A detailed analysis of the origin of deep-decoupling oscillations

The paper defines the adjusted Welander model x' = 1 − x − Kεx, y' = μ − Kεy with Kε(y−x−η) = κ1 + 1/2(κ2 − κ1)(1 + tanh((y − x − η)/ε)) and studies oscillations via numerical continuation. It establishes: (i) the region of periodic solutions bounded mainly by a Hopf curve H and a saddle-node curve S; (ii) tangency curves T± that partition the existence region into PS, P1, P2, W; (iii) Welander oscillations (region W) vanish well before the global oscillation threshold ε* ≈ 0.147; specifically W disappears already by ε ≈ 0.085; and (iv) for ε beyond ≈ 0.147 no oscillations exist anywhere in parameter space (citing prior work) . The candidate model solution derives, analytically, the Jacobian, the Hopf trace condition τ = 0, and the determinant on τ = 0, obtaining Δdet|τ=0 = (K′ − (1 + K)^3)/(1 + K). This yields a sharp necessary-and-sufficient criterion for the existence of Hopf points: K′(Δ) > (1 + K(Δ))^3, and hence an explicit 1D maximization formula for ε* that matches the paper’s quoted value ≈ 0.147 for κ1 = 0.1, κ2 = 1. It also formalizes T± via tangency functionals and explains why W disappears for ε near 0.085 while periodic orbits persist in PS/P1/P2, consistent with the paper’s continuation diagrams . Minor gaps: the model’s Step C argument that “no Hopf ⇒ no periodic orbit” is heuristic (other global mechanisms could, in principle, generate cycles), though the paper’s prior result settles nonexistence at ε ≥ ε*. Overall, the paper’s numerical-structural claims and the model’s analytic derivations are consistent and complementary.