  Difference equations, such as a Ricker map, for an increased value of the
parameter, experience instability of the positive equilibrium and transition to
deterministic chaos. To achieve stabilization, various methods can be applied.
Proportional Feedback control suggests a proportional reduction of the state
variable at every $k$th step. First, if $k \neq 1$, a cycle is stabilized
rather than an equilibrium. Second, the equation can incorporate an additive
noise term, describing the variability of the environment, as well as
multiplicative noise corresponding to possible deviations in the control
intensity. The present paper deals with both issues, it justifies a possibility
of getting a stable blurred $k$-cycle. Presented examples include the Ricker
model, as well as equations with unbounded $f$, such as the bobwhite quail
population models. Though the theoretical results justify stabilization for
either multiplicative or additive noise only, numerical simulations illustrate
that a blurred cycle can be stabilized when both multiplicative and additive
noises are involved.
