The construction of traveling wave solutions for concentration waves of chemotactic bacteria E. coli results from a delicate interplay between several scales: a macroscopic view where bacteria navigate in a heterogeneous environment, searching for food and for relatives, and a microscopic view about the details of individual locomotion. The mesoscopic scale is suitable to encompass both viewpoints. Hundred of thousands of bacteria are described as a cloud with statistical properties. Bacteria move collectively while modifying their chemical environment (communication signals, food). The main mathematical challenge consists in making the cloud of bacteria and the chemical environment moving at the same speed which is a good candidate for being the collective speed. This work concludes a long term research program initiated 10 years ago to decipher collective motion of bacteria.
This analysis initiated a long term study about front propagation in kinetic equations. The standard approximation of geometric optics for reaction-diffusion equations can be extended to kinetic models, leading to a new type of "kinetic" Hamilton-Jacobi equations in the limit of short wavelength. This enables to compute the exact rate of front propagation, and possibly front acceleration. Appropriate numerical schemes can be designed to capture the various regimes at once.
There is a unique case study of front acceleration in the field: the invasion of cane toads in Northern Australia (1930-today). Mathematical tools dedicated to kinetic equations can be adapted to cope with this biological problem. The exact rate of expansion in a minimal model can be calculated with good accuracy, revealing the subtle role of competition in the range expansion, on the contrary to standard results in this theory where competition can be safely neglected to calculte the rate of propagation in most cases.
Again, the standard approximation of geometric optics can be adapted to quantitative genetics models, when the adaptation of species to evolutionary forces can be described via a continuous trait. Interestingly, this provides formulas for measuring the maladaptation of a species to a gradual change of environment. This method is so robust that it can encompass several modes of reproduction, and various details about the species life-cycle (such as age-dependent selection).
New equations emerge from this project. Our results also contain solid mathematical grounds for some of them. Also, the methodogy performed here is of considerable interest for interdisciplinary purposes because it gives access to quantitative results easily transferable to the biologists. Hence, the outcomes and the hypotheses of the models can be thoroughly discussed.
The main results were presented in several seminars, conferences and research schools, including prestigious ones (Collège de France, ECM Berlin, AIMS plenary).
