Journal of Mathematical Biology

, Volume 78, Issue 3, pp 711–737 | Cite as

Turing patterns in a predator–prey model with seasonality

  • Xiaoying WangEmail author
  • Frithjof Lutscher


Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction–diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the ‘summer’ season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington–DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).


Consumer-resource Impulsive reproduction Turing bifurcation Pattern formation Reaction–diffusion equation Discrete system 

Mathematics Subject Classification

35R12 35B36 92D25 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of BiologyUniversity of OttawaOttawaCanada

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