Einstein Paradigm for Brownian Motion for non-linear chemotactic system

We developed Einstein paradigm of Brownian motion to build chemotactic system PDE for generic chemotactic processes. In some particular cases applied model to study the movement of the living organism in a band form towards the limited 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 of the bacteria and the random motion of bacteria arising from chemotactic response, we also consider the formation of bacterial crowd via interactions within or between the bacterial community. This crowd effect can also be seen as any organism travel or migrate in a herd/group in search of food. We have shown that under specific hypnotize on expected value of free jump of living organism from Einstein approach results in the Keller- Segel model in the depletion of substrate. On the other hand, under no specific adjustment, traveling band has been yield and explained accordingly.