Laplacian matrices and Turing bifurcations: revisiting Levin 1974 and the consequences of spatial structure and movement for ecological dynamics

Abstract

We revisit a seminal paper by Levin (Am Nat 108:207–228, 1974), where spatially mediated coexistence and spatial pattern formation were described. We do so by reviewing and explaining the mathematical tools used to evaluate the dynamics of ecological systems in space, from the perspective of recent developments in spatial population dynamics. We stress the importance of space-mediated stability for the coexistence of competing species and explore the ecological consequences of space-induced instabilities (Turing instabilities) for spatial pattern formation in predator–prey systems. Throughout, we link existing theory to recent developments in discrete spatially structured metapopulations, such as our understanding of how ecological dynamics occurring on a network can be analyzed using the Laplacian matrix and its associated eigenvalue spectrum. We underline the validity of Levin’s message, over 40 years later, and suggest it has ever-growing implications in a changing and increasingly fragmented world.