Hasan, CR, Osinga, HM, Postlethwaite, CM et al. (1 more author) (2021) Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model. Nonlinearity, 34 (8). pp. 5576-5598. ISSN 0951-7715
Abstract
We study a rock–paper–scissors model for competing populations that exhibits travelling waves in one spatial dimension and spiral waves in two spatial dimensions. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. The model also has travelling fronts that are heteroclinic connections between two equilibria in a moving frame of reference, but these fronts are unstable. However, we find that large-wavelength travelling waves can be stable in spite of being made up of three of these unstable travelling fronts. In this paper, we focus on determining the essential spectrum (and hence, stability) of large-wavelength travelling waves in a cyclic competition model with one spatial dimension. We compute the curve of transition from stability to instability with the continuation scheme developed by Rademacher et al (2007 Physica D 229 166–83). We build on this scheme and develop a method for computing what we call belts of instability, which are indicators of the growth rate of unstable travelling waves. Our results from the stability analysis are verified by direct simulation for travelling waves as well as associated spiral waves. We also show how the computed growth rates accurately quantify the instabilities of the travelling waves.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 IOP Publishing Ltd & London Mathematical Society. This is an author produced version of an article published in Nonlinearity. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | stability of travelling waves, spiral waves, heteroclinic cycles, rock–paper–scissors |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Funding Information: | Funder Grant number Leverhulme Trust RF-2018-449\9 |
Depositing User: | Symplectic Publications |
Date Deposited: | 07 Jun 2021 10:47 |
Last Modified: | 12 Mar 2023 05:18 |
Status: | Published |
Publisher: | Institute of Physics Publishing |
Identification Number: | 10.1088/1361-6544/ac0126 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:174769 |