Random Walks in a Metapopulation: How Much Density Dependence is Necessary for Long-Term Persistence?

We describe analytical and simulation models of metapopulations consisting of local populations that obey a random walk between a reflecting upper boundary (population 'ceiling') and an absorbing lower boundary (local extinction). We present analytical results for the expected time to local extinction, expected size of local populations, and incidence of density dependence. The latter is defined as the frequency of hitting the ceiling per generation per population. With these models we examine the proposition that a metapopulation consisting of random walk local populations would persist without density dependence. Long-term persistence of a metapopulation is not possible without local populations occasionally becoming large and hence being affected by density dependence. But it is possible to construct examples in which a metapopulation persists for a long time with a low incidence of density dependence, in which cases local populations typically have very short expected lifetimes. We demonstrate that, paradoxically, a persisting metapopulation may consist of only 'sink' populations (negative average growth rate in the absence of migration). Contrary to some previous suggestions, increasing migration rate generally increases density dependence in persisting metapopulations.