# EvoLudoLab: Spatial Ecological PGG - Diffusion induced coexistence

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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## Diffusion induced co-existence

The initial configuration is symmetric and gives rise to a symmetrical evolutionary end state. The competing forces of cooperators (activators) and defectors (inhibitors) gives rise to spatial pattern formation reminiscent of Turing patterns. However, in contrast to Turing's diffusion induced instability, diffusion induced co-existence develops in the vicinity of an unstable fixed point. Hence, in the absence of spatial extension, the population would disappear. Spatially heterogenous distributions enables cooperators and defectors to persist.

The parameters are <math>r=2.2</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=100</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.

Note that in order to exactly reproduce the patterns emerging in the movie, the requires a finer mesh, i.e. smaller <math>dx</math>, and hence requires considerably more CPU time.

## 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.