On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems

ABSTRACT For a map T from to C of the form , the dynamical system as a population model is competitive if . A well know theorem for competitive systems, presented by Hirsch [On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn. 2(2) (2008), pp. 169–179] and proved by Ruiz-Herrera [Exclusion and dominance in discrete population models via the carrying simplex, J. Differ. Equ. Appl. 19(1) (2013), pp. 96–113] with various versions by others, states that, under certain conditions, the system has a compact invariant surface that is homeomorphic to , attracting all the points of , and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the entries of the Jacobian matrix to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of Df to be nonpositive and each is strictly decreasing in . As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

[1]  M. Hirsch.  Systems of differential equations that are competitive or cooperative. VI: A local Cr Closing Lemma for 3-dimensional systems , 1985, Ergodic Theory and Dynamical Systems.

[2]  Peter Lancaster,et al.  The theory of matrices , 1969 .

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

[4]  S. Baigent.  Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems , 2016 .

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

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

[7]  S. Elaydi,et al.  Global stability of higher dimensional monotone maps , 2017 .

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

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

[10]  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.

[11]  Jifa Jiang,et al.  On heteroclinic cycles of competitive maps via carrying simplices , 2015, Journal of Mathematical Biology.

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

[13]  Jifa Jiang,et al.  The General Properties of Discrete-Time Competitive Dynamical Systems , 2001 .

[14]  M. Gyllenberg,et al.  A note on global stability of three-dimensional Ricker models , 2019, Journal of Difference Equations and Applications.

[15]  M. Gyllenberg,et al.  On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex , 2017 .

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

[17]  S. Baigent.  Convex geometry of the carrying simplex for the May-Leonard map , 2019, Discrete & Continuous Dynamical Systems - B.

[18]  Zhanyuan Hou.  Geometric method for global stability of discrete population models , 2020, Discrete & Continuous Dynamical Systems - B.

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

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