Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system

Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system Jos´e A. Langa†, James C. Robinson‡, Antonio Su´arez† †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. ‡Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: [email protected]; [email protected]; [email protected] Abstract. Lotka-Volterra systems have been extensively studied by many authors, both in the autonomous and non-autonomous cases. In previous papers the time asymptotic behaviour as t→ ∞ has been considered. In this paper we also consider the “pullback” asymptotic behaviour which roughly corresponds to observing a system “now” that has already been evolving for a long time. For a competitive system that is asymptotically autonomous both as t→ −∞ and as t→+∞we show that these two notions of asymptotic behaviour can be very different but are both important for a full understanding of the dynamics. In particular there are parameter ranges for which, although one species dies out as t→ ∞, there is a distinguished time-dependent coexistent state that is attracting in the pullback sense. AMS classification scheme numbers: Primary 37G10, 37G35; Secondary 34D05, 34D23. Submitted to: Nonlinearity A non-autonomous Lotka-Volterra system 2 1. Introduction In this paper we consider a non-autonomous competitive Lotka-Volterra system for two species uand v, (˙u=u(λ−a(t)u−bv)u(s) = u0>0 ˙v=v(µ−cv −du)v(s) = v0>0,(1) where a(t) is continuous, lim s→+∞a(s) = 0 < a(t)< A = lim s→−∞ a(s) for all t∈R and b, c, d > 0. We will also only consider the case λ, µ > 0, since when either of these parameters are negative the behaviour is relatively simple. Note that our model equation is asymptotically autonomous†both as t→+∞and as t→ −∞. We will refer to system (1) as E[a(t)]: such a notation makes it easier to discuss comparisons of solutions of (1) with the solutions of various autonomous systems E[a] with a(t) replaced by a constant a(the behaviour of such systems is well understood, e.g. Murray [20], and we recall this as required in what follows). Since the uand vaxes are invariant and solutions of (1) are unique the positive cone P={(u, v), u ≥0, v ≥0}and its interior P={(u, v), u > 0, v > 0}are invariant sets. It is therefore consistent to consider only positive solutions, which is also natural in the light of the ecological interpretation of (1) as a model of two competing species. Various authors have considered non-autonomous versions of the equation concentrating on the asymptotic behaviour as t→+∞. However, there is another point of view which, although equivalent in the autonomous case, allows for other arguments in the non-autonomous situation. We study the solutions in terms of the corresponding process {S(t, s)}t≥s, where S(t, s)x0represents the solution of the system at time tthat was at x0at time s(x0∈R2) (cf. Sell [24]). For non-autonomous systems the initial time is as important as the final time: this is in contrast to the autonomous case in which only the time elapsed is relevant, i.e. S(t, s) = S(t−s, 0) for all t≥s. Thus for autonomous systems the behaviour of solutions S(t, s)x0for t→ ∞ is the same as for s→ −∞. However, this “pullback” behaviour (which forms the basis of the theory of attractors in non-autonomous systems, as developed by Cheban et al. [5], Kloeden and Schmalfuss [12], Schmalfuss [23], Crauel et al. [10]) is distinct from the forwards asymptotics in the non-autonomous case. In this paper we completely describe the asymptotic behaviour of (1) both as t→ ∞ and as s→ −∞. The forwards asymptotic behaviour is essentially the same as that of the autonomous system E[0], while the pullback asymptotic behaviour is essentially the same as that of E[A]. For certain parameter ranges these behaviours are distinct, †We note here that while the theory of asymptotically autonomous equations (developed by Markus [17], Thieme [28], and Mischaikow et al. [18]) gives some useful information about the behaviour of the system as t→ ∞, a full description requires the additional analysis that we present here. A non-autonomous Lotka-Volterra system 3 and the pullback procedure provides more information about the system than would be available solely by considering the forward asymptotics. The main aim of this paper is not to offer a significant advance to the theory of one particular class of equations (Lotka-Volterra models); rather, it is to illustrate the power of the pullback idea in giving a full description of the dynamics, and to highlight some of the interesting problems that can occur in non-autonomous systems. 1.1. Previous results for non-autonomous systems There are a series of papers that treat non-autonomous N-dimensional Lotka-Volterra systems in which all the coefficients are allowed to be non-autonomous (see, among others, Ahmad [2], [3], Ahmad and Lazer [4], Montes de Oca and Zeeman [19], Redheffer [21]). We believe that we could also treat this more general problem, although we have chosen, for the sake of clarity, to consider only the simplest case where the previous methods are not directly applicable. For (1) the conditions in these papers become 0< a1≤a(t)≤a2,(2) for all t∈Rfor the results in Ahmad [1] or [2], and bµ/c < λ < ·inf t∈Ra(t)¸µ/d (3) to apply those in Ahmad and Lazer [4] or Montes de Oca and Zeeman [19] (this latter in particular requires that a(t)≥a > 0 as in Redheffer [21]). Under these hypotheses the above authors can prove the existence of a strictly positive solution of the system that is globally asymptotically stable, while if λ < bµ/c or λ > ·sup t∈R a(t)¸µ/d (4) they obtain extinction of one of the species (i.e. the solution goes to zero) as time goes to infinity. In our case none of the conditions (2–4) hold. 2. Pullback attractors in non-autonomous systems We now show how we can consider the solutions of non-autonomous equations within a dynamical systems-like framework, introduce more carefully the pullback idea that we make great use of, and show that the equations in (1) give rise to an order-preserving system. 2.1. Processes We denote by S(t, s)xthe solution of (1) at time tthat takes the value xat time s. Then S(t, s) defines a process, where a general process on a complete metric space (X, d) is a family of mappings {S(t, s)}t≥s,t, s ∈Rthat satisfy: A non-autonomous Lotka-Volterra system 4 (i) S(t, t) = IdX,for all t∈R, (ii) S(t, τ)S(τ, s)u=S(t, s)ufor all s≤τ≤t, u ∈X, and (iii) u7→ S(t, s)uis continuous in X. 2.2. The pullback attractor We now give a formal definition of the “pullback attraction” idea discussed in the introduction. For A, B ⊂Xwe let dist(A, B) denote the Hausdorff semidistance, dist(A, B) = sup a∈A inf b∈Bd(a, b). Definition 2.1. A time dependent family {K(t)}t∈Ris pullback attracting if for each t∈R lim s→−∞ dist(S(t, s)D, K(t)) = 0. for every bounded set D. This concept of attraction considers a fixed final time and moves the initial time to −∞: this does not mean that we are going backwards in time, but rather that we consider the state of the system at time tarising from the same initial condition starting at earlier and earlier times s. As remarked above, in the autonomous case this notion of pullback attraction ([12], [23]) is equivalent to the standard definition. The generalisation of the autonomous concept of invariance is somewhat more straightforward: Definition 2.2. A family {B(t)}t∈Rof subsets of Xis said to be invariant with respect to the process Sif S(t, s)B(s) = B(t)for all (s, t)∈R2, s ≤t. The definition of a non-autonomous attractor combines these notions of attraction and invariance. Definition 2.3. The family of compact sets {A(t)}t∈Ris said to be the global pullback attractor associated to the process Sif it is invariant, pullback attracting, and is minimal in the sense that if {C(t)}t∈Ris another family of closed pullback attracting sets, then A(t)⊂C(t)for all t∈R. (Chepyzhov and Vishik [6] define the concept of kernel sections for non-autonomous dynamical systems: these correspond to the fibres A(t) in the above definition of a global pullback attractor.) We now give a general result similar to those that can be found in Crauel et al. [10] and Schmalfuss [23]. The particular form of the theorem given here is modelled on a result for random attractors given in Crauel [9]. Theorem 2.4. There is a global pullback attractor {A(t)}t∈Rif and only if there exists a family {K(t)}t∈Rof compact pullback attracting sets. A non-autonomous Lotka-Volterra system 5 2.3. Complete and hyperbolic trajectories The simplest example of an invariant set is a fixed point xwith S(t, s)x=xfor all t≥s. However, this is a very strong property to require of a solution of a non-autonomous equation. One of the most basic problems with the investigation of non-autonomous systems is to find a sensible generalisation of the notion of a fixed point. A potential candidate is given by the notion of a complete trajectory, a continuous map v:R→Xwith S(t, s)v(s) = v(t),for all t≥s. This is simply a solution v(t) of the equation that is defined for all t∈R. However, there are of course many such solutions, and we would ideally like to pick out certain “distinguished” solutions. In a previous paper (Langa et al. [14]) we highlighted the importance of complete trajectories with “certain well-defined stability properties” but were less than explicit about what these might be. One fruitful concept is the notion of a “hyperbolic trajectory” (see Malhotra & Wiggins [16]) that generalises the idea of a hyperbolic fixed point. We will see in our example that although this concept is useful it is not ideal without some minor modifications. For autonomous systems hyperbolicity can be characterised using the eigenvalues of the linearised equation near the fixed point. In non-autonomous systems we need to introduce the concept of an exponential dichotomy (see Coppel [8], Sacker & Sell [22]). Definition 2.5. Let A(t)be a real N×Nmatrix and Φ(t, s)be the fundamental N×N matrix solution of dX/dt=A(t)X X(s) = I(5) so that the solution of dξ/dt=A(t)ξwith ξ(s) = ξ0is given by Φ(t, s)ξ0. Then (5) has an exponential dichotomy if there is a projection operator Pand constants Kand λ > 0 such that kΦ(t, s)Pk ≤ Ke−λ(t−s)t≥s kΦ(t, s)(I−P)k ≤ Keλ(t−s)t≤s. (6) (This definition can be generalised by allowing the projection Pto depend on tin such a way that it is invariant, i.e. that Φ(t, s)P(s) = P(t)t≥s; see Siegmund [25] for details.) Definition 2.6. A complete trajectory x(t)of dx/dt=f(x, t)is said to be hyperbolic if the linearised equation dX/dt= Df(x(t), t)X has an exponential dichotomy. A non-autonomous Lotka-Volterra system 6 As with the hyperbolicity of fixed points, the hyperbolicity of the trajectory x(t) gives rise to local stable and unstable manifolds which are preserved under perturbation. We reproduce here from [16] only the results that are relevant in this paper: for more details and a proof see Yi [29]. In the statement of the theorem Nδ(x(t)) denotes the tubular neighbourhood of x(t) in the space RN×R, Nδ(x(t)) = [ t∈R (B(x(t), δ), t) (the notation B(x, δ) is the open ball in RNof radius δcentred at x). Theorem 2.7. Let x(t)be a hyperbolic trajectory in RNsuch that the projection Pfrom definition 2.5 has rank k. Then there exists a k+ 1 dimensional Crmanifold Ws loc(t), an n−k+ 1 dimensional Crmanifold Wu loc(t), and a δsuch that (i) Ws loc is positively invariant, while Wu loc is negatively invariant, and the two manifolds intersect along x(t), (ii) trajectories on Ws loc converge towards x(t)exponentially fast as time runs forwards, and leave Nδ(x(t)) as time runs backwards, while (iii) trajectories on Wu loc leave Nδ(x(t)) as time runs forwards but converge exponentially fast towards x(t)as time runs backwards, and (iv) any trajectory starting in Nδ(x(t)) that is not in Ws loc nor Wu loc will leave Nδ(x(t)) both as time runs forwards and as time runs backwards. (v) Let dx/dt=f(x, t, p)be a family of non-autonomous ODEs that vary with the parameter pin a Crmanner (uniformly on sets of the form K×Rwhere Kis any compact subset of RN). Then any hyperbolic trajectory that exists for some value p=p0persists under small perturbations, as do its stable and unstable manifolds: these depend on pin a Crfashion. We will want to compare the dynamics of our asymptotically autonomous equation with its “limit equation” E[0]. We can treat this case in the context of the above theorem by defining a C1family of functions αp(t) such that αp(t) = (a(1/p)t≤1/p a(t)t≥(1/p)+1 and αp(t) is monotonic between t= 1/p and t= (1/p) + 1. Then the family of non-autonomous systems E[αp(t)] converge towards E[0] in the appropriate fashion as p→0. Since the dynamics of the systems E[αp(t)] are the same as those of E[a(t)] for t≥(1/p) + 1 we expect that perturbations of hyperbolic trajectories of E[0] will play a role in understanding the dynamics of E[a(t)]. 2.4. Order preserving systems When a non-autonomous system is “order-preserving” (in a sense that we now make precise) it is possible to obtain more information about the structure of its pullback A non-autonomous Lotka-Volterra system 7 attractor. This programme is carried out in some detail in Chueshov [7], Langa & Su´arez [13], and Smith [26]. 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 w1¹w2then S(t, s)w1¹S(t, s)w2for all t≥s. For our equations we can define an appropriate relation ¹on R2that makes the system order-preserving: given (u1, v1),(u2, v2) we say that (u1, v1)¹(u2, v2)⇐⇒ u1≤u2and v1≥v2.(7) In fact the following slightly stronger result (cf. Hess and Lazer [11]) will be useful. We will denote the process corresponding to the system E[f(t)] by Sf(t)(t, s), reserving S(t, s) for the process corresponding to (1). Lemma 2.8. Let f(t)and ˜ f(t)be non-negative, and denote by (u(t), v(t)) the solution of E[f(t)] with (u(s), v(s)) = (us, vs)and by (˜u(t),˜v(t)) the solution of E[ ˜ f(t)] that satisfies (˜u(s),˜v(s)) = (˜us,˜vs). Then, provided that f(t)≥˜ f(t)for all t∈R, (us, vs)¹(˜us,˜vs) =⇒Sf(t)(t, s)(us, vs)¹S˜ f(t)(˜us,˜vs) for all t≥s. We write “Sf(t)¹S˜ f(t)”. Proof. Assume initially that f(t)>˜ f(t) for all t∈R. Let [s, T] be the maximal interval on which (u(t), v(t)) ¹(˜u(t),˜v(t)) for t∈[s, T] (8) and suppose that T < ∞. At time Tone of the following three possibilities occurs: (i) u(T) = ˜u(T) = ubut v(T)>˜v(T): clearly for some δ1>0 we have v(t)≥˜v(t) for t∈(T, T +δ1]. We also have d dt(˜u−u)(T) = u[(f(T)−˜ f(T))u+b(v(T)−˜v(T))].(9) Since v(T)>˜v(T) this is strictly positive and so for some δ2>0 we have u(t)<˜u(t) for t∈(T, T +δ2]. This gives (8) on [s, T +δ], contradicting the maximality of T. (ii) v(T) = ˜v(T) = vbut u(T)<˜u(T): a similar argument can be used to show that (8) holds on [s, T +δ] for some δ > 0, since we have d dt(˜v−v)(T) = vd(u(T)−˜u(T)) <0.(10) (iii) (u(T), v(T)) = (˜u(T),˜v(T)) = (u, v): when t=Twe have (from (9) and (10)) d dt(˜u−u)(T) = u(f(T)−˜ f(T)) and d dt(˜v−v)(T) = 0. Since f(T)>˜ f(T) we have ˜u(t)> u(t) on (T, T +δ1] for some δ1>0. We also have d2 dt2(˜v−v)(T) = −vd d dt(˜u−u)(T)<0, which implies that ˜v(t)< v(t) on (T, T +δ2] for some δ2>0. Once again this gives (8) on the longer interval [s, T +δ] (with δ= min(δ1, δ2)). A non-autonomous Lotka-Volterra system 8 This proves the result for f(t)>˜ f(t). If f(t)≥˜ f(t) then for each ² > 0 apply the above result with f(t) replaced by f(t) + ²; noting that the solutions of E[f(t)] depend continuously on f(t) the result follows as stated by taking the limit as ²→0. 2.5. The separatrix of an autonomous system In one of the proofs below we will need to use properties of the separatrix of the autonomous system (˙u=u(λ−au −bv)u(0) = u0>0 ˙v=v(µ−cv −du)v(0) = v0>0,(11) where ac < bd and aµ/d < λ < bµ/c. In this case there are three fixed points: two stable fixed points at (0, µ/c) and (λ/a, 0), and one saddle point in the interior at x∗ a=1 bd −ac(bµ −cλ, dλ −aµ). We include a proof since although the system is a standard example in undergraduate differential equations courses, we were unable to find any rigorous treatment in the literature. In the proof we use (x, y)≥(u, v) to mean x≥uand y≥v. Proposition 2.9. If ac < bd and aµ/d < λ < bµ/c then there is a separatrix Γa, given as the graph of a strictly increasing continuous function φa: (0,∞)→(0,∞)with lim u→0φa(u) = 0 such that if v0     < = >     φa(u0)then lim t→∞ S(t, 0)(u0, v0) =      (λ/a, 0) x∗ a (0, µ/c)     . Furthermore for each fixed uthe value of φa(u)is continuous in aand is monotonically decreasing in a. Proof. First we consider the problem for a fixed value of a. Dulac’s criterion (∇ · [1 uv f(u, v)] <0 in P, where fis the right-hand side of (11)) shows that the equation has no periodic orbits in P, and a simple computation of the time derivative of µ|u|2+λ|v|2shows that all orbits are bounded. It follows from the Poincar´e-Bendixson theorem that every orbit converges to one of the fixed points. Let Γabe the stable manifold of x∗ a, i.e. Γa={x:Sa(t, 0)x→x∗ aas t→+∞}. This is the global extension of the local stable manifold of x∗ a, which is tangent to the linear stable manifold at x∗ a; elementary considerations of the linearisation show that the linear stable manifold moves into (u, v)> x∗ aand (u, v)< x∗ a. Since ˙uand ˙vare A non-autonomous Lotka-Volterra system 9 both positive [negative] when (u, v)> x∗ a[(u, v)< x∗ a] it follows that close to x∗ athe separatrix Γais the graph of a strictly increasing continuous function φa; the continuity of φain awithin this neighbourhood is a standard result from the theory of local stable manifolds. The global stable manifold consists of two trajectories. In order to understand their behaviour outside a neighbourhood of x∗ awe consider the time-reversed flow, writing ˇu(t) for u(−t). Then since ˙ ˇuand ˙ ˇvcan be bounded below for (u, v)≥x∗ a+ (², ²), and if ˇv(t)→ ∞ then we must also have ˇu(t)→ ∞ (if ˇu(t)→c < ∞then ˙ ˇu→ ∞, a contradiction) the stable manifold Γaextends to infinity in such a way that φais defined for every u > [x∗ a]1. Since (ˇu, ˇv)< x∗ ais invariant for the time reversed flow and this region contains no periodic orbits (Dulac’s criterion again) it follows that the trajectory to the left of x∗ aconverges to the origin as t→ −∞. The convergence of S(t, 0)(u0, v0) to one of the fixed points on the axes when the initial condition lies above/below Γais now immediate, since Γais invariant and consists of all points attracted to x∗ a. The continuity of φa(u) for all values of uis a consequence of the continuous dependence of solutions on initial conditions and on the parameter a, and it only remains to show that φa(u) is monotonically decreasing. This will follow if we can show that for distinct values of a6= ˜athe separatrices Γaand Γ˜aare disjoint, since x∗ a∈Γais strictly increasing with respect to the order ¹(this follows from a simple calculation of dx∗ a/da). Without loss of generality assume that a > ˜a, and suppose that x∈Γa∩Γ˜a. Using lemma 2.8 we have Sa¹S˜a, and so x∗ a= lim t→∞ Sa(t, 0)x¹lim t→∞ S˜a(t, 0)x=x∗ ˜a. However, x∗ aÂx∗ ˜a, a contradiction. So Γa∩Γ˜a=∅as required. 2.6. A non-autonomous logistic equation Also useful will be the following simple result that gives some properties of the solutions of the non-autonomous logistic equation dx/dt=x(p(t)−l(t)x)x(s) = x0,(12) with p(t)>0 and l∈C0(R) with l(t)>0 for all t∈R. We denote the solution of this equation as θ[p(·),l(·)](t, s;x0), and note that it can be given explicitly by θ[p(·),l(·)](t, s;x0) = eRt sp(u) du x−1 0+Rt seRr sp(u) dul(r) dr.(13) From here it is easy to deduce the following properties: Lemma 2.10. The solutions of (12) have the following properties: (i) If p(t)→p > 0and l(t)→0as t→ ∞ then θ[p,l](t, s)→ ∞ as t→ ∞. A non-autonomous Lotka-Volterra system 16 4.2.2. A pullback attracting coexistent state. Finally we investigate the parameter range in which we obtain our pullback attracting coexistent state: the autonomous system E[A] has an attracting interior fixed point for this set of parameters. In two proofs of this section we will make use of the following result from Ahmad and Lazer [4] (Lemma 3), rewritten here using our order notation. Lemma 4.4. Suppose that there exist δ,δ1, δ2>0such that for some fixed t∈R δ1a(t)> δ2d+δ(20) δ2c > δ1b+δ, and that there exist solutions x1(·)and x2(·)of (1) such that a−¹xi(s)¹a+for all s≤t where a−¹a+and both are elements of P. Then x1(t) = x2(t)for all t∈R. Theorem 4.5. If Ac > bd and bµ/c < λ < Aµ/d (21) then there exists a complete trajectory (U(t), V (t)) ∈Psuch that for each u0, v0>0 and every t∈R, lim s→−∞ S(t, s)(u0, v0) = (U(t), V (t)). Proof. For any ² > 0 such that (A−²)c > bd there exists a t0(²) such that a(t)>(A−²) for all t≤t0. It follows that for s, t ≤t0we have SA(t, s)¹S(t, s)¹SA−²(t, s). Every system E[a] with A−²≤a≤Ahas an attracting interior fixed point x∗ a, and for the parameter range considered here x∗ ais decreasing (with respect to the order ¹) in a. It follows that x∗ A¹lim s→−∞ S(t, s)(u0, v0)¹x∗ A−². Since S(t0, t)xdepends continuously on x, it follows that for any t0 S(t0, t)x∗ A¹lim s→−∞ S(t0, s)(u0, v0)¹S(t0, t)x∗ A−² and theorem 2.4 ensures the existence of a non-autonomous attractor A(t). Any two trajectories x1(t) and x2(t) in A(t) must satisfy x∗ A¹xi(t)¹x∗ A−² for t≤t0. Lemma 4.4 now guarantees that x1(t) = x2(t) for all t∈R, and thus A(t) consists of a single trajectory (U(t), V (t)) as claimed. A non-autonomous Lotka-Volterra system 17 5. Conclusions We have described in some detail the dynamics of a two-dimensional non-autonomous competitive Lotka-Volterra model that is asymptotically autonomous both as t→ ∞ and as t→ −∞. While the asymptotic behaviour as t→ ∞ corresponds to that of the limiting system far in the future, the pullback asymptotic behaviour as s→ −∞ appears to correspond to that of the limiting system in the distant past. We think that at the very least this example should serve to clarify the type of information that can be picked up using the pullback idea. If it really is forward asymptotic behaviour that is of interest then we can expect to gain little from the pullback approach, but if a propensity to favour this point of view is a product only of its familiarity (and the equivalence of the two notions in the autonomous case) then the pullback procedure provides another technique that can be useful in uncovering important qualitative features of the dynamics. For example, for a certain range of parameters there is a distinguished positive “coexistent” trajectory x(t) = (u(t), v(t)) that is pullback attracting. To give this a biological interpretation, if were to arrive today (t= 0) at a remote island on which two species have been competing according to (1) for a long time, the distribution of the two species would be very close to (u(t), v(t)). However, we know that one of the species is destined to die out in the future. Some interesting mathematical questions are also raised. When Ais sufficiently small the system E[a(t)] will be C1close to E[0] over the whole line (t∈R) and the saddle point x∗of the system E[0] will become a hyperbolic trajectory for E[a(t)]: our results confirm this, so that when Ac < bd the pullback behaviour and the forwards asymptotic behaviour are similar. However, when Ac > bd the picture is different: somehow we have to “join” the pullback behaviour (an attracting coexistent trajectory) to the forwards behaviour: it is not clear that the hyperbolic trajectory emanating from the stable positive fixed point of E[A] remains hyperbolic for all t∈R. In particular it seems more natural to allow for “eventually hyperbolic” trajectories where we only require an exponentially dichotomy for t, s ≥Tor t, s ≤T(for some appropriate T). With further analysis we believe that it would have been possible to treat not only the predator-prey and cooperative cases, but also higher-dimensional systems and more general non-autonomous terms (indeed, a related infinite-dimensional problem is studied in Langa et al. [15]). However, we have preferred to keep the problem relatively simple in order to show that the pullback procedure can be a very useful tool. Acknowledgments This work has been partially supported by Proyecto D.G.I.C.Y.T. (Spain) BFM2002- 03068. JCR is a Royal Society University Research Fellow, and would like to thank the Society for all their support. He would also like to thank EDAN for their hospitality, and Iberdrola for their generosity. A non-autonomous Lotka-Volterra system 18 References [1] S. Ahmad. On almost periodic solutions of the competing species problem. Proc. Amer. Math. Soc. 102 (1988), 855–861. [2] S. Ahmad. On the nonautonomous Volterra-Lotka competition equation. Proc. Amer. Math. Soc. 117 (1993), 199–205. [3] S. Ahmad. Extinction of species in nonautonomous Lotka-Volterra systems. Proc. Amer. Math. Soc. 127 (1999), 2905–2910. [4] S. Ahmad and A.C. Lazer. On the nonautonomous N-competing species problems, App. Anal. 57 (1995), 309–323. [5] D. N. Cheban, P. E. Kloeden and B. Schmalfuss. The relationship between pullback, forwards and global attractors of non-autonomous dynamical systems. Preprint [6] V. V. Chepyzhov and M. I. Vishik. Attractors of non-autonomous dynamical systems and their dimension. J. Math. Pures Appl. 73 (1994), 279–333. [7] I. Chueshov. Order-preserving skew-product flows and non-autonomous parabolic systems. Acta Appl. Math. 65 (2001), 185–205. [8] W.A. Coppel. Dichotomies in Stability Theory (Berlin: Springer Lecture Notes in Mathematics No. 629, 1978). [9] H. Crauel. Random point attractors versus random set attractors. J. Lond. Math. Soc. 63 (2001), 413–427. [10] H. Crauel, A. Debussche and F. Flandoli. Random attractors. J. Dynamics Differential Equations 9(1997), 397–341. [11] P. Hess and A.C. Lazer. On an abstract competition model and applications. Nonlinear Anal. TMA 16 (1991), 917–940. [12] P. E. Kloeden and B. Schmalfuss. Asymptotic behaviour of non-autonomous difference inclusions. Systems & Control Letters 33 (1998), 275–280. [13] J.A. Langa, and A. Su´arez. Pullback permanence for non-autonomous partial differential equations, Electronic Journal of Differential Equations,2002 (2002) No. 72, 1–20. [14] J.A. Langa, J.C. Robinson, and A. Su´arez. Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity 15 (2002), 887–903. [15] J.A. Langa, J.C. Robinson, and A. Su´arez. Permanence in the non-autonomous Lotka-Volterra competition model, J. Diff. Eqs., to appear. [16] N. Malhotra & S. Wiggins. Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby wave flow. J. Nonlinear Sci. 8(1998), 401–456. [17] L. Markus. Asymptotically autonomous differential systems. In Contributions to the Theory of Nonlinear Oscillations III (S. Lefschetz, ed.), 17–29. Annals of Mathematics Studies 36, Princeton University Press, 1956. [18] K. Mischaikow, H. Smith, and H.R. Thieme. Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc. 347 (1995), 1669–1685. [19] F. Montes de Oca and L. Zeeman. Extinction in nonautonomous competitive Lotka-Volterra systems. Proc. Amer. Math. Soc. 124 (1996), 3677–3687. [20] J.D. Murray. Mathematical Biology (Springer-Verlag, New York 1993). [21] R. Redheffer. Nonautonomous Lotka-Volterra systems, I and II. J. Diff. Eqs 127 (1996), 519–541 and 132 (1996), 1–20. [22] R.J. Sacker and G.R. Sell. Existence of dichotomies and invariant splitting for linear differential systems I, II, III. J. Diff. Eq. 15, 22, 22 (1974, 1976, 1976) 429–458, 478–496, 497–522. [23] B. Schmalfuss. Attractors for the non-autonomous dynamical systems. Proceedings of Equadiff 99 Berlin (ed. B. Fiedler, K. Gr¨oger and J. Sprekels) 684–689 (Singapore: World Scientific, 2000). [24] G. Sell. Non-autonomous differential equations and dynamical systems. Trans. Amer. Math. Soc. 127 (1967), 241-283. A non-autonomous Lotka-Volterra system 19 [25] S. Siegmund. Dichotomoy spectrum for nonautonomous differential equations. J. Dyn. Diff. Eq. 14 (2002a), pp. 243–258. [26] H.L. Smith. Monotone Dynamical Systems (Providence, RI: AMS Math. Surveys and Monographs vol. 41, 1995). [27] R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics (New-York: Springer-Verlag, 1988). [28] H. Thieme. Asymptotically autonomous differential equations in the plane. Rocky Mountain J. Math. 24 (1994), 351–380. [29] Y. Yi. A generalized integral manifold theorem. J. Diff. Eq. 102 (1993), 153–187.