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 R″ 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). 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 R″ 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. Some recent comparison results for systems of reaction-diffusion equations fit quite naturally in our framework and are reviewed.