Pattern Formation in Keller-Segel Chemotaxis Models with Logistic Growth

Ling Jin, Qi Wang, Zengyan Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

13 Citations (Scopus)


In this paper, we investigate pattern formation in Keller-Segel chemotaxis models over a multidimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate χ increases. Then using Crandall-Rabinowitz local theory with χ being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally, we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.
Original languageEnglish
Article number1650033
JournalInternational Journal of Bifurcation and Chaos
Issue number2
Publication statusPublished - 1 Feb 2016


  • bifurcation
  • chemotaxis model
  • logistic growth
  • Pattern formation
  • steady state

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics


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