Stability of periodic travelling waves in a Rock-Paper-Scissors model

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dc.contributor.author Hasan, CR en
dc.contributor.author Osinga, Hinke en
dc.contributor.author Postlethwaite, CM en
dc.contributor.author Rucklidge, AM en
dc.date.accessioned 2020-06-05T03:50:14Z en
dc.date.issued 2019-11-24 en
dc.identifier.citation Arxiv (1911.10447v1). 24 Nov 2019. 25 pages en
dc.identifier.uri http://hdl.handle.net/2292/51358 en
dc.description.abstract We study a Rock-Paper-Scissors model that describes the spatiotemporal evolution of three competing populations in ecology, or strategies in evolutionary game theory. The dynamics of the model is determined by a set of partial differential equations (PDEs) in a reaction-diffusion form; it exhibits travelling waves (TWs) in one spatial dimension and spiral waves in two spatial dimensions. In this paper, we focus on the stability of the TWs in a one-dimensional version of this model. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. This heteroclinic cycle gives rise to a family of periodic TWs. The existence of heteroclinic cycles and associated periodic TWs can be established via the transformation of the PDE model into a system of ordinary differential equations (ODEs) under the assumption that the wave speed is constant. Determining the stability of periodic TWs is more challenging and requires analysis of the essential spectrum of the linear operator of the periodic TWs. We compute this spectrum and the curve of instability with the continuation scheme developed in [Rademacher, Sandstede, and Scheel, Physica D, Vol. 229, 2007]. We also 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 TWs. We finally show that our results from the stability analysis are verified by direct simulation of the PDE model and how the computed growth rates accurately quantify the instabilities of the travelling waves. en
dc.relation.ispartof Arxiv en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
dc.rights https://arxiv.org/licenses/nonexclusive-distrib/1.0/license.html en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.subject nlin.PS en
dc.subject nlin.PS en
dc.subject math.AP en
dc.subject math.DS en
dc.title Stability of periodic travelling waves in a Rock-Paper-Scissors model en
dc.type Report en
dc.rights.holder Copyright: The authors en
pubs.author-url http://arxiv.org/abs/1911.10447v1 en
dc.rights.accessrights http://purl.org/eprint/accessRights/OpenAccess en
pubs.subtype Working Paper en
pubs.elements-id 788683 en
pubs.org-id Science en
pubs.org-id Mathematics en
pubs.arxiv-id 1911.10447 en
pubs.number 1911.10447v1 en
pubs.record-created-at-source-date 2020-06-05 en


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