For the Moran process in structured populations the sequence of events turns out to be of crucial importance. In large, well-mixed populations it does not matter whether an individual first reproduces and then the offspring displaces a member of the population (birth-death updating) or if first an individual dies and then the remaining members of the population compete to repopulate the vacant site (death-birth updating). Since in structured populations only a small subset of the population, i.e. a focal individual and its neighbors, competes for reproduction this equivalence no longer holds.

For birth-death updating, a focal individual is randomly selected from the entire population with a probability proportional to its payoff. Then a single clonal offspring replaces one randomly chosen neighbor of the focal individual. Spatial structure matters only in the second step. If cooperators want to spread, then a cooperator adjacent to a defector must be selected for reproduction and its offspring must displace the defector. In order to reproduce, the cooperator competes with its neighboring defector, however in cooperative interactions, the cooperator supported the defector and hence cooperators nourish their competitors. This is not the for death-birth updating and it follows that cooperators are worse off for birth-death updating. Nevertheless they are better off than under replicator updating.

Effects of space - constant N

Social Dilemmas

Space promotes cooperation for a limited parameter range for public goods type interactions in region (i). In region (iv) it increases the basin of attraction of the cooperative state. This happens simply because the required threshold frequency of cooperators is easier achieved on a local rather than global scale. Region (iii) of by-product mutualism remains unaffected by spatial structure. In region (ii), limited local interactions in a spatial setting can both promote or inhibit cooperation. However, the detrimental effects are considerably smaller than under replicator updating but still larger than for death-birth updating.

Effects of space - constant c/b

Social Dilemmas

The view of the phase space for constant cost-to-benefit ratio c/b = 0.2 confirms the positive effects of space in regions (i), (iv) and parts of (ii) as well as the negative impact in the rest of region (ii). Overall the effects are weaker than for replicator updating and the parameter range with detrimental effects is reduced. In comparison, death-birth updating further supports cooperation and essentially eliminates the negative effects in region (ii).

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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).

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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.

Compared to replicator updating, the effects of space in birth-death updating are less pronounced in regions (i), (iv) and in region (ii) the parameter range with detrimental effects on cooperation is much smaller. For death-birth updating space has the most pronounced positive effects on cooperation and essentially eliminates the deleterious effects in region (ii).

Scenarios

Clicking either on one of the pictures below (or the corresponding link to the right) opens a new window with a running applet with all parameters preset to illustrate the respective scenario. You can use this as a starting point to study effects of variations of the parameters.

Legend

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

Color code:CooperatorsDefectors
 New cooperatorsNew defectors
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Region (i): public goods games (c N/b > 1, c N/b > wN-1)

Spatial structure with limited local interactions enables cooperators to persist by forming clusters and thereby reducing the exploitation by defectors. Compared to replicator updating, birth-death updating supports fewer cooperators and results in a faster evolutionary pace (the yellow and green sites indicate the activity along cluster boundaries). Conversely, for the same parameters death-birth updating supports much higher levels of cooperation. The figure on the left depicts a snapshot of a typical lattice configuration for birth-death updating.

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

Snowdrift type interactions lead to stable co-existence in well-mixed populations. Spatial structure can either increase or reduce the equilibrium frequency of cooperators (see phase planes above). For parameters that all results in equal proportions of cooperators and defectors in well-mixed populations this is nicely illustrated for replicator updating where, for the same parameters, equilibrium frequencies can be increased, unaffected or lowered. In contrast, for birth-death updating space has always positive effects on cooperation for these parameters and may even lead to the fixation of cooperation. The figure on the left depicts a snapshot of a typical lattice configuration with around 90% cooperators instead of the 50% in well-mixed populations. Note that these parameters lead to a significant decrease in cooperation for replicator updating.

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

For by-product mutualism cooperation is dominant in both well-mixed and spatial settings and thus it is only a matter of time until cooperators reach fixation. The figure on the left depicts the transient phase after 10 generations. This confirms the faster evolutionary pace of the birth-death updating as compared to replicator updating where cooperation is less spread after 50 generations. On the other hand, the spreading speed of cooperators is even faster under death-birth updating.

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

In the bi-stable situation of coordination games, the initial fraction of cooperators needs to exceed a threshold value for cooperators to thrive. In spatial settings, all that matters is whether a sufficiently large compact cluster of cooperators was present after initialization such that locally the threshold level of cooperation is exceeded. If this is the case, the cluster acts as a seed and eventually takes over the entire population - if not, cooperators disappear. In the limit of arbitrarily large populations a sufficiently large cluster of cooperators is present with certainty even at very low initial fractions of cooperators. Hence, even in finite populations, spatial structure significantly increases the basin of attraction of the cooperative state. The figure on the left illustrates the transient growth of several seed-clusters.