This thesis is concerned with introducing competition into random models. It can be observed that there are two natural mechanisms for the evolution of a random model; either by growth or by self interactions. What we do is look at two types of models and introduce competition within them. The first model, the voter model, is an example of a self interacting model and we introduce growth into it. The second model, the Hasting-Levitov model, is a random growth model and we introduce competition within the model.
In both cases we construct diffusion approximations to model these systems when the initial population is large for the first case and when the addition of incoming particles is small in the second. Once these diffusion processes have been constructed we then analyse the long term behaviour of them and find their asymptotic distribution, this is done by using the speed measure and scale function.