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... Show moreIn 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. Show less
We consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin... Show moreWe consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin coating. This coating admits regularly spaced gaps at the so-called nodes of Ranvier. Since the signal travels much faster through the coated regions, it appears to hop between the nodes of Ranvier. However, many mathematical models that describe this propagation do not take into account the discrete structure directly.More recently, a discrete version of the famous FitzHugh-Nagumo model has been proposed to capture this discrete behaviour. In this thesis, we consider several extensions to and generalisations of this discrete FitzHugh-Nagumo model. In particular, we study infinite-range interactions, periodic behaviour and spatial-temporal discretization. Our general aim is to establish the existence and, sometimes, non-linear stability of travelling wave solutions. Our main tools in this analysis are the spectral convergence method and exponential dichotomies. In addition, we extend some general mathematical theory to systems with infinite-range interactions. Show less
In this thesis, we study travelling waves in stochastic reaction-diffusion equations. We extend techniques from the deterministic theory for travelling waves to apply to the stochastic version,... Show moreIn this thesis, we study travelling waves in stochastic reaction-diffusion equations. We extend techniques from the deterministic theory for travelling waves to apply to the stochastic version, which allows us to compute the stochastic wave speed and shape, and draw conclusions on the stability of the wave. Show less
This thesis describes how complex and real-world relevant analytical solutions can be found starting from a simple Nagumo problem posed on one or two-dimensional lattices. In Chapters 2 and 3 we... Show moreThis thesis describes how complex and real-world relevant analytical solutions can be found starting from a simple Nagumo problem posed on one or two-dimensional lattices. In Chapters 2 and 3 we will discuss solutions with periodic asymptotic values, where different types of fronts can be combined. In Chapter 4 we will discuss the existence of planar defect solutions, where we analyse travelling fronts possessing corners. Show less
