n -dimensional Kolmogorov maps, carrying simplices, and bifurcations at the origin

Abstract

AbstractFor a Kolmogorov map on the positive cone of Rn or an order interval determined by the origin and a positive element of Rn, sufficient conditions are given for the existence of a carrying simplex and a modified carrying simplex. Also, for a parametrized family of Kolmogorov maps that exhibits a bifurcation at the origin and that has a parametrized curve of positive fixed points defined for parameter values close to but above a critical value, sufficient conditions are given for the existence of modified carrying simplices or carrying simplices on order intervals determined by the origin and axial fixed points. In addition, sufficient conditions are given for global attractivity of the positive fixed point and of the boundary of the carrying simplex with respect to the order interval. The results are applied to the LPA (Larvae–Pupae–Adults) model investigated by J. Cushing in 2003.KEYWORDS: Discrete competitive modelretrotone mapcarrying simplexbifurcationattractivityKolmogorov mapLPA mapLyapunov function2020 MATHEMATICS SUBJECT CLASSIFICATIONS: Primary: 37C25.Secondary: 37C7037G35 AcknowledgmentsThe authors are indebted to two anonymous reviewers who did a painstaking and thoroughly in-depth job spotting mathematical errors, unclear or unsupported statements, typos, and grammatically incorrect sentences. The quality of this paper greatly improved with the reviewers' help. The suggestions given for Lemma 3.1(b) and the proof of Theorem 2.5 were particularly helpful. Also, one of the reviewers told us about the interesting references [Citation4,Citation13,Citation23], and [Citation31].Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 Definitions of retrotone map and carrying simplex in the literature often do not require the map to be a Kolmogorov map, see [Citation31]. The same level of generality is needed in Theorem 2.2 of this work.2 Definition 2.1 in [Citation20] of modified carrying simplex Σ requires Σ to be compact, but this requirement is unnecessary since it is also required that Σ is homeomorphic to the compact set {x∈R+n:x1+⋯+xn=1}.3 The condition ‘S|Σ is a homeomorphism’ is a common requirement for carrying simplices [Citation4] [Citation31,Citation34]. However, Definition 2.1 in [Citation20] of modified carrying simplex Σ does not require it, and the author of [Citation20] did not comment on why it was left out. Theorem 2.3 in [Citation20] together with Remark 2.1(b) in [Citation20] imply this condition is satisfied by Σ, so the conclusion of Theorem 2.3 is valid if the condition is added to the definition of modified carrying simplex. Our Theorem 2.2 requires the condition, so we chose to include it in the definition of modified carrying simplex given here. The resulting definition has the condition in common with Definition 6.1 in [Citation34].4 Whenever it is convenient, vectors in Rn are displayed in row vector form. However in formulas containing a multiplication of a matrix and a vector, vectors are considered to be in column vector form.5 A justification: U can be chosen to be connected and precompact so that S is defined and C1 on a neighbourhood of closU (the closure of U) and detDS(z)≠0 on closU, and in this case Lemma 4.1 in [Citation34] applies to closU, so S is a homeomorphism from closU onto S(closU), hence S is a homeomorphism from U onto S(U).6 For this statement and the definition of nice neighbourhood, see page 7394 of [Citation31].