I stumbled upon this research article, where they define a digital organism as an abstract minimal model of an evolving predator-prey system as follows:
An organism is defined via its genome of fixed length 2048, consisting of upper- and lowercase letters, i.e. 52 possible letters. All but eight letters are inactive and do not influence the interactions between organisms. Of the eight active letters, four are offensive (A, B, C, D) and four are defensive (a, b, c, d). When an organism replicates, its genome is subject to change from point mutations, which occur at a rate $r_\mathrm{m}$ per letter and set the mutated letter to a random letter, which may be the same as the original. In this system, the complexity is taken to be the longest functional string (separated into attack and defense complexities). Thus if an organism has a particular active string of length L, there are L chances for a point mutation to decrease the complexity, and 2 chances for a point mutation to increase the complexity or more specifically: If a mutation occurs at the first letter before or after the string, there is a $\frac{1}{13}$ chance that the length of the active string increases by 1. On the other hand, if a mutation occurs anywhere within the string, there is a $\frac{12}{13}$ chance that the active length will decrease.
Now they define the average resultant length $L$ of an active string, initially of length $L_\mathrm{0}$, after a single point mutation as \begin{equation} \langle L \rangle = \frac{3}{4}L_{\mathrm{0}} - \frac{1}{2} - \frac{1}{4L_{\mathrm{0}}} \end{equation}
How exactly do they end up with the formula for $\langle L \rangle$? I would be grateful for any help/tips.
Article: Arxiv Link
