EvoLudoLab: Rock-Paper-Scissors - SDE with Mutations

Color code:	Rock	Scissors	Paper
	New rock	New scissors	New paper

Payoffs:	Low High

Note: The gradient of the payoff scale is augmented by pale shades of the strategy colours to mark payoffs that are achieved in homogeneous populations of the corresponding type.

Stochastic dynamics with mutations - Langevin equation

The interior fixed point \(\hat x\) of the replicator dynamics is an unstable focus. Even without stochasticity all trajectories spiral away from \(\hat x\) toward the boundary of the simplex \(S_3\). However, due to mutations, the boundary is repelling, which results in a stochastic analog of a stable limit cycle. For larger mutation rates the interior fixed point becomes stable again even for \(s<1\).

The parameters are \(s = 0.2\), \(N=1000\), \(\mu=0.005\) using numerical integration of the stochastic differential equation based on Euler's method with \(dt=0.01\).

Data views

Strategy - Population	Snapshot of the spatial arrangement of strategies.
Strategy 3D - Population	Snapshot of the spatial arrangement of strategies in 3D.
Strategy - Mean	Time evolution of the strategy frequencies.
Strategy - Simplex \(S_3\)	Trajectories of strategy frequencies shown in the simplex \(S_3\). Double clicks in the interior of \(S_3\) set the initial frequencies of strategies.
Fitness - Population	Snapshot of the spatial distribution of payoffs.
Fitness 3D - Population	Snapshot of the spatial distribution of payoffs in 3D.
Fitness - Mean	Time evolution of average population payoff bounded by the minimum and maximum individual payoff.
Fitness - Histogram	Payoff distribution of each strategy in population.
Degree - Histogram	Degree distribution in structured populations.
Console Log	Displays messages, warnings and errors reported by the simulation engine plus information on the applet/application.

Game parameters

The list below describes only the parameters related to the rock-scissors-paper game and the population dynamics. Follow the link for a complete list and descriptions of all other parameters such as spatial arrangements or update rules on the player and population level.

--paymatrix <rr,rs,rp;sr,ss,sp;pr,ps,pp>

general \(3\times 3\) payoff matrix for the three strategic types \(R, S, P\).

--init <r,s,p>

initial frequencies of \(R, S, P\), respectively. Frequencies that do not add up to 100% are scaled accordingly.

--inittype <type>

type of initial configuration:

frequency <f0>,<f1>...: random distribution with given trait frequencies, f0, f1,.... Note, only available for frequency based modules and models.
density <d0>,<d1>...: initial trait densities <d1,...,dn>. Note, only available for density based modules and models.
uniform: uniform random distribution, equal frequencies of all traits.
monomorphic <t>[,<v>]: monomorphic initialization with trait t. Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the monomorphic trait is initialized at its (estimated) carrying capacity.
mutant <m>,<r>[,<v>]: single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected uniformly at random (mutants arising through cosmic rays). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
temperature <m>,<r>[,<v>]: single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected proportional to the in-degree of nodes (temperature initialization, mutants arising through errors in reproduction). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
stripes: stripes of traits. Note, only available for 2D lattices.
kaleidoscopes: configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Note, only available for some modules.

Note, for modules that admit multiple species, the initialization types for each species can be specified as an array separated by ;. With more species than initialization types, they are assigned in a cyclical manner.