  Continuous-time discrete-state random Markov chains generated by a random
linear differential equation with a random tridiagonal matrix are shown to have
a random attractor consisting of singleton subsets, essentially a random path,
in the simplex of probability vectors. The proof uses comparison theorems for
Carath\'eodory random differential equations and the fact that the linear
cocycle generated by the Markov chain is a uniformly contractive mapping of the
positive cone into itself with respect to the the Hilbert projective metric. It
does not involve probabilistic properties of the sample path and is thus
equally valid in the nonautonomous deterministic context of Markov chains with,
say, periodically varying transitions probabilities, in which case the
attractor is a periodic path.
