Classical monotonicity and comparison techniques have been used to obtain stability and structure results concerning differential equations. Fifteen years ago, Morris Hirsch developed the theory of montone dynamical systems, where the dynamical system preserves some natural ordering on the state space. This theory has been simplified by Hal Smith and Horst Thieme. Their work involves developing the theory of monotone semiflows under the strong order preserving property (SOP).

In this colloquium talk, I will introduce monotone semiflows by examples and present some of the general stability theorems available. I will discuss the order preserving properties that are required to develop the theory of monotone semiflows. I will introduce new order preserving properties that suffice to develop most of the existing theory and I will provide set-theoretic examples to show that these new properties are significantly weaker than the (SOP).