Research Article

Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit

Benoît Perthame^a1 and Stephane Génieys^a2

^a1 Département de Mathématiques et Applications, Ecole Normale Supérieure, CNRS UMR 8553, 45 rue d'Ulm, F 75230 Paris cedex 05

^a2 Université de Lyon, Université Lyon1, CNRS UMR 5208 Institut Camille Jordan, F - 69200 Villeurbanne Cedex, France

Abstract

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turing instability and the interpretation refers to adaptive evolution. By analogy with other formalisms used in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sum of Dirac masses) will happen in the limit of small mutations. In the present work we study this asymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation. We prove the convergence analytically and illustrate it numerically. We also illustrate numerically how the constraint is related to the concentration points. We investigate numerically some features of these concentration points such as their weights and their numbers. We show analytically how the constrained Hamilton-Jacobi gives the so-called canonical equation relating their motion with the selection gradient. We illustrate this point numerically.

(Online publication June 15 2008)

Key Words:

• adaptive evolution;
• Turing instability;
• nonlocal Fisher equation;
• Dirac concentrations;
• Hamilton-Jacobi equation

Mathematics Subject Classification:

• 35K57;
• 35B25;
• 49L25;
• 92C15;
• 92D15

Correspondence:

^c1 genieys@math.univ-lyon1.fr