Relative nonlinearity and permanence.

We modify the commonly used invasibility concept for coexistence of species to the stronger concept of uniform invasibility. For two-species discrete-time competition and predator-prey models, we use this concept to find broad easily checked sufficient conditions for the rigorous concept of permanent coexistence. With these results, permanent coexistence becomes a tractable concept for many discrete-time population models. To understand how these conditions apply to nonpoint attractors, we generalize the concept of relative nonlinearity and use it to show how population fluctuations affect the long-term low-density growth rate ("the invasion rate") of a species when it is invading the system consisting of the other species ("the resident") at a single-species attractor. The concept of relative nonlinearity defines circumstances when this invasion rate is increased or decreased by resident population fluctuations arising from a nonpoint attractor. The presence and sign of relative nonlinearity is easily checked in models of interacting species. When relative nonlinearity is zero or positive, fluctuations cannot decrease the invasion rate. It follows that permanence is then determined by invasibility of the resident's fixed points. However, when relative nonlinearity is negative, invasibility, and hence permanent coexistence, can be undermined by resident population fluctuations. These results are illustrated with specific two-species competition and predator-prey models of generic forms.