PRACTICAL PERSISTENCE IN ECOLOGICAL MODELS VIA COMPARISON METHODS

A basic question in mathematical ecology is that of deciding whether o r not a model for the population dynamics of interacting species predi cts their long-term coexistence. A sufficient condition for coexistenc e is the presence of a globally attracting positive equilibrium, but t hat condition may be too strong since it excludes other possibilities such as stable periodic solutions. Even if there is such an equilibriu m, it may be difficult to establish its existence and stability, espec ially in the case of models with diffusion. In recent years, there has been considerable interest in the idea of uniform persistence or perm anence, where coexistence is inferred from the existence of a globally attracting positive set. The advantage of that approach is that often uniform persistence can be shown much more easily than the existence of a globally attracting equilibrium. The disadvantage is that most te chniques for establishing uniform persistence do not provide any infor mation on the size or location of the attracting set. That is a seriou s drawback from the applied viewpoint, because if the positive attract ing set contains points that represent less than one individual of som e species, then the practical interpretation that uniform persistence predicts coexistence may not be valid. An alternative approach is to s eek asymptotic lower bounds on the populations or densities in the mod el, via comparison with simpler equations whose dynamics are better kn own. If such bounds can be obtained and approximately computed, then t he prediction of persistence can be made practical rather than merely theoretical. This paper describes how practical persistence can be est ablished for some classes of reaction-diffusion models for interacting populations. Somewhat surprisingly, the models need not be autonomous or have any specific monotonicity properties.