Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 6E (Mathematical Biology)

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 givesThis is a logistic-type equation for relative concentration; the absolute concentrations need not have a stationary value.

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.

Write $ϵ=1-q$ and $b/a=1-s$. Then $\log (1-s)=-s+O(s^2)$ and $\log q=-ϵ+O({ϵ}^2)$, givingThe largest permitted integer is still selected by the strict inequality from part (b). This says that longer master sequences need a proportionately larger selective advantage or smaller per-digit copying error.