Extinction of multiple populations and a team of Die-out Lyapunov functions

Abstract

We investigate the d-dimensional model,where each z j (t) is a time fluctuating resource. This "nonautonomous" model is a generalization of an autonomous Lotka-Volterra d-dimensional model. It is nonautonomous in the sense that it is not specified how the z j (t) are determined. Write S T for the transpose of S = (s ij ). When the rank of S T is equal to d − k for k > 0, we show that at least k species will die out and will do so exponentially fast. For the proof, we invent a family of Lyapunov functions, a "team" of Lyapunov functions that work together to show that k species must die, where k is the dimension of the kernel of S. Each Lyapunov function provides constraints as to which species must die out and together they provide a picture of which are likely to die out. We present a "trophic" condition for that Lotka-Volterra systems that guarantees that there is a trapping region that is globally attracting.