Integrodifference equations in patchy landscapes : I. Dispersal Kernels.

What is the effect of individual movement behavior in patchy landscapes on redistribution kernels? To answer this question, we derive a number of redistribution kernels from a random walk model with patch dependent diffusion, settling, and mortality rates. At the interface of two patch types, we integrate recent results on individual behavior at the interface. In general, these interface conditions result in the probability density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator. Using this characterization, we illustrate the kind of (discontinuous) dispersal kernels that result from our approach, using three scenarios. First, we assume that dispersal distance is small compared to patch size, so that a typical disperser crosses at most one interface during the dispersal phase. Then we consider a single bounded patch and generate kernels that will be useful to study the critical patch size problem in our sequel paper. Finally, we explore dispersal kernels in a periodic landscape and study the dependence of certain dispersal characteristics on model parameters.