Einstein Paradigm and Application to Chemotaxis system

We Derive Einstein model of Brownian Motion with Schematic’s in form of system of PDE. We incorporate Einstein’s method of Brownian motion to deduce the chemotactic model exhibiting a traveling band. To our knowledge this was the first time that Einstein’s method has been used to motivate equations describing the mutual interaction of the chemotactic system. We have shown that in the presence of limited and unlimited substrate, traveling bands are achievable and it has been explained accordingly. We also study the stability of the constant steady states for the system. The linearized system about a steady state is obtained. We are able to find explicit conditions for linear instability and stability under homogeneous boundary conditions.