Periodic travelling waves in cyclic populations: Field studies and reaction-diffusion models

Jonathan A. Sherratt, Matthew J. Smith

Research output: Contribution to journalLiterature reviewpeer-review

155 Citations (Scopus)


Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction-diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction-diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical. © 2008 The Royal Society.

Original languageEnglish
Pages (from-to)483-505
Number of pages23
JournalJournal of the Royal Society. Interface
Issue number22
Publication statusPublished - 5 Jun 2008


  • Ecological modelling
  • Reaction-diffusion
  • Spatio-temporal patterns
  • Wavetrains


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