# EvoLudoLab: Moran process on the cycle graph

Color code: Residents Mutants New resident New mutant
 Payoff code: Residents Mutants

## Evolutionary dynamics on the complete graph

The Moran process on a cycle is best illustrated as a linear graph (1D lattice) with periodic boundaries such that every vertex is connected to its two neighbours on the left and right. The invasion process of a mutant can then be easily illustrated over time by stacking subsequent snapshots of the population state. Each row indicates the population state at a particular time such that the most recent state is at the top and and towards the bottom of the figure are population states in the increasingly distant past.

In this graphical representation it is easy to see that mutants invade by forming a single, growing cluster. Due to the structure of the graph, there will always be at most two clusters, one of residents and another of mutants. Because of the limited opportunities for mutants to spread, the invasion process is significantly slower than on the complete graph or in unstructured populations.

For the simulations, the population size is \(N=100\). The fitness of residents is set to \(1\) and that of mutants to \(2\). Thus, a single mutant has approximately a \(50\%\) chance to take over the population. Typically it takes around \(120\) generations for the mutant to reach fixation (as compared to around \(12\) on the complete graph).

### Data views

 Snapshot of the spatial arrangement of strategies. 3D view of snapshot of the spatial arrangement of strategies. Time evolution of the strategy frequencies. Snapshot of the spatial distribution of payoffs. 3D view of snapshot of the spatial distribution of payoffs. Time evolution of average population payoff bounded by the minimum and maximum individual payoff. Snapshot of payoff distribution in population. Degree distribution in structured populations. Statistics of fixation probability for each vertex where the initial mutant arose. Statistics of conditional fixation times of residents and mutants as well as absorption time for each vertex where the initial mutant arose. Message log from engine.

## Game parameters

The list below describes only the few parameters related to the evolutionary dynamics of residents and mutants with fixed fitness (constant selection). Numerous other parameters are available to set population structures or update rules on the player as well as population level.

--resident <r>
fitness of resident r.
--mutant <m>
fitness of mutant m.
--initfreqs <m:r>
initial frequencies of residents r and mutants m. Frequencies that do not add up to 100% are scaled accordingly. If either frequency is zero, the population is initialized to a homogenous state with just a single, randomly placed individual of the opposite type.