About reaction–diffusion systems involving the Holling-type II and the Beddington–DeAngelis functional responses for predator–prey models

  • F. Conforto
  • Laurent Desvillettes
  • C. Soresina


We consider in this paper a microscopic model (that is, a system of three reaction–diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to 0 towards the solutions of a reaction–cross diffusion system of predator–prey type involving a Holling-type II or Beddington–DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington–DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.


Cross-diffusion equations Predator–prey equations Turing instability Turing patterns functional responses 

Mathematics Subject Classification

35B25 35B36 35K45 35K57 35Q92 92D25 


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Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della TerraUniversità di MessinaMessinaItaly
  2. 2.Université Paris Diderot, IMJ-PRGParisFrance
  3. 3.CMAF-CIO Centro de Matemática, Aplicações Fundamentais e Investigação, Faculdade de CiênciasLisbonPortugal

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