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

We are first concerned with the discrete Beverton-Holt equationequation* xₙ₊₁ = μₙ Kₙ xₙ/Kₙ+(μₙ-1) xₙ, \ n∈ Z = \0, 1, 2, ⋯\, equation*where \μₙ = 0∞ and ₙ = 0∞ 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.