Daniel Gomez, Center for Mathematical Biology, University of Pennsylvania
Asymptotic Analysis of Singularly Perturbed Problems with Lévy Flights
What does a reaction-diffusion system where one species has an asymptotically small diffusivity have in common with the problem of finding the average time for a Brownian particle to first hit an asymptotically small target? For a variety of reaction-kinetics the former is a singularly perturbed problem in dimensions N>=1, while the latter is a singularly perturbed problem in dimensions N>=2. In this talk we consider this problem in N=1 dimension with classical diffusion replaced by Lévy flights. Due to the discontinuity of Lévy flights, the first-hitting-time problem becomes singularly perturbed in N=1 dimension. Moreover, depending on the fractional order of the Lévy flight, the asymptotic analysis of both singularly-perturbed fractional reaction-diffusion systems and of the first-hitting-time problem share many similarities with their Brownian counterparts in one-, two-, and three-dimensions. We will highlight these similarities and also summarize some of the consequences of Lévy flights on first-hitting-times to small targets, as well as on the resulting "spike" solutions to singularly perturbed fractional reaction-diffusion systems.
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