  A class of distributed systems with a cyclic interconnection structure is
considered. These systems arise in several biochemical applications and they
can undergo diffusion driven instability which leads to a formation of
spatially heterogeneous patterns. In this paper, a class of cyclic systems in
which addition of diffusion does not have a destabilizing effect is identified.
For these systems global stability results hold if the "secant" criterion is
satisfied. In the linear case, it is shown that the secant condition is
necessary and sufficient for the existence of a decoupled quadratic Lyapunov
function, which extends a recent diagonal stability result to partial
differential equations. For reaction-diffusion equations with nondecreasing
coupling nonlinearities global asymptotic stability of the origin is
established. All of the derived results remain true for both linear and
nonlinear positive diffusion terms. Similar results are shown for compartmental
systems.
