Time-periodic carrying simplex for a competitive system of Carath\'eodory ODEs

TL;DR

This paper studies time-periodic competitive ODE systems with weaker regularity assumptions, introducing a new approach to define and approximate the carrying simplex, and proves a topological conjugacy property.

Contribution

It relaxes regularity conditions for carrying simplex existence and provides a dynamical and numerical framework for its analysis in competitive systems.

Findings

Defined the carrying simplex via the attractor of an extended flow.

Proved the system on the carrying simplex is topologically conjugate to a lower-dimensional system.

Introduced a method for numerical approximation of the carrying simplex.

Abstract

We consider time-periodic competitive systems of ordinary differential equations of Kolmogorov type. However, compared with standard assumptions, we relax the regularity of the time-dependent per-capita growth rates by imposing much weaker regularity, namely Carath\'eodory conditions. An important tool in investigating such systems is the concept of carrying simplex, that is, of an unordered invariant manifold of codimension one that attracts all nonzero orbits. We define the carrying simplex via the compact attractor of compact sets of an extended flow, and that attractor can be obtained as the limit of the actions of the solution operator on some set. Compared with previous papers, our approach has more dynamical flavour, and, further, provides a method of numerical approximation of the carrying simplex. Another feature of our paper is that we prove that the system restricted to the extended carrying simplex is topologically conjugate to a system of one dimension less. This property, appearing in the path-breaking paper by Morris W. Hirsch, has been almost universally neglected in the later papers.