Journal of Mathematical Biology

, Volume 53, Issue 4, pp 617–641 | Cite as

Diffusion, Cross-diffusion and Competitive Interaction

  • Masato Iida
  • Masayasu Mimura
  • Hirokazu NinomiyaEmail author


The cross-diffusion competition systems were introduced by Shigesada et al. [J. Theor. Biol. 79, 83–99 (1979)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by the reaction-diffusion system which only includes the linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing’s instability of the corresponding reaction-diffusion system.


Reaction-diffusion systems Cross-diffusion systems Turing’s instability 

Mathematics Subject Classification (2000)

35B25 35K55 35K57 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Masato Iida
    • 1
  • Masayasu Mimura
    • 2
  • Hirokazu Ninomiya
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of Humanities and Social SciencesIwate UniversityMoriokaJapan
  2. 2.Department of Mathematics, School of Science and TechnologyMeiji UniversityKawasakiJapan
  3. 3.Department of Applied Mathematics and InformaticsRyukoku UniversityOtsuJapan

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