The evolution of dispersal in spatially varying environments

We consider the evolution of dispersal in an environment that varies spatially but that is constant in time. We allow an age structure with dispersal possible in all life-stages. We suppose that demes are large enough that kin effects can be ignored. It has previously been shown that cost-free dispersal can persist over evolutionary time. However, several studies have shown that costly dispersal must in general be selected against. Here, we establish a fundamental result about stage-structured populations with stage-specific dispersal rates - that is, at evolutionary equilibrium, over each time step, the total reproductive value of the emigrants leaving each deme must equal the total reproductive value of the immigrants entering that deme. A simple consequence of this principle is that, if migration is restricted to a single stage - the same stage for all demes - then costly dispersal cannot evolve. Another corollary is that, with a 'sequential' age structure, over a complete life-cycle, the proportionate flow of genes out of a deme must equal the flow in. Finally, we present an example to show that dispersal may be evolutionarily stable, even when costly, if individuals can disperse more than once during their life-cycle.