Invasion Speeds with Active Dispersers in Highly Variable Landscapes: Homogenization, Defecation and the Migration of Trees

The distribution of many tree species are strongly determined by the behavior and range of vertebrate dispersers, particularly birds. Many models for seed dispersal exist, and are built around the assumption that seeds undergo a random walk (simple diffusion) while they are being carried by vertebrates, either in the digestive tract or during the process of seed storage (caching). However it is very rare for animals to perform simple random walks during dispersal; the more common behavior is ecological diffusion, in which animal motility varies with habitat type and spatial variability of landscape. While digesting or caching seeds by birds, there exists some probability distribution function in time in the process of getting seeds to the ground. This distribution generates a non-consistent seed settling rate. We introduce and apply ecological diffusion and seeds settling rates into the dispersal model to capture the high spatial variability of habitat type relative to bird movement. We will apply the method of homogenization to determine leading order solutions for the seed dispersal kernel (SDK) and construct an integrodifference equation (IDE) population model for adult trees based on the resulting dispersion of seeds. The existing theory for predicting spread rates in IDE does not apply when dispersal kernels are anisotropic. However, since the homogenized SDK is isotropic on large scales, we show that speeds calculated using the harmonic average motility, accurately predict rates of invasion for the spatially variable system