QUARTERLY OF APPLIED MATHEMATICS Volume XLII April • 1984 Number 1 THRESHOLD BEHAVIOR AND PROPAGATION FOR NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEMS MOTIVATED BY MODELING MYELINATED AXONS* By JONATHAN BELL (State University of NewYork at Buffalo) AND CHRIS COSNER (Univrsity of Miami) 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. 1. Introduction. In this paper we study the long time behavior of solutions of nonlinear difference-differential systems of the form duj/dt = uj+x - 2 Uj + Uj_, + /(Uj) (j e Z) (1.1) where /(h) will be allowed to have various qualitative behaviors to be specified below. System (1.1) arises as a model in various contexts and we will consider forms of the function f(u) suggested by some of these applications. For example system (1.1) occurs in the study of population genetics where spatially discrete (i.e. isolated) populations of diploid individuals are considered. One can derive (1.1) from model-derivation arguments given in [1] if the author' continuously distributed habitat assumption is replaced by an appropriate discrete populations assumption. In [1], Aronson and Weinberger consider three possible types of /(w), specified below by (2.1)—(2.3). Our results also apply to these classes of /'s. Another application from which system (1.1) is derived concerns the propagation of nerve pulses in myelinated axons where the membrane is excitable only at spacially discrete sites. In the Appendix we give a derivation of (1.1) based on modeling myelinated nerve axons which motivated consideration of the particular questions addressed in this paper. Specifically, a question of importance for any nerve model is whether it displays * Received February 15, 1982. The first author was partially supported by NSF grant MCS-8101666. ©1984 Brown University