The sub-supertrajectory method. Application to the nonautonomous competition Lotka-Volterra model

Bol. So . Esp. Mat. Apl. n o 51(2010), 9199 THE SUB-SUPERTRAJECTORY METHOD. APPLICATION TO THE NONAUTONOMOUS COMPETITION LOTKA-VOLTERRA MODEL J.A. LANGA ∗ , A. RODRíGUEZ-BERNAL ‡ AND A. SUÁREZ † ∗† Dpto. EDAN, University of Sevilla Aptdo. 1160, 41080 SEVILLA ‡ Departamento de Matemátia Apliada, Universidad Complutense de Madrid, 28040, Instituto de Cienias Matemátias CSIC-UAM-UC3M-UCM, MADRID langaus.es arobermat.um.es suarezus.es Abstrat In this pap er we study in detail the pullbak and forwards attrations to non-autonomous omp etition Lotka-Volterra system. In partiular, under some onditions on the parameters, we prove the existene of a unique non-degenerate global solution for these mo dels, whih attrats any other omplete b ounded tra jetory. For that we present the sub-sup ertra jetory to ol as a generalization of the now lassial subsup ersolution method. Key words: Sub-supertrajetory method, Lotka-Volterra ompetition system, attrating omplete trajetories. AMS sub jet lassiations: 35B40, 35K55, 92D25, 37L05. 1 Intro dution In this pap er we ollet some results from [6℄ and [7℄ to analyze the asymptoti dynamis of the following non-autonomous Lotka-Volterra omp etition mo del        ut−∆u=u(λ(t, x)−a(t, x)u−b(t, x)v)x∈Ω, t > s vt−∆v=v(µ(t, x)−c(t, x)u−d(t, x)v)x∈Ω, t > s u=v= 0 x∈∂Ω, t > s u(s) = us, v(s) = vs. (1) ∗ Partly supp orted by grants MTM2008-0088, HF2008-0039 and PHB2006-003PC. † Partly supp orted by grant MTM2006-07932. ‡ Partly supp orted by grants MTM2006-08262, CCG07-UCM/ESP-2393 UCM-CAM Grup o de Investigaión CADEDIF and PHB2006-003PC. 91 92 J.A. Langa, A. Rodríguez-Bernal, A. Suárez Here, u and v represent the population densities of two speies within a habitat Ω, a b ounded and smo oth domain in IRN , N≥1 , whih omp ete in the habitat. λ, µ are the growth rates of the sp eies, b, c are the interation rates b etween the sp eies, a, d desrib e the limiting eets of rowding in eah population. We are assuming that Ω is fully surrounded by inhospitable areas, sine the population densities are sub jet to homogeneous Dirihlet boundary onditions. us, vs are regular and positive funtions whih implies that the solution of (1) satises u, v ≥0 . In this work we are interested in determining the asymptoti b ehaviour of solutions of the system (1). This is a very ompliated task, and only partial results are known. For example in the autonomous ase (all the oeients in (1) are onstants) and denoting by Λ0 the prinipal eigenvalue asso iated to −∆ , then if λ or µ≤Λ0 , then one of the two sp eies (or both of them) will b e driven to extintion. However, there exist two inreasing maps F, G : [Λ0,∞)7→ IR suh that if λ > G(µ) and µ > F(λ), then (1) is p ermanent and moreover there exists a positive equilibrium solution (see Cantrell et al. [2℄ and Lóp ez-Gómez [9℄). When non-autonomous terms are allowed in the equations, this is usually done under the assumption of p erio diity, quasip erio diity or almost p erio diity, and in this ase similar results an be obtained to those for autonomous equations (see Hess [4℄, Hetzer and Shen [5℄ and referenes there in). Cantrell and Cosner [1℄ assume general non-autonomous terms that are b ounded by p erio di funtions, and using a omparison metho d give onditions on λ and µ that guarantee that (1) is p ermanent. In [6℄ we show that, under a smallness ondition on the oupling oeients bc , if there exists a b ounded and b ounded away from zero omplete tra jetories of (1), it is the unique suh tra jetory, and it also desrib es the unique pullbak and forwards attrating for (1), i.e. (u∗, v∗) is a b ounded tra jetory suh that, for any s∈IR and for any p ositive solution (u(t, s), v(t, s)) of (1) dened for t > s , one has (u(t, s)−u∗(t), v(t, s)−v∗(t)) →(0,0) as t→ ∞, or s→ −∞. (2) In this work (see [7℄) we show that this tra jetory really exists. To this end we introdue the sub-sup ertra jetory method as a to ol to get existene of intermediate omplete tra jetories asso iated to (1). Note that our onstrution is indep endent of whether or not (1) has monotoniity prop erties. Note also that the usual way in previous works (for instane [6℄, [11℄) to get existene of omplete tra jetories assoiated to a partiular system is by means of the pullbak attrator. The sub-sup ertra jetory metho d adopts a dierent and, in this ase, more fruitful strategy. Moreover, we also get the existene of minimal and maximal global b ounded tra jetories asso iated to ordered systems. In Setion 2 we present the sub-sup ertra jetory to ol, Setion 3 is devoted to the logisti equation whih app ears when one sp eies is absent. Finally, in Setion 4 we show the results of system (1). Non-autonomous Lotka-Volterra ompetition model 93 2 The sub-sup ertra jetory metho d for omplete solutions Consider the general problem        ut−∆u=f(t, x, u, v)x∈Ω, t > s vt−∆v=g(t, x, u, v)x∈Ω, t > s u=v= 0 x∈∂Ω, t > s u(s) = us, v(s) = vs, (3) where f, g are bounded on b ounded sets of IR ×Ω×IR2 and are lo ally Hölder ontinuous in time. We denote the solutions of (3) as u(t, s;us, vs), v(t, s;us, vs), for t > s. Denition 1 A pair of funtions (u, v)∈C1,2 t,x (IR ×Ω) is a omplete trajetory of (3), if for al l s < t in IR , (u(t), v(t)) is the solution of (3) with initial data us=u(s) , vs=v(s) . Denition 2 A positive funtion u(t, x) is nondegenerate at ∞ (respetively −∞ ) if there exists t0∈IR suh that u is dened in [t0,∞) (respetively (−∞, t0] ) and there exists a C1 0(Ω) funtion ϕ0(x)>0 in Ω , suh that for al l x∈Ω , u(t, x)≥ϕ0(x) for al l t≥t0 (respetively for al l t≤t0 ). The use of sub-sup ertrajetory pairs to onstrut omplete solutions an be found in Chueshov [3℄ or Langa and Suárez [8℄. Both referenes use monotoniity prop erties of the equations, see Corollaries 2 and 3 b elow. In partiular this applies to salar equations. Here we use similar ideas to onstrut bounded omplete tra jetories, without suh monotoniity assumptions. Given T0≤ ∞ and two funtions w, z ∈C((−∞, T0)×Ω) with w≤z we denote [w, z] := {u∈C((−∞, T0)×Ω) : w≤u≤z}. Now we introdue the onept of omplete sub-sup ertra jetory pair. Denition 3 Let T0≤ ∞ and (u, v),(u, v)∈ X =C1,2 t,x ((−∞, T0)×Ω) . We say that (u, v)−(u, v) is a omplete sub-supertrajetory pair of (3) if 1. u(t)≤u(t) and v(t)≤v(t) in Ω , for al l t < T0 . 2. u≤0≤u and v≤0≤v on ∂Ω , for al l t < T0 . 3. For al l x∈Ω , t < T0 ut−∆u−f(t, x, u, v)≤0≤ut−∆u−f(t, x, u, v),∀v∈[v, v], vt−∆v−g(t, x, u, v)≤0≤vt−∆v−g(t, x, u, v),∀u∈[u, u]. Note that the onept of a sub-supersolution pair, dened for t > s , has b een widely used and developed, see e.g. Pao [10℄, to onstrut solutions for the initial value problem (3). The main result of this setion is: 94 J.A. Langa, A. Rodríguez-Bernal, A. Suárez Theorem 1 Assume that there exists a omplete sub-supertrajetory pair of (3), (u, v)−(u, v) , in the sense of Denition 3. Moreover, assume u , v , u and v are bounded at −∞ . Then, there exists a omplete trajetory (u∗, v∗)∈ X of (3) suh that (u∗, v∗)∈ I := [u, u]×[v, v]. When f and g have some monotoniity properties, we an go further: Corollary 2 Under the assumptions of Theorem 1, assume moreover that f is inreasing in v and g in u . Then, there exist two omplete trajetories (u∗, v∗) and (u∗, v∗) of (3) with (u∗, v∗),(u∗, v∗)∈ I := [u, u]×[v, v] suh that they are minimal and maximal in I in the fol lowing sense: for any other omplete trajetory (u, v)∈ I we have: u(t)≤u∗(t)≤u(t)≤u∗(t)≤u(t), v(t)≤v∗(t)≤v(t)≤v∗(t)≤v(t), for al l t < T0 . (4) Corollary 3 Under the assumptions of Theorem 1, assume moreover that f is dereasing in v and g in u . Then, there exist two omplete trajetories (u∗, v∗) and (u∗, v∗) of (3) with (u∗, v∗),(u∗, v∗)∈ I := [u, u]×[v, v] and suh that they are minimal-maximal and maximal-minimal in the fol lowing sense: for any other omplete trajetory (u, v)∈ I we have: u(t)≤u∗(t)≤u(t)≤u∗(t)≤u(t), v(t)≤v∗(t)≤v(t)≤v∗(t)≤v(t), for al l t < T0 . (5) 3 The non-autonomous logisti equation Note that (1) always admits semi-trivial tra jetories of the form (u, 0) or (0, v) . In this ase, when one speies is not present, the other one satises the logisti equation    ut−∆u=h(t, x)u−g(t, x)u2 in Ω, t > s u= 0 on ∂Ω , u(s) = us≥0 in Ω . (6) It is well known that if hM:= sup Q h(t, x)<∞ and gL:= inf Q g(t, x)>0, (7) then, for every non-trivial us∈C(Ω) , us≥0 , there exists a unique p ositive solution of (6) denoted by Θ[h,g](t, s;us) . On the other hand, for m∈L∞(Ω) we denote by Λ(m) , the rst eigenvalue of −∆u=λu +m(x)u in Ω , u= 0 on ∂Ω . In partiular, we denote by Λ0:= Λ(0) . It is well known that Λ(m) is a simple eigenvalue with a p ositive eigenfuntion, and a ontinuous and dereasing funtion of m . Non-autonomous Lotka-Volterra ompetition model 95 Finally, for h, g ∈L∞(Ω) with gL:= inf{g(x), x ∈Ω}>0 onsider the ellipti equation −∆u=h(x)u−g(x)u2 in Ω , u= 0 on ∂Ω . (8) It is well known that (8) p ossesses a unique p ositive solution if, and only if, Λ(h)<0 , whih we denote by ω[h,g](x) . In the following result (see [12℄, [11℄ and [7℄ for a omplete study of (6)) we show the existene and properties of a omplete nonnegative tra jetory for (6). For this we will assume heneforth that h(t, x) and g(t, x) satisfy (7) and there exist bounded funtions h± 0(x) and H± 0(x) dened in Ω suh that lim sup t→±∞ sup x∈Ωh(t, x)−H± 0(x)≤0,0≤lim inf t→±∞ inf x∈Ωh(t, x)−h± 0(x). (9) Prop osition 4 Assume (7) and (9). Then: i) There exists a maximal bounded omplete trajetory, denoted by ϕ[h,g](t) , of (6), in the sense that, for any other non-negative omplete bounded trajetory ξ(t) of (6) we have 0≤ξ(t)≤ϕ[h,g](t), t ∈IR. Moreover, if ϕ[h,g](t, x) is nondegenerate at −∞ then it is the only one of suh solutions. ii) If Λ(H− 0)>0 , then ϕ[h,g](t) = 0 for al l t∈IR . Therefore al l non-negative solutions of (6) onverge to 0 , uniformly in Ω , in the pul lbak sense. iii) If Λ(h− 0)<0 then ϕ[h,g] is the unique omplete bounded and non-degenerate trajetory at −∞ of (6), and for t in ompat sets of IR , if s7→ us≥0 is bounded and non-degenerate, then Θ[h,g](t, s;us)−ϕ[h,g](t)→0 as s→ −∞ uniformly in Ω . iv) If Λ(H+ 0)>0 , then for al l us∈C(Ω) , us≥0 , the positive solution of (6) satises Θ[h,g](t, s;us)→0 uniformly in Ω as t→ ∞ . In partiular, ϕ[h,g](t)→0 uniformly in Ω as t→ ∞ . v) If Λ(h+ 0)<0 and ϕ[h,g]6= 0 , then ϕ[h,g] is non-degenerate at ∞ and for any s and any non-trivial initial data us≥0 , Θ[h,g](t, s;us)−ϕ[h,g](t)→0 in C1(Ω) as t→ ∞. 4 Appliations to the Lotka-Volterra omp etition mo del We assume from now on that λ, µ ∈IR and aL, dL, bL, cL>0. (10) We will assume that there exist quantities a± I≤a± S , b± I≤b± S , c± I≤c± S and d± I≤d± S suh that 0< a± I≤a(t, x)≤a± S,0< b± I≤b(t, x)≤b± S, 0< c± I≤c(t, x)≤c± S,0< d± I≤d(t, x)≤d± S, (11) 96 J.A. Langa, A. Rodríguez-Bernal, A. Suárez for all x∈Ω and for all t≥t0 or t≤t0 . In the following result we show the existene of a omplete tra jetory of (1). Prop osition 5 (Comp etitive ase) There exists a omplete trajetory (u∗, v∗) of (1) with ϕ[λ−bϕ[µ,d],a](t)≤u∗(t)≤ϕ[λ,a](t), ϕ[µ−cϕ[λ,a],d](t)≤v∗(t)≤ϕ[µ,d](t), t ∈IR. Moreover, if (11) is satised for very negative t and λ > Λ(−b− Sω[µ,d− I]) and µ > Λ(−c− Sω[λ,a− I]), (12) then (u∗, v∗) is non-degenerate at −∞ . If moreover (11) is satised for large and very negative t , (12) and λ > Λ(−b+ Sω[µ,d+ I]) and µ > Λ(−c+ Sω[λ,a+ I]) (13) holds, then (u∗, v∗) is non-degenerate at ∞ . Proof . Note that in this ase f is dereasing in v and g in u . It is enough to take (u, u) = (ϕ[λ−bϕ[µ,d],a], ϕ[λ,a]) and (v, v) = (ϕ[µ−cϕ[λ,a],d], ϕ[µ,d]). Moreover, if λ and µ satisfy (12), resp. (13), then by Proposition 6 we obtain that u and v are non-degenerate at −∞ , resp. +∞ .  Now, we an summarize the results for the system (1). Theorem 6 (Comp etitive ase) 1. If λ < Λ0 and µ < Λ0 lim s→−∞(u(t, s;us, vs), v(t, s;us, vs)) = lim t→∞(u(t, s;us, vs), v(t, s;us, vs)) = (0,0). 2. If λ < Λ0 and µ > Λ0 , then lim t→∞u(t, s;us, vs) = 0, and for every nonnegative nontrivial ˜vs we have lim t→∞v(t, s;us, vs)−Θ[µ,d](t, s; ˜vs)= lim t→∞v(t, s;us, vs)−ϕ[µ,d](t)= 0. 3. If λ > Λ0 and µ < Λ0 , then lim t→∞v(t, s;us, vs) = 0, and for every nonnegative nontrivial ˜vs we have lim t→∞u(t, s;us, vs)−Θ[λ,a](t, s; ˜vs)= lim t→∞u(t, s;us, vs)−ϕ[λ,a](t)= 0. Non-autonomous Lotka-Volterra ompetition model 97 4. If λ > Λ(−b− Sω[µ,d− I]) and µ > Λ(−c− Sω[λ,a− I]), (14) there exists a omplete bounded non-degenerate at −∞ trajetory of (1) (u∗(t), v∗(t)) . Moreover, if b or c are smal l at −∞ , that is, lim sup t→−∞ kbkL∞(Ω) lim sup t→−∞ kckL∞(Ω) < ρ0 for some suitable onstant ρ0>0 , then this is the unique bounded nondegenerate at −∞ trajetory of (1) and it is pul lbak attrating, that is lim s→−∞(u(t, s;us, vs)−u∗(s), v(t, s;us, vs)−v∗(s)) = (0,0). If moreover λ > Λ(−b+ Sω[µ,d+ I]) and µ > Λ(−c+ Sω[λ,a+ I]), (15) then (u(t, s;us, vs), v(t, s;us, vs)) is non-degenerate at ∞ . If additional ly b or c are smal l at ∞ , that is, lim sup t→∞ kbkL∞(Ω) lim sup t→∞ kckL∞(Ω) < ρ0 for some suitable onstant ρ0>0 , then al l solutions of (1) have the same asymptoti behavior as t→ ∞ . If (14) is also satised, then (u∗(t), v∗(t)) is non-degenerate at ∞ and it is also forwards attrating, that is, lim t→∞(u(t, s;us, vs)−u∗(t), v(t, s;us, vs)−v∗(t)) = (0,0). Remark 1 Similar results an be presented for the prey-predator and symbiosis ases. In Figure 1 we desrib e the asymptoti dynamial regimes (pullbak -Case a)- and forwards -Case b)) when λ and µ are onstant funtions. Region A: extintion of b oth sp eies; Regions B and C: stability of semitrivial omplete tra jetories; Regions DP and DF : p ermanene regions (existene of global nondegenerate global solutions). The limiting urves are given in (14) and (15). Referenes [1℄ R. S. Cantrell and C. Cosner, Pratial p ersistene in eologial mo dels via omparison metho ds, Pro . Royal So . Edin., 126A (1996) 247-272. [2℄ R. S. Cantrell and C. Cosner, Spatial Eology via Reation-Diusion Equations, John Wiley & Sons. Ltd. 2003. [3℄ I. Chueshov, Monotone random systems theory and appliations. Leture Notes in Mathematis, 1779. Springer-Verlag, Berlin, 2002. 98 J.A. Langa, A. Rodríguez-Bernal, A. Suárez [4℄ P. Hess, Perio di-Parab oli boundary value problems and p ositivity, Pitman Researh Notes in Mathematis 247, Harlow Longman. 1991. [5℄ G. Hetzer, W. Shen, Uniform persistene, o existene, and extintion in almost p erio di/nonautonomous ompetition diusion systems, SIAM J. Math. Anal., 34 (2002) 204-221. [6℄ J. A. Langa, J.C. Robinson, A. Ro dríguez-Bernal and A. Suárez, Permanene and asymptotially stable omplete tra jetories for nonautonomous Lotka-Volterra mo dels with diusion, SIAM J. Math. Anal. 40 (2009) 21792216. [7℄ J. A. Langa, A. Ro dríguez-Bernal and A. Suárez, On the long time b ehaviour of non-autonomous Lotka-Volterra models with diusion via the sub-sup er tra jetory method, submitted. [8℄ J. A. Langa and A. Suárez, Pullbak p ermanene for non-autonomous partial dierential equations, Eletron. J. Dierential Equations 2002, 72, 20 pp. [9℄ J. López-Gómez, On the struture of the p ermanene region for ompeting sp eies models with general diusivities and transport eets,Disrete Contin. Dyn. Syst., 2 (1996) 525-542. [10℄ C. V. Pao, Nonlinear paraboli and ellipti equations, Plenum, New York, 1992. [11℄ J.C. Robinson, A. Ro dríguez-Bernal, and A. Vidal-Lóp ez, Pullbak attrators and extremal omplete tra jetories for non-autonomous reationdiusion problems, J. Dierential Equations 238 (2007) 289337. [12℄ A. Ro dríguez-Bernal and A. Vidal-Lóp ez, Existene, uniqueness and attrativity prop erties of positive omplete tra jetories for non-autonomous reation-diusion problems, Disrete Contin. Dyn. Syst., 18 (2007) 537567.