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## 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  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, doi: 10.3934/dcdsb.2018288
##### References:
 [1] S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity, 26 (2013), 1001-1029. doi: 10.1088/0951-7715/26/4/1001. [2] S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, J. Difference Equ. Appl., 22 (2016), 609-622. doi: 10.1080/10236198.2015.1125895. [3] S. Baigent, Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex, Proc. Edinb. Math. Soc. (2), 55 (2012), 53-63. doi: 10.1017/S0013091510000684. [4] M. Benaim, On invariant hypersurfaces of strongly monotone maps, J. Differential Equations, 137 (1997), 302-319. doi: 10.1006/jdeq.1997.3269. [5] P. Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math., 446 (1994), 115-135. doi: 10.1515/crll.1994.446.115. [6] P. deMottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335. doi: 10.1007/BF00276900. [7] O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52. [8] R. Goldman, Curvature formulas for implicit curves and surfaces, Comput. Aided Geom. Design, 22 (2005), 632-658. doi: 10.1016/j.cagd.2005.06.005. [9] M.W. Hirsch and H. Smith, Monotone maps: A review, J. Difference Equ. Appl., 11 (2005), 379-398. doi: 10.1080/10236190412331335445. [10] M.W. Hirsch, Systems of differential equations which are competitive or cooperative: Ⅲ Competing species, Nonlinearity, 1 (1988), 51-71. [11] M.W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179. doi: 10.1080/17513750801939236. [12] J. Jiang, J. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding, J. Differential Equations, 246 (2009), 1623-1672. doi: 10.1016/j.jde.2008.10.008. [13] J. Jiang and L. Niu, The dynamical behavior on the carrying simplex of a three-dimensional competitive system: Ⅱ. hyperbolic structure saturation, Int. J. Biomath., 07 (2014), 1450002, 14pp. doi: 10.1142/S1793524514500028. [14] J. Jiang and L. Niu, The theorem of the carrying simplex for competitive system defined on the n-rectangle and its application to a three-dimensional system, Int. J. Biomath., 7 (2014), 1450063, 11pp. doi: 10.1142/S1793524514500636. [15] J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie-Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261. doi: 10.1007/s00285-016-1052-y. [16] J. Jiang, L. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972. doi: 10.1007/s00285-015-0920-1. [17] M.R.S. Kulenović and O. Merino, Invariant curves for planar competitive and cooperative maps, J. Difference Equ. Appl., 24 (2018), 898-915. doi: 10.1080/10236198.2018.1438418. [18] B. Lemmen and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079. [19] E.J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842. doi: 10.1090/S0002-9904-1934-05978-0. [20] J. Mierczyński, The C1 property of convex carrying simplices for competitive maps, preprint, arXiv: 1801.01032. [21] J. Mierczyński, The C1 property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations, 111 (1994), 385-409. doi: 10.1006/jdeq.1994.1087. [22] J. Mierczyński, On smoothness of carrying simplices, Proc. Amer. Math. Soc., 127 (1999), 543-551. doi: 10.1090/S0002-9939-99-04887-X. [23] J. Mierczyński, Smoothness of carrying simplices for three-dimensional competitive systems: A counterexample, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 6 (1999), 147-154. [24] J. Mierczyński, The C1 property of convex carrying simplices for three-dimensional competitive maps, J. Difference Equ. Appl., 55 (2018), 1-11. [25] L. Niu and J. Jiang, On the equivalent classification of three-dimensional competitive Atkinson-Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244. doi: 10.3934/dcds.2016.36.217. [26] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1997. [27] A.M. Rubinov, Monotonic analysis: Convergence of sequences of monotone functions, Optimization, 52 (2003), 673-692. doi: 10.1080/02331930310001634425. [28] A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Difference Equ. Appl., 19 (2013), 96-113. doi: 10.1080/10236198.2011.628663. [29] H.L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194. doi: 10.1016/0022-0396(86)90086-0. [30] H.L. Smith, Planar competitive and cooperative difference equations, J. Difference Equ. Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108. [31] P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. doi: 10.1016/0022-247X(90)90040-M. [32] A. Tineo, On the convexity of the carrying simplex of planar Lotka-Volterra competitive systems, Appl. Math. Comput., 123 (2001), 93-108. doi: 10.1016/S0096-3003(00)00063-1. [33] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. [34] E.C. Zeeman and M.L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, Nonlinearity, 15 (2002), 1993-2018. doi: 10.1088/0951-7715/15/6/311. [35] M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-216. doi: 10.1080/02681119308806158.

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##### References:
 [1] S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity, 26 (2013), 1001-1029. doi: 10.1088/0951-7715/26/4/1001. [2] S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, J. Difference Equ. Appl., 22 (2016), 609-622. doi: 10.1080/10236198.2015.1125895. [3] S. Baigent, Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex, Proc. Edinb. Math. Soc. (2), 55 (2012), 53-63. doi: 10.1017/S0013091510000684. [4] M. Benaim, On invariant hypersurfaces of strongly monotone maps, J. Differential Equations, 137 (1997), 302-319. doi: 10.1006/jdeq.1997.3269. [5] P. Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math., 446 (1994), 115-135. doi: 10.1515/crll.1994.446.115. [6] P. deMottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335. doi: 10.1007/BF00276900. [7] O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52. [8] R. Goldman, Curvature formulas for implicit curves and surfaces, Comput. Aided Geom. Design, 22 (2005), 632-658. doi: 10.1016/j.cagd.2005.06.005. [9] M.W. Hirsch and H. Smith, Monotone maps: A review, J. Difference Equ. Appl., 11 (2005), 379-398. doi: 10.1080/10236190412331335445. [10] M.W. Hirsch, Systems of differential equations which are competitive or cooperative: Ⅲ Competing species, Nonlinearity, 1 (1988), 51-71. [11] M.W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179. doi: 10.1080/17513750801939236. [12] J. Jiang, J. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding, J. Differential Equations, 246 (2009), 1623-1672. doi: 10.1016/j.jde.2008.10.008. [13] J. Jiang and L. Niu, The dynamical behavior on the carrying simplex of a three-dimensional competitive system: Ⅱ. hyperbolic structure saturation, Int. J. Biomath., 07 (2014), 1450002, 14pp. doi: 10.1142/S1793524514500028. [14] J. Jiang and L. Niu, The theorem of the carrying simplex for competitive system defined on the n-rectangle and its application to a three-dimensional system, Int. J. Biomath., 7 (2014), 1450063, 11pp. doi: 10.1142/S1793524514500636. [15] J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie-Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261. doi: 10.1007/s00285-016-1052-y. [16] J. Jiang, L. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972. doi: 10.1007/s00285-015-0920-1. [17] M.R.S. Kulenović and O. Merino, Invariant curves for planar competitive and cooperative maps, J. Difference Equ. Appl., 24 (2018), 898-915. doi: 10.1080/10236198.2018.1438418. [18] B. Lemmen and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079. [19] E.J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842. doi: 10.1090/S0002-9904-1934-05978-0. [20] J. Mierczyński, The C1 property of convex carrying simplices for competitive maps, preprint, arXiv: 1801.01032. [21] J. Mierczyński, The C1 property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations, 111 (1994), 385-409. doi: 10.1006/jdeq.1994.1087. [22] J. Mierczyński, On smoothness of carrying simplices, Proc. Amer. Math. Soc., 127 (1999), 543-551. doi: 10.1090/S0002-9939-99-04887-X. [23] J. Mierczyński, Smoothness of carrying simplices for three-dimensional competitive systems: A counterexample, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 6 (1999), 147-154. [24] J. Mierczyński, The C1 property of convex carrying simplices for three-dimensional competitive maps, J. Difference Equ. Appl., 55 (2018), 1-11. [25] L. Niu and J. Jiang, On the equivalent classification of three-dimensional competitive Atkinson-Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244. doi: 10.3934/dcds.2016.36.217. [26] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1997. [27] A.M. Rubinov, Monotonic analysis: Convergence of sequences of monotone functions, Optimization, 52 (2003), 673-692. doi: 10.1080/02331930310001634425. [28] A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Difference Equ. Appl., 19 (2013), 96-113. doi: 10.1080/10236198.2011.628663. [29] H.L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194. doi: 10.1016/0022-0396(86)90086-0. [30] H.L. Smith, Planar competitive and cooperative difference equations, J. Difference Equ. Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108. [31] P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. doi: 10.1016/0022-247X(90)90040-M. [32] A. Tineo, On the convexity of the carrying simplex of planar Lotka-Volterra competitive systems, Appl. Math. Comput., 123 (2001), 93-108. doi: 10.1016/S0096-3003(00)00063-1. [33] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. [34] E.C. Zeeman and M.L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, Nonlinearity, 15 (2002), 1993-2018. doi: 10.1088/0951-7715/15/6/311. [35] M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-216. doi: 10.1080/02681119308806158.
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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