Pullback permanence in a non-autonomous competitive Lotka-Volterra model

Pullback permanence in a non-autonomous competitive Lotka-Volterra model J. A. Langa a, J. C. Robinson b,1, A. Su´arez a,2 aDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Fac. Matem´aticas, C/ Tarfia s/n, C.P. 41012, Univ. Sevilla, Spain. bMathematics Institute, University of Warwick, Coventry, CV4 7AL, U. K. Abstract The goal of this work is to study in some detail the asymptotic behaviour of a non-autonomous Lotka-Volterra model, both in the conventional sense (as t→ ∞) and in the “pullback” sense (starting a fixed initial condition further and further back in time). The non-autonomous terms in our model are chosen such that one species will eventually die out, ruling out any conventional type of permanence. In contrast we introduce the notion of “pullback permanence” and show that this property is enjoyed by our model. This is not just a mathematical artifice, but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented (and their numbers are bounded below), even if the final fate of one of them is less happy. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations. Key words: Non-autonomous differential equations, competitive diffusion system, pullback attractor, permanence. 1991 MSC: 35J55, 35B41, 35K57, 37L05, 92D25. Email addresses: [email protected] (J. A. Langa), [email protected]arwick.ac.uk (J. C. Robinson), [email protected] (A. Su´arez). 1JCR is currently a Royal Society University Research Fellow, and would like to thank the Society for their support, along with Iberdrola for their help during his visit to Seville. 2This work has been partially supported by Project CICYT MAR98-0486 and BFM2000-0797. Preprint submitted to Journal of Differential Equations 30 March 2004 1 Introduction In this paper we analyze the long-time behaviour of the non-autonomous competitive Lotka-Volterra system                    ut−∆u=u(λ−a(t)u−bv) in Ω ×(s, +∞), vt−∆v=v(µ−cv −du) in Ω ×(s, +∞), u=v= 0 on ∂Ω×(s, +∞), u(s, x) = u0(x), v(s, x) = v0(x) in Ω, (1) where Ω is a bounded domain of RN,N≥1, with a smooth boundary ∂Ω, b, c,dare positive constants, λ, µ ∈Rand 0 < a(t)≤A. Problem (1) models the interactions between two competing species inhabiting a region Ω: u(x, t) and v(x, t) represent the population densities at location x∈Ω and time t. Moreover, we are assuming that Ω is completely surrounded by inhospitable areas, because both population densities are subject to homogeneous Dirichlet boundary conditions. Here, the operator −∆ takes into account the diffusivity of the species, λand µare the growth rates of the species, band ddescribe the interaction rates between the species and finally, a(t) and care the limiting effects of crowding in each population. The starting point of this paper is the following observation, which forms the basis of the relatively recent theory of non-autonomous attractors as developed by Crauel et al. [10], Kloeden & Schmalfuss [19], and Schmalfuss [30]. Suppose that x(t;s, x0) denotes the solution of some system at time tthat is equal to x0at time s. For an autonomous system we always have x(t;s, x0) = x(t−s; 0, x0) and so considering the time asymptotic behaviour as t→+∞is exactly the same as considering what happens as s→ −∞. However, in a non-autonomous system the initial time is as important as the final time, and these two different types of “time asymptotic behaviour” are not equivalent. We do not aim here to assert the primacy of one of these approaches over the other, but rather to demonstrate that the “pullback” procedure (considering the behaviour as s→ −∞) is a useful tool that can add to our understanding of non-autonomous systems. Similar ideas are applied to the ordinary differential equation version of (1) in Langa et al. [21] for which more detailed results are possible. In population dynamics, a basic question is to determine whether the two species will survive in the long term. This has been formalized as the criterion of permanence (see Hale and Waltman [12], Hutson and Schmitt [16] and 2 references therein). The system (1) is said to be permanent if for any positive initial data u0and v0, the solution (u(t, s;u0, v0), v(t, s;u0, v0)) enters in finite time into a compact set strictly bounded away from zero in each component. In the autonomous case, that is when a(t) = a > 0, results about permanence have been obtained using various techniques. These results depend on the value of λand µwith respect to certain principal eigenvalues of associated linear elliptic problems. We need some notation in order to state these results. Given f∈L∞(Ω), we denote by λ1(f) the principal eigenvalue of the problem      −∆w+f(x)w=σw in Ω, w= 0 on ∂Ω. We write λ1:= λ1(0). On the other hand, given γ, e ∈Rand e > 0, we denote by w[γ,e]the unique positive solution of      −∆w=γw −ew2in Ω, w= 0 on ∂Ω. Observe that wis related to the stationary solution when only one species is present. It is well-known that w[γ,e]exists if, and only if, λ1< γ, and w[f,e]≡0 if λ1≥γ. On the other hand, if λ≤λ1or µ≤λ1, then one of the two species (or both of them) will be driven to extinction. This extinction region was enlarged by L´opez-G´omez and Sabina in [25] (Corollary 4.5) to a region in the (λ, µ)-plane delimited by the curves λ=λ0(µ) and µ=µ0(λ). However, if λand µsatisfy λ > ϕ(µ) and µ > ψ(λ) (2) where ϕ(µ) = λ1(bw[µ,c]) and ψ(λ) = λ1(dw[λ,a]), then (1) is permanent (see Cantrell et al. [2], [4], [5] and L´opez-G´omez [24]). We would like to point out that λ=λ1(bw[µ,c]) and µ=λ1(dw[λ,a]) define two curves in the (λ, µ)- plane whose behaviour is analyzed in detail in [2] and [24]. In Figure 1 we have summarized the autonomous case for particular values of the parameters. In this Figure we have denoted by P:= {(λ, µ) : λ,µsatisfy (2)}and by E:= {(λ, µ) : λ < λ0(µ) or µ < µ0(λ)}. In the non-autonomous case, previous work focuses on nonlinearities that are periodic in time, or that are bounded by periodic functions. In the first case, the spectral theory still works and similar results to the autonomous case can be obtained, see Hess [14] and Hess and Lazer [15]. The second case was studied 3 λ1 E λ µ λ=ϕ(µ) λ=λ (µ) µ=ψ(λ) µ=µ (λ) 0 0 P λ1 Fig. 1. Autonomous case. E: extinction region, P: permanent region. by Cantrell and Cosner [3]. In [3] the authors assume that 0 < a0≤a(t)≤A for all t≥0, and using a comparison method, they show that if λand µsatisfy λ > λ1+µb a0 and µ > λ1+λd c, then (1) is permanent (Corollary 3.1 in [3]). In this work, we do not assume that a(t) is bounded below by a positive constant and in fact we are mainly interested in the case a(t)→0 as t→+∞. We prove in this case that there is no bounded absorbing set for (1), and so the system is not “permanent” in any conventional sense. In fact, we analyse the forward behaviour in time of (1) in detail and we show that one or both species are driven to extinction when λ < λ1or λ > ϕ(µ).(3) See Figure 2 where we have represented this case. We have denoted by E= {(λ, µ) : λ,µsatisfy (3)}. The idea of pullback convergence from the theory of random and non-autonomous attractors (cf. Crauel et al. [10], Kloeden and Schmalfuss [19], Schmalfuss [30]) allows us to ask (and answer) other questions about the behaviour of our model (1). In particular we define here a notion of pullback permanence: we say that (1) is pullback permanent if there exists a time-dependent family of (bounded) absorbing sets that are bounded away from zero in each component. This idea is not intended to replace the standard notion of permanence, but rather to complement it. This definition has an interesting biological interpretation: if we arrive at an island on which two species have already been 4 λ1λ µ λ=ϕ(µ) λ1 EE E Fig. 2. Forward behaviour in time when a(t)→0 as t→ ∞. competing (according to our model) for a long time then we can guarantee that neither species will have died out (and their numbers are bounded below in a uniform way, no matter how long this ecology has been running). This is new information, not available by considering the behaviour as t→+∞: indeed, one might expect from the inevitability of extinction as t→ ∞ that such behaviour would not occur. We get here pullback extinction if λ < λ1or µ < λ1. Moreover, assuming that a(t)→a0>0 as t→ −∞ and λand µsatisfy λ > ϕ(µ) and µ > ψ(λ, a0),(4) where ψ(λ, a0) = λ1(dw[λ,a0]), then (1) is pullback permanent. We have summarized this in Figure 3, where E={(λ, µ) : λ≤λ1or µ≤λ1}. In fact we can give a bit more information about the structure of the pullback attracting states (“the non-autonomous attractor”) by using the orderpreserving property of (1) (we define an appropriate order in section 3, cf. [15], for example): a result due to Langa and Su´arez [22] shows that (1) possesses two trajectories, maximal and minimal, that are globally stable from above and below respectively. Finally, we should mention the use of skew product flows (Hale [11], Sell [29]) in studying non-autonomous problems, particularly in the periodic, quasiperiodic, or almost periodic case. The idea is to construct an autonomous semiflow S(t) on the product space H× F, where His the natural phase space where the dynamics take place (here the dynamics of uand v) and F is the hull (see [29]) of all the time dependent terms of the equation. Provided that Fis compact in some appropriate topology the general theory of dissipative dynamical systems can be applied to study S(t) on the space 5 Pullback Permanence λ µ λ λ1 1 λ=ϕ(µ) µ=ψ(λ, a ) 0 Fig. 3. Non-autonomous case. E: pullback extinction region. H× F. However, with an entirely general non-autonomous term there is no clear choice of topology on Fthat will make it compact, a property crucial to this approach 3. This is highlighted in the theory of attractors for nonautonomous equations developed by Chepyzhov & Vishik (see, for example, [6] and [7]): while their strongest results require almost periodicity, precisely in order to obtain a compact F, they study general non-autonomous terms without appealing to skew product flows using the concepts of a “kernel” and “kernel sections”, the latter corresponding exactly to the time slices A(t) of the non-autonomous attractor whose definition we recall below. An outline of this paper is as follows: in Section 2 we introduce the concept of a process, give the definition of a non-autonomous attractor and state conditions that guarantee its existence. In Section 3 we study properties of order-preserving processes and in particular recall a result about their stability. In Section 4 we study in detail a non-autonomous logistic equation which governs the behaviour of one of the species in absence of the other: this section plays a crucial role throughout all that follows. In Section 5 we analyse both the forwards and pullback behaviour of system (1), and finish in section 6 with the existence of a non-autonomous attractor for (1) and conditions for 3Using uniform convergence on Rrequires almost periodicity. An interesting extension should be possible under the assumption that the non-autonomous terms enjoy a uniform modulus of continuity over R, for then the topology of uniform convergence on compact subsets of Rwill make Fcompact, cf. Johnson & Kloeden [17], and the recent monograph by Chepyzhov & Vishik [8]. 6 pullback permanence. 2 Non-autonomous attractors In this section we introduce the definitions of a non-autonomous attractor and of pullback permanence. Let (X, d) be a complete metric space (with metric d) and {S(t, s)}t≥s,t, s ∈R be a family of mappings satisfying: a) S(t, s)S(s, τ)u=S(t, τ)u, for all τ≤s≤t, u ∈X, b) S(t, τ)uis continuous in t,τand u. c) S(t, t) is the identity in Xfor all t∈R. Such a map is called a process. Usually S(t, τ)uwill arise as the value of the solution of a non-autonomous equation at time twith “initial condition” u at time τ. As remarked in the introduction, for an autonomous equation the solutions only depend on t−τ, and we can write S(t, τ) = S(t−τ, 0). Let Dbe a non-empty set of parameterized families of non-empty bounded sets {D(t)}t∈R. In particular, D(t)≡B∈ D, where B⊂Xis a bounded set. In what follows, we will consider a fixed base of attraction Dand throughout our analysis the concepts of absorption and attraction will be referred to this fixed base. For A, B ⊂Xdefine the Hausdorff semidistances as, dist(A, B) = sup a∈A inf b∈Bd(a, b) Dist(A, B) = inf a∈Ainf b∈Bd(a, b). Definition 1 a) Given t0∈R, we say that K(t0)⊂Xis attracting at time t0if for every {D(t)} ∈ D lim τ→−∞ dist(S(t0, τ)D(τ), K(t0)) = 0. A family {K(t)}t∈Ris attracting if K(t0)is attracting at time t0,for all t0∈R. b) Given t0∈R, we say that B(t0)⊂Xis absorbing at time t0if for every {D(t)} ∈ D there exists T=T(t, D)∈Rsuch that S(t0, τ)D(τ)⊂B(t0),for all τ≤T. A family {B(t)}t∈Ris absorbing if B(t0)is absorbing at time t0,for all t0∈R. Note that every absorbing set at time t0is attracting. As discussed in the introduction, this notion takes the final time as fixed 7 and moves the initial time backwards towards −∞. We are not evolving one trajectory backwards in time, but rather we consider the current state of the system (at the fixed time t0) which would result from the same initial condition starting at earlier and earlier times. This is called pullback attraction in the literature (cf. [18], [19], [30]). Definition 2 Let {B(t)}t∈Rbe a family of subsets of X. This family is said to be invariant with respect to the process Sif S(t, τ)B(τ) = B(t),for all (τ, t)∈R2, τ ≤t. Note that this property is a generalization of the classical property of an invariant set for a semigroup. However, in this case we have to define the invariance with respect to a family of sets depending on a parameter. Definition 3 The family of compact sets {A(t)}t∈Ris said to be the global non-autonomous (or pullback) attractor associated to the process Sif it is invariant, attracts every {D(t)} ∈ D (for all t0∈R) and minimal in the sense that if {C(t)}t∈Ris another family of closed attracting sets, then A(t)⊂C(t) for all t∈R. The general result on the existence of non-autonomous attractors is a generalization of the abstract theory for autonomous dynamical systems (Temam [32], Hale [11]): Theorem 4 (Crauel et al. [10], Schmalfuss [30]) Assume that there exists a family of compact absorbing sets. Then, the family {A(t)}t∈Rdefined by A(t) = ∪D∈DΛ(D, t) is the global non-autonomous attractor, where Λ(D, t)is the omega-limit set at time tof D≡ {D(t)} ∈ D, Λ(D, t0) = ∩s≤t0∪τ≤sS(t0, τ)D(τ). Using the pullback idea introduced above we can now give the following definition of “pullback permanence”. As in [5] we suppose that X=X0∪∂X0, where X0is open, and X0,∂X0are invariant with respect to the process S. In our application, ∂X0will be the set of solutions with at least one component identically zero. Definition 5 We say that a system has the property of pullback permanence (or that it is permanent in the pullback sense) if there exists a time-dependent 8 family of bounded sets U:R7−→ X, satisfying a) U(t)absorbs every bounded set D⊂X(cf. Definition 1). b) Dist(U(t), ∂X0)>0for all t∈R. Following Definition 3, we can define a global attractor A+⊂X0that attracts every bounded set in X0: its existence follows using Theorem 4. 3 Order-preserving non-autonomous differential equations In this section we define what it means for a process to be order-preserving. For such a process we can determine some of the structure of the non-autonomous attractor and prove the existence of a minimal and maximal trajectory on the attractor with some particular stability properties. Definition 6 We say that the process {S(t, s) : X→X}t≥sis order-preserving if there exists an order relation ‘¹’ in Xsuch that, if w1¹w2,then S(t, s)w1¹ S(t, s)w2,for all t≥s. The next definition generelizes the concept of equilibria in Hess [14], (see also Arnold and Chueshov [1] in the stochastic case and Chueshov [9] in the nonautonomous case under stronger conditions). Definition 7 Let Sbe an order-preserving process. We call the continuous map w:R→Xa complete trajectory if, for all s∈R,we have S(t, s)w(s) = w(t),for t≥s. From (w,w) such that w(t)¹w(t),for all t∈R,we can define the “interval” Iw w(t) = {w∈X:w(t)¹w¹w(t)}. The following result was proved in [22] and it gives sufficient conditions for the existence of upper and lower asymptotically stable complete trajectories, and provides some information about the structure of the non-autonomous attractor. Theorem 8 Let Sbe an order-preserving process and A(t)its associated pullback attractor attracting time-dependent families of sets in a base of attraction D. Let w, w ∈ D be such that w(t)¹w(t),for all t∈R,and assume that A(t)⊂Iw w(t),∀t∈R. 9 PROOF. If λ<λ1, then observe that λ1(−λ) = λ1−λ > 0. Hence, from (14) and Proposition 10 c) we get that u(t, s;u0, v0)→0 as t→ ∞. Similarly, when µ≤λ1we get that v(t, s;u0, v0)→0 as t→ ∞. Now, we assume µ > λ1. Let δ > 0 be such that µ > λ1+dδ. For such δthere exists t0∈Rsuch that ku(t, s;u0, v0)k∞< δ for any t≥t0. On the other hand, using the definition of θ[q,b]we obtain u=θ[λ−bv,a]and v=θ[µ−du,c].(15) Then, by (14) and Proposition 10 a), we have θ[µ−dδ,c]≤θ[µ−du,c]=v≤θ[µ,c],for t≥t0. It is sufficient to apply Proposition 10 b) and the continuity of the map f7→ w[f,e].¤ The following result shows that the system is not permament when λand µ satisfy an easily verifiable condition. The system is not permanent because one species (u) increases indefinitely and drives the other to extinction. We note here that although under the condition a(t)→0 the equation is “asymptotically autonomous” in the sense of Markus [26] (see also more recent works by Thieme [33], Mischaikow et al. [27]) the general results that are available for such systems are not sufficiently detailed to give us all the information we need: for example, it is known that if all the solutions of the limit equation are unbounded then so are the solutions of the non-autonomous equation [26], but we wish to show that while one species grows without bound the other is driven to extinction. Proposition 13 Suppose a(t)→0as t→ ∞. If λ > λ1(bw[µ,c]), then (u(t, s;u0, v0), v(t, s;u0, v0)) →(∞,0) as t→ ∞. Observe that w[µ,c]= 0 if µ≤λ1, so λ > λ1(bw[µ,c]) means λ > λ1when µ≤λ1. PROOF. Assume µ≤λ1, then by Proposition 10 c) we have that v≤θ[µ,c]→ 0 as t→ ∞. Moreover, since λ > λ1, we can obtain that λ−bθ[µ,c]> λ1for t≥t1. 16 Hence, λ1(−λ+bθ[µ,c])< λ1(−λ1) = 0, and so, by Proposition 10 d) θ[λ−bθ[µ,c],a]→ ∞, and the result follows by (14). Now, suppose µ > λ1and λ > λ1(bw[µ,c]). We define ε:= λ−λ1(bw[µ,c]) 2b Since v≤θ[µ,c]→w[µ,c]as t→ ∞, then there exists tεsuch that for t≥tε v≤w[µ,c]+ε, and so, by (15) u=θ[λ−bv,a]≥θ[λ−b(w[µ,c]+ε),a],for t≥tε. Since, a(t)→0 as t→ ∞, given δ∈(0,1] there exists tδsuch that for t≥tδ we have a(t)≤δ, and so, u≥θ[λ−b(w[µ,c]+ε),a]≥θ[λ−b(w[µ,c]+ε),δ], t ≥max{tε, tδ}.(16) Now, observe that λ1(−λ+bw[µ,c]+bε) = λ1(bw[µ,c])−λ+bε =−λ−λ1(bw[µ,c]) 2<0.(17) Taking account (16) and (17), a similar argument to the used in the proof of Proposition 10 d) shows that given a small positive σ > 0 there exists tσsuch that for t≥tσ, we have u≥Φ := λ−λ1(bw[µ,c]) 2δϕ1(−λ+b(w[µ,c]+ε)) −σ. (18) Taking σsuch that 0< σ < λ−λ1(bw[µ,c]) 4≤λ−λ1(bw[µ,c]) 4δ(19) 17 we get that kuk∞≥ kΦk∞≥λ−λ1(bw[µ,c]) 4δ. Hence, it is sufficient to take δsufficiently small in order to show that u approaches infinity. Finally, observe that by (18) we get v=θ[µ−du,c]≤θ[µ−dΦ,c], t ≥tσ, and if we can take µ < λ1(dΦ), by Proposition 10 b) we obtain that vgoes to 0. But, µ < λ1(dΦ) is equivalent to µ < λ1Ãλ−λ1(bw[µ,c]) 2δϕ1(−λ+b(w[µ,c]+ε))!−σ, which is true by (19) and (7) taking δsufficiently small. This completes the proof. ¤ 5.2 Pullback asymptotic behaviour The next two results show “pullback” extinction for some values of λand µ. The first one is similar to Proposition 12 and so we omit the proof. Proposition 14 Suppose λ < λ1. a) If µ≤λ1, then (u(t, s;u0, v0), v(t, s;u0, v0)) →(0,0) as s→ −∞. b) If µ > λ1, then (u(t, s;u0, v0), v(t, s;u0, v0)) →(0, w[µ,c])as s→ −∞. Hereafter, we denote A:D(A)7→ C0(Ω) the linear operator associated to the Laplacian. Proposition 15 Given t∈R,λ > λ1and µ≤λ1, then (u(t, s;u0, v0), v(t, s;u0, v0)) →(θ[λ,a](t, s;u0),0) as s→ −∞. PROOF. Since µ≤λ1, then v≤θ[µ,c]→0 as s→ −∞. Now, given δ > 0 there exists sδsuch that v(t, s;u0, v0)≤δfor s≤sδ. 18 Hence, by (15), we get θ[λ−bδ,a]≤θ[λ−bv,a]=u≤θ[λ,a],for s≤sδ, and so, θ[λ−bδ,a]−θ[λ,a]≤u−θ[λ,a]≤0,for s≤sδ. Thus, it suffices to prove that wδ:= θ[λ−bδ,a]−θ[λ,a]→0,as δ→0. (20) It is not hard to prove that wδsatisfies (wδ)t−∆wδ=λwδ−bδθ[λ−bδ,a]−a(t)wδ(θ[λ−bδ,a]+θ[λ,a]). Now, if we denote by gδ(r, s) = λ−a(r)(θ[λ−bδ,a](r, s;u0) + θ[λ,a](r, s;u0)) and writing wδfrom the variation of constants formula, we obtain wδ(t, s;u0) = t Z s e−A(t−r)(gδ(r, s)wδ(r, s;u0)−bδθ[λ−bδ,a](r, s;u0)) dr, and so, since ° ° °e−A(t−r)° ° °op ≤1, we get kwδ(t, s;u0)k∞≤ t Z s kgδ(r, s)k∞kwδ(r, s;u0)k∞dr+bδ t Z s kθ[λ−bδ,a](r, s;u0)k∞dr, and by Gronwall’s lemma we obtain kwδ(t, s;u0)k∞≤bδ t Z s kθ[λ−bδ,a](r, s;u0)k∞dr·eRt skgδ(r,s)k∞dr.(21) On the other hand, by Proposition 10 we have kθ[λ−bδ,a](t, s;u0)k∞≤ kθ[λ,a](t, s;u0)k∞≤r(t) for s≤T(t), for some T(t) and r(t) independent of δ. Now, (20) follows by taking δto zero in (21). ¤ The next result shows that for a fixed final time t0, the positive solution of (1) is bounded away by positive functions for ssufficiently small. 19 Proposition 16 Fix t0∈R. Assume that inf s∈(−∞,t0]a(s) = α(t0)>0, λ > λ1(bw[µ,c]),and µ > λ1(dw[λ,α(t0)]). Then, there exist s0≤t0and ei∈C0(Ω) positive functions (depending on t0), such that for all s≤s0: u(t0, s;u0, v0)≥e1and v(t0, s;u0, v0)≥e2. PROOF. Since α(t0)≤a(t)≤Afor all t≤t0, we have θ[λ,A](t0, s;u0)≤θ[λ,a](t0, s;u0)≤θ[λ,α(t0)](t0, s;u0) for s≤t0. Since λ > λ1(bw[µ,c]), µ > λ1(dw[λ,α(t0)]), we can choose ε > 0 sufficiently small such that λ > λ1(b(w[µ,c]+ε)),and µ > λ1(d(w[λ,α(t0)] +ε)).(22) For such ε > 0, and by Proposition 10 b), we obtain w[λ,A]−ε≤θ[λ,a](t0, s;u0)≤w[λ,α(t0)] +εfor s≤s0, for some s0. Using again Proposition 10 a) and (14), we get θ[µ−d(w[λ,α(t0)]+ε),c]≤v, for s≤s0. (23) On the other hand, by Proposition 10 a) θ[λ−bθ[µ,c],A]≤θ[λ−bθ[µ,c],a]≤u and by part b), w[µ,c]−ε≤θ[µ,c](t0, s;u0)≤w[µ,c]+εfor s≤s0, and so, θ[λ−b(w[µ,c]+ε),A]≤u. (24) 20 Now, by Proposition 10 b), we have that as s→ −∞, θ[µ−d(w[λ,α(t0)]+ε),c]→w[µ−d(w[λ,α(t0)]+ε),c], θ[λ−b(w[µ,c]+ε),A]→w[λ−b(w[µ,c]+ε),A]. Proposition 10 b), (22), (23) and (24) complete the proof. ¤ Assuming that a(t) tends to a positive constant as t→ −∞, we obtain a similar result to Proposition 16 but where the conditions on λand µdo not depend on t. Corollary 17 Assume a(t)→a0>0as t→ −∞, for each t∈R inf s∈(−∞,t]a(s) = α(t)>0, λ > λ1(bw[µ,c])and µ > λ1(dw[λ,a0]). Then, for all t∈R, there exist s0(t)≤tand fi∈C0(Ω) positive functions (depending on t), such that for all s≤s0it holds: u(t, s;u0, v0)≥f1and v(t, s;u0, v0)≥f2. PROOF. Since µ > λ1(dw[λ,a0]) and from the continuity of the map e7→ w[λ,e], there exists ε > 0 such that µ>λ1(dw[λ,a0−ε]). On the other hand, since a(t)→a0as t→ −∞, there exists T∈Rsuch that for all t≤T, a0−ε≤α(t)≤a(t)≤A. Then, for any t0≤Twe have that µ > λ1(dw[λ,α(t0)]), and so by Proposition 16, we get that there exist two positive functions ei such that u(t0, s;u0, v0)≥e1and v(t0, s;u0, v0)≥e2. Furthermore, for all t≥t0we have u(t, s;u0, v0) = u(t, t0;u(t0, s;u0, v0), v(t0, s;u0, v0)) from which, by the strong maximum principle, we obtain the result. ¤ 21 6 Existence of a non-autonomous attractor and pullback permanence for the Lotka-Volterra competition model We define X:= C0(Ω) ×C0(Ω) and the following process in X: for t, s ∈R, t≥s, S(t, s) : X7→ X;S(t, s)(u0, v0) = (u(t, s;u0, v0), v(t, s;u0, v0)), where (u(t, s;u0, v0), v(t, s;u0, v0)) is the unique positive solution of (1) for u0, v0∈P. Moreover, in Xwe define the following order: given (u1, v1),(u2, v2)∈ X, (u1, v1)¹(u2, v2) if, and only if, u1≤u2and v1≥v2, where “≤” is the order defined by Pin C0(Ω). It is well-known, see [15], that S(t, s) is an order-preserving process, that is, if (u1, v1)¹(u2, v2), then S(t, s)(u1, v1)¹S(t, s)(u2, v2). Moreover, we consider the norm |(u, v)|∞= kuk∞+kvk∞in X. In the next two sections we will prove the existence of a non-autonomous attractor for (1). 6.1 Absorbing set in X Let D⊂Xbe bounded, i.e., supd∈D|d|∞≤M, for M > 0,and (u0, v0)∈D. By (14) and Proposition 10 e), there exists T(t, u0, v0)∈Rsuch that ku(t, s;u0, v0)k∞≤° ° °θ[λ,a](t, s;u0)° ° °∞≤rλ(t) for s≤T(t), (25) where rλ(t) = 2eλt Rt −∞ eλτ a(τ) dτ. Similarly, kv(t, s;u0, v0)k∞≤rµ(t) for s≤T(t), (26) where rµ(t) = 2eµt cRt −∞ eµτ dτ=2µ c. 22 Clearly, this means that the ball in Xwith radius r1(t) = rλ(t) + rµ(t), BX(0, r1(t)),is absorbing for the process S(t, s). 6.2 Absorbing set in C1 0(Ω) ×C1 0(Ω) In order to obtain a family of absorbing sets in C1 0(Ω) we need the following result from [28], see also Lemma 3.1 in [4]. Here, for a Banach space Y, Y β will denote the usual fractional power spaces with norm |·|β. Recall that A: D(A)7→ C0(Ω) is the linear operator associated to the Laplacian. Lemma 18 The operator Agenerates an analytic semigroup on Y=Ck 0(Ω) for k= 0,1.Moreover Yβ,→Ck+q 0(Ω) for q= 0,1and 2β > q. Given D⊂Xbounded, we define for r≥s h(r, s) = λu(r, s;u0, v0)−a(r)u2(r, s;u0, v0)−bu(r, s;u0, v0)v(r, s;u0, v0). Then, writing ufrom the variation of constants formula, we obtain u(t, s;u0, v0) = e−A(t−s)u0+ t Z s e−A(t−r)h(r, s) dr. Hence, between t−1 and t, we get u(t, s;u0, v0) = e−Au(t−1, s;u0, v0) + t Z t−1 e−A(t−r)h(r, s) dr. Hence, |u(t, s;u0, v0)|β=° ° °Aβu(t, s;u0, v0)° ° °∞≤° ° °Aβe−A° ° °op ku(t−1, s;u0, v0)k∞+ supr∈[t−1,t]kh(r, s)k∞Rt t−1° ° °Aβe−A(t−r)° ° °op dr. Now, using the estimate ° ° °Aβe−A(t−r)° ° °op ≤Cβ(t−r)−βe−δ(t−r) 23 for some constants Cβ, δ > 0 (cf. Henry [13]), and the estimates (25) and (26), we obtain the existence of M(t) and T0(t) such that |u(t, s;u0, v0)|β≤M(t) for all s≤T0(t), with β < 1−ε, and any ε∈(0,1).Applying now Lemma 18 with q= 1 and β > 1/2,we obtain ku(t, s;u0, v0)kC1≤R1(D, t) for all s≤T0(t). Similarly, it can be proven that kv(t, s;u0, v0)kC1≤R2(D, t) for all s≤T0(t), for some R2(D, t), and so the ball in C1 0(Ω) ×C1 0(Ω), B(0, R(t)) is absorbing in C1 0(Ω) ×C1 0(Ω),for R(t) = R1(t) + R2(t), where again we have used the norm |(u, v)|C1(Ω) =kukC1(Ω) +kvkC1(Ω) in C1 0(Ω) ×C1 0(Ω). We can repeat the argument taking Y=C1 0(Ω) and Da bounded set in Y×Y. In this case, using Lemma 18 again, we obtain ku(t, s;u0, v0)kC2≤N(D, t) for all s≤T1(t), and hence, the existence of an absorbing set that is bounded in C2 0(Ω)×C2 0(Ω), and so compact in X. Analogously we can show the existence of the global attractor A+attracting every bounded set in X0. 6.3 On the structure of the pullback attractor and pullback permanence In this section we apply the results of Section 3 to our model. We take w(t) = (0, rµ(t)) and w(t) = (rλ(t),0). Firstly, observe that w(t)¹w(t). On the other hand, by (25) and (26) it follows that A(t)⊂Iw w(t). 24 Finally, we define the base of attraction in our model as D:= {w:R7→ Xcontinuous, such that, lim s→−∞ eγs kw(s)k∞ = 0} where γ= min{λ, µ}. Note, that given w= (u, v)∈ D, lim s→−∞ dist(S(t, s)(u(s), v(s)),A(t)) = 0.(27) Indeed, we have that for ssmall enough ku(t, s;u(s), v(s))k∞≤ kθ[λ,a](t, s;u(s))k∞≤eλt eλs ku(s)k∞ +Rt seλτ a(τ) dτ≤rλ(t). Moreover, it is clear that (w, w)∈ D. So, applying Theorem 8, there exist complete trajectories w∗(minimal) and w∗(maximal) that are stable in the sense of Theorem 8. In a similar way, for A+we can also apply Theorem 8 for w(t) = (f1(t), rµ(t)), w(t) = (rλ(t), f2(t)), so that, for strictly positive initial data, the non-autonomous attractor is bounded above and below by strictly positive bounds. Finally, we can conclude the pullback permanence of our model. Theorem 19 Assume that a(t)→a0>0as t→ −∞, for each t∈R inf s∈(−∞,t]a(s) = α(t)>0, λ > λ1(bw[µ,c])and µ > λ1(dw[λ,a0]). Then (1) is permanent in the pullback sense. PROOF. We write X=X0∪∂X0, where X0= (intP)2and ∂X0=X\X0. The permanence follows with U(t) = {w∈X: (f1(t), rµ(t)) ¹w¹(rλ(t), f2(t))}, where f1, f2are defined in Corollary 17 and rλand rµin (25) and (26), respectively. By Section 6.1, U(t) is absorbing and by Corollary 17 Dist(U(t), ∂X0)> 0. This completes the proof. ¤ 25