 Solution

For $0<q<1$, a finite positive equilibrium exists exactly when $a{q}^L>b$, and thenThe derivative of the ratio's vector field at $z_{\ast }$ is $-(a{q}^L-b)<0$, 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.