This thesis develops a mathematical model of a diploid population with frequency and density selection and then considers spatial effects. The inclusion of these assumptions on the standard Evolutionary Stable Strategy (ESS) definition is investigated. The organisms have two alleles and two pure behaviours. In both the continuous and discrete dynamic, all the temporally stable equilibria are investigated for the possibility of Turing Instabilities. The needed general theory concerning Turing Instabilities in discrete and continuous dynamics is also developed. An alternative to the standard interpretation of discrete diffusion is proposed and its advantages over the standard interpretation are demonstrated. There are simulations of the discrete dynamics which illustrate the theoretical conclusions.