# EvoLudoLab: Ecological Public Goods Game - Unstable limit cycle

Along the bottom of the applet there are several buttons to control the execution and the speed of the simulations - for details see the EvoLudo GUI documentation. Of particular importance are the parameters button and the data views pop-up list along the top. The former opens a panel that allows to set and change various parameters concerning the game as well as the population structure, while the latter displays the simulation data in different ways.

 Color code: Cooperators Defectors
 Payoff code: Low High

## Stable focus Q - unstable limit cycle (sub-critical Hopf bifurcation)

For non-zero birthrates simple sub-critical Hopf bifurcations are observed. The fixed point \(Q\) is stable and surrounded by an unstable limit cycle.

The parameters are \(r = 2.376\), \(N = 8\), \(c = 1\), \(b = 1\), \(d = 1.2\) using numerical integration of the replicator equation. Note that for \(b\geq 1\), super-critical Hopf bifurcations accompanied by stable limit cycles are no longer observed.

### Data views

Snapshot of the spatial arrangement of strategies.
Time evolution of the strategy frequencies.
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).
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
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.

## 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
multiplication factor \(r\) of public good.
Cost
cost of cooperation \(c\) (investment into common pool).
Lone cooperator's payoff
payoff for a cooperator if no one else joins the public goods interaction.
Lone defector's payoff
payoff for a defector if no one else joins the public goods interaction.
Base birthrate
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
constant death rate of all individuals.
Init Coop, init defect, init empty
initial densities of cooperators, defectors and empty space. If they do not add up to 100%, the values will be scaled accordingly.