On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

Abstract

For a C 1 map T from C = [0, +∞) N to C of the form Ti(x) = xifi(x), the dynamical system x(n) = T n (x) as a population model is competitive if ∂f i ∂x j ≤ 0 (i = 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 Σ ⊂ C that is homeomorphic to ∆ N−1 = {x ∈ C : x1 + • • • + xN = 1}, attracting all the points of C \ {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 = (∂f i ∂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 fi is strictly decreasing in xi. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

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

“…Then, by Theorem 2.4, we have α(p) = {0} so p ∈ B(0), a contradiction to p ∈ Σ = B(0) \ ({0} ∪ B(0)). This shows our claim (13).…”

“…(a) Under the assumption that Γ i ∩ [0, r] is strictly below Γ j for all j ∈ I N \ {i}, we first claim that (13) were not true then we would have a point p ∈ (Γ − i ∩ Σ). As 0 ∈ Σ, we have p = 0 and a nonempty…”

“…Since Σ attracts all the points of C \ {0}, the dynamics of (1) on C is essentially described by the dynamics on Σ. The carrying simplex theory was originally established by Hirsch [10] (see [13] for latest update) for competitive Kolmogorov systems of differential equations. Since then the idea of a carrying simplex for discrete systems gradually appeared in literature (see [19], [20], [14] for example).…”

Global Attraction and Repulsion of a Heteroclinic Limit Cycle in Three Dimensional Kolmogorov Maps

“…The carrying simplex theory and its various applications is one of the important and influential developments. This theory was originally established by Hirsch [10] (see [13] and [14] for latest update) for competitive Kolmogorov systems of differential equations. Since then the idea of a carrying simplex for discrete systems gradually appeared in literature (see [20], [21], [16], [5] for example).…”

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

“…This can be thought of as modelling competition for resources. In differential equation models for competition, these assumptions are typically sufficient for a carrying simplex to exist [8,11], but not in the discrete time case. We follow other authors [23,9,20,12] and add further conditions that render the map retrotone.…”