• coevolution;
  • frequency-dependent selection;
  • metapopulation;
  • sympatry;
  • time-lagged cycles

In host-parasite coevolutionary arms races, parasites probably have an evolutionary advantage. Parasite populations should be locally adapted, having higher mean fitness on sympatric than allopatric hosts. Here we assess evidence for local parasite advantage. Further we investigate how adaptation and counter-adaptation of parasites and hosts, necessarily occurring in sympatry, can generate a pattern of local adaptation. Already simple frequency-dependent selection models generate complex patterns of parasite performance on sympatric and allopatric populations. In metapopulations, with extinction, recolonization, and gene flow, variable selection pressure and stochasticity may obscure local processes or change the level at which local adaptation occurs. Alternatively, gene flow may introduce adaptive variation, so differential migration rates can modify the asymmetry of host and parasite evolutionary rates. We conclude that local adaptation is an average phenomenon. Its detection requires adequate replication at the appropriate level, that at which the local processes occur.