EvoLudoLab: Ecological Public Goods Game - Stable limit cycle

Color code:	Cooperators	Defectors

Payoff code:	Low High

Unstable focus Q, stable limit cycle - oscillations

For slightly higher \(r\) the interior fixed point \(Q\) remains an unstable focus but now surrounded by a stable limit cycle - the hallmark of a super-critical Hopf bifurcation. Cooperators and defectors co-exist in never ending periodic oscillations.

The parameters are \(r = 2.76\), \(N = 8\), \(c = 1\), \(b = 0\), \(d = 0.5\) using numerical integration of the replicator equation. The Hopf-bifurcation occurs at \(r_\text{Hopf} = 2.7672\).

Data views

Strategies - Structure	Snapshot of the spatial arrangement of strategies.
Strategies - Structure 3D	3D view of snapshot of the spatial arrangement of strategies.
Strategies - Mean	Time evolution of the strategy frequencies.
Simplex \(S_3\)	Strategy frequencies plotted in the simplex \(S_3\). If no calculation is running, mouse clicks set the initial frequencies of strategies and stops the calculations otherwise (for the ODE solver it switches to backwards integration).
Phase plane 2D	Frequencies plotted in the phase plane spanned by the population density (\(u + v = 1 - w\)) and the relative frequency of cooperators (\(f = u / (u + v)\)). Again, mouse clicks set the initial frequencies of strategies, stop the simulations or switch to backward integration
Fitness - Structure	Snapshot of the spatial distribution of payoffs.
Fitness - Structure 3D	3D view of snapshot of the spatial distribution of payoffs.
Fitness - Mean	Time evolution of average population payoff bounded by the minimum and maximum individual payoff.
Fitness - Histogram	Snapshot of payoff distribution in population.
Structure - Degree	Degree distribution in structured populations.

Game parameters

The list below describes only the parameters related to the public goods 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.

--interest <r>

multiplication factor \(r\) of public good.

--cost <c>

cost of cooperation \(c\) (investment into common pool).

--lonecooperator <l>

payoff for a cooperator if no one else joins the public goods interaction.

--lonedefector <l>

payoff for a defector if no one else joins the public goods interaction.

--basefit <b>

baseline reproductive rate of all individuals. The effective birthrate is affected by the individual's performance in the public goods game and additionally depends on the availability of empty space.

--deathrate <d>

constant death rate of all individuals.

--init <d,c,e>

initial frequencies of defectors d, cooperators c and vacant space e. Frequencies that do not add up to 100% are scaled accordingly.

--inittype <type>

type of initial configuration:

frequency

random distribution with given frequency

uniform

uniform random distribution

monomorphic

monomorphic initialization

mutant

single mutant in homogeneous population of another type. Mutant and resident types are determined by the types with the lowest and highest frequency, respectively (see option --init).

stripes

stripes of traits

kaleidoscopes

(optional) configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Not available for all types of games.