(In)stability of Travelling Waves in a Model of Haptotaxis

Kristen E. Harley, Peter Van Heijster, Robert Marangell, Graeme J. Pettet, Timothy V. Roberts, Martin Wechselberger

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We examine the spectral stability of travelling waves of the haptotaxis model studied in [K. Harley et al., SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 366-396]. In the process we apply Lienard coordinates to the linearized stability problem and use a Riccati-Transform/Grassmannian spectral shooting method \a la [K. Harley et al., Math. Biosci., 266 (2015), pp. 36-51; V. Ledoux et al., SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 480-507; V. Ledoux, S. J. A. Malham, and V. Th\"ummler, Math. Comp., 79 (2010), pp. 1585-1619] in order to numerically compute the Evans function and point spectrum of a linearized operator associated with a travelling wave. We numerically show the instability of nonmonotone waves (type IV) and the stability of the monotone ones (types I-III) to perturbations in an appropriately weighted space.

Original languageEnglish
Pages (from-to)1629-1653
Number of pages25
JournalSIAM Journal on Applied Mathematics
Volume80
Issue number4
DOIs
Publication statusPublished - 2020
Externally publishedYes

Keywords

  • Evans function
  • haptotaxis
  • Lienard coordinates
  • stability of travelling waves

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