Travelling waves on trees and square lattices

In this thesis we study bistable reaction-diffusion equations on lattice domains. The power of reaction-diffusion equations is that they can successfully model various natural and social phenomena with their intuitive and relatively simple (mathematical) representation. One of the main features of reaction-diffusion equations, both on discrete and continuous domains, is that they admit special solutions, so-called ‘travelling waves’, which we can describe as fixed profiles that move in a particular direction with some speed. Depending on their shape, we can roughly divide waves into three categories: pulses or solitons, periodic pulses (wave trains), and monotone wave fronts that connect two constant states. In this thesis we focus on the latter type of wave and we study their existence, propagation and long term behaviour on two type of discrete domains - infinite trees, and two-dimensional square lattices.