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Analysis of delayed bifurcations in reaction-diffusion systems Tzou, Justin


We analyze examples of delayed bifurcations in reaction-diffusion systems in both the weakly and fully nonlinear regimes. The delay effect results as the system passes slowly from a stable to unstable regime, and was previously analyzed in ODEs in [P.Mandel and T.Erneux, J.Stat.Phys 48(5-6) pp.1059-1070, 1987]. For spike solutions in the fully nonlinear regime, we demonstrate that delay can be quantified for a special class of problems in which the linear stability problem is explicitly solvable. In the weakly nonlinear regime, in the context of a simplified Klausmeier model for vegetation patterns, we analyze how addition of random noise can affect the magnitude of delay. In both regimes, we show that delay can play a critical role in determining the eventual fate of the system. Joint works with Yuxin Chen, Chunyi Gai, Theodore Kolokonikov, and Michael Ward.

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