Abstract
This thesis is concerned with reaction-diffusion systems in one-dimensional infinite domains.Using the theory of spatial dynamics, in which time is represented by the spatial coordinates
of a state, it can be analysed whether or not localised patterned states exist and are stable. In
essence, this dissertation addresses two major objectives.
Our first objective is to illustrate the relation between the theory of homoclinic snaking and
other theories of the from-equilibrium formation of localised patterns; so called semi-strong
interaction asymptotic analysis. We demonstrate how these different paradigms fit together
within a class of activator inhibitor-systems that include the commonly studied examples such as
the Brusselator and Schnakenberg models. An essential contribution is a demonstration that in
the limit of an infinite domain, using a semi-strong asymptotic analysis allows the calculation to
provide a large amplitude exponentially localised pattern arising from the bifurcation at a fold.
As part of this work, we also demonstrate a universal two-parameter bifurcation diagram that
emerges for a wide class of models of the activator-inhibitor type. We also investigate the stability
of these localised patterned states and the instability mechanisms. Moreover, we illustrate that
bifurcation diagrams that appear in Schnakenberg-like models can also appear in other models
with more complex nonlinear terms, including Gray-Scott, Gierer-Meinhardt and predator-prey
models.
| Date of Award | 12 May 2022 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Alan R Champneys (Supervisor) |