On bifurcations and chaos in predator-prey models with delay

Abstract

The stability of the fixed-points of general predator-prey models with Volterra-type distributed delays in the interspecies interaction terms is considered. For general functional forms of prey birth rate and predator death rate and the weak generic kernel or memory function a exp (- at), a supercritical Hopf bifurcation is shown to occur at a critical value a 0 of the parameter a dependent on the system parameters. For four different models, parameter regimes for dissipativity (contraction of phase-space volume) and stable/unstable ranges of a are determined. The four models are integrated numerically, and chaotic regimes are characterized by computing power spectra, autocorrelation functions, and fractal dimensions.