Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics

TLDR

Ecology exhibits spatial variation across scales, and partial differential equation models link organism movement with population dynamics, yet recent advances in these models have largely been omitted from standard mathematical ecology texts. The authors aim to make this literature accessible to experimental ecologists. The paper reviews how PDEs capture dispersal, invasions, critical patch size, dispersal‑mediated coexistence, and diffusion‑driven patterning in ecological systems. The models demonstrate that simple organism movement can generate striking large‑scale patterns in homogeneous environments, and that in heterogeneous settings, movement of multiple species can shift competition or predation outcomes.

Abstract

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.