EvoLudoLab: Spatial Ecological PGG - Diffusion induced coexistence in 1D
| Cooperator density: | Low High
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| 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]\displaystyle{ \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]\displaystyle{ N=8 }[/math], [math]\displaystyle{ b=1 }[/math], [math]\displaystyle{ d=1.2 }[/math], [math]\displaystyle{ r=2.3 }[/math], which yield equilibrium densities of cooperators [math]\displaystyle{ \hat u=0.058 }[/math] and defectors [math]\displaystyle{ \hat v=0.066 }[/math]. Diffusion coefficients are [math]\displaystyle{ D_C=1 }[/math] and [math]\displaystyle{ D_D=10 }[/math], spatial extension is [math]\displaystyle{ L=100 }[/math] and numerical integration uses [math]\displaystyle{ dx=0.5 }[/math] and time increments of [math]\displaystyle{ dt=0.1 }[/math].
Data views
| Snapshot of the spatial arrangement of strategies. | |
| 3D view of 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. | |
| 3D view of 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. | |
| Degree distribution in structured populations. |
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 <r>
- multiplication factor \(r\) of public good.
- --cost <c>
- cost of cooperation \(c\) (investment into common pool).
- --lonecooperator <l>
- payoff for a cooperator if no one else joins the public goods interaction.
- --lonedefector <l>
- payoff for a defector if no one else joins the public goods interaction.
- --basefit <
b> - 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 <d>
- constant death rate of all individuals.
- --init <d,c,e>
- initial frequencies of defectors d, cooperators c and vacant space e. Frequencies that do not add up to 100% are scaled accordingly.
- --inittype <type>
- type of initial configuration:
- frequency
- random distribution with given frequency
- uniform
- uniform random distribution
- monomorphic
- monomorphic initialization
- mutant
- single mutant in homogeneous population of another type. Mutant and resident types are determined by the types with the lowest and highest frequency, respectively (see option --init).
- stripes
- stripes of traits
- kaleidoscopes
- (optional) configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Not available for all types of games.
