Mathematical Modeling of Carbon Dioxide Emissions with GDP Linkage: Sensitivity Analysis and Optimal Control Strategy

The paper’s interior (coexistence) equilibrium 4E is analyzed for the four-dimensional CO2–GDP–Forest–Population model (5), with the GDP equation dG/dt = μ − ε G implying G4 = μ/ε at equilibrium, not μ = ε; see the equilibrium calculation G* = μ/ε and the numerical value G4 = 26.8125 produced from μ = 0.02145 and ε = 0.0008 in the paper’s simulations . The candidate solution assumes instead dG/dt = (μ − ε) G and thus claims that an interior equilibrium with G4 > 0 forces μ = ε, which is inconsistent with the paper’s model and results. Because G is governed by dG/dt = μ − ε G in the paper, the hyperplanes {G ≡ g0} are not invariant, and the proof cannot be restricted to Ω = {G = G4} as the candidate does; the cross term (C − C4)(G − G4) must be handled (e.g., by Young’s inequality) together with the negative terms −p(C − C4)^2 and −ε(G − G4)^2. The paper’s Lyapunov candidate, the structure of dV/dt with cross terms, the (ψ + ε)(ψ^3 + A1ψ^2 + A2ψ + A3) factorization used for local stability, and the form of the global-stability condition (38) are all presented in the PDF and match the standard route (with minor typographical glitches in the choice of coefficients m1, m2) . In short: the paper’s argument is essentially correct (modulo typos), whereas the candidate proof depends crucially on an incorrect GDP equation and an inadmissible restriction to G = G4.