{"id":"886938df-f59b-413c-891b-af762951c343","arxiv_id":"1908.03685","paper_version":5,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper proposes a corrected Adams-Bashforth scheme for a Caputo-Fabrizio fractional Lotka-Volterra system, but the derivation and stability conditions are flawed.","lead":"This paper applies Caputo and Caputo-Fabrizio fractional derivatives to a three-species Lotka-Volterra model and proposes a corrected Adams-Bashforth numerical scheme. The stability analysis and numerical method contain mathematical errors that undermine the central claims.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (14), the foundation of the proposed Caputo-Fabrizio predictor-corrector, is undefined for 0<α<1 (n=1) and omits the non-integral (1-α)/M(α) g term, so the numerical method and the simulation-based stability claims are unsupported.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: Eq. (14), the basis of the proposed numerical method, is invalid for the case 0<α<1 used throughout the paper. My independent check confirms that the factorial (n-2)! is undefined at n=1 and, more substantively, that the true inverse of the Caputo-Fabrizio derivative (2) contains a non-integral term (1-α)/M(α) g(x) that is missing from Eq. (14) and from the discretized scheme (16)-(18). Because the paper's headline contribution is the new numerical method and the numerical evidence for operator-dependent stability, this single flaw invalidates the central claim. The analytical stability section, while also questionable in its application of a linear stability theorem to nonlinear equilibria and in its algebraic presentation, would not rescue the paper even if fully corrected, since the numerical verification is an advertised part of the contribution. Therefore the reader's REJECT verdict is appropriate and no adjustment is needed.","tokens_in":14262,"tokens_out":5847,"duration_ms":54597,"concrete_test":"Independently re-derive the equivalent integral equation for CF D^α f = g from definition (2) for 0<α<1 by differentiating the integral equation once and integrating back. If the derivation yields a non-integral (1-α)/M(α) g term and no singular (x-t)^{-1} integral, then Eq. (14) and the discretization (16)-(18) are incorrect. As a supplementary computational check, apply the proposed scheme to the scalar test problem CF D^{0.5} f = -f, f(0)=1, with h=0.01, and compare with the exact solution of that CF linear equation; if the scheme does not converge to it, the numerical claim is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 builds the new numerical method on the claimed equivalent integral equation (14). For the range used in the simulations, 0<α≤1, we have n=⌈α⌉=1, so (14) contains the undefined factorial (n-2)!=(-1)! and a singular integral with kernel (x-t)^{-1}. More importantly, (14) is not the correct inverse of the Caputo-Fabrizio derivative defined in (2). Direct inversion of (2) for CF D^α f = g gives f(x)=f(0)+(1-α)/M(α)(g(x)-g(0))+α/M(α)∫_0^x g(s)ds, which includes a non-integral (1-α)/M(α) g term. The discretized predictor-corrector in Eqs. (15)-(18) uses only the integral term α/(M(α)(n-1)!) Σ b g and omits this term, so it solves a different, incorrect equation. Consequently, all numerical results in Section 5, including the stability comparisons of Caputo vs. CF operators, are produced by an algorithm that is not solving the stated model. The claimed 'new numerical method with enhanced stability' is therefore unsupported. The analytical stability discussion in §3.2 is also not a substitute, because it applies a linear-operator stability theorem (Theorem 2.2) to nonlinear equilibria without the required local reduction proof and contains apparent algebraic typos in Table 1; however, the failure of Eq. (14) alone is decisive for the paper's central numerical claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a three-species Lotka-Volterra system with Caputo and Caputo-Fabrizio (CF) fractional derivatives. It proposes a predictor-corrector Adams-Bashforth scheme for the CF equation, derives local stability conditions for the equilibria, and reports numerical comparisons that are claimed to show that the type of differential operator and the value of the fractional order significantly influence stability and dynamics. The paper's central numerical claim rests on Eq. (14), which is presented as the equivalent integral form of the CF initial value problem; the stability analysis and the simulations in Section 5 are built on that representation.","tokens_in":14698,"tokens_out":10129,"duration_ms":102073,"significance":"The motivation is reasonable: a correct, stable numerical method for CF Lotka-Volterra systems would be a useful contribution, and the paper correctly identifies that earlier CF predictor-corrector schemes [30] are flawed. The paper also attempts a direct comparison of Caputo and CF stability regions, which is an interesting and potentially useful goal. However, the central derivation is invalidated by a fundamental error in Eq. (14), so the proposed numerical method does not solve the stated model and the numerical comparisons in Section 5 are not evidence about the model's behavior. The analytical stability section also lacks a valid justification for applying linear eigenvalue conditions to nonlinear CF systems. The contribution, as it stands, cannot support the paper's claims.","major_comments":[{"comment":"Eq. (14) is not the equivalent integral form of the CF initial value problem for 0<α<1. For n=1 the first term contains (n-2)! = (-1)!, and the kernel (x-t)^{-1} is singular. Directly inverting Eq. (2) gives f(t) = f(0) + ((1-α)/M(α)) g(t) + (α/M(α)) ∫_0^t g(s) ds, which includes a non-integral (1-α)/M(α) g(t) term that Eq. (14) omits. Since the predictor-corrector scheme in Eqs. (15)-(18) is derived from Eq. (14), it solves a different equation, and all numerical results in Section 5 are therefore not simulations of the stated CF Lotka-Volterra model.","section":"Sec. 4, Eq. (14)"},{"comment":"Theorem 3.2 asserts local asymptotic stability from the eigenvalues of the Jacobian J(ε*) with the proof described as 'straightforward with Theorem 2.2 and [26]'. No linearization theorem for nonlinear systems with the CF operator is stated or proved, and eigenvalue conditions for linear fractional systems do not automatically transfer to nonlinear equilibria. Consequently, the stability classifications in Table 1 and the interpretations of Examples 1-3 are not supported by the arguments given.","section":"Sec. 3.2, Theorem 3.2"},{"comment":"The stability region depicted in Fig. 1 is inconsistent with Theorem 2.2 of the paper. The figure labels the disk centered at (0, 1/(2(1-α))) with radius 1/(2(1-α)) as the CF unstable region, but points in the left half-plane inside this disk satisfy Re(λ)<0, which Theorem 2.2 condition 3 declares asymptotically stable. In Sec. 3.2.5 the discriminant cases are also misstated: the trigonometric formulas (9)-(11) apply to Δ<0, not Δ>0, and Eq. (7) ends with -q/3 rather than -a/3.","section":"Sec. 3.2.5 and Fig. 1"},{"comment":"Table 1 contains repeated and inconsistent conditions for ε1. The Caputo row and the CF row both list 'a1a2 < a2a3 - a2' twice, whereas the text in Sec. 3.2.2 uses a1a4 and a1a6; the CF row also mixes 'a1a2 < ...' with the conditions 'a1a4 - a2a3/a2 > α/(1-α)' and 'a1a6 - a2a5/a2 > α/(1-α)'. As a result, the table cannot be used to check the asserted stability regions, and some of its entries are internally inconsistent.","section":"Table 1"}],"minor_comments":[{"comment":"The normalization function M(α) is only constrained by M(0)=M(1)=1 and is never specified for other α; since it appears in Eq. (14) and in the discretization, the numerical simulations are not reproducible without stating the choice made.","section":"Sec. 2.1"},{"comment":"The sentence 'Since a1 > 0, 1 - a3 < 0, 1 - a4 < 0' appears to contain a typo: the third condition should likely be 1 - a5 < 0, since the eigenvalue is 1 - a5.","section":"Sec. 3.2.2"},{"comment":"The notation 'Adam-Bashforth' in the keywords and text is a misspelling of 'Adams-Bashforth', and the paper contains several other typographical errors such as 'asymptoialy' and 'stablitity'.","section":"Sec. 4"},{"comment":"The axis labels and legends in Fig. 2 (right) and Fig. 5 (right) are difficult to read; the three-dimensional trajectories would benefit from clearer labeling of which curve corresponds to which species.","section":"Sec. 5"}],"recommendation":"reject","confidential_remarks":"The paper contains load-bearing errors in the derivation of the numerical method and in the stability analysis. The most decisive issue is Eq. (14), which invalidates the entire numerical section; this cannot be fixed by minor revision. I would not recommend inviting a revision unless the authors are prepared to rework the numerical method, re-run all experiments, and provide a valid linearization theorem for nonlinear CF systems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about arXiv:1908.03685. First, the central numerical method—the corrected Adams-Bashforth scheme for Caputo-Fabrizio derivatives—is invalid as written: the integral equation it starts from, Eq. (14), contains an undefined factorial for the 0<α<1 case used in all simulations, and omits the non-integral term that appears in the true inverse of the CF derivative. Second, the stability comparison section has internal inconsistencies, notably Fig. 1 drawing the CF unstable region as a circle that contradicts the cited theorem.\n\nThat said, this is not a random pile of assertions. The paper applies the known CF stability conditions from Li et al. to a three-species Lotka-Volterra model, gives five equilibria with explicit Jacobians, and runs three numerical examples to illustrate operator-dependent behavior. The topic is real: modellers do use CF operators and need working numerical schemes. The authors also engage honestly with prior work, citing Toh et al. as flawed and attempting a correction. The equilibrium analysis, while standard linearization, is done carefully for the most part.\n\nThe soft spots are load-bearing, though. The mistake in Eq. (14) is not a typo; it changes the equation being solved. The correct inverse of the CF derivative in (2) for 0<α<1 includes a non-integral (1−α)/M(α)(g(x)−g(0)) term, and the scheme in (15)–(18) uses only the integral part, so the simulations labeled “Caputo-Fabrizio” solve a different model. This alone invalidates the abstract’s claim of a new method with enhanced stability. The stability section also carries errors: applying linear-operator theorems to nonlinear equilibria without a local reduction argument, using inequalities like λ > 1/(1−α) for complex eigenvalues, and a figure that marks a region unstable where the cited theorem says the real part is negative. These are the kind of problems a referee would catch, but they are fundamental enough that a desk reject is defensible.\n\nWho gets value from this? Someone working on CF numerical methods might find the corrected inversion formula worth thinking about, but they would get more from the source papers [27,29,30]. The examples are illustrative but not reliable. I would not cite this paper or bring it to reading group. If the authors redo the numerics with the correct integral equation, the stability comparison might become interesting, but as it stands the central claims are unsupported.\n\nIf I were the editor, I would send it to a referee rather than desk reject, because the error is subtle enough that a specialist’s report would help the authors, and the topic is appropriate for the journal. But I would expect a clear reject recommendation.","headline":"A well-intentioned but mathematically broken numerical scheme invalidates the central claims, though the stability comparison and examples are also shaky.","tokens_in":15148,"tokens_out":3889,"would_cite":false,"duration_ms":40262,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34A08","65L06","92D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the type of fractional differential operator—Caputo versus Caputo-Fabrizio—and the value of the fractional order can flip which equilibria of a three-species predator-prey system are stable, and it supplies a…","keywords":["Caputo-Fabrizio fractional derivative","three-species Lotka-Volterra model","predator-prey dynamics","predictor-corrector method","Adams-Bashforth method","fractional stability","fractional differential equations"],"falsifier":"Run the proposed scheme on a scalar linear CF equation with a known exact solution at decreasing step sizes $h$; if the numerical solution does not converge to the direct integral solution of the CF initial-value problem, the claimed stability gain is an artifact of equation (14).","tokens_in":14130,"feed_emoji":"🐺","tokens_out":17312,"duration_ms":154607,"temperature":0.7,"pith_summary":"The paper claims that replacing the singular-kernel Caputo derivative with the non-singular exponential-kernel Caputo-Fabrizio operator changes the stability decisions of a three-species Lotka-Volterra predator-prey system, and that the fractional order $\\alpha$ matters as much as the model parameters. To make Caputo-Fabrizio simulations reliable, it derives a corrected predictor-corrector scheme based on the Adams-Bashforth method, claiming enhanced stability over an earlier Caputo-Fabrizio scheme. Stability is analyzed through the eigenvalues of the Jacobian at the five equilibria: Caputo stability follows the wedge condition $|\\arg \\lambda| > \\alpha\\pi/2$, while Caputo-Fabrizio stability is governed by a bounded circular region in the complex plane. Three numerical examples show that the same parameter set can be asymptotically stable under one operator and unstable under the other, and that lowering $\\alpha$ can turn previously unstable equilibria stable. If correct, the practical message is that the choice of fractional operator is not a cosmetic modeling detail but a decisive factor in whether populations persist or collapse.","feed_headline":"Derivative type flips stability of three-species predator-prey model","feed_subtitle":"With the same equations and parameters, which equilibrium survives depends on the derivative type and fractional order.","key_machinery":"The load-bearing object is the Caputo-Fabrizio fractional derivative, $\\mathrm{CF}D_0^\\alpha f(t)=\\frac{M(\\alpha)}{1-\\alpha}\\int_0^t \\exp(-\\frac{\\alpha}{1-\\alpha}(t-\\tau))f'(\\tau)\\,d\\tau$, whose exponential kernel removes the singularity of the Caputo kernel $(t-\\tau)^{-\\alpha}$. Around this operator the paper builds two tools: the linear stability criterion for CF systems, which replaces the Caputo wedge with a bounded circular boundary in the complex plane, and a corrected predictor-corrector (Adams-Bashforth-type) scheme with weights $b_{i,k+1}$ and $d_{i,k+1}$ derived from the equivalent integral equation (14). The scheme is what allows CF trajectories to be simulated; the eigenvalue criterion decides which equilibrium those trajectories should approach.","core_discovery":"On the paper's own terms, the discovery is that the Caputo-Fabrizio derivative produces a different stability geometry for fractional Lotka-Volterra systems. For the linearized system $D^\\alpha u = Au$, the Caputo stability condition is $|\\arg(\\mathrm{spec}(A))| > \\alpha\\pi/2$, an unbounded wedge in the complex plane; the Caputo-Fabrizio condition, taken from Theorem 2.2, is the disjunction $|\\lambda| \\geq 1/(1-\\alpha)$ (with $\\lambda\\neq 1/(1-\\alpha)$), $\\operatorname{Re}\\lambda > 1/(1-\\alpha)$, $\\operatorname{Re}\\lambda < 0$, or $|\\operatorname{Im}\\lambda| > 1/(2(1-\\alpha))$, and Fig. 1 draws the unstable region as a bounded circle centered at $(0, 1/(2(1-\\alpha)))$ with radius $1/(2(1-\\alpha))$. The paper classifies eigenvalues into four classes—stable under both operators, stable only under Caputo, unstable under both, and stable only under Caputo-Fabrizio—and constructs a corrected Adams-Bashforth predictor-corrector method for the CF operator. Simulating the three-species model with this scheme reproduces the theoretical stability classification at each equilibrium. The central message is that operator type and order jointly determine which equilibrium the system approaches.","pith_inferences":["The bounded-circle geometry implies that eigenvalues with large imaginary parts can be stable under CF while unstable under Caputo, so the operator could be chosen according to the spectral character of the linearization.","The examples' sensitivity to initial conditions suggests that domains of attraction, not only local stability, decide the fate of trajectories; mapping those domains for each equilibrium and each operator would be a natural next test.","Because the CF operator is interpreted in the paper as modeling distributed lag rather than memory, the stability differences could be probed with real predator-prey time series: if lag-based models fit better, CF would be the appropriate operator.","For incommensurate orders (different $\\alpha$ for prey and predators), the stability boundaries should be intersections of the individual regions, producing new equilibrium classifications; the paper lists incommensurate orders as a future direction."],"forward_implications":["For the same three-species interaction parameters, switching from the Caputo to the Caputo-Fabrizio operator can turn a stable equilibrium into an unstable one, so the choice of derivative is a substantive modeling decision.","The value of the fractional order $\\alpha$ can determine whether a given equilibrium is asymptotically stable; in the reported examples, lowering $\\alpha$ expands the stability region and can allow the system to settle at equilibria that are unstable at higher $\\alpha$.","A reliable numerical study of Caputo-Fabrizio Lotka-Volterra systems needs the corrected predictor-corrector scheme rather than the earlier CF scheme, which the paper identifies as flawed.","The four-class eigenvalue classification supplies a practical rule before simulating: locate the Jacobian eigenvalues to see whether Caputo, CF, both, or neither will stabilize the system.","Because local stability of a nonlinear system is determined by its Jacobian eigenvalues, the same stability conditions and the same corrected scheme apply to other polynomial-type CF systems, not only to the three-species Lotka-Volterra model."],"supporting_citations":[{"why":"gives the original Caputo-Fabrizio derivative definition with exponential kernel that the paper adopts as its central operator.","marker":"[8]"},{"why":"supplies the properties and normalization function $M(\\alpha)$ used in the CF definition.","marker":"[9]"},{"why":"provides the wedge condition $|\\arg(\\mathrm{spec}(A))|>\\alpha\\pi/2$ that defines Caputo stability and serves as the comparison baseline.","marker":"[26]"},{"why":"supplies the four CF stability conditions in Theorem 2.2 and the circular stability-region picture of Fig. 1.","marker":"[27]"},{"why":"supplies the Caputo stability conditions for the three-species Lotka-Volterra system that Table 1 compares against the CF conditions.","marker":"[28]"},{"why":"gives the fractional Adams-Bashforth predictor-corrector method whose structure the corrected CF scheme extends.","marker":"[29]"},{"why":"is the earlier CF predictor-corrector scheme that the paper identifies as flawed and that motivates the corrected method.","marker":"[30]"}],"fun_headline_variants":["Fractional operator choice reshapes predator-prey stability","Caputo vs Caputo-Fabrizio: which equilibrium survives?","Stability wedge vs circle in fractional predator-prey","New Adams-Bashforth for Caputo-Fabrizio dynamics","Operator type decides fate of 3-species model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole numerical method rests on equation (14), the paper's integral-equation form of a Caputo-Fabrizio initial-value problem; for the case $0<\\alpha<1$ used in the examples, that formula invokes an undefined factorial and does not match the known integral representation of the Caputo-Fabrizio derivative.","fun_headline_variants_meta":{"raw":{"variants":["Fractional operator choice reshapes predator-prey stability","Caputo vs Caputo-Fabrizio: which equilibrium survives?","Stability wedge vs circle in fractional predator-prey","New Adams-Bashforth for Caputo-Fabrizio dynamics","Operator type decides fate of 3-species model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000649,"raw_usage":{"total_tokens":2973,"prompt_tokens":936,"completion_tokens":2037,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":1954}},"tokens_in":552,"tokens_out":2037,"duration_ms":14515,"temperature":1.0,"reasoning_tokens":1954,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:06:05.978665+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the proposed scheme on a scalar linear CF equation with a known exact solution at decreasing step sizes $h$; if the numerical solution does not converge to the direct integral solution of the CF initial-value problem, the claimed stability gain is an artifact of equation (14).","supporting_citations":[{"cited_title":"A new definition of fractional derivat ive without singular ker- nel,","cited_arxiv_id":null,"evidence_quote":"gives the original Caputo-Fabrizio derivative definition with exponential kernel that the paper adopts as its central operator."},{"cited_title":"Properties of a new fractional derivat ive without singular kernel,","cited_arxiv_id":null,"evidence_quote":"supplies the properties and normalization function $M(\\alpha)$ used in the CF definition."},{"cited_title":"Stability results for fractional differential equa tions with applications to control processing,","cited_arxiv_id":null,"evidence_quote":"provides the wedge condition $|\\arg(\\mathrm{spec}(A))|>\\alpha\\pi/2$ that defines Caputo stability and serves as the comparison baseline."},{"cited_title":"Stability analysis of a f ractional-order linear system described by the caputo-fabrizio derivative,","cited_arxiv_id":null,"evidence_quote":"supplies the four CF stability conditions in Theorem 2.2 and the circular stability-region picture of Fig. 1."},{"cited_title":"Analysis of a fractional order pre y-predator model (3-species),","cited_arxiv_id":null,"evidence_quote":"supplies the Caputo stability conditions for the three-species Lotka-Volterra system that Table 1 compares against the CF conditions."},{"cited_title":"Control of a fractional-order econ omical system via slid- ing mode,","cited_arxiv_id":null,"evidence_quote":"gives the fractional Adams-Bashforth predictor-corrector method whose structure the corrected CF scheme extends."},{"cited_title":"New predictor-correc tor scheme for solving nonlinear differential equations with caputo-fabrizio ope rator,","cited_arxiv_id":null,"evidence_quote":"is the earlier CF predictor-corrector scheme that the paper identifies as flawed and that motivates the corrected method."}],"review_version":1}