Theoretical Background
General information about a Moran Process are easily available on the internet. The following series of short lectures[1] provides an excellent introduction to the topic:
Having the formulas written down for two species we can easily generalise the model for any given number of individuals’ types. Let’s imagine we have a population \(N = (N_1,\dotsc,N_n)\) composed of distinct types of individuals \(T = (T_1,\dotsc,T_n)\) and the whole system is defined by a \(M_{n,n}\) payoff matrix. For each subpopulation \((N_x, T_x)\) we calculate average payoffs (\(\overline{\pi}\)) and fitness (\(f\)) as below:
\[\begin{split}\overline{\pi}_x &= \frac{(N_x-1) \times M_{x,x} + \sum_{i\neq x} N_i \times M_{x,i}}{N-1} \\
f_x &= 1 - w + w \times \overline{\pi}_x\end{split}\]
These values might be later used in the simulations during fitness-proportional selection of individuals for the Birth process.