K Wayne Nagata
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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - Nov 2020)
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.
Much attention has been given to oscillatory reaction-diffusion predator-prey systems recently because, in the wake of predator invasions, they can exhibit complex spatiotemporal patterns, notably wave trains and associated irregular spatiotemporal oscillations, thought to occur in natural systems. This thesis considers the generation and stability of spatiotemporal patterns behind invasion in these models and an extension that includes non-local intraspecific prey competition. In the first part, we study the mechanism by which a single member is selected from a continuous family of wave train solutions behind the invasion. This was first studied by Sherratt (1998), where the author develops a selection criterion that is valid near a supercritical Hopf bifurcation in the kinetics and when the predator and prey diffuse at equal rates. We formulate a ``pacemaker" selection criterion that generalizes the criterion of Sherratt (1998), but does not depend on these assumptions. We test this pacemaker criterion on three sample systems and show that it provides a more accurate approximation and can apply to unequal diffusion coefficients. In the second part of the thesis, we study the effect of including non-local intraspecific prey competition in these systems. We first study the qualitative effect of non-local competition on the spatiotemporal patterns behind predator invasions in these models, and in a related caricature system. We find that non-local prey competition increases the parameter range for spatiotemporal pattern formation behind invasion, and that this effect is greater for lower kurtosis competition kernels. We also find that sufficiently non-local competition allows the formation of stationary spatially periodic patterns behind invasion. Second, we revisit the selection and stability of wave train solutions. We modify the selection criterion from the first part, also applying it to the non-local system, and study how the properties of selected wave trains vary with the standard deviation of the non-local prey competition kernel. We find that the wavelength of selected wave trains decreases with the standard deviation of the non-local kernel and also that unstable wave trains are selected for a larger parameter range, suggesting that spatiotemporal chaos may be more common in highly non-local systems.