Singular Bifurcations in a Modified Leslie–Gower Model

The uploaded paper establishes, for the same slow–fast system and parameter regime C = −AMQ with A − M + AM > 1/Q, the existence of two families of stable periodic orbits for ε = S ≪ 1: (i) a relaxation oscillation when the fold P is generic, and (ii) a canard-induced cycle at Q = Qc(ε) ≈ QH when P is a canard point. This is stated explicitly in Theorem 4 and proved via a composition of local maps obtained by blow-up at the fold P and the degenerate transcritical point TC, together with Fenichel persistence and contraction arguments (see the statement and proof around Figure 14 and Proposition 12). These results match the candidate solution’s outline based on Fenichel theory, the Exchange Lemma, and blow-up/normal form analysis at folds and transcritical contacts, including the canard locus Qc(ε) = QH + O(ε). Moreover, the geometric ingredients used by the model (critical manifold M0 = M0_0 ∪ M1_0, unique fold P, location and stability change on M0_0 at TC, desingularized slow flow, and the role of the condition A − M + AM > 1/Q) are all developed in the paper’s Sections 4–6 with the same conclusions. Therefore, the paper’s argument and the model’s argument are consistent and essentially use the same geometric singular perturbation toolkit, differing mainly in the level of detail (the paper provides a full blow-up construction and explicit contraction maps, while the candidate gives a high-level but correct sketch). Key support in the paper includes: system and slow–fast structure (eqs. (16)–(19) with h(u) and g(u,v)), geometry of M0 and the unique fold P (formulas (20)–(23)), slow flow on u=0 (eq. (24)), desingularized slow flow on M1_0 and the L’Hospital evaluation at a canard point (eqs. (26)–(28)), blow-up at TC and the two regimes split by A − M + AM ≶ 1/Q (Figure 13 and Proposition 12), and the main existence/stability theorem for the two cycles (Theorem 4).