Postlethwaite, CM and Rucklidge, AM orcid.org/0000-0003-2985-0976 (2019) A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock-Paper-Scissors. Nonlinearity, 32 (4). pp. 1375-1407. ISSN 0951-7715
Abstract
One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019 IOP Publishing Ltd & London Mathematical Society. This is an author produced version of a paper published in Nonlinearity. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Rock–Paper–Scissors, robust heteroclinic cycles, travelling waves |
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) |
Depositing User: | Symplectic Publications |
Date Deposited: | 04 Dec 2018 13:34 |
Last Modified: | 21 Mar 2020 01:38 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1361-6544/aaf530 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:139485 |