently small r > 0, K ⊂ Kr ⊂ O. According to (3.10), the quotients there exists νε,1 > 0 such that

γ(ν,δ) m(ν,δ)

converge uniformly to

γ(ν,δ) ε γ+ ε γ ≤ + ≤ + m(ν,δ) m 2 m 2

γ m

in Kr as (ν, δ) → (0, 0), (ν, δ) ∈ J. Thus,

in Kr for ν, δ < νε,1 , (ν, δ) ∈ J.

(3.12)

¯ ) exists such that ξ(x) ∈ [0, 1] for all x ∈ Ω ¯, On the other hand, by Urysohn’s Lemma, a function ξ ∈ C(Ω γ γ + ¯ \ Kr . Thanks to (3.10) and (3.11), we have that ξ = 0 in K, and ξ = 1 on Ω m ≤ M in Kr . Thus, m ≤ M

50

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

γ

in Kr , because M > 0 and so, m+ = 0 ≤ M if γ < 0. Hence, for the previous choice of ξ, the auxiliary γ ¯ ) satisfies non-negative function ψ := m+ (1 − ξ) + M ξ + 2ε ∈ C(Ω ⎧ γ ε + ⎪ + in K, = ⎪ ⎪ ⎪ m 2 ⎨ γ ε ε ε + ¯ + in Kr \ K, ψ ≤ M (1 − ξ) + M ξ + = M + in Ω and ψ ≥ 2 ⎪ m 2 2 ⎪ ⎪ ε ⎪ ¯ \ Kr . ⎩= M + in Ω 2 Consequently, by (3.11) and (3.12), we find that ψ≥

γ(ν,δ) m(ν,δ)

in Ω for ν, δ < νε,1 , (ν, δ) ∈ J.

(3.13)

¯ ) such that BΨ ≥ 0 on ∂Ω and Reasoning as in the proof of Lemma 3.2, there exists Ψ ∈ C 2 (Ω ψ+ Since ψ > 0, we have that Ψ ≥

ε 4

Ψ≤

ε ε ≤Ψ ≤ψ+ 4 2

¯. in Ω

> 0, and, owing to (3.13), γ+ + ε in K, m

Ψ≥

γ(ν,δ) ε + in Ω , m(ν,δ) 4

(3.14)

for every ν, δ < νε,1 with (ν, δ) ∈ J. If necessary, νε,1 > 0 can be shortened so that νε,1 < ( 4ε )2 ∥LΨk ∥∞ . Then, ¯ , we obtain for every ν, δ < νε,1 , (ν, δ) ∈ J, taking into account (3.14) and the hypothesis m(ν,δ) ≥ k in Ω that ( ) ( ε )2 γ(ν,δ) ε 2 γ(ν,δ) Ψ − m(ν,δ) Ψ = m(ν,δ) Ψ − Ψ ≤ −kΨ ≤ −k < −ν∥LΨ ∥∞ ≤ νLΨ in Ω . m(ν,δ) 4 4 Therefore, for such range of values of the parameters, Ψ is a strict supersolution of { νLu = γ(ν,δ) u − m(ν,δ) u2 in Ω , Bu = 0 on ∂Ω , ¯ for all ν, δ < νε,1 with (ν, δ) ∈ J. and hence, by the maximum principle, θ{ν,γ(ν,d) ,m(ν,d) } ≤ Ψ in Ω Consequently, thanks to (3.14), we also have that θ{ν,γ(ν,d) ,m(ν,δ) } ≤

γ+ + ε in K m

θ{ν,γ(ν,d) ,m(ν,δ) } ≤ ψ +

and

ε ¯, ≤ M + ε in Ω 2

for these values of the parameters, which provides us with the upper estimate and the global bound of the last assertion of the theorem. Note that, since ( γ )−1 γ+ + θ{ν,γ(ν,d) ,m(ν,δ) } ≥ 0 ≥ − ε in K ∩ [0, ε], m m γ

the lower estimate of the theorem holds in K ∩ ( m+ )−1 [0, ε] for every ν, δ < νε,1 with (ν, δ) ∈ ∩J. So, it γ remains to get this estimate on the compact set K0 := K ∩ ( m+ )−1 [ 2ε , +∞). ¯0 ⊂ O, a To apply the available comparison results in a regular open neighborhood of K0 , O0 , with O −1 + little bit more of technical work is needed. Since K ⊂ O ⊂ Ω ∪ ΓR ∪ γ+ (0), it becomes apparent that −1 K0 ⊂ O \ γ+ (0) ⊂ Ω ∪ ΓR+ and hence, for sufficiently small r > 0, the open set ( ) Or := {x ∈ O : dist x, ∂O \ (O ∩ ΓR+ ) > r} \ (O ∩ ΓR+ ) satisfies K0 ⊂ Or ∪ (O ∩ ΓR+ ) and ∂Or consists of two types of components, O ∩ ΓR+ and ∂Or ∩ Ω . Note that if ∂O ⊂ O ∩ ΓR+ , then ∂O \ (O ∩ ΓR+ ) = ∅ and hence, Or = O \ ∂O is an open set of class C 2 .

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

51

Should this be the case, then we can define O0 := O \ ∂O. Otherwise, let n ≥ 1 denote the number of components of ∂Or \ (O ∩ ΓR+ ). Note that, for sufficiently small r > 0, ∂Or and ∂O have the same number of components. In such case, Or/2 \ Or consists of n (open connected) components, Oj , 1 ≤ j ≤ n. Next, for every j ∈ {1, . . . , n}, let Mj be any (N − 1)-dimensional compact manifold without boundary of class C ∞ such that Mj ⊂ Oj ,

Or ⊂ int Mj ,

RN \ Or/2 ⊂ ext Mj ,

where int Mj and ext Mj stand for the two components of RN \ Mj . Lastly, in this case, we consider O0 := ∩nj=1 int Mj ∩ Or/2 . Then, O0 is an open subset of RN of class C 2 with K0 ⊂ O0 ∪ (O ∩ ΓR+ ) and ∂O0 consists of two types of ¯0 is a compact components. Namely, those of O ∩ ΓR+ and those of ∂O0 ∩ Ω = ∪nj=1 Mj ⊂ Or/2 . Thus, O ¯0 subset of O and hence, due to (3.10), γ(ν,δ) and m(ν,δ) converge uniformly to γ and m, respectively, in O as (ν, δ) → (0, 0) in J. Consequently, applying Theorem 3.4 in O0 provides us with a νε,2 > 0, νε,2 ≤ νε,1 , such that, for every ν, δ < νε,2 with (ν, δ) ∈ J, B0 ,O0 θ{ν,γ

(ν,δ) ,m(ν,δ) }

≥

γ+ −ε m

in K0 ,

where B0 stands for the boundary operator on ∂O0 defined by B0 := B on O ∩ ΓR+ and by B0 := D on ∂O0 ∩ Ω . On the other hand, owing to Lemma 2.5, B0 ,O0 θ{ν,γ

(ν,δ) ,m(ν,δ) }

B,Ω ≤ θ{ν,γ

(ν,δ) ,m(ν,δ) }

in O0 for all (ν, δ) ∈ J.

Therefore, for this range of values of the parameters, B,Ω θ{ν,γ

which ends the proof.

(ν,δ) ,m(ν,δ) }

≥

γ+ −ε m

in K0 ,

□

4. A singular perturbation theorem for competing species models The main result of this section establishes that, given any family of coexistence states of (1.2), they must approximate as d1 , d2 → 0 the global hyperbolic attractor of the non-spatial model on every patch of the inhabiting territory where it exists. It provides us with a substantial extension of Theorem 4.1 of [18], Theorem 1 of [11] and Theorem 5.1 of [12], established for the very special case when L1 = L2 = −∆ under either non-flux, or Dirichlet, boundary conditions. As in this section we are dealing with general boundary conditions of mixed type, its main result is completely new in its greatest generality. Since the proof uses the (new) singular perturbation results established in Section 3 for the scalar equation, it is far from immediate. Actually, it is rather elaborated. u v Besides the regions Ωext , Ωper , Ωbi , Ωdo , Ωdo and Ωjunk already defined in (1.7), in order to state the main result of this section it is imperative to differentiate some important areas within Ωjunk . Precisely, we per,u will denote by Ωjunk the set of points for which the non-spatial model can be perturbed to exhibit either permanence or dominance of the species u, i.e., per,u ¯ : λ(x) > 0, λ(x)d(x) > µ(x)b(x), µ(x)a(x) = λ(x)c(x)}. Ωjunk := {x ∈ Ω

By symmetry, we also define per,v ¯ : µ(x) > 0, λ(x)d(x) = µ(x)b(x), µ(x)a(x) > λ(x)c(x)}. Ωjunk := {x ∈ Ω

52

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

Finally, the remaining part of Ωjunk is added to Ωbi by considering ( ) ( ) per,u per,v per,u per,v ⋆ u v ¯ \ Ωext ∪ Ωper ∪ Ωdo Ωbi := Ωbi ∪ Ωjunk \ Ωjunk ∪ Ωjunk =Ω ∪ Ωjunk ∪ Ωdo ∪ Ωjunk ¯ : λ(x), µ(x) > 0, λ(x)d(x) ≤ µ(x)b(x), µ(x)a(x) ≤ λ(x)c(x)}, = {x ∈ Ω ¯ \ Ωext such that the non-spatial model can be perturbed to exhibit which consists of the set of values of Ω founder control competition. Using these notations, it is easily seen that, for every per,u per,v ⋆ u v ¯ \ Ωbi x ∈ Ωmax := Ω = Ωext ∪ Ωper ∪ Ωdo ∪ Ωjunk ∪ Ωdo ∪ Ωjunk ,

the non-spatial model possesses a steady state which is a global attractor with respect to the component-wise positive solutions. It is worth-emphasizing that such a steady state might not be of hyperbolic type, and that, for every x ∈ Ωmax , is given by ⎧ (0, 0) if x ∈ Ωext , ⎪ ⎪ ⎪( ) ⎪ ⎪ λ(x)d(x) − µ(x)b(x) µ(x)a(x) − λ(x)c(x) ⎪ ⎪ ⎪ , if x ∈ Ωper , ⎪ ⎪ ⎨ a(x)d(x) − b(x)c(x) a(x)d(x) − b(x)c(x) ) ( (u∗ (x), v∗ (x)) = λ(x) per,u u ⎪ , 0 if x ∈ Ωdo ∪ Ωjunk , ⎪ ⎪ a(x) ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ µ(x) per,v v ⎪ ⎩ 0, if x ∈ Ωdo ∪ Ωjunk . d(x) In agreement with the notations introduced in Section 3, we will denote by ΓR1,+ the union of the components of ΓR1 where λ > 0 everywhere, while ΓR2,+ stands for the union of the components of ΓR2 such that µ > 0 in the whole component. Similarly, we will denote by ΓRper the union of the components of ΓR1 ∩ ΓR2 for which the non-spatial model exhibits permanence everywhere. In particular, ΓRper ⊂ ΓR1,+ ∩ ΓR2,+ ∩ Ωper . We are ready to state the main result of this section. Theorem 4.1. Consider J ⊂ (0, +∞)2 with (0, 0) ∈ J¯ and a family {(u(d1 ,d2 ) , v(d1 ,d2 ) )}(d1 ,d2 )∈J of coexistence states of (1.2). Then, lim (d1 ,d2 )→(0,0)

(u(d1 ,d2 ) , v(d1 ,d2 ) ) = (u∗ , v∗ )

⋆ ¯ and uniformly on compact subsets of Ωmax ∩ (Ω ∪ ΓRper ) = (Ω \ Ωbi ) ∪ ΓRper . In particular, if Ωper = Ω per 1 2 ΓD = ΓD = ∅, i.e., ΓR = ∂Ω , then

lim (d1 ,d2 )→(0,0)

¯. (u(d1 ,d2 ) , v(d1 ,d2 ) ) = (u∗ , v∗ ) uniformly in Ω

The proof of this result follows from a series of technical lemmas. The monotone scheme introduced by the next result goes back to [18]. v(d1 ,d2 ,n) }n≥0 and Lemma 4.2. Fix d1 , d2 > 0 and consider the families {¯ u(d1 ,d2 ,n) }n≥0 , {u(d1 ,d2 ,n) }n≥0 , {¯ {v (d1 ,d2 ,n) }n≥0 defined recursively by v (d1 ,d2 ,0) := 0, u ¯(d1 ,d2 ,0) := θ{d1 ,λ,a} , v (d1 ,d2 ,n) := θ{d2 ,µ−c¯u(d1 ,d2 ,n−1) ,d} , u ¯(d1 ,d2 ,n) := θ{d1 ,λ−bv(d ,d ,n) ,a} , n ≥ 1, 1 2 u(d1 ,d2 ,0) := 0, v¯(d1 ,d2 ,0) := θ{d2 ,µ,d} , u(d1 ,d2 ,n) := θ{d1 ,λ−b¯v(d1 ,d2 ,n−1) ,a} , v¯(d1 ,d2 ,n) := θ{d2 ,µ−cu(d ,d ,n) ,d} , n ≥ 1. 1

2

Then, ¯(d1 ,d2 ,n+1) ≤ u ¯(d1 ,d2 ,n) and u(d1 ,d2 ,n) ≤ u(d1 ,d2 ,n+1) ≤ u v (d1 ,d2 ,n) ≤ v (d1 ,d2 ,n+1) ≤ v¯(d1 ,d2 ,n+1) ≤ v¯(d1 ,d2 ,n) for every n ≥ 0. Moreover, if the elliptic model (1.2) admits a coexistence state, (u(d1 ,d2 ) , v(d1 ,d2 ) ), then u(d1 ,d2 ,n) ≤ u(d1 ,d2 ) ≤ u ¯(d1 ,d2 ,n)

and

v (d1 ,d2 ,n) ≤ v(d1 ,d2 ) ≤ v¯(d1 ,d2 ,n)

for all n ≥ 0.

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

53

Proof . We will only prove the estimates for the first component, u, as those of v follow the same patterns. ¯ ) ∋ h ↦→ θ{d ,λ−bh,a} and C(Ω ¯ ) ∋ h ↦→ θ{d ,µ−ch,d} are non-increasing. Thus, By Lemma 2.5, the maps C(Ω 1 2 ¯ ) → C(Ω ¯) F(d1 ,d2 ) : C(Ω h ↦→ θ{d1 ,λ−bθ{d

2 ,µ−ch,d}

,a}

is a non-decreasing, or order preserving, operator such that ¯(d1 ,d2 ,0) u(d1 ,d2 ,0) = 0 ≤ F(d1 ,d2 ) [h] ≤ θ{d1 ,λ,a} = u

¯ ). for all h ∈ C(Ω

(4.1)

The last estimate follows from Lemma 2.5, while the first one holds by definition. Hence, for any n ≥ 1, applying n times the operator F(d1 ,d2 ) produces n+1 n+1 n n 0 ≤ F(d [u(d1 ,d2 ,0) ] ≤ F(d [u(d1 ,d2 ,0) ] ≤ F(d [¯ u(d1 ,d2 ,0) ] ≤ F(d [¯ u(d1 ,d2 ,0) ] ≤ u ¯(d1 ,d2 ,0) . 1 ,d2 ) 1 ,d2 ) 1 ,d2 ) 1 ,d2 )

On the other hand, the iterates {¯ u(d1 ,d2 ,n) }n≥0 and {u(d1 ,d2 ,n) }n≥0 can be recursively defined in terms of F(d1 ,d2 ) by u(d1 ,d2 ,0) = 0,

u(d1 ,d2 ,n) = F(d1 ,d2 ) [u(d1 ,d2 ,n−1) ],

n ≥ 1,

u ¯(d1 ,d2 ,0) = θ{d1 ,λ,a} ,

u ¯(d1 ,d2 ,n) = F(d1 ,d2 ) [¯ u(d1 ,d2 ,n−1) ],

n ≥ 1.

The last assertion follows from (4.1) taking into account that u(d1 ,d2 ) is a fixed point of F(d1 ,d2 ) . This ends the proof. □ ¯ For every (d1 , d2 ) ∈ J let {¯ Lemma 4.3. Consider J ⊂ (0, +∞)2 with (0, 0) ∈ J. u(d1 ,d2 ,n) }n≥1 , {u(d1 ,d2 ,n) }n≥1 , {¯ v(d1 ,d2 ,n) }n≥1 and {v (d1 ,d2 ,n) }n≥1 be the sequences introduced in Lemma 4.2. Then, the ¯n }n≥1 , {U }n≥1 , {V¯n }n≥1 , {V }n≥1 ⊂ C(Ω ¯ ) defined by sequences {U n n ( ) b λ+ 1 ¯ ¯ ¯ U 0 := 0, , U n := λ − (µ − cU n−1 )+ , U0 := a a d + n ≥ 1, ( ) µ 1 c + ¯ ¯ ¯ , V n := V 0 := 0, V0 := µ − (λ − bV n−1 )+ , d d a + satisfy lim (d1 ,d2 )→(0,0)

¯n , U , V¯n , V ) (¯ u(d1 ,d2 ,n) , u(d1 ,d2 ,n) , v¯(d1 ,d2 ,n) , v (d1 ,d2 ,n) ) = (U n n

uniformly on compact subsets of Ω ∪ ΓRper . Proof . We will restrict ourselves to prove the assertions for u ¯(d1 ,d2 ,n) by induction. The limiting behaviors of u(d1 ,d2 ,n) , v¯(d1 ,d2 ,n) and v (d1 ,d2 ,n) can be obtained similarly. By definition, ΓRper ⊂ ΓR1,+ ∩ ΓR2,+ and λ(x)d(x) − µ(x)b(x) > 0 ¯n ≤ Moreover, 0 ≤ U n ≥ 0,

λ+ a

and

µ(x)a(x) − λ(x)c(x) > 0

for all x ∈ ΓRper .

¯ for all n ≥ 0. Thus, since λ(x) > 0 for each x ∈ Γ per , we find that, for every in Ω R ¯n (x) ≥ µ(x) − c(x) λ(x) > 0 for all x ∈ Γ per , µ(x) − c(x)U R a(x)

(4.2)

b(x) ( d(x) µ(x)

(4.3)

and hence, λ(x) −

) ¯n (x) ≥ λ(x) − − c(x)U +

b(x) d(x) µ(x)

> 0 for all x ∈ ΓRper .

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

54

In the case n = 0, according to Theorem 3.1,

lim (d1 ,d2 )→(0,0)

u ¯(d1 ,d2 ,0) =

lim (d1 ,d2 )→(0,0)

θ{d1 ,λ,a} =

λ+ a

¯0 =U

uniformly on compact subsets of Ω ∪ ΓR1,+ ∪ λ−1 + (0). In particular, this limit holds on compact subsets of Ω ∪ ΓRper . So, we are done in this case. As induction hypothesis, assume that, for some n ≥ 0, lim (d1 ,d2 )→(0,0)

¯n u ¯(d1 ,d2 ,n) = U

uniformly on compact subsets of Ω ∪ ΓRper .

(4.4)

¯n )−1 (0), where Γ 2,∗ stands for the union of the Then, by (4.2), Ω ∪ΓRper is an open subset of Ω ∪ΓR2,∗ ∪(µ−cU + R per 2 ¯ components of Γ such that µ − cUn > 0 everywhere. Note that Γ consists of finitely many components R

of

ΓR2,∗ .

R

Moreover, by definition, u ¯(d1 ,d2 ,n) ≥ 0 for all (d1 , d2 ) ∈ J. Thus, ¯ µ − c¯ u(d1 ,d2 ,n) ≤ max µ in Ω ¯ Ω

and hence, it follows from (4.4) and Theorem 3.5 that

lim (d1 ,d2 )→(0,0)

v (d1 ,d2 ,n+1) =

lim (d1 ,d2 )→(0,0)

θ{d2 ,µ−c¯u(d

1 ,d2 ,n)

,d}

¯n )+ = d1 (µ − cU

uniformly on compact subsets of Ω ∪ ΓRper . Similarly, thanks to (4.3), Ω ∪ ΓRper is an open subset of ( )−1 ¯ n )+ Ω ∪ ΓR1,∗ ∪ λ − db (µ − cU (0), +

where ΓR1,∗ stands for the union of the components of ΓR1 such that λ − above, ΓRper consists of finitely many components of ΓR1,∗ . Since ¯, in Ω

λ − bv (d1 ,d2 ,n+1) ≤ max λ ¯ Ω

b d

(

¯n µ − cU

) +

> 0 everywhere. As

(d1 , d2 ) ∈ J,

it follows from Theorem 3.5 that

lim (d1 ,d2 )→(0,0)

u ¯(d1 ,d2 ,n+1) =

lim (d1 ,d2 )→(0,0)

θ{d1 ,λ−bv(d

1 ,d2 ,n+1)

,a}

=

uniformly on compact subsets of Ω ∪ ΓRper . This ends the proof. □

The next result is Lemma 11 of [11].

1 a

(

¯ n )+ λ − db (µ − cU

) +

¯n+1 =U

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

55

∑n ( bc )j Lemma 4.4. Setting Sn := j=0 ad in Ω , n ≥ 0, the continuous functions defined in Lemma 4.3 are given by ⎧ 0 if λ(x) ≤ 0, ⎪ ⎪ ⎪ ⎪ c(x) λ(x) ⎪ ⎪ if λ(x) > 0 and µ(x) ≤ λ(x), ⎪ ⎪ a(x) a(x) ⎪ ⎪ ( ) ⎪ ⎪ ⎪ b(x) c(x) 1 ⎪ ⎪ λ(x)S (x) − µ(x)S (x) if λ(x) > 0, µ(x) > λ(x) and n n−1 ⎪ ⎨ a(x) d(x) a(x) ¯ Un (x) = b(x) ⎪ λ(x)Sn (x) > µ(x)Sn−1 (x), ⎪ ⎪ d(x) ⎪ ⎪ ⎪ c(x) ⎪ ⎪ ⎪ 0 if λ(x) > 0, µ(x) > λ(x) and ⎪ ⎪ a(x) ⎪ ⎪ ⎪ b(x) ⎪ ⎪ ⎩ µ(x)Sn−1 (x), λ(x)Sn (x) ≤ d(x) ⎧ λ+ (x) ⎪ ⎪ if µ(x) ≤ 0, ⎪ ⎪ a(x) ⎪ ⎪ ⎪ ⎪ ⎪ b(x) ⎪ ⎪0 if µ(x) > 0 and λ(x) ≤ µ(x), ⎪ ⎪ d(x) ⎪ ⎪ ) ( ⎪ ⎪ ⎪ 1 b(x) b(x) ⎪ ⎪ µ(x) Sn (x) if µ(x) > 0, λ(x) > µ(x) and λ(x) − ⎨ a(x) d(x) d(x) U n+1 (x) = ⎪ c(x) ⎪ ⎪ λ(x)Sn−1 (x), µ(x)Sn (x) > ⎪ ⎪ a(x) ⎪ ⎪ ⎪ ⎪ λ(x) b(x) ⎪ ⎪ ⎪ if µ(x) > 0, λ(x) > µ(x) and ⎪ ⎪ a(x) d(x) ⎪ ⎪ ⎪ ⎪ c(x) ⎪ ⎪ ⎩ µ(x)Sn (x) ≤ λ(x)Sn−1 (x), a(x) ⎧ 0 if µ(x) ≤ 0, ⎪ ⎪ ⎪ ⎪ µ(x) b(x) ⎪ ⎪ if µ(x) > 0 and λ(x) ≤ µ(x), ⎪ ⎪ d(x) d(x) ⎪ ⎪ ( ) ⎪ ⎪ ⎪ 1 c(x) b(x) ⎪ ⎪ µ(x)Sn (x) − λ(x)Sn−1 (x) if µ(x) > 0, λ(x) > µ(x) and ⎪ ⎪ a(x) d(x) ⎨ d(x) V¯n (x) = c(x) ⎪ µ(x)Sn (x) > λ(x)Sn−1 (x), ⎪ ⎪ a(x) ⎪ ⎪ ⎪ ⎪ b(x) ⎪ ⎪ 0 if µ(x) > 0, λ(x) > µ(x) and ⎪ ⎪ d(x) ⎪ ⎪ ⎪ ⎪ c(x) ⎪ ⎪ ⎩ µ(x)Sn (x) ≤ λ(x)Sn−1 (x), a(x) ⎧ µ+ (x) ⎪ ⎪ if λ(x) ≤ 0, ⎪ ⎪ d(x) ⎪ ⎪ ⎪ ⎪ ⎪ c(x) ⎪ ⎪ 0 if λ(x) > 0 and µ ≤ λ(x), ⎪ ⎪ a(x) ⎪ ⎪ ( ) ⎪ ⎪ ⎪ 1 c(x) c(x) ⎪ ⎪ µ(x) − λ(x) Sn (x) if λ(x) > 0, µ(x) > λ(x) and ⎨ d(x) a(x) a(x) V n+1 (x) = ⎪ b(x) ⎪ ⎪ λ(x)Sn (x) > µ(x)Sn−1 (x), ⎪ ⎪ d(x) ⎪ ⎪ ⎪ ⎪ µ(x) c(x) ⎪ ⎪ ⎪ if λ(x) > 0, µ(x) > λ(x) and ⎪ ⎪ d(x) a(x) ⎪ ⎪ ⎪ ⎪ b(x) ⎪ ⎪ ⎩ λ(x)Sn (x) ≤ µ(x)Sn−1 (x) d(x)

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

56

¯ and n ≥ 1. for every x ∈ Ω By the monotone character of these sequences of iterates, they must have a point-wise limit. The next result characterizes it. ¯ , the limit limn→+∞ (U ¯n (x), V (x)) equals Lemma 4.5. For every x ∈ Ω n ⎧ (0, 0) ⎪ ⎪ ⎪ ) ( ⎪ ⎪ µ(x) ⎪ ⎪ ⎪ 0, ⎪ ⎪ d(x) ⎨ ¯∗ (x), V (x)) := ( λ(x) ) (U ∗ ⎪ ,0 ⎪ ⎪ a(x) ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ λ(x)d(x) − µ(x)b(x) µ(x)a(x) − λ(x)c(x) ⎪ ⎩ , a(x)d(x) − b(x)c(x) a(x)d(x) − b(x)c(x) while limn→+∞ (U n (x), V¯n (x)) is given by ⎧ (0, 0) ⎪ ⎪( ⎪ ) ⎪ ⎪ µ(x) ⎪ ⎪ 0, ⎪ ⎪ ⎪ d(x) ⎨ ) ( (U ∗ (x), V¯∗ (x)) := λ(x) ⎪ , 0 ⎪ ⎪ a(x) ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ λ(x)d(x) − µ(x)b(x) µ(x)a(x) − λ(x)c(x) ⎪ ⎩ , a(x)d(x) − b(x)c(x) a(x)d(x) − b(x)c(x)

if x ∈ Ωext , per,v v if x ∈ Ωdo ∪ Ωjunk , per,u u ⋆ if x ∈ Ωdo ∪ Ωjunk ∪ Ωbi ,

if x ∈ Ωper ,

if x ∈ Ωext per,v v ⋆ if x ∈ Ωdo ∪ Ωjunk ∪ Ωbi , per,u u if x ∈ Ωdo ∪ Ωjunk ,

if x ∈ Ωper .

¯ \ Ω⋆ . Moreover, these limits are uniform on any compact subset of Ωmax = Ω bi ¯ . Note that Sn (x) = 1 for all n ≥ 1 if b(x)c(x) = 0. If b(x)c(x) > 0, then Sn (x) is Proof . Fix x ∈ Ω increasing and ⎧ if a(x)d(x) ≤ b(x)c(x), ⎨+∞ a(x)d(x) lim Sn (x) = if a(x)d(x) > b(x)c(x). n→∞ ⎩ a(x)d(x) − b(x)c(x) Moreover, Sn = and hence,

Sn (x) Sn−1 (x)

bc ad Sn−1

+ 1,

n ≥ 1,

is decreasing, with

lim Sn (x) n→∞ Sn−1 (x)

{ } b(x)c(x) = max 1, a(x)d(x) =

{

b(x)c(x) a(x)d(x)

1

if a(x)d(x) ≤ b(x)c(x), if a(x)d(x) > b(x)c(x).

(4.5)

¯n , V ) as n → ∞, we will differentiate several different cases. First, To ascertain the limiting profile of (U n ¯n (x), V (x)) = (0, 0) for every suppose that x ∈ Ωext . Then, λ(x), µ(x) ≤ 0 and hence, by Lemma 4.4, (U n per,v v n ≥ 2. Now, suppose that x ∈ Ωdo ∪ Ωjunk . Then, µ(x) > 0,

λ(x)d(x) ≤ µ(x)b(x),

µ(x)a(x) > λ(x)c(x).

When, in addition, λ(x) ≤ 0, then the first rows of the developments given by Lemma 4.4 provide us with ( ) ¯n (x), V (x)) = 0, µ , (U n ≥ 2. n d

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

57

So, assume that λ(x) > 0. If λ(x)d(x) = µ(x)b(x), then a(x)d(x) = a(x) b(x)µ(x) > b(x)c(x) λ(x) and hence, Sn /Sn−1 decreases towards 1, or equals 1, and consequently, b(x) µ(x) d(x) = λ(x) ≤

Sn (x) Sn−1 (x) λ(x)

for all n ≥ 1.

Thus, by Lemma 4.4, ¯n (x), V (x)) = (U n

(

1 a(x)

(

λ(x)Sn (x) −

)

b(x) d(x) µ(x)Sn−1 (x)

( 1 , d(x) µ(x) −

c(x) a(x) λ(x)

)

Sn−1 (x)

)

for all n ≥ 2 and therefore, ¯n (x), V (x)) = lim (U n

(

n→∞

λ(x)d(x)−µ(x)b(x) µ(x)a(x)−λ(x)c(x) a(x)d(x)−b(x)c(x) , a(x)d(x)−b(x)c(x)

)

( ) = 0, µ(x) d(x) .

In the case when λ(x)d(x) < µ(x)b(x), since λ(x)c(x) < µ(x)a(x), we have that λ(x) <

b(x) d(x) µ(x)

and

b(x)c(x) a(x)d(x) λ(x)

b(x)a(x) a(x)d(x) µ(x)

<

=

b(x) d(x) µ(x).

Thus, { } b(x)c(x) max 1, a(x)d(x) λ(x) <

b(x) d(x) µ(x).

Therefore, thanks to (4.5), there exists n0 ≥ 1 such that { } b(x)c(x) n (x) λ(x) ≤ max 1, a(x)d(x) λ(x) ≤ λ(x) SSn−1 (x) <

b(x) d(x) µ(x)

for all n ≥ n0 . ( ) ¯n (x), V (x)) = 0, µ(x) for all n > n0 . Consequently, owing to Lemma 4.4, it becomes apparent that (U n d(x) per,u u ⋆ If x ∈ Ωdo ∪ Ωjunk ∪ Ωbi , then λ(x) > 0 and µ(x)a(x) ≤ λ(x)c(x), ( and so, ) thanks to the second line of λ(x) ¯ , 0 for all n ≥ 2. the developments in the statement of Lemma 4.4, (Un (x), V (x)) = n

a(x)

Finally, suppose that x ∈ Ωper . Then, a(x)d(x) > b(x)c(x) and λ(x), µ(x) > 0,

λ(x)d(x) > µ(x)b(x),

µ(x)a(x) > λ(x)c(x).

Thus, b(x) µ(x) d(x) < λ(x) ≤

Sn (x) Sn−1 (x) λ(x)

for all n ≥ 1.

Therefore, by the third row of the developments of Lemma 4.4, it is apparent that ( ( ) ( ) ) b(x) c(x) 1 1 ¯n (x), V (x)) = (U λ(x)S (x) − µ(x)S (x) , µ(x) − λ(x) S (x) , n n−1 n−1 n a(x) d(x) d(x) a(x) ¯∗ , V ). The uniform convergence is an for all n ≥ 2. Letting n → ∞ in this identity, provides us with (U ∗ ¯∗ , V are continuous in Ωmax . The easy consequence from Dini’s criterion, because the point-wise limits, U ∗ convergence of (U n , V¯n ) follows similarly. The proof is complete. □ We are ready to prove the singular perturbation result for the system. Proof of Theorem 4.1. We will only obtain the limit of the first component, as the limit of the second one follows similarly. Thanks to Lemmas 4.2 and 4.3, for every n ≥ 0, we have that Un =

lim (d1 ,d2 )→(0,0)

u(d1 ,d2 ,n) ≤

lim inf (d1 ,d2 )→(0,0)

u(d1 ,d2 ) ≤

lim sup (d1 ,d2 )→(0,0)

u(d1 ,d2 ) ≤

lim (d1 ,d2 )→(0,0)

¯n u ¯(d1 ,d2 ,n) = U

uniformly on compact subsets Ω ∪ ΓRper . Hence, by Lemma 4.5, letting n → ∞ yields U∗ ≤

lim inf (d1 ,d2 )→(0,0)

u(d1 ,d2 ) ≤

lim sup (d1 ,d2 )→(0,0)

¯∗ u(d1 ,d2 ) ≤ U

( ) per ¯∗ , this ends the proof. □ uniformly on compact subsets of Ωmax ∩ Ω ∪ Γjunk . Since U ∗ = U

58

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

5. The strong maximum principle for quasi-cooperative systems Throughout this section, for every d1 , d2 > 0 we denote ( ) ( d1 L1 0 d1 L 1,2 L(d1 ,d2 ) := and L(d1 ,d2 ) := 0 d2 L2 0

0 d2 L

) if L := L1 = L2 .

The next result establishes the existence of the principal eigenvalue for any operator of quasi-cooperative type. The proof of Theorem 6.5 of [29] can be easily adapted to get the first part. The strict dominance can be obtained arguing as in the proof of Theorem 3.9 of [26]. So, we will omit the technical details here in. ¯ )) be such that m12 (x) > 0 and m21 (x) > 0 for all x ∈ Ω . Then, the Theorem 5.1. Let M ∈ M2 (C(Ω boundary value problem ⎧( ( ) ) (ϕ ) ϕ ⎨ 1,2 L(d1 ,d2 ) + M =σ in Ω , ψ ψ ⎩ B1 ϕ = B2 ψ = 0 on ∂Ω , has a unique (real) eigenvalue, denoted by [ ] σ1 := σ1 L1,2 + M ; (B , B ), Ω , 1 2 (d1 ,d2 ) associated to an eigenfunction, (ϕ, ψ), such that ϕ ≫ 0 and ψ ≪ 0. Furthermore, it is strictly dominant in the sense that any other eigenvalue σ satisfies Re σ > σ1 . In order to state the remaining results of this section, the next concepts are needed. They slightly generalize Definitions 6.1 and 6.2 of [29]. ¯ )) be such that m12 (x) > 0 and m21 (x) > 0 for all x ∈ Ω . A pair Definition 5.2. Let M ∈ M2 (C(Ω 2,p 2,p (ϕ, ψ) ∈ W (Ω ) × W (Ω ), p > N , is said to be a supersolution of [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ] if ⎧( ⎨

)( ) ϕ ≥0 +M ψ ≤0 ⎩ B1 ϕ ≥ 0, B2 ψ ≤ 0 L1,2 (d1 ,d2 )

in Ω , on ∂Ω .

If any of these inequalities is strict, then (ϕ, ψ) is said to be a strict supersolution. ¯ )) be such that m12 (x) > 0 and m21 (x) > 0 for all x ∈ Ω . Then: Definition 5.3. Let M ∈ M2 (C(Ω (i) The operator [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ] is said to satisfy the maximum principle if ϕ ≥ 0 and ψ ≤ 0 for every supersolution, (ϕ, ψ). (ii) The operator [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ] is said to satisfy the strong maximum principle if ϕ ≫ 0 and ψ ≪ 0 for every supersolution, (ϕ, ψ) ̸= (0, 0), and, in particular, for any strict supersolution. The next result slightly sharpens Theorem 6.3 of [29]. It is necessary for the proof of Lemma 5.7. ¯ )) be such that m12 (x) > 0 and m21 (x) > 0 for all x ∈ Ω . Then, the Theorem 5.4. Let M ∈ M2 (C(Ω following conditions are equivalent: (i) The principal eigenvalue σ1 [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ] is positive, 1,2 (ii) [L(d1 ,d2 ) + M ; (B1 , B2 ), Ω ] admits a strict supersolution, (ϕ, ψ), with ϕ ⪈ 0 and ψ ⪇ 0, (iii) [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ] satisfies the strong maximum principle,

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

59

(iv) [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ] satisfies the maximum principle. Proof . For (i) implies (ii), let (ϕ, ψ) be a principal eigenfunction associated to σ1 > 0. Then ϕ ≫ 0, ψ ≪ 0 and, since σ1 > 0, we have that ⎧( )( ) ( ) ( ) ϕ ϕ ϕ >0 ⎨ d1 L1 0 +M = σ1 in Ω , 0 d2 L2 ψ ψ ψ <0 ⎩ B1 ϕ = B2 ψ = 0 on ∂Ω . Hence, (ϕ, ψ) is a strict supersolution with ϕ ≫ 0 and ψ ≪ 0. Next, we will show that (ii) implies (iii). Let (ϕ, ψ) be a strict supersolution with ϕ ⪈ 0 and ψ ⪇ 0. Then, { { (d1 L1 + m11 )ϕ ≥ −m12 ψ ⪈ 0 in Ω , (d2 L2 + m22 )(−ψ) ≥ m21 ϕ ⪈ 0 in Ω , and (5.1) B1 ϕ ≥ 0 on ∂Ω , B2 (−ψ) ≥ 0 on ∂Ω . Hence, ϕ and −ψ are positive strict supersolutions of [d1 L1 +m11 , B1 , Ω ] and [d2 L2 +m22 , B2 , Ω ], respectively. Thus, by Theorem 7.10 of [26], σ1 [d1 L1 + m11 ; B1 , Ω ] > 0

and σ1 [d2 L1 + m22 ; B2 , Ω ] > 0,

ϕ ≫ 0, ψ ≪ 0, and the resolvents, (d1 L1 + m11 )−1 and (d2 L2 + m22 )−1 , subject to the boundary operators B1 and B2 , are strongly positive and compact. Subsequently, we consider ϕ˜ := (d1 L1 + m11 )−1 (−m12 ψ) ≫ 0

and ψ˜ := −(d2 L2 + m22 )−1 (m21 ϕ) ≫ 0.

˜ are supersolutions of Then, ϕ − ϕ˜ and (−ψ) − (−ψ) { (d1 L1 + m11 )w = 0 in Ω , and B1 w = 0 on ∂Ω ,

{

(d2 L2 + m22 )w = 0 in Ω , B2 w = 0 on ∂Ω ,

(5.2)

˜ ψ ≤ ψ. ˜ Moreover, since one respectively, one of them being strict. Thus, Theorem 7.10 of [26] yields ϕ ≥ ϕ, ˜ ˜ of them is strict, either ϕ ≫ ϕ or ψ ≪ ψ. Hence, ˜ ϕ˜ = (d1 L1 + m11 )−1 (−m12 ψ) ≥ (d1 L1 + m11 )−1 (−m12 ψ) ( ) −1 = (d1 L1 + m11 )−1 m12 (d2 L2 + m22 ) (m21 ϕ) ( ) −1 ˜ , ≥ (d1 L1 + m11 )−1 m12 (d2 L2 + m22 ) (m21 ϕ)

(5.3)

with one of these inequalities being strict. Similarly, ˜ −ψ˜ = (d2 L2 + m22 )−1 (m21 ϕ) ≥ (d2 L2 + m22 )−1 (m21 ϕ) ( ) −1 = (d2 L2 + m22 )−1 m21 (d1 L1 + m11 ) (−m12 ψ) ( ) −1 ˜ , ≥ (d2 L2 + m22 )−1 m21 (d1 L1 + m11 ) (−m12 ψ)

(5.4)

with one of these inequalities strict. Now, we introduce the strongly positive compact operators defined by [ ] −1 T1 := (d1 L1 + m11 )−1 m12 (d2 L2 + m22 ) (m21 ·) , [ ] −1 T2 := (d2 L2 + m22 )−1 m21 (d1 L1 + m11 ) (m12 ·) ˜ > 0 and subject to the boundary operators B1 and B2 . We already know that ϕ˜ > 0, −ψ˜ > 0, ϕ˜ − T1 (ϕ) ˜ ˜ ˜ ˜ (−ψ) − T2 (−ψ) > 0 in Ω , and B1 (ϕ) = B2 (−ψ) = 0 on ∂Ω . Hence, by Theorem 3.2 (iv) of [1], spr T1 < 1

and

spr T2 < 1.

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

60

Now, consider a supersolution, (u, v) ̸= (0, 0). Then, { (d1 L1 + m11 )u ≥ −m12 v in Ω , and B1 u ≥ 0 on ∂Ω ,

{

(d2 L2 + m22 )(−v) ≥ m21 u in Ω , B2 (−v) ≥ 0 on ∂Ω ,

and setting u ˜ := (d1 L1 + m11 )−1 (−m12 v),

v˜ := −(d2 L2 + m22 )−1 (m21 u),

we have that u − u ˜ and −v − (−˜ v ) are supersolutions of (5.2). Thus, since the respective principal eigenvalues are positive, it follows from Theorem 7.10 of [26] that either u ≫ u ˜ or u = u ˜, and either v ≪ v˜ or v = v˜ in Ω . Reasoning as in (5.3) and (5.4), the estimates u ˜ ≥ T1 (˜ u) and (−˜ v ) ≥ T2 (−˜ v ) hold in Ω . Since spr T1 , spr T2 < 1, by Theorem 6.3 (d) of [26], the resolvent operators (I − Ti )−1 , i = 1, 2, are strongly positive. Hence, either u ˜ ≫ 0 or u ˜ = 0, and either v˜ ≪ 0 or v˜ = 0. Consequently, either u ≫ 0 or u = 0, and either v ≪ 0 or v = 0. It remains to show that neither u = 0 nor v = 0. If, for example, u = 0, then 0 ≥ −m12 v in Ω and so v ≥ 0. Hence v = 0, which contradicts (u, v) ̸= (0, 0). This shows that (ii) implies (iii). The fact that (iii) implies (iv) is immediate. For the proof of (iv) implies (i), let (ϕ, ψ) be an eigenfunction associated to the principal eigenvalue σ1 and suppose that σ1 ≤ 0. Then, since ϕ ≫ 0 and ψ ≪ 0, we have that ⎧( )( ) ( ) ( ) ϕ ϕ ϕ ≤0 ⎨ d1 L1 0 +M = σ1 in Ω , 0 d2 L2 ψ ψ ψ ≥0 ⎩ B1 ϕ = B2 ψ = 0 on ∂Ω . Hence, multiplying by −1, we have that (−ϕ, −ψ) is a supersolution of [L1,2 (d1 ,d2 ) + M ; (B1 , B2 ), Ω ]. Since 1,2 [L(d1 ,d2 ) + M ; (B1 , B2 ), Ω ] satisfies the maximum principle, −ϕ ≥ 0 and −ψ ≤ 0, which contradicts the assumption. □ Next, we will derive from Theorem 5.4 some of the main properties of the principal eigenvalue, σ1 . These results extend Lemma 2.2 to cover the case of quasi-cooperative systems. To state them, it is convenient to introduce the following ordering. ¯ )) be two matrices with continuous coefficients. Then, F is said to be Definition 5.5. Let F, G ∈ M2 (C(Ω greater than G in Ω , F ≻ G in Ω , if F ̸= G and fii ≥ gii

in Ω for all i ∈ {1, 2}

and

fij ≤ gij

in Ω for all i, j ∈ {1, 2}, i ̸= j.

Then, the next comparison result holds. ¯ )) be such that F ≻ G in Ω and f12 (x), f21 (x) > 0 for all x ∈ Ω . Then, Lemma 5.6. Let F, G ∈ M2 (C(Ω [ ] [ ] 1,2 σ1 L1,2 + F ; (B , B ), Ω > σ L + G; (B , B ), Ω . 1 2 1 1 2 (d1 ,d2 ) (d1 ,d2 ) Proof . Since F ≻ G in Ω , 0 < f12 (x) ≤ g12 (x)

and

0 < f21 (x) ≤ g21 (x)

for all x ∈ Ω .

Thus, thanks to Theorem 5.1, the principal eigenvalues are well defined. Subsequently, we denote them by σ1,F and σ1,G , respectively. Then, any principal eigenfunction, (ϕ, ψ), associated to σ1,G , satisfies ϕ ≫ 0, ψ ≪ 0, and ⎧ ⎨d1 L1 ϕ + g11 ϕ + g12 ψ − σ1,G ϕ = 0 in Ω , d2 L2 ψ + g21 ϕ + g22 ψ − σ1,G ψ = 0 ⎩ B1 ϕ = B2 ψ = 0 on ∂Ω .

S. Fernández-Rincón, J. López-Gómez / Nonlinear Analysis 165 (2017) 33–79

61

Since F ≻ G in Ω , we find from ϕ ≫ 0 and ψ ≪ 0 that 0 = d1 L1 ϕ + g11 ϕ + g12 ψ − σ1,G ϕ ≤ d1 L1 ϕ + f11 ϕ + f12 ψ − σ1,G ϕ, 0 = d2 L2 ψ + g21 ϕ + g22 ψ − σ1,G ψ ≥ d2 L2 ψ + f21 ϕ + f22 ψ − σ1,G ψ, with some of these inequalities strict. Thus, ⎧ ⎨d1 L1 ϕ + f11 ϕ + f12 ψ − σ1,G ϕ ≥ 0 d2 L2 ψ + f21 ϕ + f22 ψ − σ1,G ψ ≤ 0 ⎩ B1 ϕ = B2 ψ = 0

in Ω , on ∂Ω ,

with some of these inequalities strict. Therefore, by Theorem 5.4, we may conclude that [ ] 0 < σ1 L1,2 + F − σ ; (B , B ), Ω = σ1,F − σ1,G , 1,G 1 2 (d1 ,d2 ) which ends the proof.

□

Lemma 5.7. Let Ω0 be a subdomain of class C 2 of Ω such that ∂Ω0 ∩ ∂Ω consists of finitely many components of ∂Ω , if it is non-empty. For each i ∈ {1, 2}, let Bi,0 be any boundary operator of the type { h on ∂Ω0 ∩ Ω , Bi,0 h := ˜ for every h ∈ W 2,p (Ω ), p > N, Bi h on ∂Ω0 ∩ ∂Ω , where, on each component of ∂Ω0 ∩ ∂Ω , either B˜i h = h, or B˜i h = Bi h. If (B1 , B2 , Ω ) ̸= (B1,0 , B2,0 , Ω0 ), ¯ )), with m12 (x), m21 (x) > 0 for all x ∈ Ω , we have that then, for every d1 , d2 > 0 and M ∈ M2 (C(Ω [ ] [ ] 1,2 σ1 := σ1 L1,2 + M ; (B , B ), Ω < σ := σ L + M ; (B , B ), Ω . 1 2 1,0 1 1,0 2,0 0 (d1 ,d2 ) (d1 ,d2 ) Proof . Let (ϕ, ψ) be a principal eigenfunction associated to σ1 with ϕ ≫ 0, ψ ≪ 0. Then, ⎧ on ∂Ω0 ∩ Ω , ⎨ϕ > 0 on ∂Ω0 ∩ ∂Ω if B˜1 ̸= B1 , B1,0 ϕ = B˜1 ϕ = ϕ > 0 ⎩˜ B1 ϕ = B1 ϕ = 0 on ∂Ω0 ∩ ∂Ω if B˜1 = B1 , and ⎧ on ∂Ω0 ∩ Ω , ⎨ψ < 0 on ∂Ω0 ∩ ∂Ω if B˜2 ̸= B2 , B2,0 ψ = B˜2 ψ = ψ < 0 ⎩˜ B2 ψ = B2 ψ = 0 on ∂Ω0 ∩ ∂Ω if B˜2 = B2 . Since (B1 , B2 , Ω ) ̸= (B1,0 , B2,0 , Ω0 ), either B1,0 ϕ > 0, or B2,0 ψ < 0, on ∂Ω0 . Hence, (ϕ|Ω0 , ψ|Ω0 ) is a strict supersolution of ⎧( ( ) ( ) )( ) Φ Φ 0 ⎨ 1,2 L(d1 ,d2 ) + M − σ1 = in Ω0 , Ψ Ψ 0 ⎩ B1,0 Φ = 0, B2,0 Ψ = 0 on ∂Ω0 , with ϕ|Ω0 > 0 and ψ|Ω0 < 0. Therefore, by Theorem 5.4, [ ] σ1,0 − σ1 = σ1 L1,2 (d1 ,d2 ) + M − σ1 ; (B1,0 , B2,0 ), Ω0 > 0.

□

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6. Perturbation from a kinetic equilibrium which is somewhere unstable in Ω The main result of this section establishes the Principle of Parabolic Instability stated in Section 1, according with it any family of non-trivial states of (1.2) perturbing from any steady state, (u∗ (x), v∗ (x)), of the non-spatial model (1.6) must be linearly unstable provided (u∗ (x), v∗ (x)) is linearly unstable for x varying on some open subset, Ωun , of Ω . Such equilibria satisfy { [λ(x) − a(x)u∗ (x) − b(x)v∗ (x)]u∗ (x) = 0, [µ(x) − d(x)v∗ (x) − c(x)u∗ (x)]v∗ (x) = 0, µ(x) and hence, either (u∗ (x), v∗ (x)) = (0, 0), or (u∗ (x), v∗ (x)) = ( λ(x) a(x) , 0) if λ(x) > 0, or (u∗ (x), v∗ (x)) = (0, d(x) ) if µ(x) > 0, or else ) ( µ(x)a(x)−λ(x)c(x) (u∗ (x), v∗ (x)) = λ(x)d(x)−µ(x)b(x) a(x)d(x)−b(x)c(x) , a(x)d(x)−b(x)c(x) ,

if both components are well defined and positive. The fact that this ‘general principle’ holds independently of the size of the instability region Ωun is a rather astonishing feature. The next lemma is the main technical tool to prove such result. Throughout this section we are assuming that b(x) > 0 and c(x) > 0 for all x ∈ Ω . ¯ and Lemma 6.1. Assume that L := L1 = L2 and B := B1 = B2 . Let J ∈ (0, +∞)2 be with (0, 0) ∈ J, consider a family of matrices, {H(d1 ,d2 ) }(d1 ,d2 )∈J ⊂ M2 (R), such that (i) the off-diagonal entries of H(d1 ,d2 ) are positive for every (d1 , d2 ) ∈ J, (ii) H(d1 ,d2 ) converges to some matrix H∗ ∈ M2 (R) as J ∋ (d1 , d2 ) → (0, 0). If σlow [H∗ ] stands for the lower eigenvalue of H∗ , then lim (d1 ,d2 )→(0,0)

σ1 [L(d1 ,d2 ) + H(d1 ,d2 ) ; (B, B), Ω ] = σlow [H∗ ].

Proof . Note that any matrix H = (hij ) ∈ M2 (R) with h12 h21 ≥ 0 has real eigenvalues. Indeed, they are given by √ √ h11 +h22 ± (h11 +h22 )2 −4(h11 h22 −h12 h21 ) h11 +h22 ± (h11 −h22 )2 +4h12 h21 σ± [H] = = . 2 2 Hence, such a property holds for every H(d1 ,d2 ) , (d1 , d2 ) ∈ J, and, by the assumptions, also for H∗ . On the other hand, by (i), Theorem 5.1 provides us with the existence and uniqueness of the principal eigenvalue σ(d1 ,d2 ) := σ1 [L(d1 ,d2 ) + H(d1 ,d2 ) ; (B, B), Ω ], associated with it there is an eigenfunction (ϕ(d1 ,d2 ) , ψ(d1 ,d2 ) ) with ϕ(d1 ,d2 ) ≫ 0 and ψ(d1 ,d2 ) ≪ 0, unique up to a multiplicative nontrivial constant. Let σ0 and ϕ0 ≫ 0 denote the principal eigenpair of L in Ω , with ϕ0 normalized so that ∥ϕ0 ∥∞ = 1. Now, let us show that (ξ(d1 ,d2 ) ϕ0 , ζ(d1 ,d2 ) ϕ0 ) provides us with a principal eigenfunction associated to σ(d1 ,d2 ) for appropriate values of ξ(d1 ,d2 ) > 0 and ζ(d1 ,d2 ) < 0. By definition, ( ) ( ) ( ) d1 ξ(d1 ,d2 ) Lϕ0 ξ(d1 ,d2 ) ϕ0 ξ(d1 ,d2 ) ϕ0 + H(d1 ,d2 ) = σ(d1 ,d2 ) , d2 ζ(d1 ,d2 ) Lϕ0 ζ(d1 ,d2 ) ϕ0 ζ(d1 ,d2 ) ϕ0 which can be equivalently expressed as [( ) ( )] ( ) d1 ξ(d1 ,d2 ) σ0 ξ ξ ϕ0 + H(d1 ,d2 ) (d1 ,d2 ) = ϕ0 σ(d1 ,d2 ) (d1 ,d2 ) . ζ(d1 ,d2 ) d2 ζ(d1 ,d2 ) σ0 ζ(d1 ,d2 )

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63

Thus, dividing by ϕ0 ≫ 0, it becomes apparent that (ξ(d1 ,d2 ) , ζ(d1 ,d2 ) ) must satisfy ) [ ( ) ( ) ]( d1 0 ξ(d1 ,d2 ) ξ(d1 ,d2 ) , σ0 = σ(d1 ,d2 ) + H(d1 ,d2 ) ζ(d1 ,d2 ) 0 d2 ζ(d1 ,d2 ) i.e., (ξ(d1 ,d2 ) , ζ(d1 ,d2 ) ) is an eigenvector of the matrix ( ˜ (d ,d ) := σ0 d1 H 1 2 0

0 d2

) + H(d1 ,d2 )

˜ (d ,d ) admits associated with the eigenvalue σ(d1 ,d2 ) . It remains to show that the lower eigenvalue of H 1 2 an eigenvector with components of opposite sign. But this is also a consequence from the fact that the ˜ 1 , d2 ), are positive. Indeed, for any matrix H ∈ M2 (R) with off-diagonal entries of H(d1 ,d2 ) , and so of H(d h12 , h21 > 0 √ h +h − (h11 −h22 )2 +4h12 h21 σlow [H] = 11 22 2 and any associated eigenvector, (ξ, ζ), satisfies (h11 − σlow [H]) ξ = −h12 ζ

and

(h22 − σlow [H]) ζ = −h21 ξ.

Thus, ξζ < 0 if and only if h11 , h22 > σlow [H]. But, since h12 h21 > 0, we have that σlow [H] <

h11 +h22 −|h11 −h22 | 2

= min{h11 , h22 }.

˜ (d ,d ) admits an eigenfunction with components of opposite sign associated to its lower Therefore, H 1 2 eigenvalue. By the uniqueness of the principal eigenvalue, σ(d1 ,d2 ) , necessarily ˜ (d ,d ) ]. σ1 [L(d1 ,d2 ) + H(d1 ,d2 ) ; (B, B), Ω ] = σ(d1 ,d2 ) = σlow [H 1 2 ˜ (d ,d ) converge to those of H∗ . So, it does the lower eigenvalue of Finally, thanks to (ii), the entries of H 1 2 these matrices. □ The next result establishes the Principle of Parabolic Instability already stated in Section 1 when the perturbed steady states are coexistence states. Theorem 6.2. Suppose that L := L1 = L2 and B := B1 = B2 . Let {(u(d1 ,d2 ) , v(d1 ,d2 ) )}(d1 ,d2 )∈J be a ¯ such that, for some open subset sequence of coexistence states of (1.2), with J ⊂ (0, +∞)2 and (0, 0) ∈ J, Ωun ⊂ Ω , ( ) u(d1 ,d2 ) , v(d1 ,d2 ) = (u∗ , v∗ ) uniformly in Ωun lim (d1 ,d2 )→(0,0)

with (u∗ (x), v∗ (x)) linearly unstable for all x ∈ Ωun , as a nontrivial steady-state solution of (1.6). Then, δ > 0 exists such that (u(d1 ,d2 ) , v(d1 ,d2 ) ) is linearly unstable for all d1 , d2 < δ, (d1 , d2 ) ∈ J. Proof . The linear instability follows from the negativity of the principal eigenvalue of the linearization of (1.2) at the coexistence state, i.e., σ1 [L(d1 ,d2 ) ; (B, B), Ω ] < 0, where ( ) ( ) d1 L 0 −λ + 2au(d1 ,d2 ) + bv(d1 ,d2 ) bu(d1 ,d2 ) L(d1 ,d2 ) := + . (6.1) 0 d2 L cv(d1 ,d2 ) −µ + 2dv(d1 ,d2 ) + cu(d1 ,d2 ) Since the off-diagonal entries of L(d1 ,d2 ) , bu(d1 ,d2 ) and cv(d1 ,d2 ) , are positive, the existence of ¯ε (x0 ) ⊊ Ωun . σ1 [L(d1 ,d2 ) ; (B, B), Ω ] follows from Theorem 5.1. Consider x0 ∈ Ωun and ε > 0 such that B Then, by Lemma 5.7, σ1 [L(d1 ,d2 ) ; (B, B), Ω ] < σ1 [L(d1 ,d2 ) ; (D, D), Bε (x0 )].

(6.2)

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Subsequently, we set { } α(ε,d1 ,d2 ) := max −λ + 2au(d1 ,d2 ) + bv(d1 ,d2 ) , ¯ ε (x0 ) B { } γ(ε,d1 ,d2 ) := min cv(d1 ,d2 ) , ¯ ε (x0 ) B

β(ε,d1 ,d2 ) := min

{

¯ ε (x0 ) B {

ρ(ε,d1 ,d2 ) := max

¯ ε (x0 ) B

According to Lemma 5.6, the next estimate holds [( ) ( d1 L 0 α(ε,d1 ,d2 ) σ1 [L(d1 ,d2 ) ; (D, D), Bε (x0 )] ≤ σ1 + 0 d2 L γ(ε,d1 ,d2 )

} bu(d1 ,d2 ) ,

} −µ + 2dv(d1 ,d2 ) + cu(d1 ,d2 ) .

) ] β(ε,d1 ,d2 ) ; (D, D), Bε (x0 ) ρ(ε,d1 ,d2 )

(6.3)

for every d1 , d2 > 0, (d1 , d2 ) ∈ J. Since u(d1 ,d2 ) and v(d1 ,d2 ) converge to u∗ and v∗ , respectively, uniformly in Ωun as (d1 , d2 ) → (0, 0), the following limits are well defined α(ε,∗) := γ(ε,∗) :=

lim (d1 ,d2 )→(0,0)

lim (d1 ,d2 )→(0,0)

α(ε,d1 ,d2 ) ,

β(ε,∗) :=

γ(ε,d1 ,d2 ) ,

ρ(ε,∗) :=

lim (d1 ,d2 )→(0,0)

lim (d1 ,d2 )→(0,0)

β(ε,d1 ,d2 ) ,

ρ(ε,d1 ,d2 ) .

Moreover, letting ε ↓ 0 yields lim α(ε,∗) = −λ(x0 ) + 2a(x0 )u∗ (x0 ) + b(x0 )v∗ (x0 ), lim β(ε,∗) = b(x0 )u∗ (x0 ), ε↓0

ε↓0

lim ρ(ε,∗) = −µ(x0 ) + c(x0 )u∗ (x0 ) + 2d(x0 )v∗ (x0 ).

lim γ(ε,∗) = c(x0 )v∗ (x0 ), ε↓0

ε↓0

Note that −

( ) −λ(x0 ) + 2a(x0 )u∗ (x0 ) + b(x0 )v∗ (x0 ) b(x0 )u∗ (x0 ) c(x0 )v∗ (x0 ) −µ(x0 ) + c(x0 )u∗ (x0 ) + 2d(x0 )v∗ (x0 )

provides us with the linearization of the non-spatial model at (u∗ (x0 ), v∗ (x0 )), which is linearly unstable because x0 ∈ Ωun . Thus, this matrix has a positive eigenvalue. Therefore, for sufficiently small ε > 0, the matrix ( ) α(ε,∗) β(ε,∗) γ(ε,∗) ρ(ε,∗) possesses a negative eigenvalue. Consequently, owing to Lemma 6.1, for sufficiently small ε > 0 and d1 , d2 > 0, with (d1 , d2 ) ∈ J, we obtain that [( ) ( ) ] d1 L 0 α(ε,d1 ,d2 ) β(ε,d1 ,d2 ) σ1 + ; (D, D), Bε (x0 ) < 0. 0 d2 L γ(ε,d1 ,d2 ) ρ(ε,d1 ,d2 ) According to (6.2) and (6.3), we find that σ1 [L(d1 ,d2 ) ; (B, B), Ω ] < 0, which ends the proof. □ Although in Theorem 6.2 the steady state (u∗ , v∗ ) might have some component vanishing, or both, the next result shows that actually the coexistence steady states of (1.1) cannot perturb from (0, 0) uniformly in some open subset Ω0 ⊂ Ω as d1 , d2 → 0 if (0, 0) is linearly unstable in Ω0 as a steady state of (1.6). ¯ is a sequence Proposition 6.3. Assume that {(u(d1 ,d2 ) , v(d1 ,d2 ) )}(d1 ,d2 )∈J , with J ⊂ (0, +∞)2 and (0, 0) ∈ J, of coexistence states of (1.2) such that lim (d1 ,d2 )→(0,0)

(

) u(d1 ,d2 ) , v(d1 ,d2 ) = (0, 0) uniformly in Ω0

for some subdomain Ω0 ⋐ Ω of class C 2 . Then, (0, 0) cannot be linearly unstable at any x ∈ Ω0 .

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Proof . On the contrary, suppose that (0, 0) is linearly unstable with respect to (1.6) at some x0 ∈ Ω0 . Then, the linearization at (0, 0) of (1.6) for x = x0 , which is given by ( ) λ(x0 ) 0 , 0 µ(x0 ) has a positive eigenvalue. Thus, either maxΩ¯ 0 λ ≥ λ(x0 ) > 0 or maxΩ¯ 0 µ ≥ µ(x0 ) > 0. Suppose maxΩ¯ 0 λ > 0. Then, the monotonicity with respect to the domain established by Lemma 2.2 yields σ1 [d1 L1 − λ + au(d1 ,d2 ) + bv(d1 ,d2 ) ; B1 , Ω ] < σ1 [d1 L1 − λ + au(d1 ,d2 ) + bv(d1 ,d2 ) ; D, Ω0 ]

(6.4)

for all (d1 , d2 ) ∈ J, while, thanks to Theorem 2.1, the uniform convergence in Ω0 provides us with lim (d1 ,d2 )→(0,0)

σ1 [d1 L1 − λ + au(d1 ,d2 ) + bv(d1 ,d2 ) ; D, Ω0 ] = min (−λ) = − max λ < 0. ¯0 Ω

(6.5)

¯0 Ω

But, since (u(d1 ,d2 ) , v(d1 ,d2 ) ) is a coexistence state, σ1 [d1 L1 − λ + au(d1 ,d2 ) + bv(d1 ,d2 ) ; B1 , Ω ] = 0

for all (d1 , d2 ) ∈ J,

which contradicts (6.4) y (6.5) and ends the proof. □ Theorem 6.2 admits the next counterpart for semitrivial solutions of (1.2). So, the Principle of Parabolic Instability holds. Proposition 6.4. Suppose that there exists an open subset Ωun ⋐ Ω of class C 2 such that, for every ¯ un , ( λ+ (x) , 0) is linearly unstable as a steady-state solution of (1.6). Consider the family of semitrivial x∈Ω a(x) λ

solutions of (1.2), (θ{d1 ,λ,a} , 0), for sufficiently small d1 > 0, which, due to Theorem 3.1, converge to ( a+ , 0) ¯ un as d1 ↓ 0. Then, δ > 0 exists such that (θ{d ,λ,a} , 0) is linearly unstable for every d1 , d2 < δ. uniformly in Ω 1 ¯ un , (0, µ+ (x) ) By symmetry, if there is an open subset Ωun ⋐ Ω of class C 2 such that, for every x ∈ Ω d(x) is linearly unstable as a steady-state solution of (1.6), then δ > 0 exists such that (0, θ{d2 ,µ,d} ) is linearly unstable for every d1 , d2 < δ. Proof . As the second assertion follows by symmetry, we will only prove the first one. The linearization of (1.2) at (θ{d1 ,λ,a} , 0) can be easily determined from (6.1) and provides us with the eigenvalue problem ⎧ ⎨(d1 L1 − λ + 2aθ{d1 ,λ,a} )ϕ(d1 ,d2 ) + bθ{d1 ,λ,a} ψ(d1 ,d2 ) = σ(d1 ,d2 ) ϕ(d1 ,d2 ) in Ω , (d2 L2 − µ + cθ{d1 ,λ,a} )ψ(d1 ,d2 ) = σ(d1 ,d2 ) ψ(d1 ,d2 ) in Ω , (6.6) ⎩ B1 ϕ(d1 ,d2 ) = B2 ψ(d1 ,d2 ) = 0 on ∂Ω . It suffices to establish the existence and negativity of one eigenvalue associated to an eigenfunction (ϕ(d1 ,d2 ) , ψ(d1 ,d2 ) ), with ϕ(d1 ,d2 ) ≫ 0 and ψ(d1 ,d2 ) ≪ 0. Actually, this eigenvalue must be the principal one of d2 L2 − µ + cθ{d1 ,λ,a} . Let ψ(d1 ,d2 ) ≪ 0 be a principal eigenfunction associated to Σ(d1 ,d2 ) := σ1 [d2 L2 − µ + cθ{d1 ,λ,a} ; B2 , Ω ]. Then, the monotonicity with respect to the domain established by Lemma 2.2 provides us with the estimate Σ(d1 ,d2 ) = σ1 [d2 L2 − µ + cθ{d1 ,λ,a} ; B2 , Ω ] < σ1 [d2 L2 − µ + cθ{d1 ,λ,a} ; D, Ωun ].

lim (d1 ,d2 )→(0,0)

λ+ a

of θ{d1 ,λ,a} as d1 → 0, it follows from Theorem 2.1 that ( ) λ σ1 [d2 L2 − µ + cθ{d1 ,λ,a} ; D, Ωun ] = min −µ + c a+ < 0,

By the uniform convergence to

¯ un Ω

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because

(

)

λ+ a ,0

¯ un . Therefore, δ > 0 exists such that is linearly unstable in Ω Σ(d1 ,d2 ) < 0

for all d1 , d2 < δ.

It remains to solve the first equation of (6.6), i.e., (d1 L1 − λ + 2aθ{d1 ,λ,a} − Σ(d1 ,d2 ) )ϕ(d1 ,d2 ) = −bθ{d1 ,λ,a} ψ(d1 ,d2 )

in Ω ,

subject to B1 ϕ(d1 ,d2 ) = 0 on ∂Ω . Thus, by the monotony with respect to the potential established by Theorem 2.1, σ1 [d1 L1 − λ + 2aθ{d1 ,λ,a} − Σ(d1 ,d2 ) ; B1 , Ω ] > σ1 [d1 L1 − λ + aθ{d1 ,λ,a} ; B1 , Ω ] = 0. So, the operator (d1 L1 − λ + 2aθ{δ1 ,λ,a} − Σ(d1 ,d2 ) )−1 is strongly positive. Therefore, the previous equation has a unique solution, ϕ(d1 ,d2 ) ≫ 0, because −bθ{d1 ,λ,a} ψ(d1 ,d2 ) > 0. □ As a direct consequence from Proposition 6.4, the following substantial extension of Theorem 2.1 of [14] and Corollary 4.6 of [12] holds. Note that we are dealing with a general class of linear second order self-adjoint elliptic operators under general non-classical mixed boundary conditions, where the weight functions βi of the boundary operators Bi are allowed to change sign. Therefore, the result should be considered new in its greatest generality. Subsequently, the model (1.1), or (1.6), is said to be permanent if its trivial and semitrivial steady states are linearly unstable. In this case the theory of [17,22] and, in particular, [29, Ths. 3.1 and 4.1], shows that the model possesses a linearly stable coexistence steady state, which is a global attractor with respect to the component-wise positive solutions of the model if it is unique. Note that the results in all those references can be easily adapted to cover our general framework here in. per,u per,v u v Corollary 6.5. Suppose that either Ωper ̸= ∅, or Ωdo ∪ Ωjunk ̸= ∅ and Ωdo ∪ Ωjunk ̸= ∅. Then, δ > 0 exists such that the parabolic problem (1.1) is permanent for all d1 , d2 < δ.

) ( ) ( µ(x) , 0 and 0, Proof . Suppose Ωper ̸= ∅. Then, the semitrivial solutions of (1.6), λ(x) a(x) d(x) , are linearly unstable for all x ∈ Ωper . Thus, thanks to Proposition 6.4, δ > 0 exists such that (θ{d1 ,λ,a} , 0) and (0, θ{d2 ,µ,d} ) are linearly unstable for d1 < δ and d2 < δ, respectively. Therefore, (1.1) ( is permanent. ) λ+ (x) per,u per,v u v Now, suppose Ωdo ∪ Ωjunk ̸= ∅ and Ωdo ∪ Ωjunk ̸= ∅. Then, , 0 is linearly unstable for all a(x) ( ) µ+ (x) per,v per,u v u x ∈ Ωdo ∪ Ωjunk , while 0, d(x) is linearly unstable for all x ∈ Ωdo ∪ Ωjunk . Proposition 6.4 ends the proof as in the previous case. □ per,u per,v u v . Note that Corollary 6.5 holds independently of the size and the shape of Ωper , Ωdo ∪Ωjunk and Ωdo ∪Ωjunk Essentially, this entails that (1.1) is permanent for sufficiently small diffusivities for many non-spatial kinetic patterns. This is an extremely surprising feature at the light shared by the following simple examples. Suppose Ω = Ωext , i.e., λ(x) ≤ 0 and µ(x) ≤ 0 for all x ∈ Ω . Then, not only the non-spatial model (1.6) exhibits extinction, but also the spatial model (1.1). Suppose, in addition, that λ(x0 ) = 0, µ(y0 ) = 0, λ(y0 ) < 0 and µ(x0 ) < 0 for some x0 , y0 ∈ Ω , x0 ̸= y0 . Then, by slightly perturbing λ and µ, for instance taking λ + ε and µ + ε, for sufficiently small ε > 0, we can get λ > 0 and µ < 0 on some small ball around x0 , u and µ > 0 and λ < 0 on some small ball around y0 . In these balls we have that Ωdo ̸= ∅, respectively v Ωdo ̸= ∅. Therefore, by Corollary 6.5, (1.1) is permanent for sufficiently small d1 and d2 . Similarly, if λ(x0 ) = µ(x0 ) = 0 and a(x0 )d(x0 ) > b(x0 )c(x0 ) for some x0 ∈ Ω , we can perturb λ and µ nearby x0 in such a way that Ωper ̸= ∅ around x0 . This simple example tells us how a very small perturbation of the coefficients

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can provoke dramatic changes on the dynamics of the diffusive model, at least for small diffusion rates. The independence of these permanence results on the size of the regions where the non-spatial model (1.6) is permanent and on the sizes of the regions where there is dominance of u and v, is utterly attributable to the Principle of Parabolic Instability established by Proposition 6.4, according with it the semitrivial solutions of the spatial model are linearly unstable for small diffusion rates as soon as the semitrivial steady-state solutions of the non-spatial model are linearly unstable somewhere in Ω . Similarly, from Theorem 6.2, the next multiplicity result holds. Corollary 6.6. Suppose (1.1) is permanent for sufficiently small diffusion rates, d1 , d2 > 0, Ωbi ̸= ∅ and the non-spatial coexistence steady-state solution of (1.6), (u∗ (x), v ∗ (x)), x ∈ Ωbi , admits a perturbed coexistence steady state of (1.1), (u(d1 ,d2 ) , v(d1 ,d2 ) ), for sufficiently small d1 , d2 > 0, in the sense that lim (d1 ,d2 )→(0,0)

(u(d1 ,d2 ) , v(d1 ,d2 ) ) = (u∗ , v ∗ ) uniformly on compact subsets of Ωbi .

Then, δ > 0 exists such that (1.1) possesses at least two coexistence states for each d1 , d2 < δ. Proof . By Theorem 6.2, the perturbation (u(d1 ,d2 ) , v(d1 ,d2 ) ) must be linearly unstable for small diffusion rates. According to Theorem 3.1 of [29], (1.1) possesses a stable coexistence state for sufficiently small d1 , d2 > 0, because it is permanent. This ends the proof. □ It remains an open problem to ascertain whether or not such a perturbation from (u∗ , v ∗ ) exists. However, the next section will provide us with an example exhibiting this behavior. 7. Multiplicity in the symmetric problem when Ωbi ̸= ∅ This section studies a family of symmetric problems possessing at least three coexistence steady states for small diffusion rates. Precisely, we consider (1.1) under the assumptions L := L1 = L2 , B := B1 = B2 , ν := d1 = d2 > 0, λ = µ with maxΩ¯ λ > 0, a = d with minΩ¯ a > 0, and b = c with b(x) > 0 for all x ∈ Ω . Hence, its elliptic counterpart is given by ⎧ in Ω , ⎨νLu = λ(x)u − a(x)u2 − b(x)uv (7.1) νLv = λ(x)v − a(x)v 2 − b(x)uv in Ω , ⎩ Bu = Bv = 0 on ∂Ω . u v Under the previous assumptions, Ωdo = Ωdo = ∅ and thus,

¯ = Ωper ∪ Ωbi ∪ Ωext ∪ Ωjunk , Ω which allows Ωper and Ωbi to be nonempty. Thanks to the symmetry of the problem, for every solution of (7.1), (u, v), with u ̸= v, we have that (v, u) also is a solution. Moreover, (7.1) admits a solution with u = v, as shown by the next result. Lemma 7.1. Assume that maxΩ¯ λ > 0. Then, there exists ν0 > 0 such that for every ν ∈ (0, ν0 ) the problem (7.1) admits a unique coexistence state, (u, v), with u = v, given by (wν , wν ) with wν = θ{ν,λ,a+b} . Moreover, ( ) λ+ λ+ it converges to a+b , a+b uniformly on compact subsets of Ω ∪ ΓR+ ∪ (λ+ )−1 (0) as ν ↓ 0. Proof . The pair (w, w) is a component-wise positive solution of (7.1) if and only if w satisfies { νLw = λ(x)w − (a(x) + b(x))w2 in Ω , Bw = 0 on ∂Ω .

(7.2)

By Corollary 2.4, ν0 > 0 exists such that, for every ν ∈ (0, ν0 ), (7.2) admits a unique positive solution, wν := θ{ν,λ,a+b} . Its limiting behavior as ν ↓ 0 follows from Theorem 3.1. □

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Note that the coexistence state whose existence has been established by Lemma 7.1 actually exists if and only if σ1 [νL − λ; B, Ω ] < 0, which is the same condition guaranteeing the existence of the semitrivial states. By a rather standard comparison argument, it readily follows that, in general, the existence of the semitrivial states is necessary for the existence of coexistence states. The multiplicity result of this section requires the analysis of the attractivity properties of the coexistence states. The next result provides us with the instability of (wν , wν ) for sufficiently small ν > 0 when Ωbi ̸= ∅. Lemma 7.2. If Ωbi ̸= ∅, then there exists νun > 0 such that (wν , wν ) is linearly unstable for all ν ∈ (0, νun ). Proof . Since Ωbi ̸= ∅, maxΩ¯ λ > 0. Thus, Lemma 7.1 can be applied to infer that (wν , wν ) is a coexistence λ+ λ+ state for ν < ν0 that converges uniformly on compact subsets of Ω to ( a+b , a+b ) as ν → 0. As there is a λ+ (x) λ+ (x) ¯ ¯ , a(x)+b(x) ) is linearly smooth open subset of Ω ∩ Ωbi , D, with D ⊂ Ω , such that, for every x ∈ D, ( a(x)+b(x) unstable as a coexistence state of the non-spatial model, it follows from Theorem 6.2 that (wν , wν ) must be linearly unstable for sufficiently small ν > 0. This ends the proof. □ Combining this result with Corollary 6.5 provides us with the next theorem. Theorem 7.3. Assume that Ωper ̸= ∅ and Ωbi ̸= ∅. Then νm > 0 exists such that, for every ν ∈ (0, νm ), (7.1) admits 2k + 1 coexistence states, for some k ≥ 1; two of them linearly stable and another one linearly unstable and perturbing from the coexistence steady state of the non-spatial problem in the region Ωbi ∪ Ωper . Proof . Since Ωper ̸= ∅, by Corollary 6.5, the problem (7.1) is permanent for sufficiently small ν > 0. Thus, it admits a linearly stable coexistence state, (uν , vν ). Moreover, since Ωbi ̸= ∅, according to Lemma 7.2, for sufficiently small ν, the coexistence state (wν , wν ) is linearly unstable. Hence, (uν , vν ) ̸= (wν , wν ). Since (wν , wν ) provides us with the unique coexistence state, (u, v), such that u = v, we find that uν ̸= vν and therefore, (vν , uν ) provides us with a third coexistence state with the same stability character. Since, by similar reasons, the remaining coexistence states must appear by pairs, the proof is complete. □ In the context of Theorem 7.3, Theorem 4.1 provides us with the limiting profiles of all the coexistence steady states of (7.1) in Ωmax ∩ (Ω ∪ ΓRper ) = (Ωper ∪ Ωext ) ∩ (Ω ∪ ΓRper ) ) ( λ+ λ+ as ν ↓ 0, where the three coexistence states constructed in Theorem 7.3 approximate a+b , a+b as ν ↓ 0. ( ) λ+ λ+ Note that a+b , a+b is a coexistence state (if λ(x) > 0), or the trivial solution (if λ(x) ≤ 0), of the nonspatial model. The analysis of the precise behavior of the stable coexistence states constructed in Theorem 7.3 in the region Ωbi remains open in this paper, though we conjecture that they should perturb from each of the semi-trivial steady state solutions in the region Ωbi as ν > 0 perturbs from zero. Nevertheless, according to Theorem 6.2 and Proposition 6.3, they cannot perturb uniformly from the trivial solution or the coexistence steady state of the non-spatial model. ¯ = Ωper 8. Uniqueness with Ω This section shows that under Robin boundary conditions the problem (1.2) admits a unique coexistence ¯ = Ωper . Although the first version state for sufficiently small diffusion coefficients, d1 > 0 and d2 > 0, if Ω of this result was given in [19], our proof here is substantially simpler than the original one of [19] and, as a result of its simplicity, it can be easily adapted to get the result for wider classes of differential operators in

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divergence form subject to Robin boundary conditions. Once established the main uniqueness result of this section, its optimality is discussed at the light of the multiplicity result established in Section 7. As in the proof of the main theorem we are using the uniform convergence of the coexistence states as (d1 , d2 ) → (0, 0) established by Theorem 4.1, the condition ΓD1 = ΓD2 = ∅ must be imposed. ¯ = Ωper and Γ 1 = Γ 2 = ∅. Then, δ > 0 exists such that for all Theorem 8.1. Assume that Ω D D d1 , d2 ∈ (0, δ) the diffusive model exhibits a unique coexistence state which is a global attractor with respect to the component-wise positive solutions. Proof . By Corollary 6.5, δ > 0 exists such that the model is permanent for d1 , d2 ∈ (0, δ). Thus, the existence of at least a linearly stable coexistence state is ensured by previous results (see, e.g. [17,22] and [29]), which are easily adaptable to cover the case of general operators subject to general boundary conditions of mixed type. Furthermore, it is well known that it is a global attractor if it is unique. Thus, it suffices to establish the uniqueness of the coexistence state for small diffusion rates. According to Corollary 9.6 of [29], this is an easy consequence from the fact that all the coexistence states are linearly stable. Actually, the fact that any coexistence steady state is linearly stable entails that it is non-degenerate, in the sense that (0, 0) is the unique solution of its linearized system. Therefore, setting b(x) = γB(x), where γ is regarded as a parameter ranging in [0, 1], by a rather standard global continuation argument involving the implicit function theorem the uniqueness holds for all γ ∈ [0, 1]. Let (u(d1 ,d2 ) , v(d1 ,d2 ) ) be a coexistence steady state of (1.1) for diffusion rates d1 , d2 > 0. Then, its linear stability follows from the positivity of the principal eigenvalue, σ1 , of the associated eigenvalue problem ⎧ ⎨(d1 L1 − λ + 2au(d1 ,d2 ) + bv(d1 ,d2 ) )ϕ + bu(d1 ,d2 ) ψ = σϕ in Ω , cv ϕ + (d2 L2 − µ + 2dv(d1 ,d2 ) + cu(d1 ,d2 ) )ψ = σψ (8.1) ⎩ (d1 ,d2 ) B1 u = B2 v = 0 on ∂Ω , obtained by linearizing (1.2) at the given coexistence state. Since b(x)u(d1 ,d2 ) (x) > 0 and c(x)v(d1 ,d2 ) (x) > 0 for all x ∈ Ω , (8.1) is a problem of quasi-cooperative type as those analyzed in Section 5. Thus, the existence of the principal eigenvalue, σ1 , follows from Theorem 5.1. As a consequence of Theorem 5.4, σ1 > 0 if, and only if, ( ) d1 L1 − λ + 2au(d1 ,d2 ) + bv(d1 ,d2 ) bu(d1 ,d2 ) L(d1 ,d2 ) := , cv(d1 ,d2 ) d2 L2 − µ + 2dv(d1 ,d2 ) + cu(d1 ,d2 ) subject to the boundary operator (B1 , B2 ), admits a strict supersolution, (ϕ(d1 ,d2 ) , ψ(d1 ,d2 ) ), with ϕ(d1 ,d2 ) , ψ(d1 ,d2 ) ∈ W 2,p (Ω ), p > N , and ϕ(d1 ,d2 ) ⪈ 0, ψ(d1 ,d2 ) ⪇ 0 in Ω . ¯ = Ωper , we have that b(x)c(x) < a(x)d(x) for all x ∈ Ω ¯ . So, τ ∈ C(Ω ¯ ) exists such that Since Ω d(x) b(x) < τ (x) < a(x) c(x)

¯. for all x ∈ Ω

Indeed, we can take τ = (b + ξ)/a for some ξ > 0 small enough. And for sufficiently small ν > 0, let us 1 2 denote by θν,τ and θν,1 the (unique) positive solutions of { { νL1 θ = τ θ − θ2 in Ω , νL2 θ = θ − θ2 in Ω , and B1 θ = 0 on ∂Ω , B2 θ = 0 on ∂Ω , ¯ as ν ↓ 0. Hence, ν0 > 0 respectively, which, according to Theorem 3.1, converge to τ and 1 uniformly in Ω exists such that θν1 ,τ (x) d(x) b(x) < 20 < a(x) θν0 ,1 (x) c(x)

¯. for all x ∈ Ω

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So, ε > 0 exists such that a(x)θν10 ,τ (x) − b(x)θν20 ,1 (x) > ε > 0

and d(x)θν20 ,1 (x) − c(x)θν10 ,τ (x) > ε > 0

¯. for all x ∈ Ω

Next, we will show that, for sufficiently small d1 , d2 > 0, the pair (ϕ(d1 ,d2 ) , ψ(d1 ,d2 ) ) := (θν10 ,τ , −θν20 ,1 ) provides us with the desired supersolution of [L(d1 ,d2 ) ; (B1 , B2 ), Ω ]. Indeed, thanks to Theorem 4.1, ) ( µa−λc ¯, , uniformly in Ω lim (u(d1 ,d2 ) , v(d1 ,d2 ) ) = (u∗ , v∗ ) = λd−µb ad−bc ad−bc (d1 ,d2 )→(0,0)

and hence, applying L(d1 ,d2 ) yields (d1 L1 − λ + 2au(d1 ,d2 ) + bv(d1 ,d2 ) )θν10 ,τ − bu(d1 ,d2 ) θν20 ,1 = d1 L1 θν10 ,τ − (λ − au(d1 ,d2 ) − bv(d1 ,d2 ) )θν10 ,τ + (aθν10 ,τ − bθν20 ,1 )u(d1 ,d2 ) > d1 L1 θν10 ,τ − (λ − au(d1 ,d2 ) − bv(d1 ,d2 ) )θν10 ,τ + εu(d1 ,d2 ) d ,d →0

1 2 −− −−−→ −(λ − au∗ − bv∗ )θν10 ,τ + εu∗ = εu∗ > 0

and, similarly, cv(d1 ,d2 ) θν10 ,τ − (d2 L2 − µ + 2dv(d1 ,d2 ) + cu(d1 ,d2 ) )θν20 ,1 = −d2 L2 θν20 ,1 + (µ − dv(d1 ,d2 ) − cu(d1 ,d2 ) )θν20 ,1 − (dθν20 ,1 − cθν10 ,τ, )v(d1 ,d2 ) < −d2 L2 θν20 ,1 + (µ − dv(d1 ,d2 ) − cu(d1 ,d2 ) )θν20 ,1 − εv(d1 ,d2 ) d ,d →0

1 2 −− −−−→ (µ − dv∗ − cu∗ )θν20 ,1 − εv∗ = −εv∗ < 0,

¯ . By the choice of ϕ(d ,d ) and ψ(d ,d ) , they are independent of (d1 , d2 ) and with uniform convergence in Ω 1 2 1 2 satisfy B1 ϕ(d1 ,d2 ) = B2 ψ(d1 ,d2 ) = 0 on ∂Ω . This ends the proof. □ Naturally, the technical device introduced in the proof of Theorem 8.1 can be also adapted to derive a substantial generalization of Theorem 1.1 of [19], which was established for the −∆ operator under non-flux boundary conditions. Note that here we are dealing with a more general class of differential operators under mixed boundary conditions of Robin type. Precisely, Theorem 8.1 can be extended to cover the next class of reaction–diffusion systems ⎧ ∂u ⎪ + d1 L1 u = uf (u, v, x) ⎪ ⎪ ⎪ ∂t ⎪ in Ω × (0, +∞), ⎨ ∂v (8.2) + d L v = vg(u, v, x) 2 2 ⎪ ∂t ⎪ ⎪ ⎪ on ∂Ω × (0, +∞), ⎪ ⎩B1 u = B2 v = 0 u(·, 0) = u0 > 0, v(·, 0) = v0 > 0, in Ω , under the following general assumptions on f and g emphasizing the fact that the system must be of competitive type: ¯ → R are of class C 1 in each variable. (H1) f, g : R × R × Ω (H2) There exists a positive constant M such that f (M, 0, x) < 0, f (0, M, x) < 0, g(M, 0, x) < 0 and ¯. g(0, M, x) < 0 for every x ∈ Ω ¯ (H3) For every u, v ≥ 0 and x ∈ Ω , ∂w f (u, v, x) < 0 and ∂w g(u, v, x) < 0 where w ∈ {u, v}. ¯ , there exists a unique (u∗ (x), v∗ (x)) in the non-negative cone {(u, v) ∈ R2 : u, v ≥ 0} (H4) For every x ∈ Ω ¯. such that f (u, v, x) = 0 and g(u, v, x) = 0. Moreover, u∗ (x) > 0 and v∗ (x) > 0 for all x ∈ Ω

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¯ , (∂u f ∂v g − ∂v f ∂u g)|(u (x),v (x),x) > 0. (H5) For every x ∈ Ω ∗ ∗ ¯ , the linearization of the non-spatial model at (u∗ (x), v∗ (x)), According to (H3) and (H5), for every x ∈ Ω ( ) u∗ (x)∂u f (u∗ (x), v∗ (x), x) u∗ (x)∂v f (u∗ (x), v∗ (x), x) , v∗ (x)∂u g(u∗ (x), v∗ (x), x) v∗ (x)∂v g(u∗ (x), v∗ (x), x) has two negative eigenvalues. In addition, by (H2), the remaining steady states of the non-spatial model are linearly unstable. This entails (u∗ (x), v∗ (x)) to be a global hyperbolic attractor for the non-spatial model ¯. with respect to the positive cone. Consequently, the previous conditions are actually imposing that Ωper = Ω Theorem 8.2. Suppose (H1)–(H5) and ΓDi = ∅ for i = 1, 2. Then, δ > 0 exists such that, for every d1 , d2 < δ, the reaction–diffusion system (8.2) possesses a unique coexistence steady state, which is a global attractor with respect to the positive solutions. Proof . The proof follows the same patterns as the proof of Theorem 8.1. Let (u(d1 ,d2 ) , v(d1 ,d2 ) ) be a coexistence steady state of (8.2). Its linear stability is a consequence of the positivity of the principal eigenvalue of the operator ( ) d1 L1 − f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·) − u(d1 ,d2 ) ∂u f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)

−u(d1 ,d2 ) ∂v f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)

−v(d1 ,d2 ) ∂u g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)

d2 L2 − g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·) − v(d1 ,d2 ) ∂v g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)

,

subject to the boundary conditions (B1 , B2 ), which will be subsequently denoted by L(d1 ,d2 ) . Since ∂v f 2 ¯ (see (H3)), the off-diagonal entries of L(d ,d ) are positive functions. and ∂u g are negative in [0, +∞) × Ω 1 2 Thus, L(d1 ,d2 ) is of quasi-cooperative type and hence satisfies Theorems 5.1 and 5.4. Therefore, its principal eigenvalue is positive if it admits a strict supersolution, (ϕ, ψ), with ϕ > 0 and ψ < 0, i.e., ⎧ [d1 L1 − f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·) − u(d1 ,d2 ) ∂u f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)]ϕ ⎪ ⎪ ⎪ ⎪ ⎪ −u(d1 ,d2 ) ∂v f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)ψ > 0 ⎪ ⎨ in Ω , −v(d1 ,d2 ) ∂u g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)ϕ + [d2 L2 − g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·) (8.3) ⎪ ⎪ ⎪ −v(d1 ,d2 ) ∂v g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)]ψ < 0 ⎪ ⎪ ⎪ ⎩ B1 ϕ = B2 ψ = 0 on ∂Ω . ¯ and ∂Ω = Γ 1 = Γ 2 , the singular In the special case covered by this theorem, i.e., when Ωper = Ω R R perturbation results of Sections 3 and 4 can be easily adapted to cover the slightly more general Kolmogorov system of competitive type (8.2) in order to obtain that ( ) ¯. u(d1 ,d2 ) , v(d1 ,d2 ) = (u∗ , v∗ ) uniformly in Ω lim (d1 ,d2 )→(0,0)

As the corresponding proofs follow mutatis mutandis the general patterns of the proofs in Sections 3 and 4, their technical details are omitted here. Thus, thanks to (H5), it follows that ( ) ∂u f ∂v g − ∂v f ∂u g (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x) > 0 ¯ and sufficiently small d1 , d2 > 0. Hence, by (H3), there exists a function τ ∈ C(Ω ¯ ) such that for every x ∈ Ω ( ) ( ) ∂v f ∂v g (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x) < τ (x) < (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x). ∂u f ∂u g Subsequently, we consider the same positive solutions, θν10 ,τ and θν20 ,1 , as in the proof of Theorem 8.1, with ν0 small enough so that ) ) ( ( θν10 ,τ ∂v g ∂v f (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x) < 2 (x) < (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x) ∂u f θν0 ,1 ∂u g

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¯ and sufficiently small d1 , d2 > 0. Therefore, owing to (H3), ε > 0 exists such that for all x ∈ Ω ∂v f (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x)θν20 ,1 (x) − ∂u f (u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x)θν10 ,τ (x) > ε > 0, ∂u g(u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x)θν10 ,τ (x) − ∂v g(u(d1 ,d2 ) (x), v(d1 ,d2 ) (x), x)θν20 ,1 (x) > ε > 0. It remains to show that, for sufficiently small d1 , d2 > 0, the vectorial function ( ) ( ) ϕ(d1 ,d2 ) , ψ(d1 ,d2 ) := θν10 ,τ , −θν20 ,1 , satisfies (8.3). Indeed, by substituting, we find that [ ] d1 L1 − f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·) − u(d1 ,d2 ) ∂u f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·) θν10 ,τ + u(d1 ,d2 ) ∂v f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν20 ,1 = d1 L1 θν10 ,τ − f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν10 ,τ [ ] + u(d1 ,d2 ) ∂v f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν20 ,1 − ∂u f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν10 ,τ d ,d →0

1 2 −−−→ −f (u∗ , v∗ , ·)θν10 ,τ + εu∗ = εu∗ > 0 > d1 L1 θν10 ,τ − f (u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν10 ,τ + εu(d1 ,d2 ) −−

¯ . Similarly, uniformly in Ω [ ] − v(d1 ,d2 ) ∂u g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν10 ,τ − d2 L2 − g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·) − v(d1 ,d2 ) ∂v g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·) θν20 ,1 [ = −d2 L2 θν20 ,1 + g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν20 ,1 − v(d1 ,d2 ) ∂u g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν10 ,τ ] − ∂v g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν20 ,1 d ,d →0

1 2 −−−→ g(u∗ , v∗ , ·)θν20 ,1 − εv∗ = −εv∗ < 0 < −d2 L2 θν20 ,1 + g(u(d1 ,d2 ) , v(d1 ,d2 ) , ·)θν20 ,1 − εv(d1 ,d2 ) −−

¯ . Note that, by (H4), u∗ and v∗ are positive and separated away from zero on Ω ¯ . This ends uniformly in Ω the proof. □ ¯ fails to be true in an arbitrarily small Theorem 8.1 is optimal in the sense that if condition Ωper = Ω ball, Bε (x0 ), centered at some x0 ∈ Ω with radius ε > 0, where the non-spatial model exhibits a founder control competition (in other words, Bε (x0 ) ⊂ Ωbi ), then, owing to Theorem 7.3, (1.2) might admit at least three coexistence states for sufficiently small diffusion rates. A possible strategy to realize what is going on consists in modeling this change of behavior through some additional parameter incorporated to the setting of the model, in order to mimic such transition in a continuous way. Let a(x), d(x), b(x) and c(x) be four ¯ such that for some x0 ∈ Ω positive continuous functions on Ω λ(x0 ) b(x0 ) a(x0 ) = = , c(x0 ) µ(x0 ) d(x0 )

(8.4)

while a(x) λ(x) b(x) > > c(x) µ(x) d(x)

¯ \ {x0 }. for all x ∈ Ω

(8.5)

For instance, the one-dimensional choices Ω = (−1, 1),

a(x) = d(x) = 2,

λ(x) = µ(x) = 1,

b(x) = c(x) = 2 − x2 ,

x ∈ [−1, 1],

(8.6)

satisfy (8.4) and (8.5) with x0 = 0. Next, we consider (1.1) for these choices of λ(x), µ(x), a(x), d(x), and (b, c) := (bρ , cρ ), where bρ (x) := ρb(x),

cρ (x) := ρc(x),

¯ , ρ > 0, x∈Ω

(8.7)

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Fig. 1. An admissible bifurcation diagram.

where ρ is regarded as a parameter measuring the intensity of the aggressions between the antagonist species, u and v. According to (8.4) and (8.5), it becomes apparent that, for every ρ ∈ (0, 1), a(x) a(x) λ(x) b(x) ρb(x) = > > = ρc(x) c(x) µ(x) d(x) d(x)

¯. for all x ∈ Ω

¯ and, owing to Theorem 8.1, (1.1) possesses a unique Consequently, in such range of values of ρ, Ωper = Ω (linearly stable) coexistence steady-state, which is a global attractor with respect to the positive solutions ¯ fails to be true at ρ = 1, of (1.1) for sufficiently small d1 and d2 . By construction, the condition Ωper = Ω ¯ where Ωper = Ω \ {x0 }, as well as for ρ > 1 sufficiently close to 1, where there are a maximal ε1 := ε1 (ρ) > 0 and a minimal ε2 := ε2 (ρ) > ε1 such that Bε1 (x0 ) ⊂ Ωbi

¯ \ Bε (x0 ) ⊂ Ωper . and Ω 2

Actually, ε2 (ρ) can be taken arbitrarily small by choosing ρ > 1 sufficiently close to 1, i.e., limρ↓1 ε2 (ρ) = 0. In other words, for ρ > 1 sufficiently close to 1, the main assumption of Theorem 8.1 is ‘almost’ satisfied, except for a small ball centered at x0 , Bε2 (x0 ), where Ωbi ̸= ∅. Thus, thanks to the multiplicity result Theorem 7.3, at least for the symmetric choice (8.6) with d1 = d2 , the problem (1.2) might exhibit, in general, at least three coexistence states for sufficiently small diffusion rates. Fig. 1 shows an admissible bifurcation diagram of coexistence states in terms of the parameter ρ. Fig. 1 plots, using a continuous line, the components u(d1 ,d2 ,ρ) of the coexistence states (u(d1 ,d2 ,ρ) , v(d1 ,d2 ,ρ) ) of (1.2) versus the continuation parameter ρ, together with the u-components of the unique coexistence state and the semitrivial positive solutions of the associated non-spatial model, (1.6), with dashed lines, i.e., ( ) ( ) ( ) ( µ) λd − µb µa − λc λd − µρb µa − λρc λ , = , , , 0 and 0, , ad − bc ad − bc ad − ρ2 bc ad − ρ2 bc a d respectively. As one cannot represent the entire functions on the ordinate axis, Fig. 1 plots the values of each of these components particularized at the single point x0 versus the parameter ρ. The values of d1 and d2 are supposed to be fixed and small. As ρ < 1, the assumptions of Theorem 8.1 hold and the unique coexistence state of (1.2) is a global attractor of (1.1). Similarly, the coexistence steady state of the non-spatial model is a global attractor for (1.6), and each of the semitrivial positive steady states is linearly unstable. Note that, according to Theorem 4.1, u(d1 ,d2 ,ρ) can be taken as close as we wish to λd−µb ad−bc by choosing d1 and d2 sufficiently small. According to Theorem 7.3, as ρ crosses some critical value, ρ0 (d1 , d2 ) ∼ 1, the principal eigenvalue, σ1 (d1 , d2 , ρ), of the linearized system at the coexistence state that perturbs from the coexistence steady state of the non-spatial model (1.6) crosses zero and becomes negative, provoking a pitchfork bifurcation to, at least, two additional

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coexistence states of (1.2), which, according to the linearized stability principle, [6], should be linearly stable. As d1 , d2 → 0 we conjecture that these stable coexistence states approximate each of the (linearly stable) semitrivial positive steady states of the non-spatial model, and that lim

d1 ,d2 →0

ρ0 (d1 , d2 ) = 1.

From a technical point of view, the fact that σ1 (d1 , d2 , ρ) changes sign as ρ crosses ρ0 (d1 , d2 ) for sufficiently ¯. small d1 and d2 shows how the proof of Theorem 8.1 works out exclusively when Ωper = Ω ¯ to Summarizing, when the coefficients of the model move away from their original values where Ωper = Ω ¯ and Ωbi is non-empty, the principal eigenvalue any other situation case such that Ωper is a proper subset of Ω of the global attractor looses positivity crossing zero just when Ωbi becomes non-empty. Since the fact that lim

d1 ,d2 →0 ρ↑1

σ1 (d1 , d2 , ρ) = 0

is exclusively based on the values of the coefficients of the model for ρ ≤ 1, it becomes apparent that actually this is the main technical difficulty for getting general uniqueness results in truly spatially heterogeneous u v landscapes, where permanence and dominance regions, i.e., Ωper , Ωdo and Ωdo , can coexist within the same habitat, in the sense that the proof Theorem 8.1 cannot be adapted to treat these general situations. 9. Global uniqueness through Picone’s identity Despite all technical difficulties just described in the previous discussion, one can get the following uniqueness result for general spatially heterogeneous landscapes, which might be the first available theorem of this type in the literature. Naturally, by the multiplicity result of Section 7, Theorem 7.3, Ωbi should be empty for its validity, and the proof should follow completely different patterns to those of Theorem 8.1. In this occasion, the proof is based on a very sharp version of Picone’s identity, [32], going back to Lemma 4.1 of [24]. A similar device was used to get Theorem 3.4(iii) in [15], where the very special case of constants coefficients, with a = d = 1 and bc < 1, was treated. The main uniqueness result of this section can be stated as follows. Theorem 9.1. Assume that b(x)c(x) ≤ a(x)d(x) for all x ∈ Ω , with bc ̸= ad, and that either 2 c3 or cdb , or ad 2 , is a positive constant in Ω . Then, the following assertions hold:

b2 ac ,

or

b3 , a2 d

(a) Should it exist, any coexistence state of (1.2) must be linearly stable and hence, unique. (b) If (θ{d1 ,λ,a} , 0) and (0, θ{d2 ,µ,d} ) are linearly unstable, then (1.2) admits a (unique) coexistence state, which is a global attractor with respect to the component-wise positive solutions. (c) The semitrivial solutions (θ{d1 ,λ,a} , 0) and (0, θ{d2 ,µ,d} ) cannot be, simultaneously, linearly stable. u v ̸= ∅ and Ωdo ̸= ∅, then (b) holds for sufficiently small d1 , d2 > 0. (d) If Ωper ̸= ∅, or Ωdo In this theorem the domain Ω is allowed to be truly spatially heterogeneous, in the sense that any of the u v regions Ωper , Ωdo , Ωdo , Ωext and Ωjunk can be nonempty. Naturally, since bc ⪇ ad, we are actually requiring that Ωbi = ∅. But far from being a restriction, owing to Theorem 7.3, Ωbi = ∅ must hold in order to expect uniqueness. Astonishingly, Theorem 9.1 does not impose any restriction on the size and the nature of the differential operators L1 and L2 , nor on the boundary operators B1 and B2 , which can be taken of general mixed type, even with ΓDi nonempty for i ∈ {1, 2} (which is not allowed by Theorem 8.1). Moreover, it is also independent of the values of λ(x) and µ(x), as soon as they can guarantee the existence of the semitrivial steady states, as well as of the size of the diffusion rates d1 and d2 . However, in order to make sure that (1.2) possesses some coexistence state, we should impose d1 and d2 to be sufficiently small as established

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75

in Part (d). Actually, since λ(x) and µ(x) are free, we can fix an arbitrary spatial configuration for Ω and adjust the several coefficients arising in the setting of the model to get it. ¯ As a consequence of Theorem 9.1, the maximal open rectangle, R, in the (d1 , d2 )-plane, with (0, 0) ∈ R, where both semitrivial positive solutions exist can be divided into three supplementary regions according to the linearized stability of the semitrivial solutions. Rather surprisingly, the global dynamics of (1.1) in each of these regions is characterized by the dynamics of the non-spatial model (1.6). The curves of change of stability of (θ{d1 ,λ,a} , 0) and (0, θ{d2 ,µ,d} ) are given by the set of values of d1 and d2 such that [ ] S1 (d1 , d2 ) := σ1 d2 L2 − µ(x) + c(x)θ{d1 ,λ,a} ; B2 , Ω = 0, [ ] S2 (d1 , d2 ) := σ1 d1 L1 − λ(x) + b(x)θ{d2 ,µ,d} ; B1 , Ω = 0, respectively. Indeed, (θ{d1 ,λ,a} , 0) (resp. (0, θ{d2 ,µ,d} )) is linearly stable if, and only if, S1 (d1 , d2 ) > 0 (resp. S2 (d1 , d2 ) > 0), while it is linearly unstable if, and only if, S1 (d1 , d2 ) < 0 (resp. S2 (d1 , d2 ) < 0). Although our setting here is substantially more general than the previous one of [12], the analysis already carried out there in the global structure of these curves can be adapted mutatis mutandis to cover the present situation here. According to Theorem 9.1(c), there cannot exist (d1 , d2 ) ∈ R such that, simultaneously, S1 (d1 , d2 ) > 0 and S2 (d1 , d2 ) > 0. Therefore, these curves divide the rectangle R into three supplementary regions. Namely, the permanence region, Θper , which is the portion of R in between the two curves of change of stability, u where both semitrivial steady states are linearly unstable; the dominance region of u, Θdo , which is the portion of R where (0, θ{d2 ,µ,d} ) is linearly unstable and (θ{d1 ,λ,a} , 0) is linearly stable, and the dominance v region of v, Θdo , where (θ{d1 ,λ,a} , 0) is linearly unstable and (0, θ{d2 ,µ,d} ) is linearly stable. The next general result holds. Theorem 9.2. Under the hypothesis of Theorem 9.1 the following assertions are true: (a) For every (d1 , d2 ) ∈ Θper , the model possesses a unique coexistence steady state which is a global attractor for the component-wise positive solutions of (1.1). u (b) For every (d1 , d2 ) ∈ Θdo , the semitrivial solution (θ{d1 ,λ,a} , 0) is a global attractor for the componentwise positive solutions of (1.1). In particular, (1.1) cannot admit a coexistence steady state. v (c) For every (d1 , d2 ) ∈ Θdo , the semitrivial solution (0, θ{d2 ,µ,d} ) is a global attractor for the componentwise positive solutions. In particular, (1.1) cannot admit a coexistence steady state. The proof of Theorem 9.1 is based on the following generalized version of Picone’s identity, [32,24]. ¯ ) and Lu, Lv ∈ C(Ω ¯ ), where Lemma 9.3. Let u, v ∈ W 2,p (Ω ), for some p > N , be such that uv ∈ C 1 (Ω 1 L := −div(A∇·) + C ∈ {L1 , L2 }. Then, for every β ∈ C(∂Ω ) and f ∈ C (R), the following identity holds ∫ ∫ ∫ (v) ( ) ( ) 2 ′ v v v f u [uLv − vLu] = u f u ⟨∇ u , A∇ u ⟩ − f uv [DuRv − DvRu], (9.1) Ω

Ω

∂Ω

where (Du, Ru) := (u, ⟨n, A∇u⟩ + βu) on ∂Ω . Thus, if, in addition, (B, L) ∈ {(B1 , L1 ), (B2 , L2 )} and Bu = Bv = 0 on ∂Ω , then ∫ ∫ ( ) ( ) f uv [uLv − vLu] = u2 f ′ uv ⟨∇ uv , A∇ uv ⟩. Ω

Ω

Proof . Expanding the integrand on the left hand side, it is easily seen that ( ) ( ) ( ) f uv [uLv − vLu] = f uv [v div(A∇u) − u div(A∇v)] = f uv div(vA∇u − uA∇v) [ ( ) ] ( ) = div f uv (vA∇u − uA∇v) − ⟨∇f uv , vA∇u − uA∇v⟩ [ ( ) ] ( ) = div f uv (vA∇u − uA∇v) − f ′ uv ⟨∇ uv , A(v∇u − u∇v)⟩ [ ( ) ] ( ) = div f uv (vA∇u − uA∇v) + u2 f ′ uv ⟨∇ uv , A∇ uv ⟩.

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On the other hand, integrating by parts yields ∫ ∫ [ ( ) ] ( ) div f uv (vA∇u − uA∇v) = f uv ⟨vA∇u − uA∇v, n⟩ Ω ∫∂Ω ( ) = f uv [⟨vA∇u − uA∇v, n⟩ + vβu − uβv] ∂Ω ∫ ( ) =− f uv [DuRv − DvRu]. ∂Ω

Therefore, integrating the identity above in Ω , (9.1) holds.

□ 2

3

b , or ab2 d , is constant Proof of Theorem 9.1. We will only prove the result in the special case when either ac in Ω , as the proof in the other two cases follows by symmetry. Let (u, v) be a coexistence state of (1.2). In particular, u ≫ 0 and v ≫ 0. The local attractive character of (u, v) as a steady state of (1.1) is determined by the sign of the principal eigenvalue of the linearization of (1.2) at (u, v), i.e., by the unique eigenvalue of the linear eigenvalue problem ⎧ ⎨[d1 L1 − λ + 2au + bv]ϕ + buψ = σ1 ϕ in Ω , [d2 L2 − µ + 2dv + cu]ψ + cvϕ = σ1 ψ in Ω , (9.2) ⎩ B1 ϕ = B2 ψ = 0 in ∂Ω ,

to an eigenfunction, (ϕ, ψ), with ϕ ≫ 0 and ψ ≪ 0, whose existence has been established by Theorem 5.1, since b(x)u(x) > 0 and c(x)v(x) > 0 for all x ∈ Ω . Note that the left hand side of these equations can be, equivalently, written down as [d1 L1 − λ + 2au + bv]ϕ + buψ = d1 L1 ϕ − [λ − au − bv]ϕ + u(aϕ + bψ) = d1 L1 ϕ − d1 ϕu L1 u + u(aϕ + bψ), [d2 L2 − µ + 2dv + cu]ψ + cvϕ = d2 L2 ψ − [µ − dv − cu]ψ + v(dψ + cϕ) = d2 L2 ψ − d2 ψv L2 v + v(dψ + cϕ). Thus, multiplying the equations of (9.2) by ϕ2 /u and ψ 2 /v, respectively, and integrating in Ω yields ∫ ∫ ∫ ϕ3 d1 ( ϕu )2 (uL1 ϕ − ϕL1 u) + ϕ2 (aϕ + bψ) = σ1 u , ∫Ω ∫Ω ∫Ω ψ3 d2 ( ψv )2 [vL2 ψ − ψL2 v] + ψ 2 (dψ + cϕ) = σ1 v . Ω

Ω

Ω

2

On the other hand, applying Lemma 9.3 with f (t) = t yields ∫ ∫ ∫ 2 ( ϕu )2 [uL1 ϕ − ϕL1 u] = 2 ϕu ⟨∇ ϕu , A1 ∇ ϕu ⟩ ≥ 2α1 ϕu|∇ ϕu | , ∫Ω ∫Ω ∫Ω 2 ψ 2 ψ ψ ψv ⟨∇ v , A2 ∇ v ⟩ ≤ 2α2 ψv|∇ ψv | , ( v ) [vL2 ψ − ψL2 v] = 2 Ω

Ω

Ω

where α1 , α2 > 0 are the ellipticity constants of L1 and L2 , respectively. Hence, ∫ ∫ ∫ ∫ ϕ3 ϕ 2 2 σ1 uϕ|∇ u | + ϕ (aϕ + bψ) ≥ ϕ2 (aϕ + bψ), u ≥ 2d1 α1 Ω Ω Ω Ω ∫ ∫ ∫ ∫ ψ 2 ψ3 2 σ1 ≤ 2d α vψ|∇ | + ψ (dψ + cϕ) ≤ ψ 2 (dψ + cϕ). 2 2 v v Ω

Consequently, when

Ω

Ω

Ω

b2 ac

is a (positive) constant, we find that (∫ ) ∫ ∫ ϕ3 ψ3 b2 σ1 − ≥ ϕ2 (aϕ + bψ) − u ac v Ω

Ω

Ω

=

b2 ac

∫ [ ϕ2 (aϕ + bψ) − Ω

∫

ψ 2 (dψ + cϕ)

Ω

b2 2 ac ψ (dψ

(9.3) ] + cϕ) .

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Since ad ⪈ bc by hypothesis, then d ⪈ bc a and so ∫ [ ] ∫ [ ( )] b2 2 b2 2 bc ψ (dψ + cϕ) > ψ a ψ + cϕ ϕ2 (aϕ + bψ) − ac ϕ2 (aϕ + bψ) − ac Ω ∫Ω [ ] 2 = ϕ2 (aϕ + bψ) − ab 2 ψ 2 (bψ + aϕ) ∫Ω ( ) 2 = ϕ2 − ab 2 ψ 2 (aϕ + bψ) ∫Ω 1 = (aϕ − bψ) (aϕ + bψ)2 ≥ 0. a2 Ω

Thus, since ϕ ≫ 0 and ψ ≪ 0, it follows from (9.3) that σ1 > 0. 3 Similarly, when ab2 d is a positive constant, we find from c ⪇ ad b that (∫ ) ∫ ∫ ∫ ϕ3 ψ3 2 b3 b3 σ1 ϕ (aϕ + bψ) − a2 d ψ 2 (dψ + cϕ) ≥ u − a2 d v Ω Ω Ω Ω ∫ [ ] 3 ϕ2 (aϕ + bψ) − ab2 d ψ 2 (dψ + cϕ) = ∫Ω [ ( )] 3 > ϕ2 (aϕ + bψ) − ab2 d ψ 2 dψ + ad b ϕ ∫Ω [ 2 2 ] 1 a ϕ (aϕ + bψ) − b2 ψ 2 (bψ + aϕ) = a2 ∫Ω 1 = (aϕ − bψ)(aϕ + bψ)2 ≥ 0 a2 Ω

and therefore, σ1 > 0. This shows the first claim of Part (a). The uniqueness can be obtained with the global continuation argument already used to complete the proof of Theorem 8.1. It should be noted that 2 2 b2 if, e.g., ac is constant, then γacb is also constant for all γ ∈ [0, 1]. This ends the proof of Part (a). Suppose (θ{d1 ,λ,a} , 0) and (0, θ{d2 ,µ,d} ) are linearly unstable. Then, the problem is compressive, as discussed by P. Hess [17], and hence, (1.2) admits a coexistence state. By Part (a), the coexistence state is unique. The fact that the system is compressive entails the global attractive character of the unique coexistence state (see, e.g., [17], or the proof of Theorem 4.1 in [29]). This ends the proof of Part (b). The proof of Part (c) can be accomplished by using the fixed point index with respect to the cone of component-wise positive functions, P , as discussed by H. Amann [1], E.N. Dancer [7], P. Hess [17] and J. L´ opez-G´ omez [22]. Note that, for sufficiently large m > 0, the component-wise non-negative solutions of (1.2) are the fixed points of the compact positive operator H defined by ( ) (d1 L1 − λ + m)−1 (mu − au2 − buv) H(u, v) := . (9.4) (d2 L2 − µ + m)−1 (mv − dv 2 − cuv) Let B denote any bounded open set containing all non-negative solutions of (1.2) and suppose (θ{d1 ,λ,a} , 0) and (0, θ{d2 ,µ,d} ) are linearly stable. Then, by Step (i) of the proof of Theorem 4.1 in [22], the global index of H in B is given by iP (H, B) = 1.

(9.5)

On the other hand, by Steps (ii) and (iii) of the proof of [22, Th. 4.1], we have that iP (H, (0, 0)) = 0,

iP (H, (θ{d1 ,λ,a} , 0)) = iP (H, (0, θ{d2 ,µ,d} )) = 1.

Moreover, since any coexistence state, (u, v), of (1.2) is linearly stable, iP (H, (u, v)) = 1. As these facts are clearly incompatible with (9.5), the proof of Part (c) is complete. Part (d) is a direct consequence from Part (b) and Corollary 6.6. □

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Finally, we will accomplish the proof of Theorem 9.2. u Proof of Theorem 9.2. Part (a) is Part (b) of Theorem 9.1. Consider (d1 , d2 ) ∈ Θdo . Then, (θ{d1 ,λ,a} , 0) is linearly stable, while (0, θ{d2 ,µ,d} ) is linearly unstable. Thus, arguing as in the proof of the last part of Theorem 9.1, we find that (9.5) holds and that

iP (H, (θ{d1 ,λ,a} , 0)) = 1,

iP (H, (0, θ{d2 ,µ,d} )) = 0,

iP (H, (0, 0)) = 0.

On the other hand, since any coexistence state, (u, v), of (1.2) is linearly stable, we also have that iP (H, (u, v)) = 1. Therefore, by the additivity properties of the index, it becomes apparent that (1.2) cannot admit a coexistence state. Since the system is quasicooperative, the ω-limit of any component-wise non-negative solution consists of equilibria. Since the model is dissipative and (1.2) cannot admit a coexistence state, and, in addition, (0, θ{d2 ,µ,d} ) and (0, 0) are linearly unstable, the semitrivial positive solution (θ{d1 ,λ,a} , 0) must be a global attractor. Another proof of this feature, going back to E. N. Dancer [8], was sketched in Theorem 4.4 of [29]. Part (c) follows by symmetry. □ References

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