  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.
