Existence of the carrying simplex for a $C^1$ retrotone map

We present a dynamical approach to the study of decreasing, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of decreasing manifolds over the probability simplex coupled with distances between these manifolds measured by way of the Harnack and Hausdorff metrics. We establish Kuratowski convergence of radial representations of decreasing manifolds to a unique function which then provides the locally Lipschitz radial representation of the carrying simplex.

[1]  Horst R. Thieme,et al.  Dynamical Systems And Population Persistence , 2016 .

[2]  M. Zeeman,et al.  Extinction in nonautonomous competitive Lotka-Volterra systems , 1996 .

[3]  Jifa Jiang,et al.  On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points , 2015 .

[4]  S. Baigent,et al.  Global Stability of Interior and Boundary Fixed Points for Lotka–Volterra Systems , 2012 .

[5]  Alfonso Ruiz-Herrera.  Exclusion and dominance in discrete population models via the carrying simplex , 2013 .

[6]  Zhanyuan Hou,et al.  On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems , 2021, 2102.09276.

[7]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[8]  Ulrich Krause,et al.  Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications , 2015 .

[9]  S. Baigent,et al.  Global stability of discrete-time competitive population models , 2017 .

[10]  Jifa Jiang,et al.  Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding , 2009 .

[11]  Jifa Jiang,et al.  Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems $ , 2002 .

[12]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[13]  P. de Mottoni,et al.  Competition systems with periodic coefficients: A geometric approach , 1981 .

[14]  M. Hirsch.  On existence and uniqueness of the carrying simplex for competitive dynamical systems , 2008, Journal of biological dynamics.

[15]  O. Diekmann,et al.  Carrying simplices in discrete competitive systems and age-structured semelparous populations , 2007 .

[16]  Zhanyuan Hou.  On existence and uniqueness of a carrying simplex in Kolmogorov differential systems , 2020, Nonlinearity.

[17]  Katja Gruenewald,et al.  Topics On Analysis In Metric Spaces , 2016 .

[18]  Hal L. Smith.  Periodic competitive differential equations and the discrete dynamics of competitive maps , 1986 .

[19]  Jifa Jiang,et al.  On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex , 2017, Journal of mathematical biology.

[20]  D. Gale,et al.  The Jacobian matrix and global univalence of mappings , 1965 .

[21]  M. Gyllenberg,et al.  On the dynamics of multi-species Ricker models admitting a carrying simplex , 2019, Journal of Difference Equations and Applications.