A Graph Theoretic Approach to Strong Monotonicity with respect to Polyhedral Cones

Abstract

Consider the flow ϕ_t for the system of differential equations $\dot x\left( t \right) = f\left( x \right)$, xεΩ, Ω⊂$\mathbb{R}$ ⁿ, Ω open. Let K(t) be an expanding polyhedral cone of constant dimension, k be a unit vector in K(0), and x ₀εΩ. A sufficient condition for $\frac{{\partial \phi _t }}{{\partial k}}\left( {x_0 } \right)$εK(t) for t≥0 is that there exists an l so that Df(ϕ_t(x₀))+lI leaves K(t) invariant for all t≥0. If in addition (Df(ϕ_t(x₀))+lI)^n-1 takes k into the relative interior of K(t) for all t>0 then $\frac{{\partial \phi _t }}{{\partial k}}\left( {x_0 } \right)$ is in the relative interior of K(t) for all t>0. The latter condition for strong monotonicity may be cumbersome to check; a graph theoretic condition which can replace it is presented in this paper. Knowledge of the facial structure of K(t) is required. The results contained in this paper are extensions of the Kamke-Müller theorem and Hirsch's theorem for strong monotone flows. Applications from chemical kinetics and epidemiology are considered.