Convergence in strongly monotone systems with an increasing first integral

TL;DR

This paper extends a convergence result for strongly monotone systems with increasing first integrals to more general state spaces and orderings, showing that all bounded orbits converge and equilibria are unique on level sets.

Contribution

It generalizes Mierczynski's result to broader state spaces and orderings, demonstrating convergence and uniqueness properties in these more general settings.

Findings

All bounded orbits in the generalized setting converge.

Each equilibrium attracts its entire level set.

There can be at most one equilibrium per level set.

Abstract

In this paper we generalise a useful result due to J. Mierczynski which states that for a strictly cooperative system on the positive orthant, with increasing first integral, all bounded orbits are convergent. Moreover any equilibrium attracts its entire level set, and there can be no more than one equilibrium on any level set. Here, more general state spaces and more general orderings are considered. Let Y subset K subset R^n be any two proper cones. Given a local semiflow phi on Y which is strongly monotone with respect to K, and which preserves a K-increasing first integral, we show that every bounded orbit converges. Again, each equilibrium attracts its entire level set, and there can be no more than one equilibrium on any level set. An application from chemical dynamics is provided.