Periodic dynamics of a single-species population model based on the discrete Beverton-Holt equation

We are first concerned with the discrete Beverton-Holt equation$ \begin{equation*} x_{n+1} = \frac{\mu_n K_n x_n}{K_n+(\mu_n-1) x_n}, \ n\in \mathbb{Z} = \{0, 1, 2, \cdots\}, \end{equation*} $where $ \{\mu_n\}_{n = 0}^{\infty} $ and $ \{K_n\}_{n = 0}^{\infty} $ are two positive $ p $-periodic sequences. We find a sufficient condition that ensures if the origin is unstable, there exists a unique positive $ p $-periodic solution that globally attracts all positive solutions. This confirms the Cushing-Henson conjecture (a) under weaker conditions. A necessary and sufficient condition on the local stability of the origin is also provided. Then, based on this discrete equation, a single-species population model is proposed to describe the sterile insect technique for mosquito suppression. Some sufficient conditions on bistability are obtained. There is a stable extinction equilibrium and exactly two periodic orbits. One orbit is a repellor, and the other is an attractor.