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

Christopher Henderson (1) , Benoît Perthame (2, 3, 4, 5) , Panagiotis E Souganidis (1)

1 Department of Mathematics [Chicago]
2 UPD7 - Université Paris Diderot - Paris 7
3 SU - Sorbonne Université
4 MAMBA - Modelling and Analysis for Medical and Biological Applications
5 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We investigate a general, local version of the cane toads equation, which 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.

Mots clés: Traveling wave; Mathematical biology; Reaction diffusion equations; long range; long time limits; motility; mutation; spatial sorting; Cane toads equation; Front propagation

Domaines: Equations aux dérivées partielles [math.AP]

Christopher Henderson, Benoît Perthame, Panagiotis E Souganidis. Super-linear propagation for a general, local cane toads model. Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, 2018, 20 (4), pp.483-509. ⟨10.4171/IFB/409⟩. ⟨hal-01520145v2⟩