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SUMMARY:Concentration in parabolic Lotka-Volterra equations : an asymptoti
 c-preserving scheme   Helene Hivert - Helene  Hivert  (École Centrale de 
 Lyon)
DTSTART:20220412T123000Z
DTEND:20220412T131500Z
UID:TALK171554@talks.cam.ac.uk
DESCRIPTION:We consider a population structured in phenotypical trait\, wh
 ich influcences the adaptation of individuals to their environment. Each i
 ndividual has the trait of his parent\, up to small mutations. We consider
  the problem in a regime of long time and small mutations. The distributio
 n of the population is expected to concentrate at some dominant traits. Do
 minant traits can also evolve in time\, thanks to mutations. From a techni
 cal point of view\, the concentration phenomenon is described thanks to a 
 Hopf-Cole transform in the parabolic model. The asymptotic regime is a con
 strained Hamilton-Jacobi equation [G. Barles\, B. Perthame\, 2008 & G. Bar
 les\, S. Mirrahimi\, B. Perthame\, 2009]. The uniqueness of the solution o
 f the limit equation has been adressed very recently [V. Calvez\, K.-Y. La
 m\, 2020]. Because of the lack of regularity of the constraint\, it can in
 deed have jumps\, the numerical approximation of the constrained Hamilton-
 Jacobi equation must be treated with care.&nbsp\;\nWe propose an asymptoti
 c preserving scheme for the problem transformed with Hopf-Cole. We show th
 at it converges outside of the asymptotic regime\, and that it enjoys stab
 ility properties in the transition to the asymptotic regime. Eventually\, 
 we show that the limit scheme is convergent for the constrained Hamilton-J
 acobi equation.
LOCATION:Seminar Room 2\, Newton Institute
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