A Fractional Model of Abalone Growth using Adomian Decomposition Method

Adem Kilicman; Marliadi Susanto; Nadihah Wahi

REVIEW 3 major objections 5 minor 29 references

A Fractional Model of Abalone Growth using Adomian Decomposition Method

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A fractional-order McKendrick growth model, solved by Adomian decomposition, predicts abalone length best at order 0.5.

desk verdict 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. read the letter →

arxiv 2412.00035 v1 pith:ZA6YNWYM submitted 2024-11-21 math.GM

Marliadi Susanto , Nadihah Wahi , Adem Kilicman This is my paper · ORCID

classification math.GM MSC 26A3335R1192D25

keywords fractionalMcKendrickequationabalonegrowthAdomiandecompositionmethodCaputoderivativeTaylorseriesfractional-ordermodelmeanabsoluteerror

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 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.

What carries the argument

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 $\eta-r^\beta$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

To 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.

The 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.

For 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section 4, Eq. (20)] 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 5, Table 1] 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 5, system after Figure 1] 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.

minor comments (5)

[Section 4, after Eq. (23)] The text says 'the solution of Eq.(15)' but the model is Eq. (17); this equation number should be corrected.
[Section 5, text before Table 1] The text refers to 'Table 2' when comparing errors, but the manuscript contains only Table 1; the reference should be to Table 1.
[Definition 2.3, Eq. (8)] Equation (8) contains a typographical error: 'm − beta − 1' should be 'm − β − 1'.
[Definition 2.3, Eq. (9)] 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).
[Throughout] 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.

Circularity Check

2 steps flagged · score 6.0 of 10

Empirical 'prediction' is in-sample: monthly η values are taken from the same abalone data later used for MAE, and β=0.5 is chosen on that same error.

fitted input called prediction [Section 5 (Applications), Table 1, sentence defining η]
"We determine the growth rate of abalone length based on the data that were also used in [28], [29]. We calculate the rate of growth using a formula, η = ∆ h/∆t, where h denotes the abalone length."

Each monthly η_m is the observed length increment of the real data series. These η_m are then substituted into Eq. (24) to generate the Table 1 lengths for the same months. The predicted length at month m is therefore a function of the observed length change at month m; the later comparison with the real data is an in-sample check of quantities constructed from those very data, not an independent prediction.
fitted input called prediction [Section 5, last paragraph (MAE comparison and β selection)]
"By comparing the lengths of abalone on the Table 2 with the real data, we find that the mean absolute error of h0,5, h0.6, h0.7, h0.8, h0.9, and h1 are 0.2622; 0.5373; 0.7517; 0.9155; 1.0382; and 1.1294 respectively. Hence, the optimal result is achieved with a fractional order of β = 0.5."

The fractional order β is selected as the one with the smallest mean absolute error computed on the very same 24 data points that supplied η. This makes the conclusion that the fractional-order model 'provides more accuracy' a report of which in-sample fitted curve is closest, rather than a validated predictive advantage. The β=0.5 'prediction' is the best fit to the fitting data by construction.

full rationale

The only genuine circularity is in the empirical validation. The paper computes the monthly growth rate η from the same abalone length data that it later uses to compute mean absolute error, and it chooses β=0.5 as the smallest in-sample error. Hence the claimed 'prediction' is a fitted reproduction of its own inputs, not an out-of-sample forecast. The mathematical derivation in Eqs. (20)-(24) is not circular, but it is invalid: the paper implicitly uses D_s^β(M e^{rs}) = r^β M e^{rs}, which does not follow from the Caputo definition in Eq. (7) for 0<β<1; this is a correctness error outside the circularity rubric. The self-citations [28] and [29] are used only as sources of an empirical parameter r=0.04305 and data; they are not invoked as theorems that force the model. Therefore the derivation chain itself is not circular, but the central empirical comparison is partially circular, giving a score of 6.

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

The derivation rests on a false Caputo derivative identity, and the validation uses parameters fitted to the same target data. The model also assumes the biological variable can be identified with a population density without further justification.

free parameters (4)

r (initial growth rate) = 0.04305
Taken from prior study [28] by the same authors; fitted to abalone length data and used in the exponential solution (24).
η_m (monthly growth rates) = Values in Table 1, e.g., 0.4936 for month 2
Computed from real data via η = Δh/Δt; these are per-month fitted rates used directly in the model.
β (fractional order) = 0.5 (selected from grid 0.5, 0.6, ..., 1)
Chosen post hoc by minimizing mean absolute error against the same real data used to compute η; not derived.
M (initial value / initial length) = Not explicitly stated; table suggests 0.5322
The initial value M or h(month 1) is effectively calibrated to the data; not given in the paper.

assumptions (4)

ad hoc to paper Caputo fractional derivative of e^{rs} equals r^β e^{rs} for β∈(0,1).
Used implicitly in Eq. (20) to compute L_s(w0); false for Caputo derivatives, so the solution (24) is not a solution of (17).
standard math The Adomian decomposition series converges and the linear operators are invertible in the required sense.
Standard assumption of ADM; not problematic in itself, but it does not fix the derivative error.
domain assumption Biological length h can be identified with the model variable w(s,t) with no dimensional or structural correction.
The model variable is a population density over age and time; the paper applies it directly to shell length, with no derivation of this mapping.
domain assumption Data in refs. [28] and [29] are valid, complete, and representative for abalone growth.
The paper relies on these self-cited references for the real data and for r; no data table is given in this paper.

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Cite this review

Pith. "Pith review of A Fractional Model of Abalone Growth using Adomian Decomposition Method." pith.science (2026). https://pith.science/paper/ZA6YNWYM

@misc{pith2026241200035,
  author       = {Pith},
  title        = {Pith review of: A Fractional Model of Abalone Growth using Adomian Decomposition Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZA6YNWYM}},
  note         = {Machine review of arXiv:2412.00035}
}

abstract

This study is a modification of the McKendrick equation into a growth model with fractional order to predict the abalone length growth. We have shown that the model is a special case of Taylor's series after it was analysed using Adomian decomposition method and Caputo fractional derivative. By simulating the series with some fractional orders, the results indicate that the greater the fractional order of the model, the series values generated are greater as well. Moreover, the series that is close to the real data is the one with a fractional order of $0.5$. Therefore, the growth model with a fractional order provides more accuracy than a classical integer order.

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Pith tools

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

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Optimalization strategy of abalone (haliotis asinina) cul- tivation in Lombok-island using the matrix Leslie : INA-Rxiv 2018/3

Romdhini U.M. Optimalization strategy of abalone (haliotis asinina) cul- tivation in Lombok-island using the matrix Leslie : INA-Rxiv 2018/3

work page 2018

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M., Shikaa S

Samson L. M., Shikaa S. Mathematical modeling of Taraba State popu- lation growth using exponential and logistic models : Results in Control and Optimization 12 (2023) 100265

work page 2023

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et al., Estimated Effect of COVID-19 Lockdown on Skin Tu- mor Size and Survival: An Exponential Growth Model : Actas Dermosi- filiogr

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