Distinguished Lecture in Celebration of the 10th Anniversary of INS——Adaptive Evolution and Concentrations in Parabolic PDEs



Benoit Perthame, Laboratoire J.-L. Lions, Sorbonne Université, France


2020.09.15 16:00-17:00





Conference ID: 924-638-29357

PIN Code: 592664


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’.


Benoit PERTHAME received his PhD in 1983 and his Doctorat d’Etat in 1987 from Universite Paris-Dauphine. He was an assistant professor at Ecole Normale Suprieure from 1983 to 1988 and a professor at Universite d’Orleans from 1988 to 1993. Then, he joined the Universite Pierre et Marie Curie (now Sorbonne-Universite) as a full professor. From 1997 to 2007, he was on leave as a professor at Ecole Normale Suprieure. He founded the Inria team Bang in 1989 and served at CNRS national committee from 2005 to 2012. From 2013 to 2020 he served as a Director of Laboratoire Jacques-Louis Lions. He gave an Invited Lecture in 1994 and a Plenary Lecture in 2014 at the International Congress of Mathematicians, and was elected into the French Academy of Sciences in 2017.

He is known for inventing the semi-relaxed limits in Hamilton-Jacobi equations, kinetic formulations of conservation laws, the kinetic averaging lemma. His research activities, now organized within the ERC advanced grant Adora, are oriented towards mathematical modeling and analysis in the life sciences, including cell population self-organization, mean-field models in neuroscience, adaptative evolution theory and models of tissue growth.