## Periodicity in the May’s host parasitoid equation.(English)Zbl 1179.39017

Elaydi, Saber (ed.) et al., Advances in discrete dynamical systems. Proceedings of the 11th international conference on difference equations and applications (ICDEA 06), Kyoto, Japan, July 24–28, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-49-5/hbk). Advanced Studies in Pure Mathematics 53, 333-337 (2009).
Summary: We consider the May’s host parasitoid equation,
$x_{n+1}=\frac{\alpha x^2_n}{(1+x_n)x_{n-1}}\,,\quad \alpha>1.$
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.
For the entire collection see [Zbl 1170.39300].

### MSC:

 39A23 Periodic solutions of difference equations 39A20 Multiplicative and other generalized difference equations 92D25 Population dynamics (general)

### Keywords:

May’s host parasitoid equation; periodic solutions