A discrete-time dynamical system of wild mosquito population with Allee effects

We study a discrete-time dynamical system of wild mosquito population with parameters: $\beta$ - the birth rate of adults; $\alpha$ - maximum emergence rate; $\mu>0$ - the death rate of adults; $\gamma$ - Allee effects. We prove that if $\gamma\geq\frac{\alpha(\beta-\mu)}{\mu^2}$ then the mosquito population dies and if $\gamma<\frac{\alpha(\beta-\mu)}{\mu^2}$ holds then extinction or survival of the mosquito population depends on their initial state.