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.
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