Monotonicity properties of systems of ordinary differential equations

Abstract

A general framework for determining when a solution componnent is monotone with respect to changes in an initial component value is developed.

Conditions for monotonicity with respect to an orthant are formulated graph theoretically, and conditions for partial strong monotonicity are given.

Monotonicity with respect to a closed, convex cone,[(, is also investigated. For a system of differential equations, i = i(i), z(0) = .i0 , Ii En, the Kamke-Millier Theorem (1932/1927) is extended to closed, convex cones by imposing the essential hypothesis .. 3 l such that Di(z) + ll: [( t-+ [(,Vi EN, N compact. Strong monotonicity is achieved by further demanding that 3 m such that (Di(z) + (l + 1)/)"': K\ {O} i-+ int(K), V z EN, or, more practically, through a graph theoretic formulation. Given a cone with n generators, ~, a directed multigraph on n vertices, Ui, is constructed with a directed edge from Yi to Yi, i ::/: j, if e; is in the smallest face of the cone containing (Dj(z) + (l + 1)/)~, V i E N. The multigraph being strongly connected is a sufficient condition for strong monotonicity.

The results of this thesis are applicable to general autonomous ODEs, but the examples are drawn mostly from chemical kinetics.