A criterion for permanent coexistence of species, with an application to a two-prey one-predator system

Abstract

The question of whether all species in a multispecies community governed by differential equations can persist for all time is one of the most important in theoretical ecology. However, criteria for this property vary widely, asymptotic stability and global asymptotic stability being two of the conditions most widely used. In fact neither of these criteria appears to reflect intuitive concepts of persistence in a satisfactory manner: the first because it is only a local condition, the second because it rules out cyclic behavior. We argue here that a more realistic criterion is that of “permanent coexistence,” which essentially requires that there should be a region separated from the boundary (corresponding to a zero value of the population of at least one species) which all orbits enter and remain within. A mathematical technique for establishing permanent coexistence is illustrated by an application to the long-standing problem of predator-mediated coexistence in a two-prey one-predator community.