Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons

Abstract

Using a comparison theorem technique, we study the long time behavior of certain classes of nonlinear difference-differential systems. Zero is a solution for these systems. We are concerned in this paper with conditions forcing nonconvergence to zero of solutions as time approaches infinity; that is, we obtain threshold properties of the systems. The results parallel results by Aronson and Weinberger on reaction-diffusion equations somewhat, and the study was motivated by consideration of models for myelinated nerve axons.