The invasive dynamics of Aedes aegypti mosquito in a heterogenous environment

A reaction-diffusion-advection model is proposed and investigated to understand the invasive dynamics of Aedes aegypti mosquitoes. The free boundary is introduced to model the expanding front of the invasive mosquitoes in a heterogenous environment. The threshold $R^D_0$ for the model with Dirichlet boundary condition is defined and the threshold $R^F_0(t)$ for the free boundary problem is introduced, and the long-time behavior of positive solutions to the reaction-diffusion-advection system is discussed. Sufficient conditions for the mosquitoes to be eradicated or to spread are given. We show that, if $R^F_0(\infty)\leq 1$, the mosquitoes always vanish, and if $R^F_0(t_0)\geq 1$ for some $t_0\geq 0$, the mosquitoes must spread, while if $R^F_0(0)<1<R^F_0(\infty)$, the spreading or vanishing of the mosquitoes depends on the initial number of mosquitoes, or mosquitoes' invasive ability on the free boundary.