Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations

Alexander Lorz, Sepideh Mirrahimi, Benoît Perthame

Research output: Contribution to journalArticlepeer-review

74 Scopus citations

Abstract

Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
Original languageEnglish (US)
Pages (from-to)1071-1098
Number of pages28
JournalCommunications in Partial Differential Equations
Volume36
Issue number6
DOIs
StatePublished - Jan 17 2011
Externally publishedYes

Fingerprint

Dive into the research topics of 'Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations'. Together they form a unique fingerprint.

Cite this