Colin Macdonald

Associate Professor

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Doctoral Student Supervision (Jan 2008 - Nov 2019)
Pattern formation on curved surfaces (2019)

Patterns emerge in various growing biological organisms, like conifer embryos, often from an homogeneous preceding state. The embryo tip, initially hemispherical shaped gradually flattens as cotyledons arranged in a roughly regular pattern emerge. A common way to model these patterns is through reaction-diffusion systems of partial differential equations. This thesis relays results obtained studying such systems and provides various new results about pattern formation on curved surfaces via diffusion driven bifurcations.We first describe situations where, for certain critical values of domain or differential equation parameters, the unpatterned solution is unstable to two different linear normal modes. We use centre manifold and normal form reductions to analyze the existence and stability of pure and mixed modes of nonlinear patterned solutions of the reaction-diffusion system, for parameters near two cases of critical values. In one case, the system reduces to a well known example of mode interaction. In the other case, the mode interaction is new, due to very small quadratic terms in the normal form.We then perform a reduction of a nonautonomous Brusselator reaction-diffusion system of partial differential equations on a spherical cap with time dependent curvature using an asymptotic series expansion on the centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The non-isotropic nature of the domain evolution insert a small patterned component to the previously patternless state, which we call the `quasi-patternless solution'. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this nonautonomous normal form. The obtained reduction solutions are then compared to numerical solutions.Finally we provide an adaptation of the closet point method to evolving domains. We perform several convergence analysis experiments of the heat equation on test surfaces and obtain quadratic convergence. Then, a few reaction diffusion simulations are shown on evolving surfaces, some featuring non-isotropic evolution. The previously shown application of the Brusselator system on an evolving spherical cap are compared to a centre manifold reduction and also show quadratic convergence.

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