= Solution
Take profiles $b=b(y)$ and $g=g(y)$ with $y=x-ct$, and use a prime for <differentiation> with respect to $y$. The <chain rule> gives $b_t=-cb'$, $b_{xx}=b''$ and $g_t=-cg'$. Therefore
$$
\boxed{b''+cb'+gb=0,\qquad cg'=gb.}
$$
These are coupled <ordinary differential equations> for a right-moving <travelling wave> with speed $c>0$. A first-order formulation, using $v=b'$, is $b'=v$, $v'=-cv-gb$, $g'=gb/c$. In particular a positive profile has $g'>0$, consistent with depleted substrate behind the <travelling front> and fresh <biological growth factor> ahead.
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