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

Abstract

In this paper, we study the Ricker model x n + 1 = x n exp ⁡ [ r n ( 1 − x n ) ] , n = 0 , 1 , 2 … , ( ∗ ) where x 0 ≥ 0 , and { r n } 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.