Convergence in Strongly Monotone Systems with an Increasing First Integral

In this paper we generalise a useful result due to J. Mierczyński 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 ⊂ K ⊂ R be any two proper cones. Given a local semiflow φ 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.