Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean

Many simple one-dimensional discrete models, for example, the Ricker and the logistic model, exhibit chaotic behaviour for large values of the map parameter rr. However, if a bounded stochastic perturbation with a positive expectation is introduced, for rr large, the map has a blurred asymptotically stable two-cycle, under some restrictions on the perturbation bounds and their mean. The paper considers two general types of maps including the Ricker and the truncated logistic model. It is demonstrated that bistability and multistability are possible, however, when perturbations are independent and identically distributed random variables and rr is large enough, trajectories eventually go into the blurred two-cycle with probability one. Moreover, with probability close to 1, there is a number starting from which all the trajectories are in the blurred two-cycle.