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In many biological systems, natural selection acts simultaneously on multiplelevels of organization. This scenario typically presents an evolutionaryconflict between the incentive of individuals to cheat and the collectiveincentive to establish cooperation within a group. Generalizing previous workon multilevel selection in evolutionary game theory, we consider a hyperbolicPDE model of a group-structured population, in which members within a singlegroup compete with each other for individual-level replication; while the groupalso competes against other groups for group-level replication. We derive athreshold level of the relative strength of between-group competition such thatdefectors take over the population below the threshold while cooperationpersists in the long-time population above the threshold. Under strongerassumptions on the initial distribution of group compositions, we further provethat the population converges to a steady state density supporting cooperationfor between-group selection strength above the threshold. We further establishlong-time bounds on the time-average of the collective payoff of thepopulation, showing that the long-run population cannot outperform the payoffof a full-cooperator group even in the limit of infinitely-strong between-groupcompetition. When the group replication rate is maximized by an intermediatelevel of within-group cooperation, individual-level selection casts a longshadow on the dynamics of multilevel selection: no level of between-groupcompetition can erase the effects of the individual incentive to defect. Wefurther extend our model to study the case of multiple types of groups, showinghow the games that groups play can coevolve with the level of cooperation.

Long-Time Behavior of a PDE Replicator Equation for Multilevel Selection in Group-Structured Populations