= Solution
For $0<q<1$, a finite positive equilibrium exists exactly when $a q^L>b$, and then
$$
\boxed{z_*=(a q^L-b)/[a(1-q^L)].}
$$
The derivative of the ratio's <vector field> at $z_*$ is $-(aq^L-b)<0$, so this equilibrium is locally asymptotically stable; every positive initial ratio tends to it. The critical real length is $L_c=\log(b/a)/\log q$, and the admissible integer lengths satisfy $L<L_c$. Thus the largest positive integer length, if one exists, is $\boxed{L_{\max}=\lceil L_c\rceil-1}$. At equality the equilibrium is zero, so rounding down without accounting for the strict inequality can be wrong. If $q=1$, mutants are never produced from masters and the ratio grows without a finite positive equilibrium; the finite threshold formula concerns imperfect copying.
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