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

TL;DR

This paper relaxes the conditions for the existence and uniqueness of a carrying simplex in discrete Kolmogorov systems, showing it suffices for the Jacobian to have nonpositive entries and for functions to be strictly decreasing in their own variables.

Contribution

It introduces a less restrictive condition for the existence and uniqueness of a modified carrying simplex in competitive systems.

Findings

The Jacobian matrix entries can be nonpositive instead of strictly negative.

Existence and uniqueness of the modified carrying simplex are established under new conditions.

Applications include criteria for species extinction and dominance in population models.

Abstract

For a $C^1$ map $T$ from $C =[0, +\infty)^N$ to $C$ of the form $T_i(x) = x_if_i(x)$, the dynamical system $x(n) =T^n(x)$ as a population model is competitive if $\frac{\partial f_i}{\partial x_j}\leq 0$ $(i\not= j)$. A well know theorem for competitive systems, presented by Hirsch (J. Bio. Dyn. 2 (2008) 169--179) and proved by Ruiz-Herrera (J. Differ. Equ. Appl. 19 (2013) 96--113) with various versions by others, states that, under certain conditions, the system has a compact invariant surface $\Sigma\subset C$ that is homeomorphic to $\Delta^{N-1} =\{x\in C: x_1+ \cdots + x_N=1\}$, attracting all the points of $C\setminus\{0\}$, 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 $N^2$ entries of the Jacobian matrix $Df = (\frac{\partial f_i}{\partial x_j})$ 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 $f_i$ is strictly decreasing in $x_i$. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.