# EvoLudoLab: Ecological Public Goods Game - Bautin

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 |
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Payoff code: | Low | High |
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## Stable focus *Q* - multiple stable and unstable limit cycles (Bautin bifurcation)

For larger group sizes (here \(N = 12\)) more complex dynamical scenarios occur. In this example, a pair of stable and unstable limit cycles occur on one side of the Hopf bifurcation and another stable limit cycle on the other side (try \(r = 3.04\)). In fact, this is a Bautin bifurcation with \(N\) as the second parameter. Increasing \(N\) turns the super-critical Hopf bifurcation into a subcritical one.

The parameters are \(r = 3.05\), \(N = 12\), \(c = 1\), \(b = 0\), \(d = 0.5\) using numerical integration of the replicator equation. The Hopf-bifurcation occurs at \(r_\text{Hopf} = 3.0414\).

## 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.