Optimality and Sustainability of Delayed Impulsive Harvesting

The paper reduces the delayed impulsive logistic model to a delayed difference equation, derives the yield Y(E), shows the optimal effort Eopt = 1 − e^{−rT/2} and N*(nT+) = Kc/(e^{rT/2}+1), and proves that the MSY is locally stable iff k = 1 or (for k ≥ 2) rT < −2 ln(1 − 2 cos(kπ/(2k+1))) (Theorem 4.3). The candidate solution follows the same reduction, obtains the same Y(E), Eopt, and periodic solution, linearizes the Poincaré map, and derives the characteristic polynomial with a = e^{−rT}/(1−E)^2, which becomes λ^{k+1} − λ^k + E at E = Eopt, yielding the same stability threshold via standard unit-circle analysis. These steps coincide with the paper’s linearization un+1 = p0 un − pk un−k with p0 = e^{−rT}/(1−E)^2 and pk = E e^{−rT}/(1−E)^2, and with Lemma 2.6 for the normalized form at Eopt. In short, the model reproduces the paper’s main results by essentially the same argument, with only notational differences and minor extra commentary on unit-circle crossings. Key facts match the paper’s equations (11), (24), (26), (6), and Theorem 4.3.