A basic question in mathematical ecology is that of deciding whether or not a model for the population dynamics of interacting species predicts their long-term coexistence. A sufficient condition for coexistence is the presence of a globally attracting positive equilibrium, but that condition may be too strong since it excludes other possibilities such as stable periodic solutions. Even if there is such an equilibrium, it may be difficult to establish its existence and stability, especially in the case of models with diffusion. In recent years, there has been considerable interest in the idea of uniform persistence or permanence, 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 techniques for establishing uniform persistence do not provide any information on the size or location of the attracting set. That is a serious drawback from the applied viewpoint, because if the positive attracting set contains points that represent less than one individual of some species, then the practical interpretation that uniform persistence predicts coexistence may not be valid. An alternative approach is to seek asymptotic lower bounds on the populations or densities in the model, via comparison with simpler equations whose dynamics are better known. If such bounds can be obtained and approximately computed, then the prediction of persistence can be made practical rather than merely theoretical. This paper describes how practical persistence can be established for some classes of reaction-diffusion models for interacting populations. Somewhat surprisingly, the models need not be autonomous or have any specific monotonicity properties.
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