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

  • Jean P. GibertEmail author
  • Justin D. Yeakel


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.


Spatial structure Ecological landscape Stability Persistence Coexistence Networks 



We thank Alan Hastings for enlightening conversations that led to the conception of this paper and two anonymous reviewers for their thoughtful suggestions. We dedicate this paper to the memory of Alan Turing, whose transformative work still has, more than half a century later, profound implications for myriad scientific disciplines. Turing endured the intolerance of many of his contemporaries, which preferred to condemn a brilliant scientist and war hero to the uttermost ignominy than accepting him for who he was. We are also indebted to Kevin, whose stubbornness makes studying movement in space all the more interesting.


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Authors and Affiliations

  1. 1.School of Natural SciencesThe University of California, MercedMercedUSA
  2. 2.Department of BiologyDuke UniversityDurhamUSA

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