A colloquium series in Mathematics and Computer Science

Organized by:


Prof. Benoit Perthame (Université Pierre et Marie Curie)
Adaptive evolution and concentrations in parabolic PDEs

April 19, 2016 - 4:00pm


Living systems are characterized by variability; in the view of C. Darwin, they are subject to constant evolution through the three processes of population growth, selection by nutrients limitation and mutations. Several mathematical theories have been proposed in order to describe the dynamics generated by the interaction between their environment and the trait selection of the `fittest'. One can use stochastic individual based models, dynamical systems, game theory considering traits as strategies. From a populational point of view, the population obeys an integro-partial-differential equation for the density number.
We will give a self-contained mathematical model of such dynamics and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the fittest trait and eventually to compute various forms of branching points which represent the cohabitation of two different populations. The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Recent developments concern non-proliferative advantages and lead to define the notion of `effective fitness'.

This talk is based on collaborations with G. Barles, O. Diekmann, M. Gauduchon, S. Genieys, P.-E. Jabin, A. Lorz, S. Mirahimmi, S. Mischler and P. E. Souganidis.

Short bio

Benoît Perthame was born in 1959 in France. He studied at the École Normale Supérieure-Ulm (ENS) and was there assistant from 1983 through 1988. He received the habilitation (Thèse d'Etat) at ENS with Pierre-Louis Lions in 1987. In 1988 he became professor at the University of Orléans and since 1993 he has been a professor at the Pierre & Marie Curie University (Paris 6) and at the Institut Universitaire de France. He has headed the Jacques-Louis Lions laboratory since 2013 and joined the Institut national de recherche en informatique et en automatique (Inria) in 1989, where he led the BANG project team (Biophysics, Numerical Analysis and Geophysics) from 2004 onwards. Perthame has received numerous distinctions and awards including the Pècot prize from the Collège de France, the Blaise Pascal prize from the European Academy of Science and the CNRS (French National Centre for Scientific Research) silver medal. He was invited speakers at the International Congress of Mathematicians in Zurich in 1994 and plenary speaker at the International Congress of Mathematicians in Seoul in 2014. Benoît has been a driving force and a source of inspiration in the field of partial differential equations training some of the best young experts of the field in the various positions that he has occupied. Benoît has been exploring many areas of partial differential equations with a distinguished quality and vision, particularly impressive at finding simple hidden ways to attack problems which seem so technical and intricate. He gained fame from his works in kinetic equations which are basic models in statistical physics of gases and plasmas. He was in particular at the origin of the so called ``velocity averaging lemmas'' establishing partial regularity for averages of solutions of transport equations and moment propagation estimates which are at the heart of the first theory of smooth solutions for the Vlasov equations. More recently, about fifteen years ago, Benoît focused his interest in the interplay between mathematics and biology, a field that has attracted an increasing attention in the mathematic community, starting with the work of Alan Turing in the fifties, but in which nearly everything still remains to be understood. Among other things, Perthame has studied mathematical modeling of chemotaxis, movement and self-organization of cells and bacterial colonies, neural networks, tumor growth and chemotherapy, kidney flows, growth of populations and evolution.
More info available at http://www.ann.jussieu.fr/~perthame/