There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations.However, there is currently no well-developed theory that treats similar questions for the nonautonomous case. Inspired in part by the theory of pullback attractors, we discuss generalisations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.

In a previous paper we introduced various definitions of stability and instability for nonautonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations.

In this work we analyze the existence, stability, and multiplicity of coexistence states for a symbiotic Lotka Volterra model with general diffusivities and transport effects. Global bifurcation theory, blowing up arguments for a priori bounds, singular perturbation results, singularity theory, and fixed point index in cones are among the techniques used to get our results and to explain the drastic change of behavior exhibited by the dynamics of the model between the cases of weak and strong mutualism between the species. Our methodology works out to treat much more general classes of symbiotic models. Academic PressKey Words: blowing up for a priori bounds in systems; local and global bifurcation theory; singularity theory; fixed point index in cones; singular perturbations.

In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t → +∞ to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t → −∞ and as t → +∞, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case

Abstract. Lotka-Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis or prey-predator behaviour involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extended to allow explicit time dependence, generally in a periodic, quasiperiodic or almost periodic fashion. The presence of more general non-autonomous terms in the equations leads to non-trivial difficulties which have stalled the development of the theory in this direction. However, the theory of non-autonomous dynamical systems has received much attention in the last decade, and this has opened new possibilities in the analysis of classical models with general non-autonomous terms. In this paper we use the recent theory of attractors for non-autonomous PDEs to obtain new results on the permanence and the existence of forwards and pullback asymptotically stable global solutions associated to non-autonomous Lotka-Volterra systems describing competition, symbiosis or prey-predator phenomena. We note in particular that our results are valid for prey-predator models, which are not order-preserving: even in the 'simple' autonomous case the uniqueness and global attractivity of the positive equilibrium (which follows from the more general results here) is new.