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Carrying simplicies of competitive maps

Baigent, S; (2019) Carrying simplicies of competitive maps. In: Difference Equations, Discrete Dynamical Systems and Applications. Springer: Cham, Switzerland. Green open access

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The carrying simplex is a finite-dimensional, attracting Lipschitz invariant manifold that is commonly found in both continuous and discrete-time competition models from Ecology. It can be studied using the graph transform and cone conditions often applied to study attractors in continuous-time finite and infinitedimensional models from applied mathematics, including chemical reaction networks and reaction diffusion equations. Here we show that the carrying simplex can also be studied from the point of view of the graph transform and cone conditions. However, unlike many of the models mentioned above, we do not use - at least directly - a gap condition that is often used to establish existence of a globally and exponentially attracting manifold. Instead we use contraction of phase volume to ‘suck’ hypersurfaces together uniformly, and ultimately onto the carrying simplex. We give a proof of the existence of the carrying simplex for a class of competitive maps, viewed here as also normally monotone maps. The result is not new, but is carried out in the framework of the graph transform to indicate how the carrying simplex relates to other well-known classes of invariant manifolds. We also discuss the relation between hypersurfaces with positive normals, unordered hypersurfaces and also the type of maps that preserve these types of hypersurfaces. Finally we review several examples from models in Ecology where the carrying simplex is known to exist.

Type: Proceedings paper
Title: Carrying simplicies of competitive maps
Event: ICDEA 23, Timişoara, Romania, July 24-28, 2017
Location: Timişoara, Romania
Dates: 24 July 2017 - 28 July 2017
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/978-3-030-20016-9
Publisher version: https://doi.org/10.1007/978-3-030-20016-9
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10078384
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