(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

doi:10.1137/19M1259705

(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 journal › Article › Academic › peer-review

8 Citations (Scopus)

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 language	English
Pages (from-to)	1629-1653
Number of pages	25
Journal	SIAM Journal on Applied Mathematics
Volume	80
Issue number	4
DOIs	https://doi.org/10.1137/19M1259705
Publication status	Published - 2020
Externally published	Yes

Keywords

Evans function
haptotaxis
Lienard coordinates
stability of travelling waves

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10.1137/19M1259705

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@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",

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",

}

TY - JOUR

T1 - (In)stability of Travelling Waves in a Model of Haptotaxis

AU - Harley, Kristen E.

AU - Van Heijster, Peter

AU - Marangell, Robert

AU - Pettet, Graeme J.

AU - Roberts, Timothy V.

AU - Wechselberger, Martin

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Evans function

KW - haptotaxis

KW - Lienard coordinates

KW - stability of travelling waves

U2 - 10.1137/19M1259705

DO - 10.1137/19M1259705

M3 - Article

AN - SCOPUS:85090557354

SN - 0036-1399

VL - 80

SP - 1629

EP - 1653

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 4

ER -

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

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Keywords

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