A core of mathematical models for biological questions has been developed which address many areas of biology, namely
- Population based Darwinian evolution
- Bacterial colonies self-organization and movement by run and tumble
- Mechanical models of living tissues
- Models for neural assemblies based voltage distribution or elapsed times distribution between spikes
Our approach to Darwinian evolution of large populations is based on integro-differential equations and the selection of the fittest is interpreted as a Dirac concentration of solutions in the trait space. From this perspective, it encompasses a large class of problems which include nonlocal parabolic equations. A remarkable outcome is the use of the Effective Hamiltonian to define the fitness, in particular in the case of the classical problem of the evolution of dispersal. Another new application has been to the selection of the recovery time for cells under stress.
The description of run and tumble movement of bacteria uses the kinetic formalism where the population is described by space, velocity and internal state of the cells. The main questions covered by Adora concern multiscale analysis and the relations between various levels of modeling (macroscopic, kinetic, kinetic with internal states). The emergence of anomalous diffusions is a fascinating issue.
From a mechanical point of view, living tissues can be seen as a porous media flow of cells through the extra-cellular matrix. Departing from a number of models from the literature, our aim has been to unify them and understand how they are connected. Here, the incompressible limit plays an important role and a major discovery is that a free boundary problem describes the tumor edge. The major difficulty is to handle systems when nutrients or chemotaxis are included. Compactness methods based on Aronson-Benilan estimates are used and allow to establish optimal regularity and the so called 'complementarity relation'.
Models for neural assemblies also lead to nonlinear integro-differential equations for the population of neurones structured by voltage, conductances, time elapsed between spikes or a combination of those. In particular voltage-conductance models are degenerate parabolic with specific boundary conditions, and regularity has been established. Also voltage equations have been interpreted as 'macroscopic limits' of voltage-conductance models in the terminology of kinetic equations.
While the main goal was the mathematical understanding of these problems, it is remarkable that they also opened the route for discussions and collaborations with biologists.
