Noise-controlled relaxation oscillations in ratio-dependent predator–prey models

A class of (N+1)-species ratio-dependent predator–prey stochastic models, which consists of one predator population and N prey populations (or N subpopulations of a prey metapopulation), is considered. The influence of a fluctuating environment on the carrying capacities of prey populations is modeled as a dichotomous noise. The study is a follow-up of a previous investigation of the above class of models subjected to a colored-noise with low intensity [Physical Review E 74, 021101 (2006)]. Relying, in case the growth rates of prey and predator are widely different, on the mean-field approach, the self-consistency equations for prey-populations mean abundance and predator-population abundance are derived. In some region of system parameters, the variations of noise amplitude or correlation time can cause transitions of the mean field from a globally asymptotically stable equilibrium to the stable limit cycle as well as in the opposite direction. The conditions for the occurrence of such a phenomenon are found and illustrated by a phase diagram. Key-words: Stochastic dynamics, predator–prey models, functional response, ratio-dependent, dichotomous noise, limit cycle, slow-fast cycles, noise-induced transitions.