Homeostasis despite instability

We have shown previously that different homeostatic mechanisms in biochemistry create input-output curves with a "chair" shape. At equilibrium, for intermediate values of a parameter (often an input), a variable, Z, changes very little (the homeostatic plateau), but for low and high values of the parameter, Z changes rapidly (escape from homeostasis). In all cases previously studied, the steady state was stable for each value of the input parameter. Here we show that, for the feedback inhibition motif, stability may be lost through a Hopf bifurcation on the homeostatic plateau and then regained by another Hopf bifurcation. If the limit cycle oscillations are relatively small in the unstable interval, then the variable Z maintains homeostasis despite the instability. We show that the existence of an input interval in which there are oscillations, the length of the interval, and the size of the oscillations depend in interesting and complicated ways on the properties of the inhibition function, f, the length of the chain, and the size of a leakage parameter.