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Interfaces and Free Boundaries

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Volume 20, Issue 4, 2018, pp. 483–509
DOI: 10.4171/IFB/409

Published online: 2018-12-13

Super-linear propagation for a general, local cane toads model

Christopher Henderson[1], Benoît Perthame[2] and Panagiotis E. Souganidis[3]

(1) The University of Chicago, USA
(2) Sorbonne Université, Université Paris Diderot, France
(3) The University of Chicago, USA

We investigate a general, local version of the cane toads equation, models the spread of a population structured by unbounded motility. We use the thin-front limit approach of Evans and Souganidis in [Indiana Univ. Math. J., 1989] to obtain a characterization of the propagation in terms of both the linearized equation and a geometric front equation. In particular, we reduce the task of understanding the precise location of the front for a large class of equations to analyzing a much smaller class of Hamilton–Jacobi equations. We are then able to give an explicit formula for the front location in physical space. One advantage of our approach is that we do not use the explicit trajectories along which the population spreads, which was a basis of previous work. Our result allows for large oscillations in the motility.

Keywords: Reaction diffusion equations; long range, long time limits; motility; cane toads equation; mutation; spatial sorting, front propagation

Henderson Christopher, Perthame Benoît, Souganidis Panagiotis: Super-linear propagation for a general, local cane toads model. Interfaces Free Bound. 20 (2018), 483-509. doi: 10.4171/IFB/409