Dynamics of an LPAA model for Tribolium growth: insights into population chaos

The paper proves global stability of the positive equilibrium by reducing the 4D LPAA map to a scalar delay equation and invoking a monotonicity theorem; the hypotheses and bounds are clearly verified and lead to Theorem 7.1 (1 < R0 < min{e, e^{c1}(1−µa)/(c2 µa)}) . The candidate solution proposes a different Lyapunov/KL approach on the 4D normalized system. Although its normalization and ΔV upper bound are correct, the sign analysis of the key term S(w,z) is flawed: near w>1, z≈0 one has S(w,0) ≈ β'(w−1) > 0, so the proposed Lyapunov function is not nonincreasing and the global convergence argument fails. Additional inequalities used to force S≤0 are also incorrect/insufficient. Consequently, the paper’s result stands; the model’s proof is wrong.