  In many biological systems, natural selection acts simultaneously on multiple
levels of organization. This scenario typically presents an evolutionary
conflict between the incentive of individuals to cheat and the collective
incentive to establish cooperation within a group. Generalizing previous work
on multilevel selection in evolutionary game theory, we consider a hyperbolic
PDE model of a group-structured population, in which members within a single
group compete with each other for individual-level replication; while the group
also competes against other groups for group-level replication. We derive a
threshold level of the relative strength of between-group competition such that
defectors take over the population below the threshold while cooperation
persists in the long-time population above the threshold. Under stronger
assumptions on the initial distribution of group compositions, we further prove
that the population converges to a steady state density supporting cooperation
for between-group selection strength above the threshold. We further establish
long-time bounds on the time-average of the collective payoff of the
population, showing that the long-run population cannot outperform the payoff
of a full-cooperator group even in the limit of infinitely-strong between-group
competition. When the group replication rate is maximized by an intermediate
level of within-group cooperation, individual-level selection casts a long
shadow on the dynamics of multilevel selection: no level of between-group
competition can erase the effects of the individual incentive to defect. We
further extend our model to study the case of multiple types of groups, showing
how the games that groups play can coevolve with the level of cooperation.
