Keller–Segel chemotaxis

We now consider Keller–Segel models of chemotaxis of the form

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u-\vnabla \cdot(\chi(u)\vnabla v)+f_u(u),\\ \pd{v}{t}&=D\nabla^2v+ f_v(u,v),\end{aligned}\]

where we take

\[\begin{aligned}\chi&=\displaystyle\frac{cu}{1+u^2},\\f_u(u)&=u(1-u),\\f_v(u,v) &= u-av.\end{aligned}\]