The influence of the type 3 (sigmoid) functional response upon the stability of predator-prey difference models

Abstract

Sigmoid functional responses are found to exert a stabilizing influence upon a discrete-generation predator-prey model in a way analogous to that found in continuous predator-prey models. The precise effect depends upon the degree to which a predator's feeding history influences its reproductive success. The time delay intrinsic in difference equation models imposes constraints not found in differential models, however, it is shown that in an otherwise unstable model the inclusion of a sigmoid functional response can result in local stability. With the addition of prey self-regulation the stabilizing influence of the functional response acts in concert with self-regulation, as it does in continuous models. These results show that the effect of the sigmoid response upon stability is not dependent upon the assumption of continuity, and reinforces the view that sigmoid responses could be an important factor stabilizing natural communities.