Let $x$ be the concentration of a binary master sequence of length $L$ and let $y$ be the total concentration of all mutant sequences. Master sequences try to self-replicate at a total rate $a x$, but each independent digit is only copied correctly with probability $q$. Mutant sequences self-replicate at a total rate $b y$, where $a>b$, and the probability of mutation back to the master sequence is negligible.

(a) The evolution of $x$ is given by

$$\frac{d x}{d t}=a q^{L} x$$

Write down the corresponding equation for $y$ and derive a differential equation for the master-to-mutant ratio $z=x / y$.

(b) What is the maximum length $L_{\max }$ for which there is a positive steady-state value of $z$ ? Is the positive steady state stable when it exists?

(c) Obtain a first-order approximation to $L_{\max }$ assuming that both $1-q \ll 1$ and $s \ll 1$, where the selection coefficient $s$ is defined by $b=a(1-s)$.