On the global dynamics of a forest model with monotone positive feedback and memory

The paper proves that F(b)=bR(b) is strictly increasing on R+ by a change-of-variables argument using H(u), θ=σ(u,φ), and, for constant histories φ≡b, arrives at a representation from which monotonicity follows; despite a typesetting ambiguity in the displayed formula, the intended integrand is θ/g(θ), yielding a strictly increasing dependence on b . The candidate solution independently rewrites F(b) via U=b/µ as F(b)=∫0U β(xm+(1/µ)∫vU g(s)/s ds) dv and then differentiates to obtain f′(U)=(g(U)/U)∫(0,U] β(·) d(v/g(v))>0, establishing strict increase (except the trivial case β≡0). Its steps (well-posedness, AC/Leibniz justification, Stieltjes integration by parts, positivity of the measure d(v/g(v))) are logically sound under the same hypotheses as the paper (g positive, strictly decreasing, continuous; β nonnegative, globally Lipschitz) . Hence both arguments are correct, but they are methodologically different.