Solution
= Solution
The probability of an exact copy is $Q=q^L$. Incorrect copies contribute mutants at rate $a(1-Q)x$, giving $\boxed{\dot y=by+a(1-Q)x}$. For $y>0$, differentiating the ratio gives
$$
\boxed{\dot z=(aQ-b)z-a(1-Q)z^2.}
$$
This is a logistic-type equation for relative concentration; the absolute concentrations need not have a stationary value.