Mathematical Model of Seed Dispersal by Frugivorous Birds and Migration Potential of Pinyon and Juniper in Utah

Abstract

Seed dispersal of juniper and pinyon is a process in which frugivorous birds play an important role. Birds either consume and digest seeds or carry and cache them at some distance from the source tree. These transported and settled seeds can be described by a dispersal kernel, which captures the probability that the seed will move a certain distance by the end of the process. To model active seed dispersal of this nature, we introduce handling time probabilities into the dispersal model to generate a seed digestion kernel. In the limit of no variability in handling time the seed digestion kernel is Gaussian, whereas for uniform variability in handling time the kernel approaches a Laplace distribution. This allows us to standardize spatial movement (diffusion) and handling time (peak settling rate) parameters for all three distributions and compare. Analysis of the tails indicates that the seed digestion kernel decays at a rate intermediate between Gaussian and Laplace seed kernels. Using this seed digestion kernel, we create an invasion model to estimate the speed at which juniper and pinyon forest boundaries move. We find that the speed of seed invasion corresponding to the digestion kernel was faster than seeds resulting from Laplace and Gaussian kernels for more rapidly digested seeds. For longer handling times the speeds are bounded between the Laplace (faster) and Gaussian (slower) speeds. Using parameter values from the literature we evaluate the migration potential of pinyon and juniper, finding that pinyon may be able to migrate up to two orders of magnitude more rapidly, consistent with observations of pine migration during the Holocene.