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

• Volume: 2, Issue: 4, page 135-151
• ISSN: 0973-5348

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## Abstract

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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.

## How to cite

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Perthame, Benoît, and Génieys, Stephane. "Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit." Mathematical Modelling of Natural Phenomena 2.4 (2010): 135-151. <http://eudml.org/doc/222429>.

@article{Perthame2010,
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. },
author = {Perthame, Benoît, Génieys, Stephane},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {adaptive evolution; Turing instability; nonlocal Fisher equation; Dirac concentrations; Hamilton-Jacobi equation; Hamilton-Jacobi equation},
language = {eng},
month = {3},
number = {4},
pages = {135-151},
publisher = {EDP Sciences},
title = {Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit},
url = {http://eudml.org/doc/222429},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Perthame, Benoît
AU - Génieys, Stephane
TI - Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/3//
PB - EDP Sciences
VL - 2
IS - 4
SP - 135
EP - 151
AB - 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.
LA - eng
KW - adaptive evolution; Turing instability; nonlocal Fisher equation; Dirac concentrations; Hamilton-Jacobi equation; Hamilton-Jacobi equation
UR - http://eudml.org/doc/222429
ER -

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