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Analysis of scaling limits of the Kinetic Chemotaxis Equations

  • Author / Creator
    Thiessen, Ryan
  • The kinetic chemotaxis equations have long been used to model biological processes. We will analyze a volume filling variant of the kinetic chemotaxis equations on the torus. Since the kinetic chemotaxis systems have well known blow-up solutions, we spend a considerable amount of time showing conditions for which solutions of the volume filling kinetic chemotaxis equations exist globally. Due to the individualist origins of the kinetic equation (velocity jump process), the system's interesting population dynamics occur at a much larger space-time scale. Various approximation methods have been developed to explore these population dynamics, like the parabolic scaling and moment closures. In this thesis, we will explore these approximation methods in the context of chemotaxis, and we will prove two new results for the convergence of the scaling limits to the parabolic limit and the hyperbolic limit, respectively. In addition to the approximation methods' mathematical consequences, we will delve into their underlying biological meaning, explain under what conditions the methods coincide, and discuss their differences. In developing these models, a few common features appear, such as anisotropic diffusion and chemotactic mixing. For the above macroscopic models, we develop a sophisticated numerical solver to investigate anisotropic pattern formation. To our surprise, we found new spatial criss-cross patterns due to competing cues, one direction given by anisotropy versus a different direction due to chemotaxis. A full analysis of these new patterns is not part of this thesis and is left for future work.

  • Subjects / Keywords
  • Graduation date
    Fall 2020
  • Type of Item
    Thesis
  • Degree
    Master of Science
  • DOI
    https://doi.org/10.7939/r3-perf-b892
  • License
    Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.