Algebraic-delay Differential Systems, State-dependent Delay, And Temporal Order Of Reactions

摘要

Systems of the formrnx'(t) = g(r(t),x_t) 0 = △(r(t),x_t)rngeneralize 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 semi-flow 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 →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 z' is impossible.