EvoLudoLab: Spatial Ecological PGG - Diffusion induced coexistence

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 $r=2.2$ , $N=8$ , $c=1$ , $b=1$ , $d=1.2$ using numerical integration of the partial differential equation with the diffusion constants $D_{C}=1$ , $D_{D}=100$ . The initial configuration is a circular disk of equal densities of cooperators and defectors ( $0.1$ ) 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 $dx$ , 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:

frequency <f0>,<f1>...: random distribution with given trait frequencies, f0, f1,.... Note, only available for frequency based modules and models.
density <d0>,<d1>...: initial trait densities <d1,...,dn>. Note, only available for density based modules and models.
uniform: uniform random distribution, equal frequencies of all traits.
monomorphic <t>[,<v>]: monomorphic initialization with trait t. Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the monomorphic trait is initialized at its (estimated) carrying capacity.
mutant <m>,<r>[,<v>]: single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected uniformly at random (mutants arising through cosmic rays). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
temperature <m>,<r>[,<v>]: single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected proportional to the in-degree of nodes (temperature initialization, mutants arising through errors in reproduction). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
stripes: stripes of traits. Note, only available for 2D lattices.
kaleidoscopes: configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Note, only available for some modules.

Note, for modules that admit multiple species, the initialization types for each species can be specified as an array separated by ;. With more species than initialization types, they are assigned in a cyclical manner.