Dr. Justin Tzou (Macquarie University)
Title: Stability analysis of localised patterns in two and three spatial dimensions
Abstract: We present a matched asymptotics framework for constructing and analysing the stability of localised patterns that arise in singularly perturbed activator-inhibitor reaction-diffusion systems. In two spatial dimensions, by way of analyses of nonlocal eigenvalue problems, we resolve two long-standing problems regarding 1) the stability of spot patterns to oscillatory instabilities, and 2) the stability of stripe patterns to break-up instabilities, the latter motivated by the persistence of striped vegetation patterns on steep hillsides. In three spatial dimensions, we calculate explicit stability thresholds for self-replication and annihilation of spots, and derive a gradient flow that governs their slow dynamics. Joint work with Theodore Kolokolnikov, Michael J. Ward, and Shuangquan Xie.
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