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Fractional step methods applied to a chemotaxis model.

R Tyson1, L G Stern, R J LeVeque

  • 1Department of Applied Mathematics, Box 352420, University of Washington, Seattle, WA 98195-2420, USA. rebecca@amath.washington.edu

Journal of Mathematical Biology
|January 11, 2001
PubMed
Summary

A new fractional step numerical method effectively models chemotaxis, including complex diffusion and reaction terms. This approach accurately simulates bacterial growth patterns by treating chemotaxis as advection, overcoming numerical instabilities.

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Area of Science:

  • Mathematical Biology
  • Computational Science
  • Numerical Analysis

Background:

  • Chemotaxis models describe collective cell movement influenced by chemical signals.
  • Existing numerical methods struggle with density-dependent diffusion and reaction terms in chemotaxis.
  • Pattern formation in biological systems like bacterial growth requires accurate simulation of these complex dynamics.

Purpose of the Study:

  • To develop a novel fractional step numerical method for nonlinear partial differential equations in chemotaxis.
  • To address challenges posed by density-dependent diffusion, reaction, and Fickian diffusion terms.
  • To accurately simulate pattern formation in bacterial growth models.

Main Methods:

  • A fractional step approach is employed, treating the chemotaxis term as an advection term.

Related Experiment Videos

  • High-resolution methods from CLAWPACK are utilized for the advection step to capture steep gradients.
  • The A-stable and L-stable TR-BDF2 method is applied for the diffusion step.

    • Main Results:

    • The developed method successfully simulates pattern formation in bacterial growth.
    • The approach demonstrates good agreement with expected biological patterns.
    • A numerical instability observed in other diffusion methods was analyzed and successfully eliminated.

    Conclusions:

    • The fractional step numerical method provides an effective and stable approach for chemotaxis models.
    • This method accurately captures complex phenomena like density-dependent diffusion and advection.
    • The technique offers a robust tool for studying pattern formation in biological systems.