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SUMMARY:Adaptive evolution and concentrations in parabolic PDEs - Benoit P
 erthame (Université Paris VI)
DTSTART:20080228T150000Z
DTEND:20080228T160000Z
UID:TALK8812@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:Living systems are subject to constant evolution through the m
 utation/selection principle discovered by Darwin. In a very simple and gen
 eral description\, their  environment can be considered as a nutrient shar
 ed by all the population. This  alllows  certain individuals\, characteriz
 ed by a 'physiological trait'\, to expand faster because they are better a
 dapted 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 variance on the trait under the effec
 t of genetic mutations. In these circumstances\, is it possible to describ
 e the dynamical evolution of the current trait?\n\nWe will give a mathemat
 ical model of such dynamics\, based on parabolic equations\, and show that
  an asymptotic method allows us to formalize precicely the concepts of mon
 omorphic or polymorphic population. Then\, we can describe the evolution o
 f the 'best adapted trait' and eventually to compute branching points whic
 h lallows for the cohabitation of two different populations.\n\nThe concep
 ts are based on the  asymptotic analysis of the scaled parabolic equations
 . This leads to concentrations of the solutions and the difficulty is to e
 valuate the weight and position of the moving Dirac masses that desribe th
 e population. We will show that a  new type of Hamilton-Jacobi equation wi
 th constraints naturally describes this asymptotic. Some additional theore
 tical questions as uniqueness for  the limiting H.-J. equation will also b
 e addressed.\n\nThis work is a collaboration with O. Diekmann\, P.-E. Jabi
 n\, S. Mischler\, S. Cuadrado\, J. Carrillo\, S. Genieys\, M. Gauduchon  a
 nd G. Barles.\n
LOCATION:MR14\, CMS
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