fourier analysis – Conjugated Diffusion Equations

I am new to mathematical-biology and I have to solve the following (diffusion-like) equation
frac{partial a(x,t)}{partial t}= D frac{partial^2 a(x,t)}{partial x^2}\
frac{partial b(x,t)}{partial t}=-v frac{partial b(x,t)}{partial x}-mu a(x,t) b(x,t) \
frac{partial c(x,t)}{partial t}= gamma frac{partial^2 c(x,t)}{partial x^2}+k b(x,t)

for the functions a(x,t), b(x,t) and c(x,t), subject to boundary conditions
$a(x,trightarrow infty)=0$, $a(xrightarrow infty,t)=0$,
$frac{partial c(x,t)}{partial(x)}|_{x=0}=frac{partial a(x,t)}{partial x}|_{x=0}$, $b(L,t)=b_L$, $c(0,t)=0$, and $frac{partial c(x,t)}{partial x}|_{x=L}=0$,
where $D, v, mu,gamma, k$ and $b_l$ are constants.

I tried to attack this problem using, first, the separation of variables method. Although I do obtain a “simple” answer for a(x,t) written in terms of exponentials, I am stuck at the second equation. More precisely, this term with $-mu a(x,t) b(x,t)$ became a nightmare to me. I would
NDSolveValue but, so far, only erros pop out from my mathematica notebook