In this thesis we examine some nonlinear advection-diffusion-reaction equations of Fisher type. First we discuss the stability properties of the equations by writing each nonlinear equation as a system of two ordinary different equations and then analyzing the stability of their equilibrium solutions and plotting their trajectories in phase portraits. We then describe the derivation of some exact expressions for travelling wave solutions which had been obtained by previous researcher using other methods. We examine some situations where the exact travelling wave solutions are perturbed. First we perturb the initial condition and then derive approximate expressions for the perturbations. The goal of the thesis is to investigate the case where the constant speed of the background medium is perturbed by a small-amplitude spatially and temporally localized perturbation.