Chaotic dynamics of the sugarcane borer-two parasitoid agroecosystem with seasonality

The paper and the candidate solution both (i) exploit the block-lower-triangular Jacobian to decouple stability into two 3×3 subsystems, (ii) derive the same Routh–Hurwitz conditions for E5 yielding α_lb < α < α_c and β > β_lb, (iii) use the same expression α_c = r m3 (m2+n2)/(γ1 n2 z K) with z the positive root of u^2 + h1 u − h2 = 0, and (iv) establish a Hopf bifurcation at α = α_c using Liu’s coefficient criterion. The paper also states a Lyapunov function that ensures global asymptotic stability of E5 under the same inequalities. Minor issues: the paper has a small inconsistency in the Hopf-section expression for c3 (missing m3) and is terse on the negative definiteness proof; the model’s write-up slightly misdescribes the Lyapunov function’s form and the matrix structure in V̇, but its conclusions align with the paper’s results.