Some Results on the Lotka–Volterra Model and its Small Random Perturbations

The Lotka–Volterra model is a nonlinear sytem of differential equations representing competing species. We show that when the system is far from its equilibrium, then most of the time one of the populations is exponentially small. We then consider random perturbations of the classical model by noise. In the case of perturbation of coefficients averaging principle applies. In the case of perturbations leading to extinction of one of the populations large deviation principle is used to find the likely path to extinction.