global existence and uniqueness of solutions to the free boundary problem in Section \[existence\]. In Section \[specific model\], we present a detailed analysis of a specific case of model . In Section \[Section4\], we study the population dynamics of infected mosquitoes in a heterogeneous environment with a free boundary condition. In order to better understand the effects of dispersal and spatial variations on the outcome of the competition, we study system over a bounded domain with Neumann boundary conditions. We summarize our results in the last section.

Global existence of smooth solutions {#existence}
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Using arguments that are similar to those in [@dusupandinf], we can establish the following result concerning the existence and uniqueness of solutions to system -.

\[Thm\_existence\] Consider system with initial conditions . Assume that $b_1$ and $b_2$ satisfy . Then, there exists $T>0$ such that admits a unique solution $(u, v, h(t))$ satisfying

(i) $(u,v,h) \in C^{\frac{(1+\theta)}{2},1+\theta}(Q)\times C^{\frac{(1+\theta)}{2},1+\theta}(Q^\infty)\times C^{1+\frac{\theta}{2}}([0,T])$,\

(ii) $\|u\|_{C^{\frac{(1+\theta)}{2},1+\theta}(Q)}+\| v\|_{C^{\frac{(1+\theta)}{2},1+\theta}(Q^\infty)}+\|h\|_{C^{1+\frac{\theta}{2}}([0,T])}\leq K$,

where $0<\theta<1$ is the Hölder exponent in , $$\begin{array}{c}
Q=\{(t,x)\in \mathbb{R}^{2},\text{ such that }t\in[0,T] \text{ and }x\in[0,h(t)]\},\\
Q^\infty=\{(t,x)\in \mathbb{R}^{2}, \text{ such that } t\in[0,T] \text{ and }x\in[0,+\infty)\},\end{array}$$ $K$ and $T$ are constants that depend only on $h_{0}$, $\theta$, $\| u_{0}\|_{C^{2}([0,h_{0}])}$ and $\|v_{0}\|_{C^{2}([0,+\infty))}$.

The 