Kinetic models in biology and Related fields

It brings together three teams of mathematicians specializing in EDP and already collaborating with teams working in the fields of biology and medicine. Scientists propose a detailed study of the modeling and the mathematical and numerical analysis of problems coming from these fields. These problems have in common the need to study the qualitative properties of EDP systems, which are distinct from all the systems appearing in biology and related fields. 

Develop modeling using Partial Differential Equations (in particular kinetic) and numerical simulation for problems arising in biology

They want to answer questions that arise in various situations in biology (population dynamics, cell movement, oncology, biological fluids, etc.) through the systematic use of the most modern methods in PDE. Modeling is not completely stabilized in many of the problems that they intend to study. so that they participated in a project in which collaborate largely with several teams of biologists/doctors. The themes on which they wish to work are the large agent systems and their structure. The interaction between a large numbers of individuals (chemotaxis for cells, the overall movement for animals, droplets in biological aerosols) can be modeled and simulated on many scales. Typically, they use a large number of EDOs (sometimes stochastic) [microscopic approach], kinetic equations [mesoscopic approach], or hyperbolic / parabolic EDP [macroscopic approach]. - Growth, coalescence, and fragmentation. The evolution of a large number of cells is also the result of complex processes of grouping, fragmentation, and size increase. These processes lead to the writing of non-linear integro differential equations which present a broad spectrum of global behaviors that one wishes to study, the appearance of periodic solutions, explosions infinite time (gelation), etc. The addition of a spatial structure (in the form of diffusion terms) leads to an even richer structure. - New paradigms for reaction-diffusion models and new types of reaction-diffusion systems from biology have recently appeared. They include non-standard terms that radically change mathematical methods. New methods (e.g., computer-aided evidence) are also helping to renew research in this area.

Contemporary methods for the study of Partial Differential Equations and Integral Equations

The methods come from the most recent developments in PDE theory, they include

 - the existence of remarkable stationary states, due to topological methods (degree theory) but also due to computer-assisted proofs, in which the dimension part infinite of equations is treated at the theoretical level, while a finite-dimensional part is treated at the computer level (with rigorous control of all errors)

 - Entropy estimates/entropy dissipation, which allows obtaining quantitative results of convergence towards a state of equilibrium, even when one starts from an initial datum far from equilibrium.

 - Disruptive methods, which include recent developments relating to the lemmas of duality, which allow obtaining parabolic regularity even when the coefficients of the equations are not continuous.

 - Time-dependent scaling, which makes possible the detailed study of the asymptotic behavior of equations not having a non-trivial state of equilibrium, and lead to the construction of profiles.

 - renormalized solutions, which appear naturally when we consider kinetic or parabolic equations in which the a priori regularity of the solutions is not sufficient to rigorously define solutions in the sense of distributions.

 All these methods are used for the different types of equations (kinetic, hyperbolic, parabolic of infinite dimension, integro differential, etc.).

Summary

Here is a detailed study of mathematical/numerical modeling and analysis of problems from different areas of biology, including cell biology, biological fluids, population dynamics, and collective behavior in the animal.

The common point of these problems is the need to examine the qualitative properties and numerical approximations of PDE systems designed specifically to model these problems, and which are distinct from all systems appearing in physics. Most often, they have application-specific characteristics, such as an infinite number of equations for coagulation-fragmentation or cross-diffusion terms for the spatial evolution of intelligent species.

The methods that will be used come from the most recent developments in PDE theory, including the existence of remarkable non-trivial solutions, entropy/entropy dissipation estimates, linear and nonlinear stability of equilibrium, lemmas of duality, computer-assisted proofs, time-dependent scale changes, renormalized solutions, etc. Scientists studied in particular, to extract information from systems that are far from known stationary states.

 This study is structured around three main themes:

 - Large multi-agent systems and their spatial structure. These situations arise when a large number of individuals following a simple law give rise to complex phenomena, as in statistical mechanics. Many levels of modeling lead to different classes of equations (kinetic, parabolic, etc.). Biological issues considered here include swarming, chemotaxis, aerosols, and neural networks.

- Growth, coalescence, and fragmentation. These phenomena appear in biology and (bio) chemistry at different scales (molecules, cells, etc.). They are modeled by an infinite number (sometimes even a continuum) of EDPs, leading to a wide variety of behaviors. The explosion phenomenon known as gelation is particularly interesting, and its study in a spatially inhomogeneous context is a recent and very promising problem.

  - New paradigms for reaction-diffusion models. These models have been used for a long time for the modeling of living organisms. New types of reaction-diffusion equations have recently attracted attention, including cross-reaction-diffusion systems and equations, including non-local terms. Likewise, recently devised methods (such as computer-assisted proofs of existence) have shed new light on classical problems.

In a series of recent works, different models of neural networks have been analyzed, and the qualitative behavior of the solutions has been described in the case of poor connectivity between neurons. Two questions remain particularly important. First of all, what are the most relevant models and what is the qualitative behavior of the solutions beyond the weak connectivity regime. Scientists work directly on these questions.

 New results on the lemmas of duality have made it possible to progress in the knowledge of the reaction-diffusion equations of reversible chemistry and help them to understand the kinetic models in biology.

Author: Vicki Lezama