I am dealing with the following system of partial differential equations that describes the effects of random motility on bacteria that consume a diffusible substrate:
\begin{cases} b_{t} = \mu b_{xx} + [f(s) - K_e] b\\ s_{t} = D s_{xx} - \frac{f(s)}{\gamma} b\\ s(0,t) = s(1,t) = b(0,t) = b(1,t) = 0\\ s(x,0) = b(x,0) = sin(\pi x). \end{cases}
In this system, b represents the bacterial density and s the substrate concentration.
The function f(s) is the substrate-dependent growth rate of the bacterium. When f(s) is a constant, such that it doesn't depend on s, I know how to calculate a solution of the PDE system. Since the model is more realistic when f does depend on s, I want to derive a solution in which for example $$f(s) = \frac{f_{max}s}{K + s} \text{ (Monod model) }.$$ However, I don't know if it is even possible to derive a solution for the PDE system with this function f. I tried to derive a solution for the PDE system with $f(s) = s$, but I didn't succeed on that.
In addition to that, I looked up some papers to find a function f and a corresponding solution to the PDE system, but I couldn't find one.
Maybe anyone here has an idea. In any case, thanks in advance!