Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics

Most of the fundamental elements of ecology, ranging from individual behavior to species abundance, diversity, and population dynamics, exhibit spatial variation. Partial differential equation models provide a means of melding organism movement with population processes and have been used extensively to elucidate the effects of spatial variation on populations. While there has been an explosion of theoretical advances in partial differential equation models in the past two decades, this work has been generally neglected in mathematical ecology textbooks. Our goal in this paper is to make this literature accessible to experimentally ecologists. Partial differential equations are used to model a variety of ecological phenomena; here we discuss dispersal, ecological invasions, critical patch size, dispersal—mediated coexistence, and diffusion—driven spatial patterning. These model emphasize that simple organism movement can produce striking large—scale patterns in homogeneous environments, and that in heterogeneous environments, movement of multiple species can change the outcome of competition or predation.