The ANR MODEVOL is a project funded by the french funding agency "Agence Nationale de la Recherche". This project started on February 2014, and will run until July 2017. It gathers the following researchers:

- Gaël Raoul (coordinator), CMAP, Ecole Polytechnique,
- Sepideh Mirrahimi, IMT, University Paul Sabatier,
- Jérôme Coville, BIOSP, INRA Avignon,
- Nicolas Champagnat, IECL, INRIA Nancy,
- Laurent Desvillettes, IMJ, Université Paris-Diderot,
- Sylvain Gandon, CEFE Montpellier.

Summary:

Theoretical models have always played an important role in evolutionary biology, since ex- perimental results are often difficult to obtain. Thanks to the rapid evolution of experimental methods as well as new areas of interest emerging from medical, societal or ecological appli- cations, increasingly complex evolutionary problems have to be addressed, and the limitations of available models are becoming obvious. New models and original mathematical methods are necessary to move forward. This project has been built after extensive discussions with evolu- tionary biologists, and targets several important questions. Those questions have been chosen because the mathematical problems that they raise are deep and challenging, but also because the result we would build should answer biological questions.

An important difficulty faced by evolutionary biologists is dealing with spatially structured populations. The models they consider usually when studying evolutionary phenomena can only deal with basic forms of spatial structure (typically meta-populations) that are not satisfactory for problems such as evolutionary epidemiology or the effect of global warming. We will consider both asexual and sexual populations, and in both cases, the models that are able to describe the evolution of spatially structured populations present real mathematical difficulties. In the first case, the models are based on reaction-diffusion equations with non-local terms, for which some result recently appeared, but are still poorly understood. The models for sexual populations are quite different, and even defining solutions for such equations is a challenging task. However, preliminary works on those models have given us some new insights, and we believe that we can obtain interesting mathematical results.

A second problem that we would like to address is an example of transitory evolution, that is, situations where a population is away from any demographical equilibrium. Here also, the available theoretical tools fail. We focus on the problem of evolutionary rescue, when a population is submitted to a sudden change of its environment. A typical example is a population of bacteria submitted to an antibiotic. The fate of the population will depend on the few mutants that may have acquired resistance to the antibiotic. Modeling the survival probability of the population then requires precise estimates on the tails of the phenotypic distribution of the population. One of our objectives will be the analysis of those tails.

To achieve those goals, we will put together the complementary skills of specialists of kinetic equations, reaction-diffusion equations, probability theory, as well as those of an evolutionary biologist. All the members of the project have already worked on problems related to its themes, and are willing to create innovative mathematical approaches to those problems. We plan to work in tight collaboration, thanks to frequent visits and regular meetings. We will also popularize our results in the evolutionary biology community through publications in biology journals and short workshops.

List of publications part of this project:

- P. Degond, A. Frouvelle, G. Raoul, Local stability of perfect alignment for a spatially homogeneous kinetic model, Journal of Statistical Physics 157(1), 84-112 (2014).
- S. Mirrahimi and J.-M. Roquejoffre, Uniqueness in a class of Hamilton-Jacobi equations with constraints, Comptes Rendus Mathematique, 353, 489-494 (2015).
- S. Méléard and S. Mirrahimi, Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity, Communications in Partial Differential Equations, 40(5), 957-993 (2015).
- Q. Griette, G. Raoul, S. Gandon, Existence and qualitative properties of travelling waves for an epidemiological model with mutations, submitted to a biology journal.
- S. Mirrahimi and J.-M. Roquejoffre, A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach, Submitted.
- Q. Griette, G. Raoul, Existence and qualitative properties of travelling waves for an epidemiological model with mutations. Submitted.
- H. Berestycki, J. Coville, H.H. Vo, On the principal eigenvalue of nonlocal operator in unbounded domain, submitted.