Families of discrete kernels for modeling dispersal

Integer lattices are important theoretical landscapes for studying the consequences of dispersal and spatial population structure, but convenient dispersal kernels able to represent important features of dispersal in nature have been lacking for lattices. Because leptokurtic (centrally peaked and long-tailed) kernels are common in nature and have important effects in models, of particular interest are families of dispersal kernels in which the degree of leptokurtosis can be varied parametrically. Here we develop families of kernels on integer lattices with several important properties. The degree of leptokurtosis can be varied parametrically from near 0 (the Gaussian value) to infinity. These kernels are all asymptotically radially symmetric. (Exact radial symmetry is impossible on lattices except in one dimension.) They have separate parameters for shape and scale, and their lower order moments and Fourier transforms are given by simple formulae. In most cases, the kernel families that we develop are closed under convolution so that multiple steps of a kernel remain within the same family. Included in these families are kernels with asymptotic power function tails, which have provided good fits to some observations from nature. These kernel families are constructed by randomizing convolutions of stepping-stone kernels and have interpretations in terms of population heterogeneity and heterogeneous physical processes.