A Generalization of Smillie's Theorem on Strongly Cooperative Tridiagonal Systems

TL;DR

This paper generalizes Smillie's theorem on the stability of strongly cooperative tridiagonal systems by weakening conditions and incorporating an observability criterion, broadening its applicability in various scientific models.

Contribution

It extends previous stability results to cooperative systems with weaker conditions and introduces an observability-type condition using totally nonnegative differential system theory.

Findings

Generalization of Smillie's stability theorem

Inclusion of time-varying, periodic systems

Broader applicability to biological, ecological, and chemical models

Abstract

Smillie (1984) proved an interesting result on the stability of nonlinear, time-invariant, strongly cooperative, and tridiagonal dynamical systems. This result has found many applications in models from various fields including biology, ecology, and chemistry. Smith (1991) has extended Smillie's result and proved entrainment in the case where the vector field is time-varying and periodic. We use the theory of linear totally nonnegative differential systems developed by Schwarz (1970) to give a generalization of these two results. This is based on weakening the requirement for strong cooperativity to cooperativity, and adding an additional observability-type condition.