Limit Cycles in Random Environments

We analyze simple predator-prey models in stochastic environments by a perturbational approach near bifurcating regimes. We obtain the stationary probability distribution for the radial variable, even when the system is on a limit cycle. The basic technique involves an adaptation of the method of averaging to take account of random fluctuations. The Fokker-Planck equation is used to find the stationary probability distribution of the secular variables. We find: (a) The radius of the limit cycle decreases as noise increases. (b) If noise dispersion is larger than the deterministic radius, no limit cycle exists. Hence, if noise is relatively large, the stationary probability distribution of a small deterministic limit cycle may be difficult to differentiate from the distribution of a stable focus. (c) The dispersion of the angular variable increases linearly in time. We give several remarks concerning soft and hard transitions, and the phenomenon of hysteresis. Finally, we discuss the entrainment of frequencies in a random and periodic environment for a predator-prey system. In general, noise will tend to break the synchronization after a mean time which depends on the size of the fluctuations, the frequency shift between the two modes, and the strength of the coupling. The results thus obtained are applicable to more complex models with robust stability as well. A few examples of two and four predator-prey species are discussed in detail.