 Three-state cyclic conversion model

Three nonnegative amounts convert in the directed cycle $A\to B\to C\to A$ at positive first-order rates $a,b,c$. The linear system of ordinary differential equations conserves $A+B+C=N$. Its characteristic polynomial is $\lambda [{\lambda }^2+(a+b+c)\lambda +ab+bc+ca]$. The two nonzero eigenvalues have negative real part, so every trajectory converges to the unique steady state with that total:This is also a column-convention continuous-time Markov chain after normalizing by $N>0$: every component loses to one successor and receives from one predecessor. A complex-conjugate pair produces damped oscillations about equilibrium even though the amounts remain nonnegative. A repeated nonzero eigenvalue is defective and requires a Jordan chain.