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

Benoît Perthame; Stephane Génieys

Mathematical Modelling of Natural Phenomena (2010)

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

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topPerthame, 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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