Non-constant periodic solutions of the Ricker model with periodic parameters

In this paper, we study the Ricker model xn+1=xnexp⁡[rn(1−xn)],n=0,1,2…,(∗)where x0≥0, and {rn}n=0∞ is a sequence of positive ω-periodic numbers. We obtain sufficient conditions for equation (*) to have at least two non-constant periodic solutions by transforming the existence problem of non-constant ω-periodic solutions of (*) into the existence problem of positive fixed points except 1 of some function. For the special case of period-two parameters, we show that (∗) has at most two non-constant 2-periodic solutions, and give necessary and sufficient conditions for (*) to have no non-constant 2-periodic solutions, to have a unique non-constant 2-periodic solution which or the equilibrium 1 is semi-stable, and to have exactly two non-constant 2-periodic solutions, respectively. Finally, some specific examples are also provided to illustrate our theoretical results.