Any resources on the long term dynamics of reaction-diffusion pdes?

Question

I'm currently working on a reaction diffusion pde that models some problems in population genetics. To state it briefly, I have some 'super' insects that can produce 'super' insects offsprings by mating with ordinary insects in the wild. Ideally, releasing some super insects might turn a whole population of ordinary insects into super insects.This process can be described by two reaction diffusion pde like this
$\frac{\partial u}{\partial t}=D\Delta u +f(u,v)$
$\frac{\partial v}{\partial t}=D\Delta v +g(u,v)$
where $D$ is the diffusion coefficient.
Imagine I drop the 'super' insects in a certain area which is covered by ordinary insects, I want to know whether the super insects can spread to the whole arena as $t\to \infty$. Currently my finding is that there is a certain threshold for $u_0\over v_0$, that is to say, I have to release at high enough rate at the beginning so that super insects' population density reaches some threshold for them to spread. Also, I need to release above a certain amounts of super insects for it to spread, so there is a threshold for $\int_{D_0}u$ where $D_0$ is the area I release the super insects.
I want to know whether there are some books or articles that have already dealt with this type of problem, concerning the long term behavior of reaction-diffusion pdes or more general pdes. I'd be grateful if someone can share with me some thoughts or resources.