Rapid travelling waves in the nonlocal Fisher equation connect two unstable states

Matthieu Alfaro (1) , Jérôme Coville (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 BioSP - Biostatistique et Processus Spatiaux

In this note, we give a positive answer to a question addressed in \cite{Nad-Per-Tan}. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are \lq\lq rapid\rq\rq) of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with {\it fat tails}.

Mots clés: travelling waves; turing instability; integro-differential equation

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Jérôme Coville. Rapid travelling waves in the nonlocal Fisher equation connect two unstable states. Applied Mathematics Letters, 2012, 25 (12), pp.25 (2012), 2095-2099. ⟨10.1016/j.aml.2012.05.006⟩. ⟨hal-00696102⟩