Traveling wave solutions from microscopic to macroscopic chemotaxis models

Roger Lui, Zhian Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

42 Citations (Scopus)


In this paper, we study the existence and nonexistence of traveling wave solutions for the one-dimensional microscopic and macroscopic chemotaxis models. The microscopic model is based on the velocity jump process of Othmer et al. (SIAM J Appl Math 57:1044-1081, 1997). The macroscopic model, which can be shown to be the parabolic limit of the microscopic model, is the classical Keller-Segel model, (Keller and Segel in J Theor Biol 30:225-234; 377-380, 1971). In both models, the chemosensitivity function is given by the derivative of a potential function, Φ(v), which must be unbounded below at some point for the existence of traveling wave solutions. Thus, we consider two examples: Φ(v) = In v and Φ(v) = In[v/(1-v)]. The mathematical problem reduces to proving the existence or nonexistence of solutions to a nonlinear boundary value problem with variable coefficient on ℝ. The main purpose of this paper is to identify the relationships between the two models through their traveling waves, from which we can observe how information are lost, retained, or created during the transition from the microscopic model to the macroscopic model. Moreover, the underlying biological implications of our results are discussed.
Original languageEnglish
Pages (from-to)739-761
Number of pages23
JournalJournal of Mathematical Biology
Issue number5
Publication statusPublished - 1 Jan 2010
Externally publishedYes


  • Chemical kinetics
  • Chemotaxis
  • Keller-Segel model
  • Potential functions
  • Traveling wave solutions
  • Turning rate functions

ASJC Scopus subject areas

  • Modelling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

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