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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Concentration in parabolic Lotka-Volterra equation
s : an asymptotic-preserving scheme Helene Hiver
t - Helene Hivert (Ă‰cole Centrale de Lyon)
DTSTART;TZID=Europe/London:20220412T133000
DTEND;TZID=Europe/London:20220412T141500
UID:TALK171554AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/171554
DESCRIPTION:We consider a population structured in phenotypica
l trait\, which influcences the adaptation of indi
viduals to their environment. Each individual has
the trait of his parent\, up to small mutations. W
e consider the problem in a regime of long time an
d small mutations. The distribution of the populat
ion is expected to concentrate at some dominant tr
aits. Dominant traits can also evolve in time\, th
anks to mutations. From a technical point of view\
, the concentration phenomenon is described thanks
to a Hopf-Cole transform in the parabolic model.
The asymptotic regime is a constrained Hamilton-Ja
cobi equation [G. Barles\, B. Perthame\, 2008 & G.
Barles\, S. Mirrahimi\, B. Perthame\, 2009]. The
uniqueness of the solution of the limit equation h
as been adressed very recently [V. Calvez\, K.-Y.
Lam\, 2020]. Because of the lack of regularity of
the constraint\, it can indeed have jumps\, the nu
merical approximation of the constrained Hamilton-
Jacobi equation must be treated with care. \;\
nWe propose an asymptotic preserving scheme for th
e problem transformed with Hopf-Cole. We show that
it converges outside of the asymptotic regime\, a
nd that it enjoys stability properties in the tran
sition to the asymptotic regime. Eventually\, we s
how that the limit scheme is convergent for the co
nstrained Hamilton-Jacobi equation.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:
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