Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes

We compute the mean patch occupancy for a stochastic, spatially explicit patch-occupancy metapopulation model on a dynamic, correlated landscape, using a mathematically exact perturbation expansion about a mean-field limit that applies when dispersal range is large. Stochasticity in the metapopulation and landscape dynamics gives negative contributions to patch occupancy, the former being more important at high occupancy and the latter at low occupancy. Positive landscape correlations always benefit the metapopulation, but are only significant when the correlation length is comparable to, or smaller than, the dispersal range. Our analytical results allow us to consider the importance of spatial kernels in all generality. We find that the shape of the landscape correlation function is typically unimportant, and that the variance is overwhelmingly the most important property of the colonisation kernel. However, short-range singularities in either the colonisation kernel or landscape correlations can give rise to qualitatively different behaviour.