An approach to practical persistence

A basic problem in community ecology is determining whether a community of interacting species will survive in the long term. A criterion ensuring this is that of permanence (or uniform persistence), which is based on the idea that species densities for large time are above minimum non-zero levels. There are various mathematical techniques for investigating permanence, but they do not yield an estimate for the minimum levels, and these may lie below minimum viability levels in the biological sense of 'Practical Persistence'. Here we study a technique for obtaining explicit expressions for the minimum levels when one species is 'slow'. This is illustrated for a predator-prey problem governed by difference equations, and we note that the technique is applicable even when the dynamics is chaotic.