Periodicity in the May's host parasitoid equation

Abstract

In this paper we consider the May's host parasitoid equation, $$\begin{array}{}\text{(1)}& x_{n+1}=\frac{a{x}_n^2}{(1+x_n)x_{n-1}},\alpha >1.\end{array}$$ 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 $n\ge 7$. There are no non-equilibrium periodic solutions of periods 2, 3, 4, 5 or 6.