Modeling group dynamics of phototaxis: From particle systems to PDEs

This work presents a hierarchy of mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move towards light. Based on experimental observations, we conjecture that the motion of the colony towards light depends on certain group dynamics. This group dynamics is assumed to be encoded as an individual property of each bacterium, which we refer to as ’excitation’. The excitation of each individual bacterium changes based on the excitation of the neighboring bacteria. Under these assumptions, we derive a stochastic model for describing the evolution in time of the location of bacteria, the excitation of individual bacteria, and a surface memory effect. A discretization of this model results in an interacting stochastic many-particle system. The third, and last model is a system of partial differential equations that is obtained as the continuum limit of the stochastic particle system. The main theoretical results establish the validity of the new system of PDEs as the limit dynamics of the multi-particle system.