@article{b9fc831835ed4038b4919e45b705f4a7,
title = "(In)stability of Travelling Waves in a Model of Haptotaxis",
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.",
keywords = "Evans function, haptotaxis, Lienard coordinates, stability of travelling waves",
author = "Harley, {Kristen E.} and {Van Heijster}, Peter and Robert Marangell and Pettet, {Graeme J.} and Roberts, {Timothy V.} and Martin Wechselberger",
note = "Funding Information: \ast Received by the editors May 3, 2019; accepted for publication (in revised form) April 7, 2020; published electronically July 14, 2020. https://doi.org/10.1137/19M1259705 Funding: The work of the second author was supported by the Australian Research Council under grant DE140100741. The work of the sixth author was supported by the Australian Research Council under grant DP180103022. \dagger School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4000, Australia (kristen.e.harley@gmail.com, petrus.vanheijster@qut.edu.au, graeme.pettet@alumni. qut.edu.au). \ddagger School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW, 2006, Australia (robert.marangell@sydney.edu.au, trob5740@uni.sydney.edu.au, martin.wechselberger@ sydney.edu.au ). 1Note that the original model in [33] ignored diffusion (\varepsilon = 0), as it was assumed that diffusion only played a minimal role. Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",
year = "2020",
doi = "10.1137/19M1259705",
language = "English",
volume = "80",
pages = "1629--1653",
journal = "SIAM Journal on Applied Mathematics",
issn = "0036-1399",
publisher = "Society for Industrial and Applied Mathematics",
number = "4",
}