On nonapproach to equilibrium in compartmental systems

The question of convergence of a solution of a compartmental system to an equilibrium point, as
t \rightarrow \infty
, is of considerable interest [1]-[6]. Although it was not explicitly stated, the remarks in [1] seem to suggest that in the case of closed systems, of the type specified below, every solution approaches an equilibrium point. These remarks prompted the construction of counter examples, to be presented below, to show that a closed compartmental system may admit periodic solutions as well as aperiodic solutions that do not approach a constant vector.