The Lotka-Volterra models with non-local reaction terms

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Manuscript submitted to AIMS Journal THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS Willian Cintra Universidade de Bras´ılia, Departamento de Matem´atica Campus Darcy Ribeiro, 01 CEP 70910-900, Bras´ılia, DF, Brasil M´ onica Molina-Becerra Universidad de Sevilla, Escuela Polit´ecnica Superior Dpto. de Matem´atica Aplicada II, Calle Virgen de ´ Africa, 7, 41011 Sevilla, Spain Antonio Su´ arez∗ Universidad de Sevilla, Fac. de Matem´aticas Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico and IMUS, Calle Tarfia s/n, 41013 Sevilla, Spain Abstract. In this paper we consider a diffusive Lotka-Volterra system including nonlocal terms in the reaction functions. We analyze the main types of interactions between species: competition, predator-prey and cooperation. We provide existence and non-existence of positive solutions results. For that, we employ mainly bifurcation method and a priori bounds. 1. Introduction. Since the paper of Furter and Grinfeld [8] (1989) is admitted that there is no real justification for assuming that the interactions between species are local. Hence, they proposed to include non-local terms in space to consider the interactions between species giving more reasonable and more realistic models, see also Britton [1] where a population model is proposed including a intraspecific competition term that depends not simply on the population density at that point but on the average population density near the point. In this paper we consider two species inhabiting in Ω, a regular and bounded domain in IRN, whose population densities are denoted by u(x) and v(x). Specifically, we deal with the following system            −∆u=uλ−u−ZΩ a(x)u(x)dx +b1v+b2ZΩ b(x)v(x)dxin Ω, −∆v=vµ−v−ZΩ d(x)v(x)dx +c1u+c2ZΩ c(x)u(x)dxin Ω, u=v= 0 on ∂Ω, (1) where λ, µ ∈IR denote the growth rates of the species uand v, respectively, b1and c1 are the local interaction coefficients, both negative or positive imply a competition or 2020 Mathematics Subject Classification. 35J57, 35B45, 35B32. Key words and phrases. Lotka-Volterra model, non-local terms, coexistence states. ∗Corresponding author: Antonio Su´arez. 1 2 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ symbiosis interaction, respectively, and one of them positive and the other negative represent a prey-predator interaction. Observe that in (1) we have included two types of non-local terms. The first one appears when one species is not present, that is, in the absence of one species, the other follows a non-local logistic equation of the form    −∆w=wγ−w−ZΩ e(x)w(x)dxin Ω, w= 0 on ∂Ω, (2) where γ∈IR and the term −RΩe(x)w(x)dx represents the intraspecific nonlocal competition. Hence, in (1) the species take into account two negative crowding effects caused by the intraspecific competition, one local, −uand −v, and other nonlocal, −RΩa(x)u(x)dx and −RΩd(x)v(x)dx, with positive weights a, d ∈C(Ω), a, d ≥0, a, d 6≡ 0. The second non-local term is related with the interspecific non-local interactions. Indeed, b2and c2stand for the strength of nonlocal interspecific coefficients, being the nonlocal terms RΩb(x)v(x)dx and RΩc(x)u(x)dx with b, c ∈C(Ω), b, c ≥0, b, c 6≡ 0. Of course, due to the biological interpretation, b2and c2have the same signs than b1and c1, respectively, that is, b1·b2≥0, c1·c2≥0 and (b1, b2)6= (0,0) 6= (c1, c2). When b2=c2= 0 and the intraspecific nonlocal competition is not present, that is a≡d≡0, then (1) is the classical Lotka-Volterra system which has been extensively analyzed, see for instance the monograph [2]. System (1) has been studied in [12] in the competition case (that is bi<0, ci<0, i= 1,2). In this work, the nonlocal competition term is determined by a diffusion kernel function to model the movement pattern of the biological species. This allows to convert the nonlocal problem (1) into an equivalent local system with three variables. In [14] a very general reaction-diffusion system depending on the spatial average is studied. It is shown that for the two-species model, Hopf bifurcation from the constant equilibrium can occur. A nonlocal cooperative Lotka-Volterra model and a nonlocal Rosenzweig-MacArthur predator-prey model are used to demonstrate the bifurcation of spatially non-homogeneous patterns. In [17] the local terms do not appear and it is assumed that λ=µ. Under some conditions on the coefficients and in the competition case, the authors obtain the existence of positive solutions using bifurcation arguments. Different nonlocal reactions terms appear in [3], [13], [16] and references therein. In this paper, the main goal is to give existence and non-existence of positive solution results. Specifically, we are interested in determining coexistence regions, that is, subsets D ⊂ IR2such that if (λ, µ)∈ D then (1) possesses at least a coexistence state. A coexistence state is a solution (u, v) of (1) such that u(x), v(x)> 0 for all x∈Ω. On the contrary, we also provide non-existence regions N ⊂ IR2, namely if (λ, µ)∈ N then (1) does not admit positive solutions. Surely, these regions depend on the type of interactions between the species. To obtain these results, the nonlocal terms cause difficulties, mainly because classical comparison arguments do not work in general. To overcome this difficulty we have used the bifurcation arguments. In order to do it, we study in details the semitrivial solutions, (u, 0) and (0, v), and then we prove the existence of a continuum of coexistence states THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 3 Figure 1. Coexistence (D) and non-existence (N) regions of (1) in the competition case. emanating from them. Again, the behaviour of this continuum depends strongly on the interactions of the species. Now, we state our main results. For that, we need to introduce some notation. Denote by λ1the principal eigenvalue of −∆ under homogenous Dirichlet boundary conditions. Moreover, there exist two continuous functions F, G : [λ1,∞)→IR, which will be provided later, and that satisfy the following properties: 1. F(λ) is increasing (resp. decreasing) if c1, c2≤0 (resp. c1, c2≥0), (c1, c2)6= (0,0) and lim λ→+∞F(λ)=+∞(resp. −∞). 2. G(µ) is increasing (resp. decreasing) if b1, b2≤0 (resp. b1, b2≥0), (b1, b2)6= (0,0) and lim µ→+∞G(µ)=+∞(resp. −∞). Our first result is concerning to the competition case. We show a competitive exclusion principle: fixed the growth rate of a species, the two species can not coexist if the growth rate of the other species is small or large. Also, we show a coexistence result (see Figure 1). Theorem 1.1. (Competition) Assume that bi, ci≤0with i= 1,2,(b1, b2)6= (0,0) 6= (c1, c2). 1. (1) does not admit coexistence states if λ≤λ1or µ≤λ1. 2. Fix λ>λ1(resp. µ > λ1), there exists µc(λ)>0(resp. λc(µ)>0) such that (1) does not admit coexistence states if µ>µc(λ)(resp. λ>λc(µ)). 3. (1) possesses at least a coexistence state if (µ−F(λ))(λ−G(µ)) >0. With respect to the prey-predator case, we have (Figure 2): 4 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ Figure 2. Coexistence (D) and non-existence (N) regions of (1) in the predator-prey case. Theorem 1.2. (Predator-prey) Assume that ci≤0≤biwith i= 1,2and (b1, b2)6= (0,0) 6= (c1, c2). 1. (1) does not admit coexistence states if µ≤λ1. 2. Fix µ > λ1, there exist λ1p(µ)<0< λ2p(µ)such that (1) does not admit coexistence states if λ<λ1p(µ)or λ>λ2p(µ). 3. (1) possesses at least a coexistence state if µ > F(λ)and λ>G(µ). The cooperation case is more envolved. We assume in this case that b1c1<1, that is a weak local cooperation. Now, we can find two different regimes. The first one assumes that all the cooperation coefficients are small. In the second one, the nonlocal interspecific cooperation coefficients can be large. Theorem 1.3. (Symbiosis) Assume bi, ci≥0for i= 1,2,(b1, b2)6= 0 6= (c1, c2). 1. (Weak nonlocal cooperation) Assume (Hw) (b1+b2kbk1)(c1+c2kck1)<1. (a) (1) does not admit coexistence states if λ≤λ1and µ≤λ1. (b) Fix λ>λ1(resp. µ>λ1), there exists µws(λ)∈IR (resp. λws(µ)∈IR) such that (1) does not admit coexistence states if µ<µws(λ)(resp. λ < λws(µ)). (c) (1) possesses at least a coexistence state if µ > F(λ)and λ>G(µ). 2. (Strong nonlocal cooperation) Assume that b1c1<1,b(x), c(x)>0for all x∈ Ωand b2> Cb1, for some C > 0. There exists a continuous and nonincreasing function H: IR+7→ IR+such that if (Hs)c2> H(b2), then, THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 5 Figure 3. Coexistence (D) and non-existence (N) regions of (1) in the symbiosis case: weak (on left) and strong (on right) (a) Fix λ∈IR (resp. µ∈IR), there exists µss(λ)∈IR (resp. λss(µ)∈IR) such that (1) does not admit coexistence states if µ > µss(λ)(resp. λ > λss(µ)). (b) (1) possesses at least a coexistence state if µ < F(λ)or λ<G(µ).(3) Let us compare our results with the classical local Lotka-Volterra model. The obtained results in the competition, predator-prey and weak cooperation cases are rather similar to the those in the Lotka-Volterra model, although mainly due to loss of monotony results, our techniques differ from those used to obtain the results in the local case. However, the strong cooperation case is completely different to the local case. Indeed, in the local case when the cooperation is strong (b1c1>1) it is necessary to impose some restrictions in the dimension of the space, specifically, N < 5, to obtain existence of coexistence states for λand µverifying (3), see [6] and [11]. In this case, when the cooperation is strong in the non-local term (b2or c2large) we obtain existence of coexistence states for λand µverifying (3) without any restriction in the spatial dimension. An outline of the paper is as follows: In Section 2 we present some results useful throughout the paper. Section 3 is devoted to nonlocal logistic equation. In Section 4 we give some general bifurcations results. These results are applied in the last section to prove the main results. 2. Preliminaries. In this section we collect some results which will be used along the paper. We start by fixing some notations. Given m∈L∞(Ω) we denote by λ1[−∆ + m] the principal eigenvalue of the problem −∆u+m(x)u=λu in Ω, u= 0 on ∂Ω. (4) It is well-known that the map m∈L∞(Ω) 7→ λ1[−∆ + m]∈IR is continuous and increasing. (5) We denote by λ1:= λ1[−∆] and ϕ1its principal positive eigenfunction associated such that kϕ1k∞= 1. 6 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ Finally, given g∈C(Ω) we denote by gL:= min x∈Ω g(x), gM:= max x∈Ω g(x). We also need to study the classical local logistic equation −∆w=w(γ−w) in Ω, w= 0 on ∂Ω, (6) where γ∈R. In what follows, we summarize some well-known results concerning to (6), see for instance [2] and [6]. Proposition 1. There exists a classical positive solution of (6)if and only if γ > λ1. In such case, it is unique and we denote it by ωγ. Moreover: 1. Defining ωλ1≡0, the map γ∈[λ1,+∞)7→ ωγ∈C2 0(Ω) is continuous and increasing. 2. It holds that (γ−λ1)ϕ1≤ωγ≤γin Ω.(7) 3. If wis a positive supersolution of (6), then (γ−λ1)ϕ1≤ωγ≤win Ω. 4. If wis a positive subsolution of (6), then w≤ωγ≤γin Ω. 3. The nonlocal equation of one species. In this Section, we obtain results for the nonlocal logistic equation, that is, what happens when one of the species is not present, specifically of the equation    −∆w=wγ−w−ZΩ e(x)w(x)dxin Ω, w= 0 on ∂Ω, (8) where γ∈IR and e∈C(Ω), e≥0, e6≡ 0 in Ω. Before giving the main results, we will study an auxiliary linear non-local problem. Lemma 3.1. Let m∈L∞(Ω),n∈L∞(Ω), such that λ1[−∆ + m]>0and n≥0 in Ω. Thus, for each f∈L2(Ω), there exists a unique solution u∈W2,2(Ω) of    −∆u+m(x)u+n(x)ZΩ e(x)u(x)dx =f(x)in Ω, u= 0 on ∂Ω. (9) Moreover, if f∈Lp(Ω),p > 1, then u∈W2,p(Ω). Proof. Since λ1[−∆ + m]>0, by the characterization of the maximum principle, there exists a unique positive solution h∈W2,∞(Ω) of −∆h+m(x)h=e(x) in Ω, h= 0 on ∂Ω.(10) THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 7 Similarly, let w∈W2,2(Ω) be the unique solution of            −∆w+m(x)w=f(x)−ZΩ f(x)h(x)dx 1 + ZΩ h(x)n(x)dx n(x) in Ω, w= 0 on ∂Ω. (11) We claim that wis the unique solution of (9). To prove that, it is sufficient to show ZΩ e(x)w(x)dx =ZΩ f(x)h(x)dx 1 + ZΩ h(x)n(x)dx . Taking has test function in (11) yields ZΩ e(x)w(x)dx +ZΩ f(x)h(x)dx 1 + ZΩ h(x)n(x)dx ZΩ n(x)h(x)dx =ZΩ f(x)h(x)dx, which implies the result. Analogously, if uis a solution of (9), taking has test function in (9) shows that ZΩ e(x)u(x)dx =ZΩ f(x)h(x)dx 1 + ZΩ h(x)n(x)dx . Then, substituting into (9) we conclude that uis the unique solution of (11). Furthermore, if f∈Lp(Ω), by elliptic regularity, the unique solution of (10) belongs to W2,p(Ω). Along the paper, given a function e∈C(Ω), e≥0, e6≡ 0 in Ω, we denote by Ae:= ZΩ e(x)ϕ1(x)dx 1 + ZΩ e(x)ϕ1(x)dx . Now, we are ready to state and prove the main result with respect to (8). Theorem 3.2. The nonlocal logistic equation (8)possesses a classical positive solution if, and only if, γ > λ1. Moreover, it is unique if it exists, and it will be denoted by θγ. In addition, defining θλ1≡0, the map γ∈[λ1,+∞)7→ θγ∈C2 0(Ω) is continuous, derivable and increasing. Furthermore, the following inequalities hold: 1. θγ≤ωγ≤γin Ω. 2. (γ−λ1)Ae≤ZΩ e(x)θγ(x)dx ≤γkek1 1 + kek1 .(12) 3. γ 1 + kek1 −λ1ϕ1≤θγ≤γ(1 −Ae) + λ1Aein Ω. 8 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ Proof. It is known the result about existence and uniqueness of positive solution of (8), see [5]. We will show that the map γ∈[λ1,+∞)7→ θγis increasing. Fix λ∈[λ1,+∞) and let µ > λ, we claim that λ−ZΩ e(x)θλ(x)dx < µ −ZΩ e(x)θµ(x)dx. (13) Suppose the contrary: λ−ZΩ e(x)θλ(x)dx ≥µ−ZΩ e(x)θµ(x)dx. (14) Denoting Rδ:= δ−RΩe(x)θδ(x)dx, we have that θλand θµare solutions of (6) with γ=Rλand γ=Rµ, respectively, that is θλ=ωRλand θµ=ωRµ. Then, by (14), θλ≥θµ, and therefore ZΩ e(x)θλ(x)dx ≥ZΩ e(x)θµ(x)dx. We get λ−ZΩ e(x)θλ(x)dx ≤λ−ZΩ e(x)θµ(x)dx < µ −ZΩ e(x)θµ(x)dx, which is an absurd. By (13), we have Rλ< Rµand by the monotonicity of the solution of (6) we obtain that θλ≤θµ. And, thus, we get the map, λ∈[λ1,+∞)7→ θλis increasing. Using this fact, and the uniqueness of positive solution of (6), the continuity follows. Now, we prove the differentiability of the map λ7→ θλ∈C2 0(Ω) using the Implicit Function Theorem. Define the map H: IR ×C2 0(Ω) 7→ C(Ω) given by H(γ, w) := −∆w−wγ−w−ZΩ e(x)w(x)dx. It is clear that H(γ, θγ) = 0 and that DwH(γ, θγ)ξ=−∆ξ−γξ + 2θγξ+ξZΩ e(x)θγ(x)dx +θγZΩ e(x)ξ(x)dx. We denote L:= −∆−γ+ 2θγ(x) + ZΩ e(x)θγ(x)dx. Since m(x) := −γ+ 2θγ+RΩe(x)θγ(x)dx ∈L∞(Ω), the principal eigenvalue of L=−∆ + m(x) is well defined. We claim that λ1[L]>0.(15) Indeed, in view of characterization of maximum principle (see, for instance, [9, Theorem 7.5.2]), to prove (15), it is sufficient to show that there exists a strict positive supersolution of the operator L. Indeed, since θγis a positive solution of (8), it satisfies Lθγ=−∆θγ−γθγ+ 2θ2 γ+θγZΩ e(x)θγ(x)dx =θ2 γ>0 in Ω, showing the claim. Therefore, we can apply Lemma 3.1 to conclude that DwH(γ, θγ) is an isomorphism. This concludes the proof of the differentiability. Notice that θγis a subsolution of (6), then by Proposition 1 θγ≤ωγ≤γin Ω. THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 15 Figure 4. Bifurcation diagram (1) in the competition case Proof. The proof of this result proceeds by contradiction Observe that if uis a positive solution, then λ=λ1−∆ + u+ZΩ au −b1v−b2ZΩ bv> λ1−∆−b1v−b2ZΩ bv.(25) By Proposition 2, we get that v≥R2(λ, µ)ϕ1,ZΩ bv ≥R2(λ, µ)kbϕ1k1, and then by (25) λ>λ1[−∆−b1R2(λ, µ)ϕ1]−b2R2(λ, µ)kbϕ1k1, a contradiction for µlarge. We are ready to prove Theorem 1.1 (see Figure 4). Proof of Theorem 1.1.Fix λ>λ1. Regarding µas bifurcation parameter, by Theorem 4.2 from the semitrivial solution (θλ,0) emanates a continuum C+ (µ,u,0) of coexistence states of (1) at µ=F(λ)> λ1. Thanks to Corollary 1and Proposition 3, (1) does not admit coexistence states neither µ≤λ1nor µ > µc(λ). Hence, ProjIR(C+ (µ,u,0))⊂(λ1, µc(λ)), where given (µ, u, v)∈C+ (µ,u,0) we have denoted ProjIR(µ, u, v) = µ. On the other hand, thanks to Proposition 2we have that kuk∞,kvk∞≤Cfor all (µ, u, v)∈C+ (µ,u,0). By elliptic regularity, we get that kukX,kvkX≤C, and hence C+ (µ,u,0) is bounded. Hence, by Theorem 4.2 there exists a positive solution (µ∗, θµ∗) of (17) such that λ=G(µ∗) and (µ∗,0, θµ∗)∈C+ (µ,u,0). Hence, we obtain that (F(λ), G−1(λ)) ⊂ProjIR(C+ (µ,u,0)). This concludes the proof. 16 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ 5.2. Prey-predator. Proposition 4. Assume that ci≤0≤biwith i= 1,2and (b1, b2)6= (0,0) 6= (c1, c2). If (u, v)is a coexistence state of (1), then R3(λ, µ)ϕ1≤u≤λ+µ(b1+b2kbk1)and v≤ωµ≤µ, where R3(λ, µ) = 1 1 + kak1 (λ−µ(b1+b2kbk1)kak1)−λ1. Proof. It is clear that v≤µby the same argument as the above Proposition. Thus −∆u≤uλ−u−ZΩ a(x)u(x)dx +b1µ+b2kbk1µ≤u(λ−u+µ(b1+b2kbk1)). Then, uis a subsolution of (6). By Proposition 1we have that u≤λ+µ(b1+b2kbk1). Moreover, since bi≥0, if uM= max x∈Ω u(x) we conclude that u(x)≤uM≤λ−ZΩ a(x)u(x)dx +µ(b1+b2kbk1), and then, ZΩ a(x)u(x)dx ≤kak1 1 + kak1 (λ+µ(b1+b2kbk1)). Hence, −∆u≥uλ−u−ZΩ a(x)u(x)dx≥uλ 1 + kak1 −kak1 1 + kak1 (µ(b1+b2kbk1)) −u and then u≥R3(λ, µ)ϕ1. Again, we can conclude that Corollary 2. Assume that ci≤0≤biwith i= 1,2and (b1, b2)6= (0,0) 6= (c1, c2). If λ+µ(b1+b2kbk1)≤0or µ≤λ1, then (1)does not admit coexistence states. Now, we show a non-existence result. Proposition 5. Assume that ci≤0≤bifor i= 1,2and (b1, b2)6= (0,0) 6= (c1, c2). Fix µ>λ1, then there exists λp(µ)such that (1)does not admit coexistence states for λ>λp(µ). Proof. Suppose that (u, v) is a coexistence state of (1). Using Proposition 4we get −∆v=vµ−v−ZΩ d(x)v(x)dx +c1u+c2ZΩ cu ≤v(µ+c1R3(λ, µ)ϕ1+c2R3(λ, µ)kϕ1ck1). Therefore, multiplying by ϕ1and integrating, we get 0≤(µ−λ1+c2R3(λ, µ)kϕ1ck1)ZΩ vϕ1+c1R3(λ, µ)ZΩ vϕ2 1, a contradiction for λlarge. THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 17 Figure 5. Bifurcation diagram (1) in the prey-predator case Proof of Theorem 1.2.Fix µ>λ1and consider λas a parameter (see Figure 5). By Theorem 4.2 from the semitrivial solution (0, θµ) emanates a continuum C+ (λ,0,v) of coexistence states of (1) at λ=G(µ). Thanks to Corollary 2and Proposition 5, (1) does not admit positive solution neither λ<λ1p(µ) nor λ>λ2p(µ). On the other hand, thanks to the a priori bounds of positive solutions, C+ (λ,0,v)is bounded. Hence, we conclude that (G(µ), F−1(µ)) ⊂ProjIR(C+ (λ,0,v)). The proof is completed. 5.3. Cooperation. We assume that a weak local cooperation occurs, that is b1c1< 1. We are going to obtain two different regimes of cooperation. In the first one, all the cooperation coefficients are small. Proposition 6. (Weak nonlocal cooperation) Assume bi, ci≥0for i= 1,2, (b1, b2)6= (0,0) 6= (c1, c2)and (Hw) (b1+b2kbk1)(c1+c2kck1)<1. Then, if (u, v)is a coexistence state of (1), we have u≤λ+µ(b1+b2kbk1) 1−(b1+b2kbk1)(c1+c2kck1), v ≤µ+λ(c1+c2kck1) 1−(b1+b2kbk1)(c1+c2kck1). Proof. Let (u, v) be a coexistence state of (1), then there exists xM∈Ω such that u(xM) = max Ω u(x) := uM. Since −∆u(xM)≥0, we have that uM+ZΩ a(x)u(x)dx ≤λ+b1v(xM) + b2ZΩ b(x)v(x)dx ≤λ+b1vM+b2ZΩ b(x)v(x)dx ≤λ+vM(b1+b2kbk1),(26) 18 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ with vM= max Ω v(x). Analogously, vM+ZΩ d(x)v(x)dx ≤µ+c1uM+c2ZΩ c(x)u(x)dx ≤µ+uM(c1+c2kck1).(27) From these inequalities, one can derive uM≤λ+vM(b1+b2kbk1), vM≤µ+uM(c1+c2kck1),(28) and using (Hw), we obtain the result. As consequence, we obtain Corollary 3. Assume that bi, ci≥0for i= 1,2and (b1, b2)6= (0,0) 6= (c1, c2)and (Hw). Fix λ∈IR (resp. µ∈IR), then there exists µws(λ)(resp. λws(µ)) such that (1)does not admit coexistence state for µ < µws(λ)(resp. λ<λws(µ)). In the following result, we obtain a priori bounds even for b2and/or c2large. To be precise, we need to introduce the function H(x) :=      (aM+C)(dM+C)−c1(b1aMdM+xCbL) cL(xbL(1 −b1c1) + b1C)x∈hb1dM bL,dM+C c1bLi, c1aM cL x > dM+C c1bL, where C=1−b1c1 kϕ1k1 . Since HdM+C c1bL=c1aM cL , His continuous. Moreover, it is evident that His decreasing in hb1dM bL,dM+C c1bLiand Hb1dM bL=aM+C b1cL . In Figure 6we have represented in the (b2, c2)-plane the regiones defined by (Hw) and (Hs). Proposition 7. (Strong nonlocal cooperation) Assume bi, ci>0with b1c1<1, b(x), c(x)>0for x∈Ωand (Hs)b2> b1dM/bL, c2> H(b2). If (u, v)is a coexistence state of (1), we have kuk∞≤C1,kvk∞≤C2. Moreover, fix λ>λ1(resp. µ>λ1), there exists µss(λ)(resp. λss(µ)) such that for µ > µss(λ)(resp. λ>λss(µ)) system (1)does not admit coexistence states. Proof. We consider the auxiliary problem:    −∆u=u(γ1−u+b1v) in Ω, −∆v=v(γ2−v+c1u) in Ω, u=v= 0 on ∂Ω. (29) Since b1c1<1, there exists a unique positive solution (ˆuγ1,γ2,ˆvγ1,γ2) of (29) (see Theorem 5.1 in [7]). THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 19 Figure 6. Regions in the (b2, c2)-plane defined by (Hw) and (Hs). Now, we intend to find a lower bound of solution of (29). For this we consider, (u, v)=(ε1ϕ1, ε2ϕ1), ε1and ε2are positive constants such that: ε1−b1ε2=γ1−λ1, ε2−c1ε1=γ2−λ1.(30) That is,        ε1=b1(γ2−λ1)+(γ1−λ1) 1−b1c1 , ε2=c1(γ1−λ1)+(γ2−λ1) 1−b1c1 . (31) Observe that, thanks to (30) and kϕ1k∞= 1, −∆u=ε1λ1ϕ1≤ε1ϕ1(γ1−ε1ϕ1+b1ε2ϕ1) = u(γ1−u+b1v), −∆v=ε2λ1ϕ1≤ε2ϕ1(γ2−ε2ϕ1+c1ε1ϕ1) = v(γ2−v+c1u). Then, (u, v) is a subsolution of the problem (29) with ε1, ε2defined by (30). Therefore, ε1ϕ1≤ˆuγ1,γ2, ε2ϕ1≤ˆvγ1,γ2.(32) 20 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ Now, let (u, v) be a coexistence state of (1), then (u, v) is the solution of (29) with γ1=λ−ZΩ a(x)u(x)dx +b2ZΩ b(x)v(x)dx, γ2=µ−ZΩ d(x)v(x)dx +c2ZΩ c(x)u(x)dx. By (32) and (31) λ−λ1−ZΩ au dx +b2ZΩ bv dx +b1µ−λ1−ZΩ dv dx +c2ZΩ cu dx 1−b1c1 ϕ1≤u µ−λ1−ZΩ dv dx +c2ZΩ cu dx +c1λ−λ1−ZΩ au dx +b2ZΩ bv dx 1−b1c1 ϕ1≤v then, by integrating A1ZΩ u dx +B1ZΩ v dx ≤R1 B2ZΩ v dx +A2ZΩ u dx ≤R2 (33) with A1=c2b1cL−aM−C, B1=b2bL−b1dM, A2=c2cL−c1aM, B2=b2c1bL−dM−C, R1=λ1−λ+b1(λ1−µ), R2=λ1−µ+c1(λ1−λ). First, observe that since b1c1<1, we have that b1<dM+C c1dM . 1. Assume that b2∈(b1dM bL,dM+C c1bL]. Then, B2≤0< B1. We distinguish two cases: (a) Assume that c2∈(c1aM cL,aM+C b1cL]. In this case A1≤0< A2. Then, assuming that D:= B1A2−A1B2>0,(34) we can conclude that ZΩ u dx ≤T1 D,ZΩ v dx ≤T2 D,(35) where T1=R2B1−B2R1, T2=R1A2−R2A1. Observe that (34) is equivalent to c2> H(b2). (b) Assume that c2∈(aM+C b1cL,+∞). In this case, A1, A2>0. Then, we can easily conclude that ZΩ u dx ≤R1 A1 ,ZΩ v dx ≤R1 B1 . 2. Assume that b2>dM+C c1bL. Then, B1, B2>0. THE LOTKA-VOLTERRA MODELS WITH NON-LOCAL REACTION TERMS 21 Figure 7. Bifurcation diagram (1) in the symbiosis case (a) Assume that c2∈(c1aM cL,aM+C b1cL]. In this case A1≤0< A2, and from the second equation we deduce that ZΩ u dx ≤R2 A2 ,ZΩ v dx ≤R2 B2 . (b) Assume that c2>aM+C b1cL. In this case A1, A2>0, again the result is deduced easily, in fact ZΩ u dx ≤min R1 A1 ,R2 A2,ZΩ v dx ≤min R1 B1 ,R2 B2. Hence, in all the cases, we have obtained that ZΩ u≤E1,ZΩ v≤E2 and as consequence    −∆u≤u(λ+b2bME2−u+b1v) in Ω, −∆v≤v(µ+c2cME1−v+c1u) in Ω, u=v= 0 on ∂Ω, (36) whence we deduce thanks to b1c1<1 that kuk∞≤C1,kvk∞≤C2. With respect to the non-existence result, observe that, fixed λ∈IR, R1<0, R2<0, T1<0 and T2<0 for µlarge. This concludes the proof. Proof of Theorem 1.3.a) Assume that bi, ci≥0 for i= 1,2, (b1, b2)6= (0,0) 6= (c1, c2) and (Hw). Fix µ>λ1(see Figure 7a)). By Theorem 4.2 from the semitrivial solution (0, θµ) emanates a continuum C+ (λ,0,v)of coexistence states of (1) at λ=G(µ). Since F(λ)< λ1for λ > λ1, it is not possible the existence of λ∗> λ1such that µ=F(λ∗). Hence by Theorem 4.2,Cis unbounded. By Proposition 6, any coexistence state (u, v)∈C+ (λ,0,v)is bounded when λbelongs to a bounded subset of IR, and then we can conclude that (G(µ),+∞)⊂ProjIR(C+ (λ,0,v)). 22 WILLIAN CINTRA, M ´ ONICA MOLINA-BECERRA AND ANTONIO SU´ AREZ In a similar way, we can show that for λ>λ1fixed we get (F(λ),+∞)⊂ProjIR(C+ (µ,u,0)). b) Assume now that (Hs). Fix µ < λ1(see Figure 7b)). Then, by Theorem 4.3 from the semitrivial solution (θλ,0) bifurcates at λ=F−1(µ) an unbounded continuum C+ (λ,u,0) of coexistence states of (1). Thanks to Proposition 7, we have a priori bounds of the solutions and non-existence of coexistence states for λlarge, we conclude (−∞, F−1(µ)) ⊂ProjIR(C+ (λ,u,0)). In a similar way, for λ<λ1we show that (−∞, G−1(λ)) ⊂ProjIR(C+ (µ,0,v)). Acknowledgment. W. Cintra was partially supported by the project CAPES– PrInt no88887.466484/2019-00; M. Molina-Becerra and A. Su´arez were partially supported by PGC 2018-0983.08-B-I00 (MCI/AEI/FEDER, UE) and by the Consejer´ıa de Econom´ıa, Conocimiento, Empresas y Universidad de la Junta de Andaluc´ıa (US-1380740, P20-01160 and US-1381261, P20-00592, respectively). MMB was partially supported by the Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa (TIC-0130). REFERENCES [1] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66. [2] R. S. Cantrell, C. Cosner, “Spatial Ecology Via Reaction-Diffusion Equations”, Wiley Series in Mathematical and Computational Biology. 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