Systems of Ordinary Differential Equations Which Generate an Order Preserving Flow. A Survey of Results

This article consists of a survey of results concerning the qualitative behavior of solutions of systems of ordinary differential equations which generate an order preserving flow. We restrict our consideration to partial orderings on Rⁿ induced by any one of its orthants; a flow preserves ordering if any two solutions $x(t)$ and $y(t)$ are ordered, x(t) y(t), for all $t > 0$ whenever x(0) y(0). Many of the important results for such systems have only recently been obtained, principally by M. W. Hirsch, who pointed out the tendency of their solutions to converge to equilibrium. Less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria. A particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibrium. One of our main goals is to show that by allowing partial orderings on Rⁿ generated by orthants other than the positive one, the usual restrictive Kamke (quasimonotone) condition (all “off diagonal” feedbacks are positive) which results from the standard ordering is modified in such a way as to allow (selectively) some negative feedback. As a consequence, there are many interesting and nontrivial applications of the theory. Although the focus of this paper is on systems of ordinary differential equations, some recent comparison results, derived by Conway and Smoller, for systems of reaction-diffusion equations fit quite naturally in our framework and are reviewed.