Adaptive evolution and concentrations in parabolic PDEs

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• Benoit Perthame (Université Paris VI)
• Thursday 28 February 2008, 15:00-16:00
• MR14, CMS.

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Living systems are subject to constant evolution through the mutation/selection principle discovered by Darwin. In a very simple and general description, their environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a ‘physiological 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 variance on the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?

We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precicely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the ‘best adapted trait’ and eventually to compute branching points which lallows for the cohabitation of two different populations.

The concepts are based on the asymptotic analysis of the scaled parabolic equations. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that desribe the population. We will show that a new type of Hamilton-Jacobi equation with constraints naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.

This work is a collaboration with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon and G. Barles.

This talk is part of the Applied and Computational Analysis series.

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