  This paper studies the asymptotic behavior of several continuous-time
dynamical systems which are analogs of ant colony optimization algorithms that
solve shortest path problems. Local asymptotic stability of the equilibrium
corresponding to the shortest path is shown under mild assumptions. A complete
study is given for a recently proposed model called EigenAnt: global asymptotic
stability is shown, and the speed of convergence is calculated explicitly and
shown to be proportional to the difference between the reciprocals of the
second shortest and the shortest paths.
