Chaotic dynamics in a simple predator-prey model with discrete delay

The paper’s Theorem 3.1 establishes exactly the threshold and dynamical regimes the candidate states: E0 is always unstable; E1 is unstable for 0 ≤ τ < τc and globally asymptotically stable (with respect to X0) for τ > τc; the positive equilibrium E+ exists iff 0 ≤ τ < τc, is GAS when τ=0, and, for τ>0, yields uniform persistence for data in X0. These appear in the paper’s scaled model (5), its equilibrium set (7) and threshold τc = (1/s) ln(Y/s) (8), and Theorem 3.1 with its proof in Appendix A.2, which uses logistic comparison, a standard linear DDE stability estimate for y, and persistence theory à la Smith–Thieme. The candidate derives the same conclusions but uses slightly different tools (Halanay inequality; a Volterra-type Lyapunov function at τ=0; Hale–Waltman/Cao–Fan–Gard persistence templates). Hence both are correct and consistent, with different proof ingredients. Key places in the paper: the scaled system and well-posedness (Proposition 2.1) and bounds , the characteristic equation and Theorem 3.1 summary , the global stability proof of E1 for τ>τc in Appendix A.2 , and the uniform persistence argument (Appendix A.2, Part 3(b)) .