Lotka-Volterra models with fractional diffusion

Lotka-Volterra models with fractional diffusion Michele O. Alves1, Marcos T. O. Pimenta2and Antonio Su´ arez3, 1. Dpto. de Matem´atica Universidade Estadual de Londrina - UEL 86057-970 - Londrina - PR , Brazil 2. Dpto. de Matem´atica e Computa¸c˜ao Faculdade de Ciˆencias e Tecnologia, UNESP - Univ Estadual Paulista 19060-900 - Pres. Prudente - SP, Brazil 3. Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla C/. Tarfia s/n, 41012 - Sevilla, Spain, E-mail addresses: michelealv[email protected], pimen[email protected], [email protected] Abstract In this paper we study the Lotka-Volterra models with fractional Laplacian. For that, we study in detail the logistic problem and show that the sub-supersolution method works for the scalar problem and in case of systems as well. We apply this method to show existence and non-existence of positive solutions in terms of the system parameters. Key Words. Fractional Laplacian, Lotka-Volterra models, sub-supersolution method. AMS Classification. 35J25, 45M20, 92B05. 1 Introduction In this paper we study the following systems                  (−∆)αu=u(λ−u−bv) in Ω, (−∆)βv=v(µ−v−cu) in Ω, u=v= 0 on ∂Ω, (1.1) uno where Ω ⊂IRN,N≥1, is a bounded and regular domain, λ, µ, b, c ∈IR and α, β ∈(0,1). Here, uand vdenote the densities of two species inhabiting in Ω, the habitat, which is surrounded by inhospitable areas, due to the homogeneous Dirichlet boundary conditions. In (1.1) we are assuming that the species diffuse following the fractional laplacian, see Section 2 where we have defined this non-local operator. When α=β= 1, (1.1) is the classical Lotka-Volterra system with random walk, widely studied in the last years in all the cases: competition (b, c > 0), predator-prey (b > 0 and c < 0) and symbiosis (b, c < 0), see [8] and references therein. Fractional operators are used in different contexts: physics, finance and ecology; see [14] and [21] for the ecological meaning of the fractional diffusion. For many years, the nonoriented animal movement was modelled by the classical Brownian motion. However, it seems that when the species is searching for resources, the strategy based on L´evy flights (supported in long jumps) could be more appropriate in some situations. This kind of strategy is optimal for the location of targets which are randomly and sparsely distributed, but the Brownian motion is optimal where the resources are abundant. The L´evy diffusion processes are generated by fractional powers of the Laplacian (−∆)γfor γ∈(0,1). We are interested in the existence of non-negative solutions of (1.1). It is clear that (1.1) possesses the trivial solution (u, v) = (0,0) for all λ, µ ∈IR, since when u≡0 (resp. v≡0) then v(resp. u) verifies an equation of type        (−∆)γw+c(x)w=w(σ−w) in Ω, w= 0 on ∂Ω, (1.2) log where γ∈(0,1), σ∈IR and c∈L∞(Ω). This is the classical logistic equation, studied in [19] and [20] with homogeneous Dirichlet and Neumann boundary conditions, respectively, 2 with γ= 1/2 in both papers. To study this equation, we previously analyze the eigenvalue problem        (−∆)γw+c(x)w=λw in Ω, w= 0 on ∂Ω. (1.3) eigenintro We study the existence of a principal eigenvalue, the unique eigenvalue of (1.3) having a positive eigenfunction, denoted by λ1[γ;c]. This problem has been analyzed in [1] and [19] (for γ= 1/2) and in [20] for the Neumman case. We study some properties of this eigenvalue and of its eigenfunction associated. Then, we prove that (1.2) possesses a positive solution if and only if σ > λ1[γ;c]. Moreover, it is unique and we denote it by θ[γ,σ−c]. Moreover, we try to give an ecological interpretation of the result, comparing our results with the obtained in local operator case, in which the fractional Laplacian is substituted by the classical Laplacian operator. For the existence, we employ the sub-supersolution method. Let us point some remarks. The sub-supersolution method has been used previously in non-linear fractional diffusion problem, see for instance [3] and [9]. In both papers, the method is consequence of a maximum principle and a classical iterative argument. However, we present a different proof which is also valid, with minor technical changes, for systems. Once studied in detail (1.2), we analyse the existence of solutions with both positive components of (1.1). For that, we apply the sub-supersolution method. We first show that this method works for systems, and then we apply it to (1.1). For that, we have to find appropriate sub-supersolutions of (1.1) using the results obtained for the logistic equations. We prove the following results: a) If b, c > 0 or b, c < 0 and bc < C(α, β) for some positive constant (detailed in Section 6) and λand µverify λ>λ1[α;bθ[β,µ]] and µ>λ1[β;cθ[α,λ]],(1.4) condigeneralintro b) or b > 0 and c < 0 and λand µverify λ>λ1[α;bθ[β,µ−cθ[α,λ]]]] and µ>λ1[β;cθ[α,λ]]],(1.5) condigeneralppintro 3 then there exists at least a positive solution of (1.1). We show that conditions (1.4) and (1.5) define regions on the λ−µplane. The paper is organized as follows. In Section 2 we present the functional setting necessary for the remainder of the work. Section 3 is devoted to the eigenvalue problem. We study the existence and main properties of the principal eigenvalue. In Section 4 we study equation (1.2). The sub-supersolution method for systems is shown in Section 5. Finally, in the last Section we study the existence of positive solution of (1.1). 2 Preliminaries In this section we begin introducing the functional framework necessary to develop the theory, and recover some known results about the different forms to define the fractional power of the Laplacian with Dirichlet boundary condition. 2.1 Functional setting Consider a smooth bounded domain Ω ⊂RN. Since in bounded domains there are some non-equivalent definitions of the fractional laplacian operator, let us explain what we mean by the symbol (−∆)α. For u∈C∞ 0(Ω) such that u=P∞ k=1 bkϕk, where λk, ϕkare the eigenpairs of (−∆, H1 0(Ω)), (λkrepeated as much as its multiplicity and {ϕk}forming an ortonormal basis of L2(Ω)), we define (−∆)αu:= ∞ X k=1 λα kbkϕk. Then the operator (−∆)αis defined on D((−∆)α) = {u∈L2(Ω); P∞ k=1 λα kb2 k<+∞} by density. Now, let us consider the half cylinder with base Ω, C:= Ω ×(0,+∞), and denote its lateral boundary by ∂LC:= ∂Ω×[0,+∞). 4 We denote (x, y)∈ C,x∈Ω and y > 0 and define Hα(C) := v∈H1(C); kvkα<+∞, Hα 0(C) := {v∈ Hα(C); v= 0 on ∂LC}, where kvkα:= k−1 αZC y1−2α|∇v|2dxdy1 2, kα=21−2αΓ(1 −α) Γ(α),α∈(0,1) and Γ is the Gamma function. It is not difficult to see that Hα 0(C) is a Hilbert space when endowed with the norm k·kα, which comes from the following inner product hv, wiα=k−1 αZC y1−2α∇v·∇wdxdy. Consider the following subspace of the fractional Sobolev space Hα(Ω), Vα 0(Ω) := {trΩv;v∈ Hα 0(C)} which is a Banach space when endowed with the norm kukVα 0(Ω) := kuk2 L2(Ω) +ZΩZΩ |u(x)−u(y)|2 |x−y|N+2αdxdy1 2 , where trΩis the trace operator defined by trΩv=v(·,0) for v∈ Hα 0(C). Moreover, by Trace Theorem (see Proposition 2.1 in [9]) and embeddings for fractional Sobolev spaces (see Theorem 6.7 in [12]) it follows that ktrΩvkLp(Ω) ≤Ckvkα,∀v∈ Hα 0(C),where p∈(1,2α) (2.1) tracetheorem where 2α=2N N−2α. By Proposition 2.1 in [9] it holds that Vα 0(Ω) = (u∈L2(Ω); u= ∞ X k=1 bkϕksatisfying ∞ X k=1 b2 kλα k<+∞). As far as the following scalar nonlocal problem is concerned,        (−∆)αu=f(x, u) in Ω, u= 0 on ∂Ω, (2.2) P2 5 the approach we are going to follow is by associating to (2.2) a one-more dimensional local problem in C. This can be made by considering the procedure to get a local realization of (−∆)αdescribed beneath. As proved in [9] [Section 2.1], for each u∈ Vα 0(Ω), there exists a unique v∈ Hα 0(C), called its α−harmonic extension such that                  −div(y1−2α∇v) = 0 in C, v= 0 on ∂LC, v(·,0) = uon Ω. Moreover, if u= ∞ X k=1 bkϕkis its spectral decomposition, then v(x, y) = ∞ X k=1 bkϕk(x)ψ(λ 1 2 ky),∀(x, y)∈ C,(2.3) harmonicextension where ψsolves the Bessel equation                  ψ00 +(1 −2α) sψ0=ψ s > 0 −lim s→0+s1−2αψ0(s) = kα ψ(0) = 1. (2.4) besselequation Let u∈ Vα 0(Ω) and v∈ Hα 0(C) its α−harmonic extension. Define the functional 1 kα ∂v ∂yαΩ×{0}∈ V0(Ω)∗by 1 kα ∂v ∂yα(·,0), g:= 1 kαZC y1−2α∇v.∇˜gdxdy, where ˜gis the α−harmonic extension of g∈ Vα 0(Ω) and ∂v ∂yα(x, 0) = −lim y→0+y1−2α∂v ∂y(x, y),∀x∈Ω. Then we can define an operator Aα:Vα 0(Ω) → Vα 0(Ω)∗such that Aαu:= 1 kα ∂v ∂yαΩ×{0} , 6 where vis the α−harmonic extension of uto C. Let us prove that the operators Aαand (−∆)αare in fact the same, i.e., that for all u∈ Vα 0(Ω), Aαu= ∞ X k=1 bkλα kϕk,where u= ∞ X k=1 bkϕk. By linearity, it is enough to prove that for all ϕk, 1 kα ∂v ∂yα(·,0), ϕk=h(−∆)αu, ϕkiL2(Ω) ,for all k∈N, where vis the α−harmonic extension of u. For u∈ Vα 0(Ω) and k∈N, let vand ˜ϕkbe the α−harmonic extensions of uand ϕk, respectively. By (2.3), v(x, y) = P∞ k=1 bkϕk(x)ψ(λ1/2 ky) and ˜ϕk(x, y) = ϕk(x)ψ(λ1/2 ky). Now, integration by parts and properties of ϕkimply that for each y > 0, it holds ZΩ y1−2α∇xv(x, y)·∇x˜ϕk(x, y)dx =y1−2αbkλkψ(λ 1 2 ky)2+ψ0(λ 1 2 ky)2. Then, by (2.4) 1 kα ∂v ∂yα(·,0), ϕk=1 kαZC y1−2α∇v·∇ ˜ϕkdxdy =1 kαZ+∞ 0 y1−2αbkλkψ(λ 1 2 ky)2+ψ0(λ 1 2 ky)2dy =1 kα lim η→0+y1−2αλ 1 2 kbkψ0(λ 1 2 ky)ψ(λ 1 2 ky)y=η =bkλα k =h(−∆)αu, ϕkiL2(Ω) . Hence, in (2.2) we are going to understand (−∆)αas Aα. For simplicity, without loss of generality, we can assume throughout this paper that kα= 1. Then, we define debil Definition 2.1. u∈ V0(Ω) is a weak solution of (2.2) if u=trΩvwhere v∈ Hα 0(C)is a weak solution of          −div(y1−2α∇v)=0 in C, ∂v ∂yα(x, 0) = f(x, v(x, 0)) on Ω. In this case, vis such that ZC y1−2α∇v·∇ψdxdy =ZΩ f(x, v(x, 0))ψ(x, 0)dx, ∀ψ∈ Hα 0(C).(2.5) weakform 7 2.2 Maximum principle Along the paper, the following maximum principle will be very useful, see Lemma 2.5 in [9] for a related result. Proposition 2.2. Let d∈L∞(Ω) and v∈ Hα(C)such that v≥0in ∂LCand          −div(y1−2α∇v)≥0in C, ∂v ∂yα(x, 0) + d(x)v(x, 0) ≥0on Ω. a) Assume that d≥0in Ω, then v≥0in C. b) Assume that v≥0in C. Then, either v≡0or v > 0in C. MaximumPrinciple Proof. a) The proof follows just by using −v−as test function, where v=v++v−. b) In this paragraph we follow the proof of Lemma 4.9 in [6]. Define w(x, y) := eAy2αv(x, y). Then, wsatisfies          −div(y1−2α∇(e−Ay2αw)) ≥0 in C, ∂w ∂yα(x, 0) + (d(x)+2Aα)w(x, 0) ≥0 on Ω. We can choose Asuch that d(x)+2Aα ≤0 in Ω, and so ∂w ∂yα(x, 0) ≥0 in Ω. Take R > 0, consider now the even extension of win Ω ×(−R, R), defined by ˜w(x, y) =        w(x, y) if y > 0, w(x, −y) if y≤0. We can show that −div(|y|1−2α∇(e−A|y|2α˜w)) ≥0 in Ω ×(−R, R). 8 Define now the problem        −div(|y|2α∇(e−A|y|2αh) = 0 in Ω ×(−R, R), h= ˜won (Ω ×{−R})∪(Ω ×{R}). The above problem possesses a solution by [13] (see also Theorem 3.2 in [6]) and by the maximum principle we get that h≤˜win Ω ×(−R, R). On the other hand, by the strong maximum principle, see Lemma 2.3.5 in [13], we conclude that h > 0. This finishes the proof. Remark 2.3. Observe that Proposition 2.2 can be stated in an equivalent way: Assume d∈L∞(Ω) and (−∆)αu+d(x)u≥0in Ωand u≥0on ∂Ω. Then, a) If d≥0in Ω, then u≥0in Ω. b) Assume that u≥0in Ω. Then, either u≡0or u > 0in Ω. 2.3 Regularity results The following result follows by Lemma 3.3 in [10], see also Proposition 5.1 in [2]. cotas1 Lemma 2.4. Assume that f∈C(Ω ×IR) and that there exists a constant Cand p∈ (2,2N/(N−2α)) such that |f(x, t)| ≤ C(1 + |t|p−1), x ∈Ω, t ∈IR. If v∈ Hα 0(C)is a solution of (2.5) and u=trΩv, then v∈L∞(C)∩Cσ(C)and u∈Cσ(Ω) for some σ∈(0,1). Consider now the linear problem        (−∆)αu=g(x) in Ω, u= 0 on ∂Ω. (2.6) P2line The following result is also taken from [10] (Lemma 3.2), see also [7]. 9 where α∈(0,1) and c∈L∞(Ω) or equivalently the equation                  −div(y1−2α∇v) = 0 in C, v= 0 on ∂LC, ∂v ∂yα(x, 0) + c(x)v(x, 0) = λv(x, 0) −v(x, 0)2on Ω. (4.2) logis2 Theorem 4.1. Equation (4.1) possesses a positive solution if and only if λ > λ1[α;c]. Moveover, if it exists, this is the unique positive solution and we denote it by θ[α,λ−c]. Furthermore, θ[α,λ−c]∈C2,σ(Ω) for some σ∈(0,1) and the following property holds: if we denote by ϕ1the principal eigenfunction associated to λ1[α;c]such that kϕ1k∞= 1, then (λ−λ1[α;c])ϕ1(x)≤θ[α,λ−c](x)≤λ−cL,∀x∈Ω.(4.3) ine theorem4.1 Remark 4.2. A similar result holds for (4.2). In this case, we denote by Θ[α,λ−c]the unique positive solution of (4.2), that is, θ[α,λ−c]=trΩΘ[α,λ−c]. Moreover, Θ[α,λ−c]∈ C2,σ(C)∩L∞(C). In the proof of Theorem 4.1 we are going to apply the well known sub-supersolution method. Despite of the definitions and results about this subject in the fractional setting are a rather standard adaptation of the sub-supersolution method to second order operators, we present them here for the sake of completeness. Let us consider the problem (2.2) which is associated to the extension problem                  div(y1−2α∇v) = 0 in C, v= 0 on ∂LC, ∂v ∂yα(x, 0) = f(x, v(x, 0)) on Ω, (4.4) subsuper where f∈C(Ω ×IR). Recall the definition of solution of (4.4), Definition 2.1. Definition 4.3. We say that (v, v)is a sub-supersolution of (4.4) if v, v ∈ Hα(C),u:= trΩv, u := trΩv∈L∞(Ω) and: a) v≤vin Cand v≤0≤von ∂LC. 16 b) For all ψ∈ Hα 0(C),ψ≥0, it holds ZC y1−2α∇v·∇ψdxdy ≤ZΩ f(x, v(x, 0))ψ(x, 0)dx (4.5) subsolution and ZC y1−2α∇v·∇ψdxdy ≥ZΩ f(x, v(x, 0))ψ(x, 0)dx. (4.6) supersolution Theorem 4.4. Assume that (v,v)is a sub-supersolution of (4.4). Then, there exists a solution vof (4.4) such that v≤v≤vin C. In consequence, there exists a solution u∈ Vα 0(Ω) of (2.2) such that u=trΩv≤u≤u=trΩvin Ω. teoremasubsuper Proof. Let v,vbe such that (4.5) and (4.6) hold, respectively. Let us define for x∈Ω and t∈IR ˜ f(x, t) :=                  f(x, u(x)) if t≤u(x), f(x, t) if u(x)≤t≤u(x), f(x, u(x)) if t≥u(x), and consider the problem                  div(y1−2α∇v) = 0 in C, v= 0 on ∂LC, ∂v ∂yα(x, 0) = ˜ f(x, v(x, 0)) on Ω. (4.7) T Observe that by the definition of ˜ fwe have that ZΩ ˜ f(x, u(x, 0))ψ(x, 0)dx≤Ckψ(x, 0)kL2(Ω),(4.8) clave for some positive constant C, for all u∈ Hα(C) and ψ∈ Hα 0(C). Here, we have used that u, u ∈L∞(Ω) and f∈C(Ω ×IR) First, we show that (4.7) possesses at least a solution. Define the operator T:Hα 0(C)7→ (Hα 0(C))0 17 given by (Tu, v) = ZC y1−2α∇u·∇vdxdy −ZΩ ˜ f(x, u(x, 0))v(x, 0)dx, ∀u, v ∈ Hα 0(C). We study some properties of the map T. •T is a bounded map. It is clear, using (4.8), that if ubelongs to a bounded set of Hα 0(C), then T(u) is also bounded in (Hα 0(C))0. •Tis pseudomonotone: given a sequence un* u in Hα 0(C) such that lim sup(Tun, un−u)≤0, we have to show that lim inf(Tun, un−v)≥(Tu, u −v)∀v∈ Hα 0(C).(4.9) claim Observe that from (4.8) we have that ZΩ ˜ f(x, un(x, 0))(un(x, 0) −u(x, 0))dx≤Ckun−ukL2(Ω) →0, hence using that un* u in Hα 0(C) 0≥lim sup(Tun, un−u) = lim sup ZC y1−2α∇un·∇(un−u) = lim sup kunk2 α−kuk2 α. We can conclude that kuk2 α≥lim sup kunk2 α≥lim inf kunk2 α≥ kuk2 α, and then lim kunk2 α=kuk2 α. Consequently, un→uin Hα 0(C) and we get that lim inf(Tun, un−v) = lim inf{(Tun, un−u)+(Tun, u −v)} ≥ (Tu, u −v). •Tis coercive, that is, lim kvkα→∞ (T(v), v) kvkα =∞. It is clear that (T(v), v)≥ kvk2 α−Ckvk2 L2(Ω), 18 whence it follows that Tis coercive. Then, we can conclude from Theorem 2.7 in Chapter 2 of [15] that there exists a solution of (4.7), that is, T(v) = 0. Now, we show that v∈[v, v], and hence vis solution of (4.4). Indeed, define ˜v:= v−v. Note that, for all ψ∈ Hα 0(C), ψ≥0, ZC y1−2α∇˜v·∇ψdxdy ≤ZΩf(x, v(x, 0)) −˜ f(x, v(x, 0))ψ(x, 0)dx. Taking ψ= (v−v)+, we have that ZC y1−2α|∇˜v+|2dxdy ≤0. Then v≤vin Cand in a similar way one can prove that v≤v. Now let us present the proof of the Theorem 4.1. Proof of Theorem 4.1. First consider a positive solution u∈ Vα 0(Ω) of (4.1), and consider v∈ Hα 0solution of (4.2). If λ−cL≤0, then by the maximum principle it follows that v≤0. So, assume that λ−cL>0. Taking in (4.2) ψ= (v−(λ−cL))+,we can show that v≤λ−cLin C. By Lemma 2.4, we have that u∈L∞(Ω); and then, using Lemma 2.5 we arrive that u and vare regular functions. Now, suppose that there exists a positive solution u∈ Vα 0(Ω) of (4.1) for some λ∈R. Then note that uis a positive solution of (3.1) with c(x) substituted by (c(x) + u(x)). Then by Theorem 3.1 λ=λ1[α;c+u]> λ1[α;c]. Now let us prove that λ > λ1[α;c] is sufficient to the existence of a positive solution. Let Ω⊂⊂ Ω0, Ω0an open bounded set, C0= Ω0×(0,+∞) and E∈ Hα 0(C0) the unique positive 19 solution of                  div(y1−2α∇v) = 0 in C0, v= 0 on ∂LC0, ∂v ∂yα(x, 0) = 1 in Ω0. (4.10) e Denote by e(x) := trΩ0E. Observe that from the regularity results, e∈L∞(Ω0) and by Proposition 2.2 we get that E > 0. Note in particular that for ψ∈ Hα 0(C), we can extend it in such a way that ψ∈ Hα 0(C0) and then, it holds ZC y1−2α∇E·∇ψdxdy =ZΩ ψ(x, 0)dx. Let us take v=KE where Kis a positive constant to be chosen. Note that vis a supersolution of (4.2) if and only if for all ψ∈ Hα 0(C), ψ≥0 ZC y1−2α∇E·∇ψdxdy +KZΩ c(x)E(x, 0)ψ(x, 0)dx ≥ZΩ (λE(x, 0)−KE(x, 0)2)ψ(x, 0))dx, this is equivalent to ZΩ ψ(x, 0) Ke(x)2+e(x)(c(x)−λ)+1dx ≥0∀ψ∈ Hα 0(C), ψ ≥0. It suffices that Ke(x)2+e(x)(cL−λ) + 1 ≥0 a.e. in Ω, which is possible by choosing K large enough. For the subsolution, let us take v=Ψ1where  > 0 is a constant to be chosen and Ψ1∈ Hα 0(C) is a positive eigenfunction associated to λ1[α;c]. Then, for all ψ∈ Hα 0,ψ≥0, writing λ1=λ1[α;c] we have ZC y1−2α∇v·∇ψdxdy +ZΩ c(x)v(x, 0)ψ(x, 0)dx =ZΩ λ1ϕ1ψ(x, 0)dx ≤ZΩ ϕ1ψ(x, 0)(λ−ϕ1)dx if and only if ϕ1≤(λ−λ1) in Ω, (4.11) cond1 20 where we have denoted ϕ1=trΩΨ1. Since ϕ1∈ Vα 0(Ω), ϕ1∈L∞(Ω) and ϕ1>0 in Ω, (4.11) is possible and it follows that we have a sub-supersolution pair if  > 0 is small enough. Now Theorem 4.4 implies the existence of solution if λ>λ1[α;c]. To prove the uniqueness of positive solution, all the arguments of [4] (see also [5]) can be adapted to the fractional setting, see Lemma 5.2 in [3] or Proposition 4.2 in [19]. Then, there exists a solution θ[α,λ−c]∈ Vα 0(Ω) of (4.1) if and only if λ>λ1[α;c]. We prove now (4.3). The first inequality follows since ϕ1is a subsolution for all ∈(0, λ −λ1[α;c]]. For the second, note that θ[α,λ−c]≤λ−cL. To compare different solutions of the logistic equation we need the following result: compa Proposition 4.5. Assume that vis a bounded subsolution of (4.2), then trΩv≤θ[α,λ−c]. Proof. Since vis bounded, it is clear that we can choose K > 0 such that KE is supersolution of (4.2) and v≤KE. By uniqueness, we conclude that v(x, 0) ≤θ[α,λ−c]. As a direct consequence of Proposition 4.5, we deduce Corollary 4.6. If λ1≤λ2and c2≤c1in Ω, then θ[α,λ1−c1]≤θ[α,λ2−c2]. Let us give an interesting biological interpretation of this result, comparing with the linear diffusion case. Recall that the classical logistic equation        −∆u+c(x)u=λu −u2in Ω, u= 0 on ∂Ω, (4.12) logiscla possesses a unique positive solution if and only if λ>λ1[1; −c]. Let us compare this result with the obtained for (4.1) in the particular case N= 1, c∈IR and Ω = Br. In Figure 1 we have represented by continuous line G1(r) := λ1[1; c;Br] and by pointed line Gα(r) := λ1[α;c;Br] with c= 0 (a similar representation can be made with c6= 0). Take Λ large (Λ >1). Then, there exist rα< r1such that Λ = G1(r1) = Gα(rα). Then, 21 r 1 rRR 1 r 1 Figure 1: We have represented in continuous line the map G1(r) = λ1[1; c;Br] and by pointed line Gα(r) = λ1[α;c;Br]. We have denoted by λ0=√λ1. a) If r < rα, for (4.1) and (4.12) the species die. b) If r > r1, the species persists in both cases. c) Assume that r∈(rα, r1). Then, the species disappears in the local diffusion and it persists in the fractional diffusion case. Assume now λsmall, (λ < 1). Then, there exist R1< Rαsuch that λ=G1(R1) = Gα(Rα). Moreover, a) If r < R1, for (4.1) and (4.12) the species die. b) If r > Rα, the species persists in both cases. c) Assume that r∈(R1, Rα). Then, the species disappears in the fractional diffusion and it persists in the local diffusion case. Hence, in the case of favourable habitats (abundant resources) there exist domains such that the species with fractional diffusion persists, while the species with linear diffusion 22 dies. In a contrary way, for unfavourable habitats, there exist domains when the opposite occurs. 5 The sub-supersolution method for systems In this section we extend the sub-supersolution method employed in the last section to the system setting. Let us consider                  (−∆)αu=f(x, u, v) in Ω, (−∆)βv=g(x, u, v) in Ω, u=v= 0 on ∂Ω, (5.1) subsupersystem where f, g ∈C0(Ω ×R2) and α, β ∈(0,1). definew Definition 5.1. We say that (u, v)∈ Vα 0(Ω) ×Vβ 0(Ω) is a solution of (5.1) if there exists (U, V )∈ Hα 0(C)×Hβ 0(C)such that trΩU:= u,trΩV:= vand (U, V )is solution of                          div(y1−2α∇U) = div(y1−2β∇V)=0 in C, U=V= 0 on ∂LC, ∂U ∂yα(x, 0) = f(x, U(x, 0), V (x, 0)) on Ω, ∂V ∂yβ(x, 0) = g(x, U(x, 0), V (x, 0)) on Ω, (5.2) subsupersystem2 Definition 5.2. We say that U, U ∈ Hα(C),V , V ∈ Hβ(C)is a pair of sub-supersolution of (5.1) if u:= trΩU, u := trΩU, v := trΩV , v := trΩV∈L∞(Ω), and a) U≤Uand V≤Vin Cand U≤0≤Uand V≤0≤Von ∂LC. b) For all (ψ, φ)∈ Hα 0(C)×Hβ 0(C),ψ, φ ≥0and (u, v)∈[U, U]×[V,V], it holds ZC y1−2α∇U·∇ψdxdy ≤ZΩ f(x, U(x, 0), v(x, 0))ψ(x, 0)dx, ZC y1−2α∇U·∇ψdxdy ≥ZΩ f(x, U(x, 0), v(x, 0))ψ(x, 0)dx, 23 ZC y1−2β∇V·∇φdxdy ≤ZΩ f(x, u(x, 0), V (x, 0))φ(x, 0)dx, ZC y1−2β∇V·∇φdxdy ≥ZΩ f(x, u(x, 0), V (x, 0))φ(x, 0)dx, where [U,U] = {w∈ Hα(C); U≤w≤Uin C} and analogous for [V,V]. Theorem 5.3. Assume that there exists a pair (U, U)-(V,V)of sub-supersolution of (5.2). Then, there exists a solution (U, V )∈ Hα 0(C)×Hβ 0(C)of (5.1) such that U≤U≤U, V ≤V≤Vin C. Moreover, there exists a solution (u, v)∈ Vα 0(Ω) ×Vβ 0(Ω) of (5.1) such that u≤u≤uin Ωand v≤v≤vin Ω. teoremasubsupersistemas Proof. The proof is similar to Theorem 4.4. Define the operators T1and T2by T1(w) =                  uif w≤u, wif u≤w≤u, uif w≥u, T2(z) =                  vif z≤u, zif v≤z≤v, vif z≥v, and the functions ˜ f(x, u, v) = f(x, T1(u), T2(v)),˜g(x, u, v) = g(x, T1(u), T2(v)). Consider the problem                          div(y1−2α∇U) = div(y1−2β∇V) = 0 in C, U=V= 0 on ∂LC, ∂U ∂yα(x, 0) = ˜ f(x, U(x, 0), V (x, 0)) on Ω, ∂V ∂yβ(x, 0) = ˜g(x, U(x, 0), V (x, 0)) on Ω. (5.3) fractionalsystem First, we prove that (5.3) has at least a solution. For that, consider the space H:= Hα 0(C)×Hβ 0(C) with the norm k(u, v)k=kukα+kvkβand the map T:H 7→ (H)0defined by (T(u, v),(w, z)) = ZC y1−2α∇u·∇wdxdy −ZΩ ˜ f(x, u(x, 0))w(x, 0)dx, ZC y1−2β∇v·∇zdxdy −ZΩ ˜g(x, v(x, 0))z(x, 0)dx. 24 Now, we can follow just the arguments of Theorem 4.4 and show that there exists (U, V ) solution of (5.3), that is, T(U, V ) = (0,0). Again, we can prove that (U, V ) is solution of (5.1), for that it suffices to show that (U, V )∈[U,U]×[V,V]. Define ˜ U=U−U, then taking T2(V) in the definition of sub-solution, we get that for all ψ∈ Hα 0,ψ≥0, ZC y1−2α∇˜ U·∇ψdxdy ≤ZΩhf(x, U, T2(V)) −˜ f(x, U, V )iψ(x, 0)dx ≤0. Taking ψ= (U−U)+we get that U≤U. The same argument can be used to the other inequalities. 6 Application to the Lotka-Volterra systems In this section we apply the above results to system (1.1), or equivalently, to the system                          div(y1−2α∇U) = div(y1−2β∇V) = 0 in C, U=V= 0 on ∂LC, ∂U ∂yα(x, 0) = U(x, 0)(λ−U(x, 0) −bV (x, 0)) in Ω, ∂V ∂yβ(x, 0) = V(x, 0)(µ−V(x, 0) −cU(x, 0)) in Ω, (6.1) uno2 First, we deduce some bounds of the solutions of (1.1). cotassol Proposition 6.1. a) Assume that b, c > 0and let (u, v)a positive solution of (1.1). Then, u≤θ[α,λ], v ≤θ[β,µ]. b) Assume that b > 0and c < 0and let (u, v)a positive solution of (1.1). Then, u≤θ[α,λ−bθ[β,µ]]≤θ[α,λ], θ[β,µ]≤v≤θ[β,µ−cθ[α,λ]]. c) Assume that b, c < 0and let (u, v)a positive solution of (1.1). Then, θ[α,λ]≤u, θ[β,µ]≤v. Proof. a) Assume that b, c > 0 and and let (u, v) a positive solution of (1.1), that is, (u, v)=(trΩU, trΩV), being (U, V ) solution of (6.1). With a similar reasoning to the 25