Hamilton's rule in multi-level selection models.

Hamilton's rule is regarded as a useful tool in the understanding of social evolution, but it relies on restrictive, overly simple assumptions. Here we model more realistic situations, in which the traditional Hamilton's rule generally fails to predict the direction of selection. We offer modifications that allow accurate predictions, but also show that these Hamilton's rule type inequalities do not predict long-term outcomes. To illustrate these issues we propose a two-level selection model for the evolution of cooperation. The model describes the dynamics of a population of groups of cooperators and defectors of various sizes and compositions and contains birth-death processes at both the individual level and the group level. We derive Hamilton-like inequalities that accurately predict short-term evolutionary change, but do not reliably predict long-term evolutionary dynamics. Over evolutionary time, cooperators and defectors can repeatedly change roles as the favored type, because the amount of assortment between cooperators changes in complicated ways due to both individual-level and group-level processes. The equation that governs the dynamics of cooperator/defector assortment is a certain partial differential equation, which can be solved numerically, but whose behaviour cannot be predicted by Hamilton's rules, because Hamilton's rules only contain first-derivative information. In addition, Hamilton's rules are sensitive to demographic fitness effects such as local crowding, and hence models that assume constant group sizes are not equivalent to models like ours that relax that assumption. In the long-run, the group distribution typically reaches an equilibrium, in which case Hamilton's rules necessarily become equalities.