In this study, two discrete mosquito population-control models incorporating the Allee effect are developed to investigate the impact of different sterile mosquito release strategies. By applying the theory of difference equations, a comprehensive analysis is conducted on the existence and stability of fixed points in scenarios with and without sterile mosquito releases. Conditions for the existence and stability of positive fixed points are rigorously derived. The findings reveal that in the absence of a positive fixed point, the wild mosquito population inevitably declines to extinction. When a single positive fixed point exists, the population dynamics exhibit dependence on the initial population size, potentially leading to either extinction or stabilization. In cases where two positive fixed points are present, a bistable dynamic emerges, indicating the coexistence of two mosquito populations.

In this paper, we employ the Nash–Moser iteration technique and the Saloff–Coste’s Sobolev inequality to study the local and global properties of positive solutions to the Trudinger equation ${\Delta }_{p,f}{u}^{{\displaystyle \frac{1}{p-1}}}+b{u}^q+c{u}^r=0$ on a complete smooth metric measure space with m-Bakry-Émery Ricci curvature bounded from below, where $b,c\in \mathbb{R}$, $p>1$, and $q\le r$ are real constants. We first give universal gradient estimates for the above equation under certain assumptions on b, c, p, q, and r. As their natural corollary, Harnack inequalities and Liouville-type theorems for positive solutions are obtained. Later, we consider the explicit global gradient estimates for such entire solutions through the global gradient estimates obtained.