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REVIEW 3 major objections 5 minor 26 references

Bifurcation Analysis of Predator-Prey System using Conformable Fractional Order Discretization

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Pith's one-line read Using a conformable-fractional discretization, the paper locates the exact parameter curves where a predator-prey coexistence state undergoes Neimark-Sacker and flip bifurcations, and gives a control term that stabilizes it.

desk verdict A known map in fractional disguise: the main theorem contradicts its own parameter domain and the Lyapunov computation that would fix it is never done; not worth referee time. read the letter →

arxiv 2501.02386 v2 pith:FLZWMLMI submitted 2025-01-04 math.DS nlin.CD

Muhammad Rafaqat , Abubakar Masha , Nauman Ahmed , Ali Raza , Wojciech Sumelka This is my paper · ORCID
classification math.DSnlin.CD MSC 39A2837G1534A0892D25
keywords conformablefractionalderivativeLotka-Volterrapredator-preymodelNeimark-Sackerbifurcationflipstabilityoffixedpointschaoscontroldiscrete-timedynamicalsysteminvariantcurve
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 sets out to show that a conformable-fractional discretization of the continuous Lotka-Volterra predator-prey system leads to a discrete map whose local dynamics can be classified completely. The positive coexistence equilibrium $E^*$ is a sink, saddle, source, or non-hyperbolic point according to sharp inequalities on the parameters, and crossing the set $b = d + \alpha/h^{\alpha}$ forces the equilibrium to undergo a Neimark-Sacker bifurcation, with an attracting invariant curve appearing on one side and a repelling one on the other. The paper also proves a flip-bifurcation surface and proposes a hybrid control term that leaves $E^*$ fixed while allowing the bifurcation threshold to be shifted. If these results are right, ecologically interesting oscillations and chaos in the fractional predator-prey model can be located, delayed, or suppressed by adjusting $b$ and the control parameter $\beta$.

What carries the argument

The central object is the discretized map (2.3), obtained by replacing the fractional derivative in (2.2) with an exponential step whose fractional order enters through the factor $h^{\alpha}/\alpha$: $x_{n+1}=x_n e^{(r(1-x_n)-b y_n)h^{\alpha}/\alpha}$, $y_{n+1}=y_n e^{(b x_n-d)h^{\alpha}/\alpha}$. The argument then runs through the Jacobian at $E^*$, the characteristic polynomial (2.6), the transversality derivative of the eigenvalue modulus, and the normal-form coefficients $\xi_{20},\xi_{11},\xi_{02},\xi_{21}$ for the first Lyapunov exponent; for the flip case it uses a center-manifold computation with the coefficients $\beta_1$ and $\beta_2$. This machinery converts bifurcation detection into algebra on explicit parameter curves.

What would settle it

Take the parameter values of Example 5.1, continue the map (2.3) through $b=d+\alpha/h^{\alpha}\approx 3.62597$, and compute the first Lyapunov exponent from the published $\xi$-coefficients: Theorem 3.1 stands only if the exponent changes sign exactly at the crossing and a closed invariant curve is visible in the phase portraits at the stated values. Separately, deriving (2.3) from (2.2) by the piecewise-constant conformable argument would settle whether the map is the true discretization.

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Extended reading notes

Core claim

The paper's central claim is that system (2.3) -- the map $x_{n+1}=x_n e^{(r(1-x_n)-b y_n)h^{\alpha}/\alpha}$, $y_{n+1}=y_n e^{(b x_n-d)h^{\alpha}/\alpha}$ -- is the conformable fractional discretization of (2.2), and that its coexistence fixed point is $E^*=(d/b,\,(b-d)r/b^2)$ for $b>d$. Its local stability is governed by the characteristic polynomial $\lambda^2+p\lambda+q=0$ with $p=-2+drh^{\alpha}/(b\alpha)$ and $q=\frac{-drh^{\alpha}(dh^{\alpha}+\alpha)+b(h^{2\alpha}dr+\alpha^2)}{b\alpha^2}$. Theorem 2.2 lists exactly when $E^*$ is a sink, saddle, source, or flip-unstable. Theorem 3.1 states that on $A_{NS}=\{0<\alpha\le 1,\, b=d+\alpha/h^{\alpha},\, 0<drh^{\alpha}/(b\alpha)<4\}$, the eigenvalue moduli cross the unit circle with nonzero speed $d|\lambda|/d\epsilon = h^{3\alpha}dr/(2\alpha^2(h^{\alpha}d+\alpha))\ne 0$, so a Neimark-Sacker bifurcation occurs at $E^*$; when the perturbed parameter is negative, an attracting invariant curve emanates, and when positive, a repelling one. A separate set $A_{PD}$ gives a period-doubling bifurcation, and the hybrid control system (4.1) preserves $E^*$ while its Jury inequalities provide a stability window.

Load-bearing premise

The load-bearing premise is that the map (2.3) really is the conformable-fractional discretization of the continuous fractional Lotka-Volterra system; the paper states this but supplies no derivation, and every stability and bifurcation result is a statement about that map.

Editorial extensions

If this is right

  • The coexistence equilibrium is stable only inside the parameter regions listed in Theorem 2.2; outside them, small perturbations produce saddle or source behavior, so the map can serve as a stability checklist for a fractional predator-prey interaction.
  • At $b = d + \alpha/h^{\alpha}$ with $0<drh^{\alpha}/(b\alpha)<4$, the system generically undergoes a Neimark-Sacker bifurcation: an invariant closed curve appears near the threshold, and the dynamics become quasi-periodic, with the sign of the perturbation deciding whether the curve attracts or repels.
  • On the flip surface $h^{\alpha}>2\alpha/r$ and $b = h^{\alpha} d r (h^{\alpha} d + 2\alpha)/(d h^{2\alpha} r + 4\alpha^2)$, the fixed point period-doubles, starting a cascade that the numerical runs connect to chaotic behavior.
  • The hybrid controller (4.1) leaves the positive fixed point unchanged and, whenever the Jury conditions (4.3) hold, stabilizes it, so the control parameter $\beta$ can postpone or suppress bifurcation-induced oscillations.

Reading between the lines

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

  • Editorial inference: since the paper declares $b>0$ throughout, the 'b negative' clause in Theorem 3.1 lies outside the stated parameter domain; reading it ecologically would flip the sign of the predation term, so the attracting-curve statement is better understood as a property of the mathematical continuation rather than a direct biological scenario.
  • Editorial inference: all thresholds depend on the product $h^{\alpha}/\alpha$, so changing the fractional order $\alpha$ and the step size $h$ together moves the bifurcation curves; this gives a direct way to test the model by checking whether the onset of oscillations shifts as predicted when $\alpha$ is varied.
  • Editorial inference: the same conformable discretization and center-manifold apparatus could be applied to fractional predator-prey models with Holling type II or III functional responses; this paper treats only the bilinear Lotka-Volterra interaction.
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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 / 5 minor

Summary. The manuscript proposes a discrete predator-prey map (2.3) as a conformable fractional-order discretization of the continuous Lotka-Volterra system (2.2). It computes the positive fixed point, gives a stability classification in Theorem 2.2, and then claims a Neimark-Sacker bifurcation in Theorem 3.1 and a period-doubling bifurcation in Section 3. A hybrid control strategy is added in Section 4, and numerical bifurcation diagrams and phase portraits are presented in Section 5. The central advertised result is Theorem 3.1, which asserts that the system undergoes a Neimark-Sacker bifurcation at the positive fixed point when the parameter b varies near the set A_NS, with attracting or repelling invariant curves depending on the sign of b.

Significance. If fully established, the paper would provide a useful discrete fractional-order bifurcation analysis of a predator-prey model. The manuscript does contain constructive components: an explicit fixed-point formula, the characteristic polynomial, a stability classification, a hybrid control law with Jury conditions, and several numerical illustrations. Those parts are standard and the numerics are suggestive. However, the main theoretical claims are not supported by the text: the first Lyapunov coefficient in Theorem 3.1 is never evaluated, the flip-bifurcation coefficients are asserted nonzero without computation, and the derivation of map (2.3) from the fractional system is omitted. As a result, the paper does not currently establish its advertised bifurcation results.

major comments (3)
  1. [Section 3, Theorem 3.1 and the set A_NS] Theorem 3.1 is internally inconsistent. The set A_NS is defined by the conditions b > 0 and b = d + α/h^α, so any point in A_NS has b > 0. The theorem nevertheless concludes that 'when b is negative, an attracting invariant curve emanates from E*' and 'when b is positive, a repelling invariant curve emanates from E*'. Since b is strictly positive on A_NS, the negative-b branch is outside the hypotheses and the statement has no content on that branch. Moreover, the first Lyapunov coefficient a is only written as a generic formula in terms of ξ20, ξ11, ξ02, and ξ21; the manuscript never substitutes the computed C and D coefficients, never evaluates a, and never verifies a ≠ 0. The transversality condition d|λ|/dϵ ≠ 0 is shown, but existence of the invariant curve and the direction of bifurcation require the sign and non-vanishing of a. Therefore Theorem 3.1 does not follow from the preceding derivation.
  2. [Section 2, Eq. (2.3)] The map (2.3) is presented as the conformable fractional discretization of the continuous fractional predator-prey system (2.2), but no derivation is given anywhere before the conclusion. The 'piecewise constant argument technique, conformal fractional case' is mentioned only in Section 6, and no theorem, lemma, or calculation connects (2.2) to (2.3). This is load-bearing because all stability and bifurcation statements in the paper concern the map (2.3); if that map is not actually obtained from the fractional system, the ecological conclusions of the paper have no basis. The authors need to state the discretization rule and show the intermediate steps, or clearly identify (2.3) as an independent discrete model.
  3. [Section 3, flip bifurcation] The period-doubling bifurcation is not proven. After the center-manifold reduction, the manuscript declares β1 = ∂²Φ/(∂e∂ϵ*) + (1/2)(∂Φ/∂ϵ*)(∂²Φ/∂e²) ≠ 0 and β2 = (1/6)(∂³Φ/∂e³) + (1/2)(∂²Φ/∂e²)² ≠ 0, but it never computes these expressions from the coefficients c_i, d_i, h_i, nor does it show that they are nonzero for any parameter range. The same section also has apparent algebra/notation slips in the center-manifold coefficient comparison (for example, the equation for the e_n ϵ* term mixes d4 and d5 and the h2 equation is not consistent with the preceding line). Since the sign and non-vanishing of β1 and β2 are precisely the nondegeneracy conditions for a flip bifurcation, the claim that the system 'observes a period-doubling bifurcation' within A_PD is unsupported.
minor comments (5)
  1. [Theorem 2.2, item 1] The denominator in the second bullet contains an undefined symbol 'c' (the expression '4bα²/[c d(-bh^α + h^α d + 2α)]'), which makes the stated sink condition unreadable.
  2. [Theorem 2.2, item 5] The phrase 'The positive fixed points E∗ are complex' is unclear; the intended meaning is presumably that the eigenvalues are complex, not that the fixed point is complex.
  3. [Section 3, A_PD definition] The set A_PD excludes h^α d r/(bα) ≠ 4, while Theorem 2.2 item 4 excludes both h^α d r/(bα) ≠ 2 and ≠ 4; the manuscript does not explain why the value 2 is no longer excluded in the flip-bifurcation set.
  4. [Section 5, Example 5.2] The parameter listing 'd = 2.5, d = 2.91667' uses d twice; the second value is evidently meant to be b, which makes the numerical flip example difficult to reproduce as written.
  5. [Figure captions and text] The caption of Figure 2 says 'Bifurcation Diagrams for the Controlled System (4.1)', but the text refers to 'figure (5)' for the controlled system; the figure numbering is inconsistent.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the main theorem has unverified nondegeneracy but that is a correctness gap, not circularity.

full rationale

The paper's stability and bifurcation analysis is self-contained for the discrete map (2.3): the Jacobian (2.4)-(2.5) is computed directly from the map, the characteristic equation (2.6) yields the eigenvalue conditions in Theorem 2.2, and the Neimark-Sacker and flip bifurcation sections perform standard center-manifold and Lyapunov-coefficient manipulations for this map. There is no fitting of parameters to data and no 'prediction' that reuses fitted values; all parameters are symbolic or chosen by hand for numerical examples. The only self-citations are [24,25] (Rahimi, Sumelka, Yang) cited for the conformable fractional discretization framework in the abstract and conclusion, but the actual map (2.3) is simply stated, and the bifurcation derivations do not rely on any theorem from those papers. The problems in Theorem 3.1 — the set A_NS requires b>0 while the theorem speaks of 'b negative', and the first Lyapunov coefficient 'a' is written but never evaluated, leaving the attracting/repelling claim unsubstantiated — are internal-consistency and proof-completeness failures, not a circular equivalence between output and input. Because no equation or fitted parameter reduces to its own definition, the circularity score is low, reflecting only the minor self-cited discretization framing.

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

No parameters are fitted to data. The system parameters r,b,d,h and fractional order alpha are either symbolic in the theorems or chosen by hand for numerical examples; they are not estimated from observations. The map itself is the main postulated object, not a fitted curve. The paper does not introduce new physical or mathematical entities.

assumptions (4)
  • domain assumption The discrete map (2.3) is the correct conformable fractional discretization of system (2.2).
    Stated between (2.2) and (2.3); no derivation or consistency proof is provided, but every stability and bifurcation result depends on this map.
  • standard math Jury stability conditions and standard bifurcation theory (center manifold, Lyapunov coefficient) apply to the map.
    Used throughout Sections 2-3 and in Theorem 3.1; standard for 2D discrete maps.
  • standard math Lemma 1 from [26] gives the stability classification for the characteristic equation (2.6).
    Invoked after Eq. (2.5); the paper relies on this external lemma without re-deriving it.
  • standard math The center manifold approximation W^c(0,0) has the assumed form (3.8) for the flip bifurcation.
    Assumed in Section 3 for the flip bifurcation analysis; no verification of the non-hyperbolic and transversality conditions beyond declared inequalities.

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Cite this review

Pith. "Pith review of Bifurcation Analysis of Predator-Prey System using Conformable Fractional Order Discretization." pith.science (2026). https://pith.science/paper/FLZWMLMI

@misc{pith2026250102386,
  author       = {Pith},
  title        = {Pith review of: Bifurcation Analysis of Predator-Prey System using Conformable Fractional Order Discretization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLZWMLMI}},
  note         = {Machine review of arXiv:2501.02386}
}
read the original abstract

In this paper, conformal fractional order discretization [20, 24, 25] is used to analyze bifurcation analysis and stability of a predator-prey system. A continuous model has been discretized into a discrete one while preserving the fractional-order dynamics. This allows us to look more closely at the stability properties of the system and bifurcation phenomena, including period-doubling and Neimark-Sacker bifurcation. Through numerical and theoretical methods, this research investigated how the modification in system parameters affects the overall dynamics, which may have implications for ecological management and conservation strategies.

Figures

Figures reproduced from arXiv: 2501.02386 by the authors.

Figure 1
Figure 1. Bifurcation Diagrams for the System(2.3) 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Bifurcation Diagrams for the Controlled System ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Graph of the Maximum Lyapunov Exponent for the System ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Phase Portrait for the System (2.3) Example 5.2. For α = 0.5, h = 1, r = 2, d = 2.5, d = 2.91667, x0 = 0.8 and y0 = 0.09 then the system (2.3) has a positive fixed point (0.857143, 0.0979592). The Jacobian of the system (2.3) at this fixed point has the eigenvalues λ1 …
Figure 5
Figure 5. Figure 5: Bifurcation Diagrams for the System (2.3) 6 Conclusion In [23], the author uses the forward Euler scheme to discretize the system (2.1) and then investigates the complex dynamics of the discrete-time predator-prey system (2.1) in the closed first quadrant R2 +. He ob￾s…

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Reference graph

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