Particle, kinetic and fluid models for phototaxis

In this work we derive a hierarchy of new mathematical models fordescribing the motion of phototactic bacteria, i.e., bacteria thatmove towards light. These models are based on recent experimentssuggesting that the motion of such bacteria depends on theindividual bacteria, on group dynamics, and on the interactionbetween bacteria and their environment. Our first model is acollisionless interacting particle system in which we followthe location of the bacteria, their velocity, and their internalexcitation (a parameter whose role is assumed to be related tocommunication between bacteria). In this model, the light sourceacts as an external force. The resulting particle system is anextension of the Cucker-Smale flocking model. We prove that whenall particles are fully excited, their asymptotic velocity tends toan identical (pre-determined) terminal velocity. Our second model is a kinetic model for the one-particle distribution functionthat includes an internal variable representing the excitationlevel. The kinetic model is a Vlasov-type equation that is derivedfrom the particle system using the BBGKY hierarchy and molecularchaos assumption. Since bacteria tend to move in areas that werepreviously traveled by other bacteria, a surface memory effect isadded to the kinetic model as a turning operator that accounts forthe collisions between bacteria and the environment. The third andfinal model is derived as a formal macroscopic limit of the kineticmodel. It is shown to be the Vlasov-McKean equation coupledwith a reaction-diffusion equation.