Reaction-diffusion systems from kinetic models for bacterial communities on a leaf surface

TL;DR

This paper derives reaction-diffusion equations from kinetic models to better understand bacterial community patterns on leaf surfaces, highlighting nonlinear and cross-diffusion effects and their relation to microscopic interactions.

Contribution

It provides a systematic derivation of macroscopic reaction-diffusion systems from kinetic Boltzmann equations, including nonlinear and cross-diffusion, for bacterial populations.

Findings

Reaction-diffusion equations can be derived from kinetic models with nonlinear effects.

The approach links microscopic parameters to macroscopic diffusion and reaction coefficients.

Pattern formation and Turing instabilities are analyzed for bacterial populations.

Abstract

Many mathematical models for biological phenomena, such as the spread of diseases, are based on reaction-diffusion equations for densities of interacting cell populations. We present a consistent derivation of reaction-diffusion equations from systems of suitably rescaled kinetic Boltzmann equations for distribution functions of cell populations interacting in a host medium. We show at first that the classical diffusive limit of kinetic equations leads to linear diffusion terms only. Then, we show possible strategies in order to obtain, from the kinetic level, macroscopic systems with nonlinear diffusion and also with cross-diffusion effects. The derivation from a kinetic description has the advantage of relating reaction and diffusion coefficients to the microscopic parameters of the interactions. We present an application of our approach to the study of the evolution of different bacterial populations on a leaf surface. Turing instability properties of the relevant macroscopic systems are investigated by analytical methods and numerical tools, with particular emphasis on pattern formation for varying parameters in two-dimensional space domains.