Einstein's Brownian motion model for chemotactic system and traveling band

TL;DR

This paper introduces a novel chemotactic model inspired by Einstein's Brownian motion, capturing the formation of traveling bands of organisms responding to chemical substrates and their crowd interactions.

Contribution

First application of Einstein's Brownian motion to derive equations for chemotactic systems with crowd effects and traveling band formation.

Findings

Traveling bands form under limited and unlimited substrate conditions.

The model incorporates crowd interactions, reflecting herd behavior.

Traveling bands are explained through the proposed PDE system.

Abstract

We study the movement of the living organism in a band form towards the presence of chemical substrate based on a system of partial differential evolution equations. We incorporate the Einstein's method of Brownian motion to deduce the chemotactic model exhibiting travelling band. It is the first time that Einstein method has been used to motivate equations describing mutual interaction of chemotactic system. In addition to considering chemotactic response and the random motion of organism, we also consider the formation of crowd by organism via interactions within or between the community. This crowd effect can also be seen as any organism travel or migrate in a herd or group in search of food. We have shown that in the presence of limited and unlimited substrate traveling bands are achievable and it has been explained accordingly.