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

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.