On the approximation of continuous dispersal kernels in discrete-space models

1. Models that represent space as a lattice have a critical function in theoretical and applied ecology. Despite their significance, there is a dearth of appropriate theoretical developments for the description of dispersal across such lattices. 2. We present a series of methods for approximating continuous dispersal in discrete landscapes (denoted as centroid-to-centroid, centroid-to-area, area-to-centroid and area-to-area dispersal). We describe how these methods can be extended to incorporate different conditions at the boundary of the simulation arena and a framework for approximating continuous dispersal between irregularly shaped patches. 3. Each approximation method was tested against a baseline of continuous Gaussian dispersal in a periodic simulation arena. The residence probabilities for an individual dispersing in each time step according to a Gaussian kernel across grids of three differing resolutions were calculated over a number of dispersal steps. In addition, the steady-state asymptotic properties for the transition matrices for each approximation method and cell resolution were calculated and compared against the uniform expectation under continuous dispersal. 4. All four methods described in this article provide a reasonable approximation to the continuous baseline (<0·03 absolute error in probability calculations) on landscapes with grid cells of length equal to the expected dispersal distance or finer, but error increases as grid cells become progressively larger than the expected dispersal distance. 5. Each approximation method exhibits a different spatial pattern of approximation error. Centroid-to-centroid dispersal overestimates residence probabilities near the origin, resulting in decreased invasion rates relative to the baseline diffusion process. All other approximation methods underestimate residence probabilities near the origin and overestimate such probabilities in the peripheries, leading to an overestimation of invasion rates. 6. The asymptotic properties of centroid-to-centroid and area-to-centroid dispersal approximation methods deviate from that which is expected under continuous dispersal. This characteristic renders these methods unsuitable for use in long-term simulation studies where the equilibrium properties of the system are of interest. 7. Centroid-to-area and area-to-area approximation methods exhibit both low approximation error and desirable asymptotic properties. These methods provide a viable mechanism for linking individual-level dispersal to larger-scale characteristics such as metapopulation connectivity.