Convex geometry of the carrying simplex for the May-Leonard map

Stephen Baigent

doi:10.3934/dcdsb.2018288

Article Contents

2019, Volume 24, Issue 4: 1697-1723. Doi: 10.3934/dcdsb.2018288

This issue Previous Article Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques Next Article Synchronization of first-order autonomous oscillators on Riemannian manifolds

Convex geometry of the carrying simplex for the May-Leonard map

Stephen Baigent^,

Department of Mathematics, UCL, Gower Street, London, WC1E 6BT, UK

^* Corresponding author
^* Corresponding author

Received: August 2017

Revised: April 2018

Early access: August 2018

Published: January 2019

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Abstract

We study the convex geometry of certain invariant manifolds, known as carrying simplices, for 3-species competitive Kolmogorov-type maps. We show that if all planes whose normal bundles are contained in a fixed closed and solid convex cone are rendered convex (concave) surfaces by the map, then, if there is a carrying simplex, it is a convex (concave) surface. We apply our results to the May-Leonard map.

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Mathematics Subject Classification: Primary: 37C70, 37C65, 34C45; Secondary: 37C05, 34C12.

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Figure 1. Carrying simplices for the May-Leonard model (8) with $r = 2$. Left: Convex carrying simplex for $\alpha = 3/4, \beta = 2/3$ (see example 11.2). Right: Concave carrying simplex $\alpha = 5/4, \beta = 7/6$ (see example 11.1)

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Figure 2. Mapping of $\Delta({\pmb{a}} )$ by ${\pmb T}$ to the new set ${\pmb T}(\Delta({\pmb{a}} ))$

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Figure 3. Bounds on the intersection of planes with the axes. Left figure: Convex surface, $0 < x_{\min} < x_{\max} < q_1$. Right figure: Concave surface, $q_1 < x_{\min} < x_{\max} $

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Figure 4. Carrying simplices for the May-Leonard model (8) with $r = 2$. Top left: $\alpha = 4/5, \beta = 3/4$. Top right: $\alpha = 2/3, \beta = 7/12$, Bottom left: $\alpha = 7/5, \beta = 4/3$. Bottom right: $\alpha = 3/2, \beta = 7/5$

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