Stochastic persistence

Abstract

A fundamental issue in mathematical ecology and population biology is to estimate the minimal conditions that ensure the long-term survival of the interacting species in the system whether they be bio-chemicals, animals, plants or even micro-organisms like viruses (see, e.g., Lotka [6], Volterra [7], Taylor and Jonker [8]). When such conditions are satisfied, the interacting populations are said to persist or coexist. A similar question, in mathematical models of epidemics, is to understand whether or not a population will survive or be erased by a disease (see, e.g., Kermack and McKendrick [9]). Traditionally, to analyze these types of questions, the interactions between the species have been modelled by nonlinear systems of differential and difference equations which led to the development of what is now known as the mathematical theory of persistence of dynamical systems.