## Evolutionary Games and Population Dynamics:

# Spatial dynamics of ecological public goods games

#### by Christoph Hauert, Version 1.0, April 2009.

- Location:
- VirtualLabs
- » Ecology
- »
*Spatial public goods games*

Spatial patterns can unfold if defectors diffuse (migrate) faster than cooperators, ` D_{D} ≥ D_{C}`. Increased migration rates of defectors are motivated by the fact that defectors deplete common resources (or are unable to sustain them) and this causes them to move elsewhere. Conversely, migration rates of cooperators should be lower in order to enable them to take advantage of the locally sustained common resource. Consequently, the dominant effect of spatial dynamics is not cooperators outrunning defectors, but instead, the defectors' relentless search of productive patches. Slow migration facilitates aggregation of cooperators, whereas fast migration supports defectors to readily locate cooperator patches, but it also impedes their ability to exploit one particular patch.

If cooperators and defectors can co-exist in a stable equilibrium in the absence of space, this does not necessarily imply that the corresponding spatially homogenous state is stable as well. In the vicinity of the equilibrium, the spatial dynamics may take on the form of an activator-inhibitor system. Any deviation from the equilibrium is amplified by cooperators (activators) but suppressed by defectors (inhibitors). If defectors migrate (diffuse) faster than cooperators, these antagonistic forces may give rise to the formation of complex patterns (Turing instability). Any small local disturbance propagates through the system and induces stable heterogeneous strategy distributions. Local disturbances may give rise to rearrangements of the patterns but then quickly relax into another qualitatively indistinguishable distribution of cooperators and defectors.

Conversely, if the co-existence equilibrium is unstable, then spatial extension and migration often prevents extinction and stabilizes co-existence of cooperators and defectors either in static spots and stripes similar to Turing patterns or in chaotic dynamics of ever changing patterns. In general, homogeneous populations near the co-existence equilibrium exhibit periodic density oscillations of increasing amplitude that eventually result in extinction. However, any small local disturbance can propagate through space and trigger stationary, heterogeneous strategy distributions. The pattern formation is again driven by the opposing forces of cooperators (activators) and defectors (inhibitors). However, the activator-inhibitor system develops in the vicinity of an unstable fixed point, which could be termed 'diffusion induced co-existence' in contrast to the classical 'diffusion induced instability' of Turing patterns. Also note that while Turing patterns rely on substantial differences in the diffusion constants of activators and inhibitors, this does not apply to diffusion induced co-existence, where dynamic patterns emerge even for ` D_{D} = D_{C}`.

Individuals consuming common resources, such as in Hardin's Tragedy of the commons, or producing common resources may alter and shape their environment in an enduring manner. This is particularly evident in microbial systems involving extracellular products such as in antibiotic resistance, biofilms or swarming and represent crucial determinants of microbial ecology. Spatial ecological public goods model concurrent spontaneous habitat diversification and species co-existence and hence suggest a mechanism to promote biodiversity.

*This tutorial complements scientific articles co-authored with Joe Yuichiro Wakano and Martin Nowak and provides interactive Java applets to visualize and explore the systems' dynamic for parameter settings of your choice.*

## Dynamical scenarios

The rich dynamics of Ecological Public Goods in spatial settings can be explored by interactive simulations or visualized through movies of high accuracy simulations. Clicking on the images loads interactive real-time VirtualLabs simulations that illustrate the characteristics of the corresponding dynamical regime. Because the simulations require significant computational power, high resolution movies are provided as an alternative (requires Quicktime 7 or higher - movies are H.264 encoded). The densities of cooperator and defectors across space are indicated by the brightness of the green and red color components, respectively. Thus, regions of co-existence appear yellow and black regions are vacant. The initial configuration is a disk of homogeneous cooperator and defector densities centered in an empty plane (no-flux boundaries).

## Legend | Evolution of cooperator and defector distributions in spatial ecological public goods games.
| ||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

## Spatial ChaosEcological Public Goods Games in spatial settings can exhibit chaotic dynamics and produce fascinating ever changing patterns. This movie illustrates the onset of chaos when starting from a symmetrical initial configuration. The deterministic dynamics should, in principle, preserve the symmetry but instead the symmetry is maintained only for some time and then breaks down due to limitations of the numerical integration. The exponential amplification of arbitrarily small disturbances is the hallmark chaotic systems. In the absence of spatial dimensions the population would be unable to persist. High resolution movie of spatial chaos (high accuracy integration, 4.1MB). | |||||||||||||||||||||||||||||||||||||||||

## Intermittent ActivityBetween the regimes of chaotic dynamics and static patterns an interesting region of intermittent activity occurs: The formation of quasi-static patterns alternates with rapid changes leading to global rearrangement and redistribution of cooperators and defectors. This movie illustrates the successive periods of stasis with intermittent bursts of activity. At first the bursts are synchronized across space but this breaks down over time generating states where large parts are largely static but get regularly stirred up by a wave of change. High resolution movie of intermittent activity (high accuracy integration, 4.1MB). | |||||||||||||||||||||||||||||||||||||||||

## Diffusion induced instability (Turing patterns)In the absence of space, cooperators and defectors can co-exist in a stable equilibrium | |||||||||||||||||||||||||||||||||||||||||

## Diffusion induced co-existenceIn the absence of space, the co-existence equilibrium High resolution movie of diffusion induced co-existence (high accuracy integration, 0.5MB). |

## Virtual lab

The applet below illustrates the different components. Along the bottom there are several buttons to control the execution and the speed of the simulations. Of particular importance are the * Param* button and the data

*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.*

**views**Cooperator density: | Low | High | |||||||
---|---|---|---|---|---|---|---|---|---|

Defector density: | Low | High |

Population density: | Low | High |
---|

Payoff code: | Low | High |
---|

Controls | |

Params | Pop up panel to set various parameters. |
---|---|

Views | Pop up list of different data presentations. |

Reset | Reset simulation |

Run | Start/resume simulation |

Next | Next generation |

Pause | Interrupt simulation |

Slider | Idle time between updates. On the right your CPU clock determines the update speed while on the left updates are made roughly once per second. |

Mouse | Mouse clicks on the graphics panels start, resume or stop the simulations. |

Data views | |

Structure - Strategy | Snapshot of the spatial arrangement of strategies. |
---|---|

Mean frequency | Time evolution of the strategy frequencies. |

Simplex S_{3} |
Frequencies plotted in the simplex S. Mouse clicks set the initial frequencies of strategies or stops the simulations.
_{3} |

Phase Plane 2D | Frequencies plotted in the phase plane spanned by the population density () and the relative frequency of cooperators (x + y = 1 - z). Mouse clicks set the initial frequencies of strategies, stop the simulations or switch to backward integration.
f = x / (x + y) |

Structure - Fitness | Snapshot of the spatial distribution of payoffs. |

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

Histogram - Fitness | Snapshot of payoff distribution in population. |

## Game parameters, PDE 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 e.g. referring to update mechanisms of players and the population.

- Interest:
- multiplication factor
of public good.*r* - Cost:
- cost of cooperation
(investment into common pool).*c* - Lone cooperator's payoff:
- payoff for a cooperator if no one else joins the public goods interaction.
- Lone defector's payoff:
- payoff for a defector if no one else joins the public goods interaction.
- Base birthrate:
- 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:
- constant death rate of all individuals.
- Init Coop, init defect, init empty:
- initial densities of cooperators, defectors and empty space. If they do not add up to 100%, the values will be scaled accordingly.