Spatial pattern formation in a chemotaxis-diffusion-growth model

Kousuke Kuto, Koichi Osaki, Tatsunari Sakurai, Tohru Tsujikawa*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

65 Citations (Scopus)


Mimura and one of the authors (1996) proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For this model, Tello and Winkler (2007) [22] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. (2008) numerically showed several spatio-temporal patterns in a rectangle. Motivated by their work, we consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints in the present paper. First we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, we exhibit several numerical results for the stationary and oscillating patterns.

Original languageEnglish
Pages (from-to)1629-1639
Number of pages11
JournalPhysica D: Nonlinear Phenomena
Issue number19
Publication statusPublished - 2012 Oct 1
Externally publishedYes


  • Bifurcation
  • Chemotaxis
  • Pattern formation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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