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## WHAT IS IT

This model investigates the evolution and maintenance of cooperation in spatially structured populations.

## HOW IT WORKS

This model is based on Nowak and May (1992), Nowak et al. (1994) and the Netlogo model PD Basic Evolutionary Wilensky, U. (2002). Each round, every individual patch plays the prisoner's dilemma with a fixed predefined number of immediate neighbors (8 or 4), and after that, each site in the lattice is occupied either by its original owner or by one of its neighbors. Deterministic and probabilistic update rules can be tested, as well as stochasticity m, and the cheating advantage b. The update is asynchronous. See the references for more details.

## HOW TO USE IT

Use the slider "cooperators" and the button "setup" to start with a random spatial distribution of cooperators in the lattice. Or use the button "setup-1-at-the-center" to start with a non-cooperator in a sea of cooperators. Click "go once" to run once (one tick/generation), or click "go" to run forever.

The user can manipulate the cheating-advantage b of non-cooperators, the number of neighbors ("n-neighbors") that each site interacts with, and the "update-rule".

If the probabilistic "update-rule" is selected, the user can manipulate the stochasticity value m. Selecting the deterministic update rule is equivalent to m = ∞.

Blue is a cooperative site (C); red a non-cooperative site (D - for defection); yellow - a change from C to D; green - a change from D to C.

The plots show the number of sites in each class, and the evolution of the populations of cooperators and defectors. One tick is the time unit corresponding to one generation.

## THINGS TO NOTICE

For particular combinations of the values of the cheating advantage and stochasticity it can be shown that spatial arrays promote the coexistence of strategies. Can you find these values?

## THINGS TO TRY

1) Setup one defector at the center, select the deterministic update rule and run with both 8 and 4 neighbors. See the beautiful patterns arising.

2) Setup 50% of cooperators, select the probabilistic update rule and run with 8 neighbors. Observe what happens during and after 200 generations using different combinations of the cheating advantage parameter and the stochasticity parameter. Sometimes defectors "win". Sometimes cooperators "win". And sometimes cooperators survive in clusters, while defectors survive in unconnected or connected short lines, surrounded or not by constantly changing strategy sites.

2.1) See what happens with null stochasticity for any value of the cheating advantage parameter.

## RELATED MODELS

PD Basic Evolutionary

## CREDITS

The present model was coded by David N. Sousa (davidnsousa@gmail.com). Feel free to contact.

## REFERENCES

Nowak, M. A., & May, R. M. (1992). Evolutionary games and spatial chaos. Nature, 359(6398), 826-829.

Nowak, M. A., Bonhoeffer, S., & May, R. M. (1994). Spatial games and the maintenance of cooperation. Proceedings of the National Academy of Sciences, 91(11), 4877-4881.

Wilensky, U. (2002). NetLogo PD Basic Evolutionary model. http://ccl.northwestern.edu/netlogo/models/PDBasicEvolutionary. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

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