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CATEGORIES:Applied and Computational Analysis
SUMMARY:Adaptive evolution and concentrations in parabolic
PDEs - Benoit Perthame (Université Paris VI)
DTSTART;TZID=Europe/London:20080228T150000
DTEND;TZID=Europe/London:20080228T160000
UID:TALK8812AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/8812
DESCRIPTION:Living systems are subject to constant evolution t
hrough the mutation/selection principle discovered
by Darwin. In a very simple and general descripti
on\, their environment can be considered as a nut
rient shared by all the population. This alllows
certain individuals\, characterized by a 'physiol
ogical trait'\, to expand faster because they are
better adapted to the environment. This leads to
select the 'best adapted trait' in the population
(singular point of the system). On the other hand
\, the new-born population undergoes small varianc
e on the trait under the effect of genetic mutatio
ns. In these circumstances\, is it possible to des
cribe the dynamical evolution of the current trait
?\n\nWe will give a mathematical model of such dyn
amics\, based on parabolic equations\, and show th
at an asymptotic method allows us to formalize pre
cicely the concepts of monomorphic or polymorphic
population. Then\, we can describe the evolution o
f the 'best adapted trait' and eventually to compu
te branching points which lallows for the cohabita
tion of two different populations.\n\nThe concepts
are based on the asymptotic analysis of the scal
ed parabolic equations. This leads to concentratio
ns of the solutions and the difficulty is to evalu
ate the weight and position of the moving Dirac ma
sses that desribe the population. We will show tha
t a new type of Hamilton-Jacobi equation with con
straints naturally describes this asymptotic. Some
additional theoretical questions as uniqueness fo
r the limiting H.-J. equation will also be addres
sed.\n\nThis work is a collaboration with O. Diekm
ann\, P.-E. Jabin\, S. Mischler\, S. Cuadrado\, J.
Carrillo\, S. Genieys\, M. Gauduchon and G. Barl
es.\n
LOCATION:MR14\, CMS
CONTACT:
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