A Characterization of the Invariance of Positivity for Functional Differential Equations

Consider a system of functional differential equations of the form x = f(t,x,), t > to, */,, = *>, (1) in which x is a real n-vector valued function of t, x denotes dx/dt, $ is an initial condition function, and x, denotes the function defined by xi(s) = x(t + s) for s 0.* (When f(t, x,) depends only on t and .x,(0), (1) reduces to a system of ordinary differential equations.) The main purpose of this paper is to give a solution to the problem of determining conditions under which (under certain typically very reasonable conditions on f ) , xU) of (1) has components that are all positive for t > to whenever is positive in, for example, the sense that

0. The problem arises in connection with the mathematical modeling and analysis of economic processes, and it comes up in several other areas as well. (An example concerning the synchronization of geographically separated oscillators is described in Section 2.6.) In some instances, the invariance of positivity in the sense described above is crucial, in that the lack of * For background material concerning equations of the type (1), see, for example, Refs. 1 and 2. 1885