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MESOPROBIO Report Summary

Project ID: 639638
Funded under: H2020-EU.1.1.

Periodic Reporting for period 1 - MESOPROBIO (Mesoscopic models for propagation in biology)

Reporting period: 2015-09-01 to 2017-02-28

Summary of the context and overall objectives of the project

This project aims at unraveling propagation phenomena arising in the life sciences by means of multi-scale models. These models incorporate several biologically relevant variables at once (position of individuals, age, dispersal ability, life-history traits to name but a few). It is a great mathematical challenge to investigate these models without separating the various scales. This is of particular relevance when evolutionary processes are at play during the propagation phenomena. We are working on several case studies (concentration waves of bacteria, invasion of the cane toads in Northern Australia). The mathematical models we are studying belong to the wide class of kinetic equations. However, they depart significantly from the equations which are found usually in mathematical physics such as the Boltzmann equation for the kinetic theory of gases. Indeed, biological problems offer new research highlights to put mathematical efforts on.

We are analyzing traveling waves, spreading phenomena, and the approximation of geometric optics in the context of structured models. Tools are inspired from classical problem in mathematical physics. But they are adapted to the biological side in a suitable, non trivial, way. By using these modern tools, we aim at developing a new theory of propagation phenomena and adaptive dynamics at the mesoscale.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We have analyzed traveling waves in velocity-, dispersion-, and age-structured models. This finds applications in the study of concentration waves of chemotactic bacteria, the rate of acceleration of the cane toads wave expansion, and also the evolution of age-structured populations, including the theory of senescence. The construction of traveling wave solutions to the kinetic model of chemotactic bacteria is based on several new mathematical ideas. On the other hand, the approximation of geometric optics was found to being relevant in several problems which were apparently not related with each other. In particular, it was surprising to make a formal correspondence between the adaptive dynamics of asexual and sexual modes of reproduction in quantitative genetics models, although the underlying mechanisms are different. This allows to make an ambitious quantitative studies, e.g. deriving analytical formulas for the adaptation of a population to environmental change. This works goes beyond the classical approximations. It enables to by-pass numerical simulations as it offers a synthetic view on a number of complex problems.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Clearly, the two main achievements which were not expected at the beginning of the project are: 1/ the construction of concentration waves of chemotactic bacteria in a coupled kinetic-parabolic model for a population of bacteria in a dynamic environment. The problem was mathematically very challenging, and the solution came after a long analytical work by the PI. This opens a wide range of problems in the mathematical community. Nonetheless, counter-intuitive phenomena were discovered thanks to a close collaboration between the PI and experts in numerical analysis ; 2/ the approximation of geometric optics for recombination processes via the Fisher infinitesimal model. This opens a new theory which make possible a comparison analysis between several kinds of models of quantitative genetics.
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