Wednesdays at 4:15 PM in 384H.
We consider an integro-PDE model for a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We show that in the limit of small mutation rate, the solution concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Hastings and Dockery et al. showed that for two competing species, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may well hold for a continuum of traits. This talk is based on joint work with King-Yeung Lam.
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) in 1d with ignition reactions f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when L is the Laplacian) as well as some non-local diffusion operators. We extend these results to general Levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of X_1 is finite - gives rise to a unique (up to translation) traveling front. We also prove that our result is sharp, showing that no traveling front exists when the first moment of X_1 is infinite.
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