= Solution
Substitute $gb=cg'$ into the <bacterial> profile equation. This gives
$$
b''+cb'+cg'=0,
$$
which integrates once to
$$
b'+c(b+g)=C.
$$
At the asymptotic constant states the profile <derivative> tends to zero. The state ahead of the <travelling wave>, where $b\to0$ and $g\to1$, therefore gives $C=c$. The state behind it, where $g\to0$ and $b\to b_0$, gives $C=cb_0$. Since $c>0$,
$$
\boxed{b_0=1,\qquad b'+c(b+g)=c.}
$$
This <first integral> expresses the matching between biomass produced and <biological growth factor> consumed after the chosen <nondimensionalization>.
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