# EvoLudoLab: Spatial Ecological PGG - Diffusion induced coexistence in 1D

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 instability in 1D

Pattern formation process in ecological public goods games through diffusion-induced instability in 1-dimensional spatial systems. Initially the strategy distribution is homogeneous with densities near the unstable equilibrium <math>\mathbf Q</math> and a perturbation in the center (increase of the densities by 10%, fixed (reflecting) boundary conditions). The figure on the left shows a space–time plot of the pattern formation process where the abscissa denotes space, and the ordinate the time (from top to bottom). Note the increasing fluctuations of the background before the pattern emerges.

The game parameters are <math>N=8</math>, <math>b=1</math>, <math>d=1.2</math>, <math>r=2.3</math>, which yield equilibrium densities of cooperators <math>\hat u=0.058</math> and defectors <math>\hat v=0.066</math>. Diffusion coefﬁcients are <math>D_C=1</math> and <math>D_D=10</math>, spatial extension is <math>L=100</math> and numerical integration uses <math>dx=0.5</math> and time increments of <math>dt=0.1</math>.

## 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.