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

Systems of the form x′(t) = g(r(t), xt) 0 = ∆(r(t), xt) generalize differential equations with delays r(t) < 0 which are given implicitly by the history xt 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 τ : t 7→ 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.