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Stabilization of two cycles of difference equations with stochastic perturbations

A map which experiences a period doubling route to chaos, under a stochastic perturbation with a positive mean, can have a stable blurred two-cycle for large enough values of the parameter. The limit dynamics of this cycle is described, and it is demonstrated that most well-known population dynamics models (e.g. Ricker, truncated logistic, Hassel and May, Bellows maps) have this stable blurred two-cycle. For a general type of maps, in addition, there may be a blurred stable area near the equilibrium.