The \(C^1\) property of convex carrying simplices for competitive maps. (English) Zbl 1441.37030
Summary: For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In this paper it is shown that its convexity implies that it is a \(C^1\) submanifold-with-corners, neatly embedded in the non-negative orthant. The proof uses the characterization of neat embedding in terms of inequalities between Lyapunov exponents for ergodic invariant measures supported on the boundary of the carrying simplex.
MSC:
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37C65 | Monotone flows as dynamical systems |
| 37D10 | Invariant manifold theory for dynamical systems |
| 37D30 | Partially hyperbolic systems and dominated splittings |
Keywords:
smooth dynamics; invariant one-codimensional manifold; Lyapunov exponents; ergodic invariant measuresReferences:
| [1] | Aliprantis, C. D. and Border, K. C.. Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin, 2006. · Zbl 1156.46001 |
| [2] | Arnold, L.. Random Dynamical Systems. Springer, Berlin, 1998. · Zbl 0906.34001 |
| [3] | Baigent, S.. Convexity preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex. Proc. Edinb. Math. Soc. (2)55(1) (2012), 53-63. · Zbl 1241.35041 |
| [4] | Baigent, S.. Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems. Nonlinearity26(4) (2013), 1001-1029. · Zbl 1270.34108 |
| [5] | Baigent, S.. Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems. J. Difference Equ. Appl.22(5) (2016), 609-622. · Zbl 1343.93055 |
| [6] | Balreira, E. C., Elaydi, S. and Luís, R.. Global stability of higher dimensional monotone maps. J. Difference Equ. Appl.23(12) (2017), 2037-2071. · Zbl 1385.37034 |
| [7] | Benaïm, M.. On invariant hypersurfaces of strongly monotone maps. J. Differential Equ.137(2) (1997), 302-319. · Zbl 0889.58013 |
| [8] | Brunovský, P.. Controlling nonuniqueness of local invariant manifolds. J. Reine Angew. Math.446 (1994), 115-135. · Zbl 0783.58061 |
| [9] | Hirsch, M. W.. Systems of differential equations which are competitive or cooperative. III. Competing species. Nonlinearity1(1) (1988), 51-71. · Zbl 0658.34024 |
| [10] | Hirsch, M. W.. On existence and uniqueness of the carrying simplex for competitive dynamical systems. J. Biol. Dyn.2 (2008), 169-179. · Zbl 1152.92026 |
| [11] | Jiang, J., Mierczyński, J. and Wang, Y.. Smoothness of the carrying simplex for discrete-time competitive dynamical systems: a characterization of neat embedding. J. Differential Equ.246(4) (2009), 1623-1672. · Zbl 1166.37009 |
| [12] | Jiang, J. and Niu, L.. On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex. J. Math. Biol.74(5) (2017), 1223-1261. · Zbl 1365.37063 |
| [13] | Jiang, J., Niu, L. and Wang, Y.. On heteroclinic cycles of competitive maps via carrying simplices. J. Math. Biol.72(4) (2016), 939-972. · Zbl 1355.37042 |
| [14] | Lian, Z. and Lu, K.. Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems on a Banach Space. American Mathematical Society, Providence, RI, 2010, p. 967. · Zbl 1200.37047 |
| [15] | Marszczewska, K.. Approximation of carrying simplices in competitive Lotka-Volterra systems (in Polish). MSc Dissertation, Institute of Mathematics and Computer Science, Wrocław University of Technology, 2005. |
| [16] | Mierczyński, J.. The C^1 property of carrying simplices for a class of competitive systems of ODEs. J. Differential Equ.111(2) (1994), 385-409. · Zbl 0804.34048 |
| [17] | Mierczyński, J.. Smoothness of carrying simplices for three-dimensional competitive systems: a counterexample. Dyn. Contin. Discrete Impuls. Syst.6(1) (1999), 149-154. · Zbl 0941.34034 |
| [18] | Mierczyński, J.. On peaks in carrying simplices. Colloq. Math.81(2) (1999), 285-292. · Zbl 0936.34036 |
| [19] | Mierczyński, J.. The C^1 property of convex carrying simplices for three-dimensional competitive maps. J. Difference Equ. Appl.24(8) (2018), 1199-1209. · Zbl 1401.37033 |
| [20] | Mierczyński, J. and Schreiber, S. J.. Kolmogorov vector fields with robustly permanent subsystems. J. Math. Anal. Appl.267(1) (2002), 329-337. · Zbl 1017.34052 |
| [21] | Niu, L. and Ruiz-Herrera, A.. Trivial dynamics in discrete-time systems: carrying simplex and translation arcs. Nonlinearity31(6) (2018), 2633-2650. · Zbl 1391.37018 |
| [22] | Ruiz-Herrera, A.. Exclusion and dominance in discrete population models via the carrying simplex. J. Difference Equ. Appl.19(1) (2013), 96-113. · Zbl 1303.92107 |
| [23] | Smale, S.. On the differential equations of species in competition. J. Math. Biol.3 (1976), 5-7. · Zbl 0344.92009 |
| [24] | Smith, H. L.. Periodic competitive differential equations and the discrete dynamics of competitive maps. J. Differential Equ.64(2) (1986), 165-194. · Zbl 0596.34013 |
| [25] | Tineo, A.. On the convexity of the carrying simplex of planar Lotka-Volterra competitive systems. Appl. Math. Comput.123(1) (2001), 93-108. · Zbl 1036.34057 |
| [26] | Wang, Y. and Jiang, J.. Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems. J. Differential Equ.186(2) (2002), 611-632. · Zbl 1026.37016 |
| [27] | Zeeman, E. C. and Zeeman, M. L.. On the convexity of carrying simplices in competitive Lotka-Volterra systems. Differential Equations, Dynamical Systems, and Control Science. Dekker, New York, 1994, pp. 353-364. · Zbl 0799.92016 |
| [28] | Zeeman, E. C. and Zeeman, M. L.. From local to global behavior in competitive Lotka-Volterra systems. Trans. Amer. Math. Soc.355(2) (2003), 713-734. · Zbl 1008.37055 |
| [29] | Zeeman, M. L.. Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. Dynam. Stab. Syst.8(3) (1993), 189-217. · Zbl 0797.92025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
