Analysis of transient dynamics motivated by a mathematical model of the inflammatory response
Judy Day (Mathematical Biosciences Institute (MBI), The Ohio State University)
(October 22, 2009 10:30 AM - 11:18 AM)
The goal of this talk is to describe the analysis of a specific aspect of transient dynamics not covered by previous theory. The question addressed is whether one component of a perturbed solution to a system of differential equations can overtake the corresponding component of a reference solution as both converge to a stable node at the origin, given that the perturbed solution was initially farther away and that both solutions are nonnegative for all time. We call this phenomenon tolerance, for its relation to a biological effect.
Using geometric arguments it is shown that tolerance will exist in generic linear systems with a complete set of eigenvectors and in excitable nonlinear systems. A notion of inhibition is also defined that may constrain the regions in phase space where the possibility of tolerance arises in general systems. However, these general existence theorems do not yield an assessment of tolerance for specific initial conditions. To address that issue, some analytical tools were developed to determine if particular perturbed and reference solution initial conditions will exhibit tolerance.