Algebraic-Delay Differential Systems, State-Dependent Delay, and Temporal Order of Reactions

Abstract

Systems of the form $$\begin{array}{ll}x'(t) = g(r(t),x_t)\\ \quad\,\, 0 = \Delta(r(t),x_t) \end{array}$$ generalize differential equations with delays r(t) < 0 which are given implicitly by the history x t of the state. We show that the associated initial value problem generates a semiflow with differentiable solution operators on a Banach manifold. The theory covers reaction delays, signal transmission delays, threshold delays, and delays depending on the present state x(t) only. As an application we consider a model for the regulation of the density of white blood cells and study monotonicity properties of the delayed argument function $${\tau:t\mapsto t+r(t)}$$ . There are solutions (r, x) with τ′(t) > 0 and others with τ′(t) < 0. These other solutions correspond to feedback which reverses temporal order; they are short-lived and less abundant. Transient behaviour with a sign change of τ′ is impossible.