[8] Cell-Invasion : Clustering : Density Dependent Diffusion (DDD) Model

It would be convenient to formulate the clustering effect in the form of a 1D Fokker-Planck equation that can be quickly solved numerically. For that purpose, remember that the main point of the clustering effect is the simultaneous presence of fast and slow diffusing particles at any depth z. Assume we know the fraction P_NC of particles at z which have no closeby partner within their interaction range r_c. Then we can define a local effective diffusion constant as

D_{eff} =  P_{NC} D_{fast} + (1-P_{NC}) D_{slow}.

Consider a small stripe of material around depth z. Let the local 3D particle density within this z-stripe be rho. We neglect any possible correlations between the positions of different particles within the stripe, i.e. we assume spatial Poisson statistics with average density rho. Then the fraction P_NC is the probability that, for an arbitrary particle, the nearest neighbor is found in a distance larger than r_c. The average number of particles in a sphere of radius r_c is given by

\overline{n} = \rho \;\frac{4}{3}\pi r^3_c.

According to Poisson statistics, the probability that n=0 is

P_{NC} = e^{-\overline{n}} = e^{ - \rho \;\frac{4}{3}\pi r^3_c} = e^{- \rho / \rho_c},

with

\rho_c = \frac{3}{4 \pi}\; r^{-3}_c.

Thus we can define a density dependent diffusion constant:

D_{eff}(\rho) \;= \; (e^{-\rho / \rho_c})  D_{fast} \;+\; (1-e^{-\rho / \rho_c}) D_{slow}.

The figure below shows an example:



Using this, we can write our model as a non-linear Fokker-Planck equation for the position and time-dependent particle density:

\frac{d}{dt} \rho(z,t)\; = \;D_{eff}(\rho(z))\; \frac{\partial^2}{\partial x^2} \;\rho(z,t).