Death-birth process

Spatial effects in social dilemmas: cooperation versus c N/b and w

Effects of spatial structure on cooperators and defectors as compared to well-mixed populations. The change in equilibrium frequencies is shown for a fixed group size N = 5 as a function of the normalized cost-to-benefit ratio c N/b and of the synergy/discount factor w. In blue regions space promotes cooperation and in red regions spatial structure reduces the equilibrium fraction of cooperators. The saturation of the colors indicates the strength of the effect. In well-mixed populations, defectors dominate in region (i) and cooperators dominate in region (iv). Along the dashed line, equal proportions of cooperators and defectors represent a fixed point, which is stable in region (ii), indicating co-existence of cooperators and defectors, but unstable in region (iv) separating the basins of attraction of the homogeneous states with all cooperators and all defectors, respectively (bi-stability).

Spatial effetcs in social dilemmas: cooperation versus N and w

Same situation as above but now the effects of space on the equilibrium fraction of cooperators are shown for a fixed cost-to-benefit ratio c/b = 0.2 and as a function of the group size N and of the synergy/discount factor w.

For the death-birth updating spatial structure almost unanimously promotes cooperation, with the exception of a tiny parameter range in region (ii). This is quite different from birth-death updating and even more from replicator updating, where spatial structure often dimishes cooperation as compared to well-mixed expectations or even eliminates it altogether.

Evolution of cooperators and defectors in well-mixed populations where individuals interacting in social dilemmas with discounted or synergistically enhanced accumulation of cooperative benefits.

Region (i): public goods games (c N/b > 1, c N/b > w^N-1)

In public goods type interactions cooperators are doomed in the absence of spatial structure. In contrast, spatial structure enables cooperators to thrive by forming clusters and therby reducing exploitation by defectors. The striking difference is illustrated in the snapshot of a typical lattice configuration with equilibrium frequencies of cooperators around 90%. The effects of space are most pronounced for death-birth updating. Birth-death updating and replicator updating both result in equilibrium frequencies of cooperators around 30%.

Region (ii): snowdrift games (1 > c N/b > w^N-1)

Snowdrift type interactions result in co-existence of cooperators and defectors in well-mixed populations. Interestingly, spatial structure can inibit cooperation to equilibrium levels below well-mixed expectations or even eliminate cooperation altogether. The extent of these detrimental effects depends on the updating rule. For death-birth updating negative effects are restricted to a tiny parameter range (see phase planes above) but birth-death updating and, in particular, replicator updating substantially supress cooperation over an extended parameter range. The snapshot on the left depicts a typical equilibrium configuration with more than 90% cooperators for parameters that result in 40% cooperators for replicator updating - in well-mixed populations these parameters lead to equal proportions of cooperators and defectors.

Region (iii): by-product mutualism (c N/b < 1, c N/b < w^N-1)

Space does not affect the evolutionary success of cooperators - it may only affect the time it takes for cooperators to take over the entire population. Similarly, the different update rules also affect the spreading speed of cooperators. Death-birth updating produces the fastest growth. Birth-death updating is slightly slower and replicator updating is much slower - to the extent that fewer cooperators are present after 50 generations than in the snapshot depicted on the left after 10 generations.

Region (iv): coordination games (1 < c N/b < w^N-1)

For the bi-stable dynamics of coordination games in well-mixed populations the only relevant quantity is the initial fraction of cooperators. Above a certain threshold cooperators thrive and vanish below. In structured populations, this threshold needs no longer to be achieved on a global level but instead it is sufficient to have sufficiently high local cooperator density. Whenever such clusters of cooperators exist, they keep expanding until all defectors are displaced. This dynamics greatly extends the basin of attraction of cooperation. In fact, in the limit of large populations even arbitrarily low initial fractions of cooperators are sufficient to reach the cooperative end state because a single sufficiently large cluster is enough to seed the successful spreading of cooperators.