Section: Research Program
Research axis 2: reaction and motion equations for living systems
Luis Almeida, Casimir Emako-Kazianou, Alexander Lorz, Benoît Perthame, Nicolas Vauchelet.
The Mamba team had initiated and is a leader on the works developed in this research axis. It is a part of a consortium of several mathematicians in France through the ANR Blanc project Kibord, which involves in particular members from others Inria team (DRACULA, REO). Finally, we mention that from Sept. 2017 on, Mamba benefits from the ERC Advanced Grant of Benoît Perthame.
We divide this research axis, which relies on the study of partial differential equations for space and time organisation of biological populations, according to various applications using the same type of mathematical formalisms and methodologies: asymptotic analysis, weak solutions, numerical algorithms.
Mathematical modelling for bacterial chemotaxis.
Chemotaxis is the phenomenon in which cells direct their motion in response to a chemical signal present in their environment. Our unique expertise is on mathematical aspects of the kinetic equations which describe the run and tumble motion of bacteria and their asymptotic analysis.
An interdisciplinary collaboration with biophysicists from Institut Curie has been successful on experimental observations concerning the interaction between two species of bacteria and emergence of travelling bands . The mathematical models used in this work are derived in  thanks to a diffusive limit of a kinetic system with tumbling modulation along the path. A numerical investigation of this limit is provided in . These works enter into the framework of the PhD of Casimir Emako-Kazianou . Recently, we have been able to derive rigorously such kinetic models from a more sophisticated equation incorporating internal variable when cells adapt rapidly to changes in their environment .
In the mathematical study of collective behaviour, an important class of models is given by the aggregation equation. In the presence of a non-smooth interaction potential, solutions of such systems may blow up in finite time. To overcome this difficulty, we have defined weak measure-valued solutions in the sense of duality and its equivalence with gradient flows and entropy solutions in one dimension . The extension to higher dimensions has been studied in . An interesting consequence of this approach is the possibility to use the traditional finite volume approach to design numerical schemes able to capture the good behaviour of such weak measure-valued solutions , .
Free boundary problems for tumour growth.
Fluid dynamic equations are now commonly used to describe tumour growth with two main classes of models: those which describe tumour growth through the dynamics of the density of tumoral cells subjected to a mechanical stress; those describing the tumour through the dynamics of its geometrical domain thanks to a Hele-Shaw-type free boundary model. The first link between these two classes of models has been rigorously obtained thanks to an incompressible limit in  for a simple model. This result has motivated the use of another strategy based on viscosity solutions, leading to similar results, in .
Since more realistic systems are used in the analysis of medical images, we have extended these studies to include active motion of cells in , viscosity in  and proved regularity results in . The limiting Hele-Shaw free boundary model has been used to describe mathematically the invasion capacity of a tumour by looking for travelling wave solutions, in , see also axis 3. It is a fundamental but difficult issue to explain rigorously the emergence of instabilities in the direction transversal to the wave propagation. For a simplified model, a complete explanation is obtained in .