{"id":"1d361d69-bd7b-46bb-9c62-d598430e8dfc","arxiv_id":"2412.00035","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A fractional-order McKendrick model solved by Adomian decomposition is claimed to fit abalone growth best at order 0.5, but the derivation uses an incorrect fractional derivative of an exponential.","lead":"This paper modifies the McKendrick growth equation into a fractional-order model and reports that order 0.5 best fits abalone length data. The derivation relies on a false Caputo derivative identity, so the model and its accuracy claim are not supported.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (20) rests on the false Caputo identity D_s^β(M e^{rs}) = r^β M e^{rs}; the true derivative is M r s^{1-β}E_{1,2-β}(rs), so the ADM series (23) and solution (24) do not solve Eq. (17).","rationale":"The reader's weakest_assumption exactly matches the central load-bearing concern. The Adomian recursion (19) is only as sound as its first application; Eq. (20) treats the Caputo derivative of an exponential as the classical exponential scaling, which is true for β=1 but not for β∈(0,1). Therefore the series (23) and the explicit solution (24) are not solutions of the stated fractional McKendrick equation. The paper's headline empirical claim about β=0.5 rests on evaluating this invalid solution, so the central claim is not established. The empirical validation is also circular, as Section 5 computes η from the same 24 observations used to report errors, but this point is secondary. Since the reader already reached REJECT for these reasons and my review confirms them, the verdict is unchanged.","tokens_in":7269,"tokens_out":6595,"duration_ms":57931,"concrete_test":"Compute the exact Caputo fractional derivative for the initial condition with β=0.5, r=0.04305, and s=1 using Eq. (8): D_s^{0.5}(e^{rs}) = (r/√π) ∫_0^1 (1-ξ)^{-1/2} e^{rξ} dξ = √r e^r erf(√r), and compare it with r^{0.5} e^r used in Eq. (20). If the two values differ, Eq. (20) is invalid. As a second check, substitute the proposed w(s,t) from Eq. (24) into Eq. (17) and evaluate the residual using the same Caputo definition; it should be nonzero for β<1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the first ADM iteration, Eq. (20). To compute L_s(w0) = ∂^β(M e^{rs})/∂s^β, the paper implicitly uses the identity D_s^β(M e^{rs}) = r^β M e^{rs}. Under the Caputo definition given in Eqs. (7)-(8), with β∈(0,1) and m=1, the correct expression is D_s^β(M e^{rs}) = M r s^{1-β} E_{1,2-β}(r s), which is not a constant multiple of e^{rs}. For example, at β=0.5, r=0.04305, and s=1, the paper's value r^β e^r is about 0.216, while the Caputo derivative is about 0.050. Because Eq. (20) is wrong, Eqs. (21)-(23) and the closed form (24) are not solutions of Eq. (17); substituting (24) leaves a nonzero residual in the fractional PDE. The identity only becomes correct at β=1, the integer-order limit. The empirical comparison in Section 5 additionally fits η to the same 24 data points used for the error report, but the mathematical flaw is already decisive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a fractional-order modification of the McKendrick equation, ∂w/∂t + ∂^β w/∂s^β = η w, with initial condition w(s,0)=M e^{rs}, and claims to solve it by the Adomian decomposition method (ADM) using the Caputo fractional derivative. The authors obtain the closed-form expression w(s,t)=M e^{rs} e^{(η−r^β)t}, which they describe as a Taylor series, and then use this expression to predict abalone length growth. They compare results for several fractional orders β against 24 months of abalone data and conclude that β=0.5 gives the best fit, with lower mean absolute error than the integer-order model.","tokens_in":7569,"tokens_out":8343,"duration_ms":72617,"significance":"If the derivation and empirical comparison were correct, the paper would offer a simple example of a fractional-order transport model improving empirical growth prediction, which could be of interest to applied fractional-calculus communities. However, the central mathematical step is invalid: the Caputo derivative of e^{rs} is not r^β e^{rs} for β∈(0,1), so the purported ADM solution does not solve the stated fractional PDE. The empirical validation is also in-sample, with the growth rates and the fractional order both selected using the same data that is later scored. Under the stated standards, the paper's central claims are not established, and the manuscript would require a fundamental reworking rather than local revision.","major_comments":[{"comment":"The step L_s(w0)=∂^β(M e^{rs})/∂s^β=r^β M e^{rs} is incorrect under the Caputo derivative defined in Eqs. (7)–(8). For β∈(0,1), the correct derivative is ∂^β/∂s^β (M e^{rs}) = M r s^{1−β} E_{1,2−β}(r s), which is not a constant multiple of e^{rs}. For example, with β=0.5, r=0.04305, and s=1, the value r^β e^r is about 0.216, while the Caputo derivative is about 0.050. Because Eq. (20) is the first ADM iteration, the subsequent terms in Eqs. (21)–(23) and the closed form in Eq. (24) do not satisfy Eq. (17); substituting Eq. (24) into Eq. (17) leaves a nonzero residual for every β∈(0,1). This invalidates the central derivation and the claimed solution.","section":"Section 4, Eq. (20)"},{"comment":"The empirical comparison is in-sample in two related ways. First, the monthly growth rates η are computed from the same 24 observations that are later used to compute the mean absolute errors, so the comparison measures the fit to the calibration data, not predictive accuracy. Second, the fractional order β=0.5 is selected by minimizing that same in-sample error; this is curve fitting, not evidence that the fractional model is intrinsically more accurate. In addition, the paper does not specify how the monthly η values and the variables s and t enter Eq. (24) to produce Table 1. With the stated parameters, the β=1 entry at month 2 is not obviously 0.8687 under any explicit choice of s; the construction of the table is therefore not reproducible as written.","section":"Section 5, Table 1"},{"comment":"The manuscript applies a population-density equation to individual abalone length without defining the relation between w(s,t) and the measured length h. The model (17) is a linear transport equation for a density, and its solution (24) describes an exponential-in-time evolution of that density, not a mechanism for individual body-length growth. The system of equations with different η_1,...,η_23 is not derived from Eq. (17), and it is unclear whether the model remains well-posed when η is allowed to change at each time step. This disconnect weakens the biological interpretation and the relevance of the numerical comparison.","section":"Section 5, system after Figure 1"}],"minor_comments":[{"comment":"The text says 'the solution of Eq.(15)' but the model is Eq. (17); this equation number should be corrected.","section":"Section 4, after Eq. (23)"},{"comment":"The text refers to 'Table 2' when comparing errors, but the manuscript contains only Table 1; the reference should be to Table 1.","section":"Section 5, text before Table 1"},{"comment":"Equation (8) contains a typographical error: 'm − beta − 1' should be 'm − β − 1'.","section":"Definition 2.3, Eq. (8)"},{"comment":"In the definition of ∂^β g(s,t)/∂t^β, the differential under the integral should be dξ, not ds, and the notation should be made consistent with Eq. (8).","section":"Definition 2.3, Eq. (9)"},{"comment":"There are several typographical and grammatical errors, including 'and and the slow growth rate', 'an essential tools', and 'Mittage-Leffler'; these should be corrected in any revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The manuscript is a math.GM preprint with a decisive mathematical error in Eq. (20), and the empirical section is an in-sample fit. The editor may also wish to check the relationship with the authors' companion paper [29], since the present manuscript does not clearly delineate what is new relative to that work. I do not see a path to acceptance within the current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my take on Susanto et al. The paper's central result, Eq. (24), is not a solution of the fractional PDE (17). The first ADM iteration replaces D_s^β(M e^{rs}) with r^β M e^{rs}, which is false under the Caputo definition the paper itself gives in Eq. (7)-(8). The correct derivative is M r s^{1-β} E_{1,2-β}(r s), so the whole series (20)-(23) and the exponential closed form do not solve the stated problem. This is not a minor slip; every later term depends on it.\n\nTo give credit where it is due: the paper is clearly organized, applies ADM to a concrete aquaculture question, uses real abalone length data, and reports its tables and error metrics transparently. The idea of a fractional-order age-structured growth model is worth exploring, and the authors do connect to the relevant literature, including their own earlier work. The data handling is easy to follow and would be reproducible if the math were sound.\n\nThe soft spots are proportional to the main error. Beyond the derivative mistake, the validation is circular: the monthly growth rates η are computed from the same 24 data points that later produce the MAE table, and β=0.5 is selected by minimizing that same in-sample error. Also, as the table's constant column ratios show, the fractional order only rescales the integer-order curve by a constant factor e^{r-r^β}; it doesn't introduce new dynamics. So even if the derivation were correct, the \"improvement\" would be a fitted constant, not a structural advantage.\n\nFor whom is this? Applied fractional-modeling readers might get a template for setting up ADM on McKendrick-type equations, but the template is currently wrong. The authors could fix the derivative identity and re-derive the solution; the Mittag-Leffler form might still be worth comparing against data. As it stands, I would not send this to peer review. A competent referee would find the same error in minutes, and the in-sample fit would fail any honest review. My recommendation is to reject/desk-reject.","headline":"The fractional model's closed form does not solve the stated equation, and the reported accuracy is in-sample; the main claim fails on two independent grounds.","tokens_in":8084,"tokens_out":3976,"would_cite":false,"duration_ms":34107,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26A33","35R11","92D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"A fractional-order McKendrick growth model, solved by Adomian decomposition, predicts abalone length best at order 0.5.","keywords":["fractional McKendrick equation","abalone growth","Adomian decomposition method","Caputo fractional derivative","Taylor series","fractional-order model","mean absolute error"],"falsifier":"Compute $D_s^\\beta(M e^{rs})$ directly from the integral definition in Eq. (7) for $\\beta=0.5$ and compare the result with $r^{0.5}M e^{rs}$; if the two expressions differ for any positive $s$, then Eq. (20), and therefore the closed-form solution (24), is not a solution of Eq. (17), and the reported errors in Table 1 do not describe a solution of the fractional model.","tokens_in":7045,"feed_emoji":"🐚","tokens_out":7641,"duration_ms":64648,"temperature":0.7,"pith_summary":"This paper tries to show that replacing the first-order space derivative in the McKendrick growth equation with a fractional derivative of order $\\beta$ produces a model that fits real abalone length data better than the classical integer-order model. Using the Adomian decomposition method with a Caputo fractional derivative, the authors derive the closed solution $w(s,t)=M e^{rs} e^{(\\eta-r^\\beta)t}$ and observe that this solution is a Taylor series. Simulating the series with orders from $0.5$ to $1$ against 24 months of observed abalone lengths, the paper reports that predicted lengths increase with $\\beta$ and that $\\beta=0.5$ has the smallest mean absolute error. The conclusion is that this fractional-order growth model is more accurate for abalone than the integer-order version.","feed_headline":"Fractional growth model predicts abalone length best at order 0.5","feed_subtitle":"An order-0.5 fractional model beats the integer-order curve on 24 months of real abalone data.","key_machinery":"The load-bearing machinery is the Adomian decomposition recursion combined with the Caputo fractional derivative. Starting from $w_0=M e^{rs}$, the operator equation $L_t w + L_s w = \\eta w$ is inverted with $L_t^{-1}$ to generate $w_{n+1}=-L_t^{-1}[L_s w_n]+\\eta L_t^{-1}[w_n]$. Treating the fractional derivative of the exponential initial condition as $r^\\beta M e^{rs}$ produces each term $w_n=(\\eta-r^\\beta)^n M e^{rs} t^n/n!$, so the infinite sum is $M e^{rs}e^{(\\eta-r^\\beta)t}$, a Taylor series in $t$ in which the fractional order $\\beta$ enters only through the exponent coefficient $\\eta-r^\\beta$.","core_discovery":"The paper claims that the fractional McKendrick growth equation $$\\frac{\\partial w}{\\partial t} + \\frac{\\partial^\\$\\beta$ w}{\\partial s^\\$\\beta$} = \\eta w, \\quad w(s,0)=M $e^{{rs}}$,$$ with the Caputo fractional derivative, is solved by the Taylor series $w(s,t)=M e^{rs} e^{(\\eta-r^\\beta)t}$ obtained through the Adomian decomposition method. It further claims that when this series is evaluated with a fixed initial growth rate $r$ and month-by-month growth rates $\\eta$ taken from real abalone length data, the order $\\beta=0.5$ gives the smallest mean absolute error, and that therefore the fractional-order model is more accurate than the classical integer-order model.","pith_inferences":["The paper does not explore this, but the closed form depends only on $\\eta-r^\\beta$, so the same expression could be fitted to other species or traits with $\\beta$ as a fitted parameter.","Beyond the paper's in-sample comparison, a split-sample or leave-one-out test would show whether $\\beta=0.5$ generalizes or simply fits the training months best.","The paper does not say so, but because $r<1$, $\\beta$ acts as a continuous tuning knob that shrinks the effective growth rate; testing intermediate orders would reveal whether 0.5 is near a genuine optimum."],"forward_implications":["For any fractional order $\\beta\\in(0,1]$, the ADM iteration yields $w(s,t)=M e^{rs} e^{(\\eta-r^\\beta)t}$, so the fractional order changes only the effective growth exponent $\\eta-r^\\beta$.","Larger $\\beta$ gives larger predicted lengths at each month, which is what Table 1 shows.","Among $\\beta=0.5,0.6,0.7,0.8,0.9,1$, the smallest mean absolute error against the real data is at $\\beta=0.5$ with error $0.2622$.","The model can be run month by month with changing growth rates $\\eta$, expressed as a system of equation copies of Eq. (17).","The paper's conclusion is that fractional order outperforms integer order for this abalone dataset."],"supporting_citations":[{"why":"Supplies the Adomian decomposition treatment of growth equations that the present model extends to fractional order.","marker":"[17]"},{"why":"Source for the Adomian decomposition method used throughout the paper.","marker":"[18]"},{"why":"Gives the Riemann-Liouville fractional definitions that frame the fractional calculus background.","marker":"[21]"},{"why":"One of the sources for the Caputo fractional derivative definition used in Eq. (7).","marker":"[22]"},{"why":"Second source for the Caputo fractional derivative definition used in Eq. (7).","marker":"[23]"},{"why":"Provides the fractional integral properties, including the power rule and inverse property, that the ADM iteration implicitly relies on.","marker":"[24]"},{"why":"Supplies the initial growth rate $r=0.04305$ and the real abalone length data used in the comparison.","marker":"[28]"},{"why":"Prior fractional growth model for abalone length that this work builds on and extends.","marker":"[29]"}],"fun_headline_variants":["Order 0.5 fractional model tops integer for abalone growth","Fractional order 0.5 best fits abalone length data: study","Adomian-decomposed fractional model improves abalone growth fit","Half-order fractional equation beats classical for abalone growth","Abalone growth: fractional McKendrick with order 0.5 wins"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes that the Caputo fractional derivative of the exponential initial condition $M e^{rs}$ is exactly $r^\\beta M e^{rs}$, the same form as an integer-order derivative; if this step does not hold, the ADM series and the $\\beta=0.5$ accuracy comparison collapse.","fun_headline_variants_meta":{"raw":{"variants":["Order 0.5 fractional model tops integer for abalone growth","Fractional order 0.5 best fits abalone length data: study","Adomian-decomposed fractional model improves abalone growth fit","Half-order fractional equation beats classical for abalone growth","Abalone growth: fractional McKendrick with order 0.5 wins"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000903,"raw_usage":{"total_tokens":3817,"prompt_tokens":807,"completion_tokens":3010,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":423,"completion_tokens_details":{"reasoning_tokens":2918}},"tokens_in":423,"tokens_out":3010,"duration_ms":19889,"temperature":1.0,"reasoning_tokens":2918,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:20:04.502321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $D_s^\\beta(M e^{rs})$ directly from the integral definition in Eq. (7) for $\\beta=0.5$ and compare the result with $r^{0.5}M e^{rs}$; if the two expressions differ for any positive $s$, then Eq. (20), and therefore the closed-form solution (24), is not a solution of Eq. (17), and the reported errors in Table 1 do not describe a solution of the fractional model.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Adomian decomposition treatment of growth equations that the present model extends to fractional order."},{"cited_title":"Solving delay differential systems with history functions by the Adomian decomposition method : Applied Mathematics and Computation 218 (2012) 5994–6011","cited_arxiv_id":null,"evidence_quote":"Source for the Adomian decomposition method used throughout the paper."},{"cited_title":"What is a fractional derivative? : Journal of Computational Physics 293 (2015) 4–13","cited_arxiv_id":null,"evidence_quote":"Gives the Riemann-Liouville fractional definitions that frame the fractional calculus background."},{"cited_title":"Caputo derivatives of fractional variable order: Numerical approximations : Commun Nonlinear Sci Nu- mer Simulat 35 (2016) 69–87","cited_arxiv_id":null,"evidence_quote":"One of the sources for the Caputo fractional derivative definition used in Eq. (7)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Second source for the Caputo fractional derivative definition used in Eq. (7)."},{"cited_title":"Generalized differential transform method for solving a spaceand time-fractional diffusion-wave equation : Physics Letters A 370 (2007) 379–387","cited_arxiv_id":null,"evidence_quote":"Provides the fractional integral properties, including the power rule and inverse property, that the ADM iteration implicitly relies on."},{"cited_title":"Logistic model of abalone’s length growth in Sekotong, West Lombok : AIP Publishing LLC-2199 (2019)-030002","cited_arxiv_id":null,"evidence_quote":"Supplies the initial growth rate $r=0.04305$ and the real abalone length data used in the comparison."},{"cited_title":"Fractional growth model of abalone length : Partial Differential Equations in Applied Mathematics 10 (2024) 100668","cited_arxiv_id":null,"evidence_quote":"Prior fractional growth model for abalone length that this work builds on and extends."}],"review_version":1}