General Approaches to Pattern Formation in Reaction-Diffusion Systems

Abstract

This thesis is concerned with the analysis of reaction-diffusion systems using a general approach, without making any strong assumptions, unless required. Specifically, four topics are treated, each of which provides a theory that can be applied to a general class of systems. First, general necessary and sufficient conditions to find Turing and Turing-wave instabilities are posed for systems with an arbitrary number of components posed on the line with a diagonal diffusion matrix. The results are illustrated in several examples with a different number of components. Furthermore, the results are extended to systems that involve nondiagonal diffusion matrices, and a general method is presented for designing systems that show any type of diffusion-driven instability. Moreover, the same theory turns out to apply to systems posed on generalized parallelograms or infinite domains in R^n.

Second, weakly nonlinear analysis is used to obtain expressions for the amplitude equations for travelling and standing waves in general reaction-diffusion systems with n components in one spatial dimension. Furthermore, a code that can be used to compute all the relevant coefficients accompanies this calculation. Moreover, an explanation for each of the coefficients in the normal form is provided in terms of the stability of each type of solution after the bifurcation and the existence of secondary bifurcations like folds.

Third, the general methodology is extended to study Turing bifurcations in a class of so-called bulk-surface reaction-diffusion equations in a ball. Although the procedure follows the same approach, the process in this context involves a more complicated analysis as the amplitude equations have extra symmetries inherited from the geometry of the problem: O(3) symmetry. The results are corroborated using a dedicated finite element simulation package called FEniCS
to study the Brusselator model together with a system derived from cell polarity that has four components in each of the bulk and the surface.

Finally, this thesis deals with the generalization of beyond-all-order asymptotics to study codimension-two bifurcation points where a Turing bifurcation changes its criticality in arbitrary n-component systems. Explicit conditions and equations to obtain Maxwell points are found, together with the determination of explicit formulae for the width of the so-called snaking region around the Maxwell point within which localised patterns exist. Furthermore, codes are implemented to make all the necessary calculations automatically. The results are checked with earlier computations that are available in the literature for specific example systems, and extended to other systems including a four-component model. The final result gives a general methodology for predicting the properties of homoclinic snaking for any reaction-diffusion system in a neighbourhood of a codimension-two Turing bifurcation.

Although all the ideas are developed in a general framework without a particular application in mind, the corresponding chapters present examples and explain how the theory can be applied to different model systems that arise in mathematical biology and the theory of pattern formation.

Date of Award	17 Jun 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Alan R Champneys (Supervisor)