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Singular bifurcations in a modified Leslie-Gower model

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Pith's one-line read In the degenerate case $C=-AMQ$, the paper proves that a degenerate transcritical point plus a fold/canard point organizes stable relaxation and canard cycles that converge to singular cycles as $\varepsilon\to0$.

desk verdict Interesting blow-up analysis of a degenerate transcritical point, but the central QH formula is algebraically wrong, so the main canard-cycle theorem is stated at the wrong parameter value. read the letter →

arxiv 2411.18059 v2 pith:CHCZXXP4 submitted 2024-11-27 math.DS nlin.CD

Roberto Albarran García , Martha Alvarez-Ramírez , Hildeberto Jardón-Kojakhmetov This is my paper · ORCID
classification math.DSnlin.CD MSC 34E1534C2337G1092D25
keywords slow-fastsystemLeslie-GowermodelweakAlleeeffectcanardcyclesrelaxationoscillationssingularHopfbifurcationblow-upmethoddegeneratetranscritical
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies a Leslie-Gower predator-prey model with a Holling type II response and a weak Allee effect on the prey, in the regime where the prey evolves much faster than the predator, so the system is slow-fast. Earlier work found canard cycles and relaxation oscillations in the generic case; this paper's contribution is the degenerate case in which the slow and fast nullclines meet on the predator axis, creating a degenerate transcritical point $TC$. It claims that, when a certain parameter inequality holds, $TC$ acts not as an attractor but as a saddle-like organizing center, and together with the fold point $P$ it produces a stable relaxation cycle and a stable transitory canard cycle for small $\varepsilon$. The proof uses blow-up desingularization, transition maps, and an intrinsic criticality formula for the singular Hopf bifurcation; the authors also numerically locate and analytically confirm a nearby codimension-two bifurcation point. If correct, the paper shows that degenerate transcritical singularities can drive sustained predator-prey oscillations rather than simple extinction.

What carries the argument

The load-bearing object is the degenerate transcritical point $TC=(0,-AM)$ that appears when $C=-AMQ$, where the slow nullcline meets the fast nullcline on the $v$-axis; at this point the vector field is nilpotent, so standard slow-fast theory does not apply directly. The machinery is the blow-up transformation $(u,v,\varepsilon)=(r\bar u,r\bar v,r\bar\varepsilon)$, which resolves the nilpotent point into charts (entry, central, exit, bottom) containing only semi-hyperbolic equilibria and center manifolds. In these charts the transitions $\Sigma_0\to\Sigma_2$ and $\Sigma_1\to\Sigma_2$ are contractions, and together with contraction away from the singularities this yields stable cycles via a contraction-map argument. A second piece of machinery is the intrinsic formula from [4] for the criticality of a slow-fast Hopf point, used to show the Hopf bifurcation at $Q=Q_H$ is supercritical without computing a normal form.

What would settle it

Take the paper's own parameter values $A=1/2$, $M=-1/10$, $S=1/20$, set $U_1=u_p$ from (21), and solve the trace equation (14) for $Q$; compare with $Q_H$ from (15). A disagreement of the size reported by a direct check (about 2.10 versus 11.45) would show the quantitative canard location is not as claimed, though it would not by itself disprove the qualitative existence of a cycle.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4: for $C=-AMQ$ and $A-M+AM>1/Q$, if the fold point $P$ is a generic jump point there is a locally stable relaxation cycle $\gamma_\varepsilon$ converging in Hausdorff distance to the singular cycle $\gamma_0$ as $\varepsilon\to0$; if instead $P$ is a canard point at $Q=Q_H$, and $Q=Q_c(\varepsilon)\approx Q_H$ is chosen for a maximal canard, there is a locally stable transitory canard cycle $\tilde\gamma_\varepsilon$ converging to $\tilde\gamma_0$. The singular cycles pass through the degenerate transcritical point $TC$, so the paper's slogan is that $TC$ organizes the oscillatory dynamics rather than being an absorbing extinction point. The paper further claims, via a normal-form-free intrinsic calculation, that the singular Hopf bifurcation at $Q_H$ is supercritical in an open parameter region, so the small cycles are attracting. It also proves existence and uniqueness of the generic relaxation oscillation when $C<-AMQ$ using an entry-exit function, and identifies the location of the codimension-two nilpotent bifurcation point analytically.

Load-bearing premise

The central calculation that pins down the exact value of the bifurcation parameter $Q_H$ at which the equilibrium sits on the fold/canard point is algebraically correct; if that value is wrong, the claimed canard cycle and its numerical simulations occur at the wrong parameter values.

Editorial extensions

If this is right

  • For $C=-AMQ$ and $A-M+AM>1/Q$, a stable relaxation oscillation exists through $TC$ when the fold is a generic jump point (Theorem 4(1)).
  • When the fold is a canard point, a stable transitory canard cycle exists for $\varepsilon$ small and $Q$ near $Q_H$, converging to the singular canard as $\varepsilon\to0$ (Theorem 4(2)).
  • The singular Hopf bifurcation at $Q_H$ is supercritical for an open set of parameters, so the bifurcating small-amplitude cycles are stable.
  • In the complementary regime $A-M+AM<1/Q$, the degenerate point $TC$ is attracting and no oscillatory behavior is organized by it.
  • A codimension-two nilpotent bifurcation point exists near $TC$, and the numerical bifurcation diagram shows canard-explosion cycles ending in a homoclinic orbit when multiple equilibria are present.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the proof of Theorem 4 is parameterized by $Q_H$, a corrected trace-zero identity would shift the quantitative location of the canard cycle; the qualitative existence argument via blow-up could still survive.
  • The same four-chart blow-up proof could be adapted to other predator-prey models whose nullclines meet degenerately on an invariant axis, potentially turning 'attractor' transcritical points into organizing centers for oscillations.
  • Ecologically, the saddle-like behavior of $TC$ implies long bouts of near-extinction of prey alternating with recovery; measuring residence time near $TC$ in simulations could serve as a direct check.
  • The intrinsic criticality formula could be applied to locate supercritical versus subcritical Hopf boundaries in higher codimension settings where normal-form computations are impractical.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies a rescaled Leslie–Gower predator–prey model with a weak Allee effect and a fast prey/slow predator structure. The main claims are: stability thresholds and Hopf values for the unique positive equilibrium in a generic and in a degenerate transcritical case; an intrinsic determination of the criticality of the singular Hopf bifurcation following De Maesschalck et al.; a blow-up desingularization of the degenerate point TC; existence and stability of relaxation oscillations and transitory canard cycles near the fold and TC; and a numerical bifurcation analysis that locates a nearby Takens–Bogdanov point. The central theoretical object is Theorem 4(2), which asserts that for C = -AMQ and A-M+AM > 1/Q, a stable canard cycle exists for Q = Qc(ε) near QH, with QH given by Eq. (15).

Significance. If the main claims were correct, the paper would make a useful contribution to the slow-fast predator-prey literature: the blow-up treatment of the degenerate transcritical point TC goes beyond the generic analysis in Zhu and Liu, the use of the intrinsic criticality formula of [4] is appropriate and avoids normal-form computations, and the numerical Takens-Bogdanov check in Appendix B is a genuine consistency test rather than a fitted result. The analysis is self-contained and transparently derives the trace and determinant expressions. However, the central quantitative claim is not supported as written because the Hopf value QH used throughout is algebraically incorrect, and the stability statement in Theorem 4(2) omits a hypothesis that the paper itself shows is necessary. The conceptual blow-up framework appears salvageable, but the current manuscript cannot be accepted in its present form.

major comments (3)
  1. [Sections 3.1 and 3.2, Eqs. (11) and (15)] The Hopf values in Eqs. (11) and (15) contain an algebraic sign error. Setting tr(J)=0 in Eq. (14) and solving for Q gives, in the generic case, QH = 3(U1+C) / [S(U1+A) + 2(A-M+AM)U1 - 3AM + (M+1-A)U1^2]; in the degenerate case C=-AMQ this reduces to QH = 3U1 / [S(U1+A) + 2(A-M+AM)U1 + (M+1-A)U1^2]. Equation (15) instead has (M-A-1)U1^2 in the denominator, and Eq. (11) has (M-1-A)U1^2; both differ from the correct coefficient (M+1-A)U1^2 by -2U1^2. For A=1/2, M=-1/10, S=1/20, the correct degenerate Hopf value near the fold is approximately 2.1, whereas Eq. (15) evaluated at the fold coordinate U1≈0.582 gives approximately 11.45. At Q≈11.45 the positive equilibrium is at U1≈0.91, far from the fold point P, so the identification of Q=QH with a canard point in Theorem 4(2) is false as stated, and the canard explosion shown in Figure 15 is parametrized around the wrong value of Q. This error is load-bearing for the thresholds in Theorems 2 and 3 and for the main canard-cycle theorem.
  2. [Section 6.5, Theorem 4(2)] Theorem 4(2) asserts the existence of a locally stable canard cycle for every parameter tuple satisfying C=-AMQ and A-M+AM>1/Q, without imposing supercriticality of the singular Hopf bifurcation. The paper's own Section 5 shows that the criticality coefficient σ in Eq. (31) changes sign (Figure 8), so the canard cycle is not stable in the subcritical regime. The proof of Proposition 12(2) explicitly depends on an 'appropriate choice of parameters' that makes the singular Hopf bifurcation supercritical. The theorem statement should include this hypothesis (for example, σ<0, or parameters in the open set of Proposition 3), and the stability conclusion should be restricted accordingly.
  3. [Section 5, Proposition 3] Proposition 3 asserts the existence of an open set of parameters for which the Hopf bifurcation is supercritical, but the supporting evidence is numerical evaluation of σ in Figure 8; no analytic computation or proof is supplied. Since the stability assertion in Theorem 4(2) depends on this existence, the manuscript should either provide a proof (for example, by evaluating σ and its derivatives at a concrete parameter point) or state the claim as a numerical observation rather than as a proposition. As written, the formal statement is not established.
minor comments (6)
  1. [Appendix A] The phrase 'solving for wJ11' should read 'solving for U1J11'; the derivation text also promises an expression equivalent to 1/Q - J11 but displays U1J11, which is confusing.
  2. [Theorems 4 and 5] 'Haussdorf' should be 'Hausdorff' in both occurrences.
  3. [Figure 14 and Theorem 4] The panel references in Theorem 4 appear inconsistent with the caption: if the left panel is the transitory canard and the right panel is the relaxation oscillation, then Theorem 4(1) should refer to the right panel and Theorem 4(2) to the left panel.
  4. [Section 3.1, after Theorem 2] The phrase 'the (complex) eigenvalues of J have negative real parts negative' contains a duplicated word and should be corrected.
  5. [Appendix B, Eqs. (65)-(66)] The notation switches between S and ε within the same displayed formulas; since S=ε is set only later, the notation should be harmonized.
  6. [Figure 17 caption] 'codimension2bifurcation' should be 'codimension-2 bifurcation'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the derivation is self-contained; the suspected QH algebra error is a correctness issue, not a self-referential reduction.

full rationale

The paper's central claims are derived from the model rather than from fitted parameters or self-citations. QH in (15) is obtained by setting tr(J)=0 in (9)/(14), i.e. from the equilibrium's Jacobian, and is then used as the slow-fast Hopf / canard parameter; this is a consistency check with the model, not a prediction forced by an input. The criticality calculation in Section 5 applies the external intrinsic formula of De Maesschalck et al. [4] and verifies conditions (30) at P; the sign of sigma is explored numerically but is not used as an input to define the Hopf value. The canard-cycle Theorem 4 rests on standard GSPT transition maps from Krupa-Szmolyan [21] and the blow-up construction, whose details are standard and cited to [24] and (for the survey) to the co-authored [19]; [19] is not load-bearing for the theorem. The Takens-Bogdanov appendix solves tr=det=0 from J(E1) and then compares with the MatCont value, which is a verification, not a circular prediction. The skeptic's algebraic objection to (15) — the sign of the U1^2 term in the denominator — is a mathematical correctness concern: if (15) is wrong, Theorem 4(2) is evaluated at the wrong Q-value, but the formula still comes from the paper's own trace expression and does not reduce to the conclusion it purports to prove. Thus no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the slow-fast scaling assumption, standard GSPT theorems, the non-degeneracy condition A-M+AM > 1/Q, and the correctness of the trace-derived formulas; the latter is where the paper fails. No free parameters are fitted to data.

assumptions (4)
  • domain assumption The prey growth rate is much larger than the predator growth rate, so S = ε is a small parameter.
    Introduced in Section 4 to formulate the slow-fast system (16).
  • domain assumption The fold point is a generic jump point or a canard point, and the equilibrium lies on the repelling branch with 0 < U1 < up in Theorem 4.
    These are the parametric hypotheses of Theorem 4.
  • ad hoc to paper Non-degeneracy A-M+AM > 0 and A-M+AM > 1/Q for the degenerate blow-up analysis.
    Introduced in Section 6 to ensure the center manifold M2 is repelling and the exit chart has a saddle; excludes the case A-M+AM = 1/Q.
  • standard math The intrinsic criticality formula of De Maesschalck et al. [4] is valid for the system.
    Cited as an external theorem in Section 5.

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Pith. "Pith review of Singular bifurcations in a modified Leslie-Gower model." pith.science (2026). https://pith.science/paper/CHCZXXP4

@misc{pith2026241118059,
  author       = {Pith},
  title        = {Pith review of: Singular bifurcations in a modified Leslie-Gower model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CHCZXXP4}},
  note         = {Machine review of arXiv:2411.18059}
}
abstract

We study a predator-prey system with a generalist Leslie-Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey's population often grows much faster than its predator, allowing us to introduce a small time scale parameter $\varepsilon$ that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc., exist. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens-Bogdanov point.

Figures

Figures reproduced from arXiv: 2411.18059 by the authors.

Figure 1
Figure 1. The intersection of the prey nullcline h(u) = (u + A)(1 − u)(u − M) (red curve) and the predator nullcline ℓ(u) = (u + C)/Q (black line) for (2) with weak (M < 0) Allee effect. The intersection indicates two, one or zero positive equilibrium points of system. (c) If A−M −1 ≥ 0 and M −A−AM + 1 Q ≥ 0, then system (2) has no positive equilibrium points. ��� ��� ��� ��� ��� ��� ��� ��� ��� (a) The case 2a. in Theorem 1 … view at source ↗
Figure 2
Figure 2. The intersection of the of the prey nullcline h(u) = (u+A)(1−u)(u−M) (red curve) and the predator nullcline ℓ(u) = (u + C)/Q (black line) for (2) with weak (M < 0) Allee effect. The intersection indicates two, one or zero positive equilibrium points of system. 3. If AM + C Q < 0 and M < 0, (see [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The intersection of the of the prey nullcline h(u) = (u+A)(1−u)(u−M) (red curve) and the predator nullcline ℓ(u) = (u + C)/Q (black line) in the (2) model with weak (M < 0) Allee effect. The intersection indicates one or up to three positive equilibrium points of the system. Remark 2. • Case 2 in Theorem 1 corresponds to a degenerate case where the two nullclines intersect along the v-axis, see [PITH_FULL_IMAGE:fig… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Dynamics of the slow-fast system (16) for (a) C < −AMQ and (b) C = −AMQ, where M0 0 and M1 0 are the critical submanifolds. The fold point P = (up, vp) and TC = (0, −AM) are shown, as well as a schematic representation of the slow flow in the different components of th…
Figure 5
Figure 5. Figure 5: Slow flow when the equilibrium point E1 is in either normally hyper￾bolic branch of M1 0 . (3) When the equilibrium point E1 coincides with the fold point P, that is U1 = up, then g(u, h(u)) = 0 and h ′ (u) = 0 for Q = QH. In this case, the right-hand side of (25) is a…
Figure 6
Figure 6. Figure 6: Left: schematic of the blow-up of a generic fold point. Right: schematic of the blow-up of a canard point. For these pictures we have used the local setup of the model under study. Hence, the right branch of the critical manifold is attract￾ing, while the left branch i…
Figure 7
Figure 7. Figure 7: Singular cycles that we focus on. The solid red cycle in (a) represents a candidate orbit for a regular relaxation oscillation, which appear when C < −AMQ. Such a cycle has already been studied in [42], but here we include its analysis for completeness, see Section 7. …
Figure 8
Figure 8. Figure 8: We present slices of (31) for the indicated values of parameters. In this way, we verify the region of parameters for which the Hopf bifurcation is super- (σ < 0) or sub- (σ > 0) critical. The black curve indicates σ = 0. Now that we have determined that the point P is…
Figure 9
Figure 9. Figure 9: Following the analysis presented in this section, we provide a sketch of the dynamics of (35) near the origin. The limit dynamics are presented in green and blue (for ε1 = 0 and r1 = 0 respectively), while a sample orbit of (35) is shown in red. 6.2. Central chart. The…
Figure 10
Figure 10. Figure 10: Phase-portrait of (42) for A − M + AM − 1 Q < 0 on the left and A − M + AM − 1 Q > 0 on the right. The phase portraits are drawn qualitatively by exploiting Proposition 6, the fact that dv2 dt > 0 for all v2 < 0 and the relative arrangement of the nullclines v2 = (A −…
Figure 11
Figure 11. Figure 11: Dynamics of (35) in the chart K3 = {u¯ = 1} for A−M +AM − 1 Q < 0 on the left and A − M + AM − 1 Q > 0 on the right. The flow in blue occurs in the (ε3, v3)-plane, while the green one in the (r3, v3)-plane. The black orbits are a sketch of a sample of orbits flowing f…
Figure 12
Figure 12. Figure 12: Dynamics of (59) near the origin. In contrast to the flow of (35), the center manifold in the plane (v4, ε4) is not unique. M0 0 M1 0 Σ0 Σ1 Σ2 A − M + AM < 1 Q M0 0 M1 0 Σ0 Σ1 Σ2 A − M + AM > 1 Q [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: The two non-equivalent blown-up dynamics of the point TC for A − M + AM < 1 Q on the left and A − M + AM > 1 Q on the right. The dotted curves depict the critical manifold. The corresponding reduced flows are indicated by the magenta arrows. Compare with [PITH_FULL_I…
Figure 14
Figure 14. Figure 14: On the left we show we show a transitory canard passing through the fold point P and the singular point TC. On the right, a singular relaxation oscillations passing through the same singularities. These cycles are obtained for different parameter regimes (recall that …
Figure 15
Figure 15. Figure 15: Simulations of (2) with S = ε = 0.05, A = 1 2 , M = − 1 10 , and C = −AMQ. The equilibrium point E1 is located close to the fold point P, and is indicated in blue. The parameter Q is chosen as Q = QH − δ where δ is shown at the top of each figure. Notice, indeed, the …
Figure 16
Figure 16. Figure 16: Sections of the flow: ∆in and ∆out . We define a flow return map Π : ∆in → ∆in, by the composition of the following two maps Φ : ∆in → ∆out , Ψ : ∆out → ∆in , namely Π = Ψ◦ Φ. For a fixed 0 < ε ≪ 1, the trajectory of system (61) starting at a point (u+, v+) on section…
Figure 17
Figure 17. Figure 17: Equilibria of (16) and their bifurcations in a codimension 2 bifurcation diagram obtained via Matcont [7]. For this diagram we fix the parameters A = 1 2 , M = − 1 10 and ε = 1 50 . The gray surface corresponds to equilibria. For fixed values of C, we show stable equi…
Figure 18
Figure 18. Figure 18: Bifurcation diagram highlighting the limit cycles (black curves) arising from Hopf bifurcations (compare with [PITH_FULL_IMAGE:figures/full_fig_p036_18.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.