On the Neimark-Sacker bifurcation in a discrete predator-prey system.

A two-parameter family of discrete models describing a predator-prey interaction is considered, which generalizes a model discussed by Murray, and originally due to Nicholson and Bailey, consisting of two coupled nonlinear difference equations. In contrast to the original case treated by Murray, where the two populations either die out or may display unbounded growth, the general member of this family displays a somewhat wider range of behaviour. In particular, the model has a nontrivial steady state which is stable for a certain range of parameter values, which is explicitly determined, and also undergoes a Neimark-Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter space and a repelling one in others.