# EvoLudoLab: Spatial Ecological PGG - Chaos

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.

Cooperator density: | Low | High | ||||||||
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Defector density: | Low | High | ||||||||

Population density: | Low | High |

Payoff code: | Low | High |
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## Spatio-temporal chaos

The initial configuration is symmetric and this symmetry is preserved for some time but then breaks down and settles in spatio-temporal chaos. In the absence of spatial extension, the population would not be able to persist. Fluctuations of increasing amplitude would drive the population to extinction.

The parameters are <math>r=2.34</math>, <math>N=8</math>, <math>c=1</math>, <math>b=1</math>, <math>d=1.2</math> using numerical integration of the partial differential equation with the diffusion constants <math>D_C=1</math>, <math>D_D=2</math>. The initial configuration is a circular disk of equal densities of cooperators and defectors (<math>0.1</math>) in the center of a square lattice with fixed (reflecting) boundary conditions.

## 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.