Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations

By Alexander Lorz, Sepideh Mirrahimi, Benoit Perthame. 2010

164 citations recorded by the source corpus.

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 co-exist? 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.

Subfields: Modeling and Simulation, Public Health, Environmental and Occupational Health, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f18d825263db38ebcb0df0158f40834dd39a9c51

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Citations in the corpus

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait (2012, background): “This work, as well as the results of [15] are ﬁrst steps in the mathematical understanding of the dynamics of those models.”
A Hamilton-Jacobi approach for a model of population structured by space and trait (2013, methodology): “Our objective is to generalize the methods developed recently on models structured only by a phenotypical trait [28, 5, 26] to spatial models, to be able to use the previous results in more general frameworks.”
Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration (2012, methodology): “ical evolution of the selected trait in the population [13,14]. Recently, PDE-based methods have been successfully used to derive such a canonical equation for continuous mutation-selection processes [15,16,17]. Here we extend this theory to the case of front invasion coupled with a basic mutation p”