Team:KU Leuven/Modeling/Top
1-D continuous model
The Keller segel model used is [1] : $$\frac{\partial A}{\partial t} = D_a \bigtriangledown^2 A + k_A A(1 - \frac{A}{k_{p}}),$$ $$\frac{\partial B}{\partial t} = D_b \bigtriangledown^2 B + \bigtriangledown (X(B,H,R) \bigtriangledown R)+ k_B B(1 - \frac{B}{k_{p}}), $$ $$ \frac{\partial R}{\partial t} = D_r \bigtriangledown^2 B + k_r A - k_{lossH} R $$ $$\frac{\partial H}{\partial t} = D_h \bigtriangledown^2 B + k_h A - k_{lossR} H . $$ With: $$ X(B,H,R) = \frac{-B K_{c1} H}{K_{c2} R}. $$ When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
References
Spatio-Temporal Patterns Generated by Salmonella typhimurium, D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. 0.Budrene,l and H. C.Berg , Biophysical Journal Volume 68 May 1995 2181-2189
