Persistence (Permanence), Compressivity, and Practical Persistence in Some Reaction–Diffusion Models from Ecology

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Abstract
In recent years mathematical ecology has become a very active field of research. There are both practical and theoretical reasons for all the activity. The practical interest is mostly based on concerns about the impact of human interference and environmental change on natural communities. The theoretical interest is based partly on the practical interest and partly on the fact that the appropriate analytic methods have recently become available. The essential applied problem is to understand how the interactions of biological species with each other and their environment influence their coexistence, extinction, population size, and other vital phenomena. One theoretical approach is to model those interactions using system of differential or difference equations and then try to analyze the models and interpret their results. A fundamental question is whether a given model predicts survival or extinction for a given population. That question raises a “metaquestion”: what is meant by survival (for one species) or coexistence (for several species)? Extinction is in a sense simpler; any reasonable definition of extinction requires a prediction that the population becomes or tends toward zero after a sufficient amount of time, at least with high probability if not deterministically. A simple definition for survival in a mathematical model of a population would be the presence of a globally attracting equilibrium with the population or population density positive. There are a few problems with that definition. What if the model has coefficients which vary with time, so that there is never any equilibrium? What if there are several equilibria? How about limit cycles? Two populations whose interactions produce stable oscillations would certainly seem to coexist, but would not be at equilibrium. A number of alternative definitions of survival or coexistence have been proposed in attempts to address such problems. We shall describe some of those and list their strengths and weaknesses. The general ideas are relevant to many sorts of models, but most of the illustrations will be 102in terms of reaction-diffusion systems.

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