Homeostatic Mechanisms in Biological Systems

The paper’s Theorem 2.1 states that near-perfect homeostasis on D=(I0,+∞) implies either a critical point z′(Ic)=0 or a sequence In→∞ with z′(In)→0; if z′ is monotone then limI→∞ z′(I)=0. The proof uses sign analysis plus dyadic intervals to bound average slopes, yielding the vanishing-derivative sequence (and the monotone corollary) . The model’s solution proves essentially the same fact via a direct mean value theorem argument on expanding intervals, which actually shows (ii) always holds; it also supplies a clean argument for the monotone case. Thus both are correct, with closely related core ideas (average-slope bounds on long intervals) and the model offering a slightly sharper statement. Minor typos in the paper’s proof do not affect correctness.