On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems
Zhanyuan Hou
Abstract
For a C1 map T from C=[0,+∞)N to C of the form Ti(x)=xifi(x), the dynamical system x(n)=Tn(x) as a population model is competitive if ∂fi∂xj≤ 0 (i≠j). 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 Σ⊂C that is homeomorphic to ΔN−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 N2 entries of the Jacobian matrix Df=(∂fi∂xj) 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.