# American Institute of Mathematical Sciences

April  2019, 24(4): 1697-1723. doi: 10.3934/dcdsb.2018288

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

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

* Corresponding author

Received  August 2017 Revised  April 2018 Published  April 2019 Early access  August 2018

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.

Citation: Stephen Baigent. Convex geometry of the carrying simplex for the May-Leonard map. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1697-1723. doi: 10.3934/dcdsb.2018288
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##### References:
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)
Mapping of $\Delta({\pmb{a}} )$ by ${\pmb T}$ to the new set ${\pmb T}(\Delta({\pmb{a}} ))$
Bounds on the intersection of planes with the axes. Left figure: Convex surface, $0 < x_{\min} < x_{\max} < q_1$. Right figure: Concave surface, $q_1 < x_{\min} < x_{\max}$
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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