K Wayne Nagata

Associate Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Spatiotemporal patterns in mathematical models for predator invasions (2010)

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.

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