EvoLudoLab: Asymmetric Donation game - limit cycles

From EvoLudo

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. In particular, the parameters button allows to set and change various simulation parameters including the game as well as the population structure, while the data views pop-up list along the top visualizes the simulation data in different ways.

Color code: Rich Cooperators Poor Cooperators Rich Defectors Poor Defectors
New rich cooperator New poor cooperator New rich defector New poor defector
Payoff code: Low High

Note: The shades of grey of the payoff scale are augmented by blueish and reddish shades indicating payoffs for mutual cooperation and defection, respectively.

Asymmetric Donation game

In the asymmetric donation game with negative environmental feedback stable limit cycles can arise if cooperation is dominant on rich patches because they exploit and degrade their patch but defection dominates on poor ones.

Parameters: benefits to co-player regardless of patch quality \(b_r=b_p=4\); costs of cooperation of rich \(c_r=-1\) and poor patches \(c_p=1\); rate of ecological changes, \(\lambda=0.3\), degradation rate of rich patches by cooperators, \(\rho=0.01\) and background fitness, \(\sigma=1\) (to avoid biologically not meaningful negative rates of reproduction).

Data views

Strategies - Structure

Snapshot of the spatial arrangement of strategies.

Strategies - 3D Structure

Snapshot of the spatial arrangement of strategies in 3D (only available in webGL compliant browsers).

Strategies - Mean

Time evolution of the strategy frequencies.

Strategies - Simplex S3

Time evolution of the strategy frequencies on the simplex S3 (requires a subset of exactly 3 active strategies).

Strategies - 2D Phase Plane

Time evolution of the strategy frequencies represented in the phase plane spanned by the frequency of cooperators \(x=x_r+x_p\) and frequency of rich patches \(\alpha=x_r+y_r\). Note this is a projection because the presentation neglects correlations \(\nu\) between strategies and patch qualities.

Fitness - Structure

Snapshot of the spatial distribution of payoffs.

Fitness - 3D Structure

Snapshot of the spatial distribution of payoffs in 3D (only available in webGL compliant browsers).

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.

Console log

Message log from engine.

Game parameters

The list below describes only the few parameters related to the evolutionary dynamics in asymmetric \(2\times2\) games with environmental feedback. Numerous other parameters are available to set population structures or update rules on the player as well as population level.

--payoffs <a:b;c:d>
\(2\times2\) or \(4\times4\) payoff matrix for cooperators (on rich and poor patches) and defectors (on rich and poor patches); default 1:0;1.65:0.
--environment <g[:b]>
background fitness on rich (poor) patches g (and optionally (b); default 0:0.
--feedback <Cp→r:Dr→p[:Cr→p:Dp→r]>
feedback between strategies and patches; p→r: restoration rates of poor patches for cooperators and defectors; r→p: degradation rates of rich patches for cooperators and defectors; default 0:0:0:0.
--asymmetry <a>
type of asymmetry; g genetic (inherited) asymmetries; e environmentally induced asymmetries; default e.
--basefit, -W <[s]b>
background fitness \(\sigma\) to prevent negative birth rates; prepending an s makes the background fitness static, i.e. independent of the selection strength; default 0.
--initfreqs <c:d>[|<cr:cp:dr:dp>]
initial frequencies of cooperators c and defectors d or, alternatively, cooperators on rich cr and poor cp patches as well as defectors on rich dr and poor dp patches. Frequencies that do not add up to 100% are scaled accordingly. If frequency c or d is zero, the population is initialized to a homogenous state with just a single, randomly placed individual of the opposite type. Default: 1:1 (equal frequencies).
--initrich <r>
initial frequencies of rich patches r; default 0.5.