The role of convection on spreading speeds and linear determinacy for reaction-diffusion-convection systems. - Ameera Nema Al-Kiffai (2015)
This thesis is concerned with spreading speeds and linear determinacy for both discretetime recursion models Un+1 = Q[un] and reaction-diffusion-convection systems (PDE) under a co-operative assumption. In this thesis we are interested in the role of convection terms in propagation and linear determinacy. Such reaction-diffusion-convection systems have monotone travelling-wave solutions of the form w(x - ct) that describe the propagation of species as a wave with a fixed speed c, connecting two equilibria, a stable equilibrium beta and an unstable equilibrium 0 of the reaction term. The concept of spreading speeds was introduced by Aronson and Weinberger in [2] as a description of asymptotic speeds of spread, and in fact they showed that this spreading speed can be characterized as a minimal travelling wave speed. We discuss a characterization theory of spreading speeds of the PDE system in terms of critical travelling wave speeds. We present sufficient conditions involving both the reaction and convection terms of the PDE system for spreading speeds to equal values obtained from the linearization of the travelling-wave problem of the PDE system about the unstable equilibrium 0. These conditions guarantee the linear determinacy for the discrete-time recursion models and the PDE systems. As a result of the asymmetry in propagation that is caused by the convection terms in the PDE system, and a corresponding lack of reflection invariance in the abstract system un+1 = Q[un], we present separate conditions for non-increasing and non-decreasing initial data, called right and left conditions respectively, and we consider right and left spreading speeds. Weinberger, Lewis and Li in [42] allowed there to be more equilibria other than 0 and beta, in which case different components may spread at different speeds. This implies the need for both slowest and fastest spreading speeds, called right and left slowest (fastest) spreading speeds corresponding, to non-increasing and non-decreasing initial data respectively. We also give sufficient conditions on the reaction and convection terms such that right (left) slowest spreading speed equals right (left) fastest spreading speed for the PDE system, which implies that the system has a right (left) single spreading speed. Examples are included that illustrate the key propositions and theorems, for instance, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied, as well as a system that is right (left) linearly determinate in absence of convection terms, but it is not left linearly determinate in the presence of a convection term.