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

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 --> infinity 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 --> -infinity and as t --> +infinity 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 --> infinity, there is a distinguished time-dependent coexistent state that is attracting in the pullback sense.