Linearization and invariant manifolds on the carrying simplex for competitive maps

• Published in: Journal of Differential Equations Vol. 267; no. 12; pp. 7385 - 7410
• Main Authors: Mierczyński, Janusz, Niu, Lei, Ruiz-Herrera, Alfonso
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.12.2019
• Subjects: Carrying simplex; Invariant foliation; Invariant manifold; Linearization; Pseudo-stable manifold; Pseudo-unstable manifold; Carrying simplex; Invariant foliation; Pseudo-unstable manifold; Invariant manifold; Pseudo-stable manifold; Linearization
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2019.08.001

A result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface of codimension one which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. Our results give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics.