Abstract
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.
Highlights
Microorganisms live in environments that are often severely limited in resources or in which vital inputs such as light and nutrients fluctuate unpredictably
For this model we prove that asymptotically, one can expect a flocking behavior for the particles, i.e., a motion for which all particles move towards the light source with identical velocities
The dynamics of the particle system is assumed to be dominated by the external force that is due to the light source time-asymptotically
Summary
Microorganisms live in environments that are often severely limited in resources or in which vital inputs such as light and nutrients fluctuate unpredictably. The work [24] was devoted to deriving a system of PDEs for describing phototaxis as the limit of a stochastically interacting many particle system. The forces that control the dynamics of the velocities and of the excitation are divided into external and internal forces For this model we prove that asymptotically, one can expect a flocking behavior for the particles, i.e., a motion for which all particles move towards the light source with identical velocities. We add collisions due to the interaction of particles with the environment These “collisions” incorporate into the model a surface memory effect (that is experimentally observed).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
