# EvoLudoLab: Spatial Ecological PGG - Diffusion induced instability 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 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

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

Population density: | Low | High |

Payoff code: | Low | High |
---|

## 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 stable coexistence equilibrium \(\mathbf Q\) 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).

The game parameters are \(N=8\), \(b=1\), \(d=1.2\), \(r=2.5\), which yield equilibrium densities of cooperators \(\hat u=0.078\) and defectors \(\hat v=0.11\). Diffusion coefﬁcients are \(D_C=1\) and \(D_D=10\), spatial extension is \(L=100\) and numerical integration uses \(dx=0.5\) and time increments of \(dt=0.1\).

## 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.