The evolution of fronts in kpp-type reaction-diffusion models with cut-off reaction rates

Tisbury, Alex (2019). The evolution of fronts in kpp-type reaction-diffusion models with cut-off reaction rates. University of Birmingham. Ph.D.

Preview

Tisbury2019PhD.pdf
Text - Accepted Version
Available under License All rights reserved.
Download (3MB) | Preview

Abstract

This thesis concerns two closely related problems. Firstly, we consider Kolmogorov--Petrovskii--Piscounov (KPP) type models in the presence of an arbitrary cut-off in reaction rate at concentration u=u_c. For each fixed cut-off value u_c in (0,1), we prove the existence of a unique permanent form travelling wave with a continuous and monotone decreasing propagation speed v^*(u_c). We extend previous asymptotic results in the limit of small u_c and present new asymptotic results in the limit as u_c approaches 1, which are obtained via the systematic use of matched and regular asymptotic expansions, respectively. We then use the theory of matched asymptotic expansions to obtain a detailed description of the evolution of the asymptotic structure of the solution where, in particular, we establish the rate of convergence of the solution onto the permanent form travelling wave structure in large-time for arbitrary u_c in (0,1). Secondly, we consider the impact of a heterogeneous environment enforced by a background shear flow on such fronts. We employ a two-scale asymptotic expansion to describe their speed of propagation in the limit of small Damköhler number (corresponding to slow reactions) and finite Péclet number (corresponding to similar strength of flow and molecular diffusivity).

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):