Diffusion of Biological Organisms: Fickian and Fokker-Planck Type Diffusions

Abstract

In this paper we derive diffiusion equations in a heterogeneous environment. We consider a system of discrete kinetic equations that consists of two phenotypes of different turning frequencies. The two phenotypes change their states according to state transition frequencies which depend on the environment. We show that the density of the total population of the two phenotypes converges to the solution of a Fokker-Planck type diffiusion equation if turning frequencies are of higher order than the state transition frequencies. If it is the other way around, i.e., if the state changes many times between each turning, the density converges to the solution of a Fickian diffiusion equation.