In structured populations, the individuals occupy the nodes of a lattice, graph or network. The performance or fitness of all individuals is determined through interactions within their neighborhood. The evolutionary dynamics can be modeled in different ways but in any case, the strategy of an individual is updated by probabilistically comparing its fitness with the fitnesses of its neighbors. This updating can be done in synchrony, e.g. reflecting an annual reproductive cycle, or in asynchrony which approximates a continuous time system. In addition, there are various meaningful ways to implement the probabilistic comparison and the derivation of the fitness.

In well-mixed populations, the replicator dynamics predicts periodic oscillations driven by the cyclic dominance of cooperators, defectors and loners. However, all trajecories are structurally unstable such that noise eventually drives the system to the boundary of the simplex S3 and thus eliminating two strategies resulting in a homogenous state (usually all loners). In particular, this also happens in the case of finite population sizes. Thus, the long-term maintenance of cooperation requires stabilization. This can be achieved in different ways, e.g. through modifications of the dynamics or by considering structured populations.

In contrast to well-mixed populations, cooperators can survive in structured populations already in compulsory public goods games by forming compact clusters. Voluntary participation and the loner option largely extends the range where cooperators persist because loners provide additional protection to cooperators by reducing exploitation. This results in four different dynamical regimes.


All collections provide more detailed information on different aspects of the dynamics of voluntary public goods games in structured populations including many preconfigured simulations illustrating and highlighting particular scenarios.


Evolutionary kaleidoscopes

For symmetrical initial configuration and deterministic updates of the population and the players fascinating spatio-temporal patterns can unfold, which ressemble dynamic persian carpets or evolutionary kaleidoscopes. Even though the scientific relevance of these findings is rather limited, they do have quite some entertainment value.


Dynamical regimes

The spatial structure of lattice populations suppresses global oscillations and replaces them with uncorrelated local ones. This prevents the build up of fluctuations that can lead to extinctions. The loner option largely extends the range where cooperators persist as compared to compulsory interactions. Depending on the multiplication factor r, four different dynamical regimes are observed.

The figure on the left shows the fraction of cooperators (blue) defectors (red) and loners (yellow) on a square lattice as a function of r.


Oscillations: synchronization versus stabilization

In lattice populations the spatial separation of the oscillators leads to uncorrelated fluctuations. This is compromised in population stuctures that include long range connections, such as small world networks. Far reaching connections can induce global synchronization of the fluctuations driven by the cyclic dominance of cooperaators, defectors and loners. This often results in diverging amplitudes of oscillations and, eventually, ends in a homogenous state, i.e. in the loss of two strategies.

The figure on the left shows the average fraction of cooperators together with the minima and maxima on random regular graphs as a function of r.

Phase transitions

Phase transitions

Voluntary public goods games in lattice populations produce four dynamical regimes that are separated by the extinction of one or two strategies. Each of these transitions exhibits a critical phase transitions and suggests interesting links to condensed matter physics.

The figure on the left shows a typical snapshot near the extinction threshold of defectors and the transition to a state with all loners. The size of cooperator and loner domains increases as defectors decrease but nevertheless it is the defectors that maintain the domain dynamics because in absence of defectors, cooperators would take over.


The buttons along the bottom control the execution and the speed of the simulations. Of particular importance are the Param button and the data views pop-up list on 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. Use the above collections to run simulations illustrating particular scenarios with all parameters preset accordingly.

Color code:CooperatorsDefectorsLoners
 New cooperatorsNew defectorsNew loners
Payoff code:Low       High

Note: The pale strategy colors are very useful to get an intuition of the activitiy in spatially structured systems. The shades of grey of the payoff scale are augmented by blueish and reddish shades, which indicate payoffs for mutual cooperation and defection, respectively.

Java applet on voluntary participation in public goods games in structured populations. Sorry, but you are missing the fun part!
ParamsPop up panel to set various parameters.
ViewsPop up list of different data presentations.
ResetReset simulation
RunStart/resume simulation
NextNext generation
PauseInterrupt simulation
SliderIdle time between updates. On the right your CPU clock determines the update speed while on the left updates are made roughly once per second.
MouseMouse clicks on the graphics panels generally start, resume or stop the simulations.
Data views
Structure - Strategy Snapshot of the spatial arrangement of strategies. Mouse clicks cyclically change the strategy of the respective site for the preparation of custom initial configurations.
Mean frequency Time evolution of the strategy frequencies.
Simplex S3 Frequencies plotted in the simplex S3. Mouse clicks set the initial frequencies of strategies.
Phase Plane 2D Frequencies plotted in the phase plane spanned by the frequency of participants (x + y = 1 - z) and the relative frequency of cooperators (f = x / (x + y)). Mouse clicks set the initial frequencies of strategies or stop the simulations.
Structure - Fitness Snapshot of the spatial distribution of payoffs.
Mean Fitness Time evolution of the mean payoff of each strategy together with the average population payoff.
Histogram - Fitness Histogram of payoffs for each strategy.

Game parameters

The list below describes only the few parameters related to the voluntary public goods game. Follow the link for a complete list and descriptions of all other parameters e.g. referring to update mechanisms of players and the population.

multiplication factor r of public good.
cost of cooperation c (investment into common pool).
payoff for loners. Typically this value is positive but smaller than r - 1 such that groups of cooperators are better of but loners are better off than groups of defectors.
Lone cooperator, lone defector:
payoffs for cooperators and defectors which are forced to act as loners because they could not find interaction partners. Usually this will be the same as the loner's payoff.
Init Coop, init defect, init loner:
initial fractions of cooperators, defectors and loners. If they do not add up to 100%, the values will be scaled accordingly. Setting the fraction of cooperators to 100% (and the others to zero) results in a symmetrical initial configuration suitable for observing evolutionary kaleidoscopes.