EvoLudoLab: Spatial Ecological PGG - Diffusion induced coexistence

From EvoLudo

Cooperator density:
Low High
Payoff code:
Low High

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]\displaystyle{ r=2.2 }[/math], [math]\displaystyle{ N=8 }[/math], [math]\displaystyle{ c=1 }[/math], [math]\displaystyle{ b=1 }[/math], [math]\displaystyle{ d=1.2 }[/math] using numerical integration of the partial differential equation with the diffusion constants [math]\displaystyle{ D_C=1 }[/math], [math]\displaystyle{ D_D=100 }[/math]. The initial configuration is a circular disk of equal densities of cooperators and defectors ([math]\displaystyle{ 0.1 }[/math]) in the center of a square lattice with fixed (reflecting) boundary conditions.

Note, in order to exactly reproduce the patterns emerging in the movie a finer mesh is required, i.e. smaller [math]\displaystyle{ dx }[/math], and hence requires considerably more CPU time.

Data views

Strategies - Structure

Snapshot of the spatial arrangement of strategies.

Strategies - Structure 3D

3D view of snapshot of the spatial arrangement of strategies.

Strategies - Mean

Time evolution of the strategy frequencies.

Simplex \(S_3\)

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).

Phase plane 2D

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

Fitness - Structure

Snapshot of the spatial distribution of payoffs.

Fitness - Structure 3D

3D view of snapshot of the spatial distribution of payoffs.

Fitness - Mean

Time evolution of average population payoff bounded by the minimum and maximum individual payoff.

Fitness - Histogram

Snapshot of payoff distribution in population.

Structure - Degree

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:
random distribution with given frequency
uniform random distribution
monomorphic initialization
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 of traits
(optional) configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Not available for all types of games.