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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 |
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.
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. |
show all references
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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