Fast dispersal simulation using bivariate quantiles

Spatial-temporal models of the spread of invasive species can require dispersal of large numbers of individuals from many locations at recurrent times, making them slow to execute. We present a fast algorithm for simulating dispersal of large numbers of individuals. The algorithm is stochastic and can be applied using any bivariate probability density function as the dispersal kernel. It achieves computational efficiency while still allowing the simulation of rare and important long-distance dispersals by combining different approaches for within and outside the tail of the dispersal kernel. The tail is specified by a given bivariate quantile, where the q-th bivariate quantile is defined to be the contour of equiprobability within which a proportion 0< q <1 of dispersing individuals will settle. We provide a method for finding bivariate quantiles that can be applied to any bivariate dispersal kernel derived from independent densities for distance and direction of dispersal. To illustrate this approach, we show how the Cauchy distribution can be used to produce isotropic and anisotropic bivariate dispersal kernels by assuming that the direction of dispersal is either random or takes a von Mises distribution. We show that the algorithm is considerably faster than generating individual random samples from a bivariate dispersal kernel. It also performs better for larger grid sizes, and when there are larger numbers of individuals to be spread, than an approach that generates samples from a Binomial distribution for each grid cell using the probability of dispersal to that cell. The degree of computational efficiency achieved by the algorithm compared to the Binomial approach depends upon the speed with which random sample scan be generated from the tail of the bivariate dispersal kernel used.