Canonical functions for dispersal-induced synchrony

Two processes are universally recognized for inducing spatial synchrony in abundance: dispersal and correlated environmental stochasticity. In the present study we seek the expected relationship between synchrony and distance in populations that are synchronized by density-independent dispersal. In the absence of dispersal, synchrony among populations with simple dynamics has been shown to echo the correlation in the environment. We ask what functional form we may expect between synchrony and distance when dispersal is the synchronizing agent. We formulate a continuous-space, continuous-time model that explicitly represents the time evolution of the spatial covariance as a function of spatial distance. Solving this model gives us two simple canonical functions for dispersal-induced covariance in spatially extended populations. If dispersal is rare relative to birth and death, then covariances between nearby points will follow the dispersal distance distribution. At long distances, however, the covariance tails off according to exponential or Bessel functions (depending on whether the population moves in one or two dimensions). If dispersal is common, then the covariances will follow the mixture distribution that is approximately Gaussian around the origin and with an exponential or Bessel tail. The latter mixture results regardless of the original dispersal distance distribution. There are hence two canonical functions for dispersal-induced synchrony.