Research Article: Consolidating Birth-Death and Death-Birth Processes in Structured Populations

Date Published: January 28, 2013

Publisher: Public Library of Science

Author(s): Joshua Zukewich, Venu Kurella, Michael Doebeli, Christoph Hauert, Yamir Moreno.


Network models extend evolutionary game theory to settings with spatial or social structure and have provided key insights on the mechanisms underlying the evolution of cooperation. However, network models have also proven sensitive to seemingly small details of the model architecture. Here we investigate two popular biologically motivated models of evolution in finite populations: Death-Birth (DB) and Birth-Death (BD) processes. In both cases reproduction is proportional to fitness and death is random; the only difference is the order of the two events at each time step. Although superficially similar, under DB cooperation may be favoured in structured populations, while under BD it never is. This is especially troubling as natural populations do not follow a strict one birth then one death regimen (or vice versa); such constraints are introduced to make models more tractable. Whether structure can promote the evolution of cooperation should not hinge on a simplifying assumption. Here, we propose a mixed rule where in each time step DB is used with probability and BD is used with probability . We derive the conditions for selection favouring cooperation under the mixed rule for all social dilemmas. We find that the only qualitatively different outcome occurs when using just BD (). This case admits a natural interpretation in terms of kin competition counterbalancing the effect of kin selection. Finally we show that, for any mixed BD-DB update and under weak selection, cooperation is never inhibited by population structure for any social dilemma, including the Snowdrift Game.

Partial Text

Evolutionary game theory was developed to model frequency-dependent selection [1]. Arguably the most captivating system that exhibits frequency dependent selection is the evolution of cooperation under social dilemmas, a problem that has puzzled researchers across disciplines for decades (for a review, see [2]). In a social dilemma, cooperators provide a benefit to a group at some cost to self, while defectors pay no cost and contribute nothing. Groups of cooperators “do better” than groups of defectors, yet in any mixed group defectors “do best” [3]. The tension in social dilemmas is that defection maximizes a given individual’s payoff while cooperation maximizes the total payoff to the group.

The conditions for selection favouring or and for or mutations being beneficial are summarized in Table 1 and Figures 1–2 for well-mixed and structured populations under BD, DB and mixed update rules. Table 1 illustrates that taking or for the mixed rule recovers the results for the BD and DB updates. The limit recovers the results for well-mixed populations – a highly connected population behaves like a well-mixed one.

We set out to resolve the disparity between Birth-Death and Death-Birth updates for structured populations demonstrated by Ohtsuki et al.[8]. They considered a simple Prisoner’s Dilemma where cooperators pay a cost to donate a benefit (subscripts used to distinguish from the used in this paper) to their interaction partner and defectors neither provide benefits nor suffer costs. Under weak selection, they showed that cooperation is favoured and beneficial under the DB update if , where indicates the average number of neighbours. However, cooperation is never favoured or beneficial under BD.