Periodicity in the May's host parasitoid equation

Abstract

In this paper we consider the May's host parasitoid equation, $$ \tag{1} x_{n+1} = \frac{ax_n^2}{(1 + x_n) x_{n-1}}, \alpha \gt 1. $$ We show that with initial conditions $x_{-1} = x_0 = 1$ there are values of $\alpha$ giving periodic solutions of prime period $n$ for all integers $n \geq 7$. There are no non-equilibrium periodic solutions of periods 2, 3, 4, 5 or 6.