Three-species predator-prey model with respect to Caputo and Caputo-Fabrizio fractional operators

Leila Eftekhari; Leo Lahti; Moein Khalighi; Soleiman Hosseinpour

REVIEW 4 major objections 4 minor 31 references

Three-species predator-prey model with respect to Caputo and Caputo-Fabrizio fractional operators

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict A well-intentioned but mathematically broken numerical scheme invalidates the central claims, though the stability comparison and examples are also shaky. read the letter →

arxiv 1908.03685 v5 pith:PHXLA5CK submitted 2019-08-10 math.NA cs.NA

Leila Eftekhari , Moein Khalighi , Soleiman Hosseinpour , Leo Lahti This is my paper · ORCID

classification math.NAcs.NA MSC 34A0865L0692D25

keywords Caputo-Fabriziofractionalderivativethree-speciesLotka-Volterramodelpredator-preydynamicspredictor-correctormethodAdams-Bashforthstabilitydifferentialequations

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

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.

What carries the argument

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.

What would settle it

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).

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

That 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.

The 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.

Who 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.

If 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Sec. 4, Eq. (14)] 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.
[Sec. 3.2, Theorem 3.2] 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.
[Sec. 3.2.5 and Fig. 1] 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.
[Table 1] 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.

minor comments (4)

[Sec. 2.1] 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.
[Sec. 3.2.2] 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.
[Sec. 4] 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'.
[Sec. 5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the paper's central claims are not derived from their own outputs, and the numerical method is an independent construction, even if equation (14) may be mathematically questionable.

full rationale

The paper's central claim is a new predictor-corrector scheme for the Caputo-Fabrizio operator, built on the asserted equivalent integral equation (14), and a stability comparison between Caputo and Caputo-Fabrizio operators. The stability criteria are imported from external references: Theorem 2.2 is attributed to Ref. [27] (Li, Cheng, Li, and Zhong), not to the present authors, and the Caputo criteria come from Matignon [26] and Ref. [28]. The numerical scheme adapts the Adams-Bashforth-multon method from Ref. [29] and claims to correct Ref. [30]; neither is the authors' own prior work. The self-citations in the introduction (Refs. [3]-[6]) are background examples of fractional calculus applications and are not used to justify the numerical method or the stability conclusions. There is no fitting of parameters to a subset of data and then calling the result a prediction, and no definition is constructed so that the target result is true by definition. Equation (14) may indeed be invalid for 0<alpha<1, since it contains (n-2)! for n=1 and omits the non-integral (1-alpha)/M(alpha) term of the true CF inverse; however, an incorrect mathematical equivalence is a correctness flaw, not circularity. The paper does not reduce its claimed result to its own assumptions or to a self-citation chain. Accordingly, the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the CF definition and stability theorem from [27], the standard local stability reduction to the Jacobian, and the integral formula in Eq. (14). The latter is invalid for n=1, and the stability theorem is misquoted or internally inconsistent.

free parameters (1)

Normalization function M(α) in the CF derivative = unspecified; typically set to 1
The CF derivative definition depends on M(α), but the paper never specifies it. Stability conditions in Theorem 2.2 implicitly assume a value; if M(α) differs, the conditions would change.

assumptions (4)

domain assumption The Caputo-Fabrizio derivative definition (Eq. 2) is a valid fractional operator and the linear stability theorem from [27] is correctly stated as Theorem 2.2.
The entire stability analysis depends on these conditions, but Theorem 2.2 as printed is incomplete and inconsistent with Fig. 1.
domain assumption Local stability of the nonlinear system (4) is determined by the eigenvalues of its Jacobian using the linear stability conditions.
Invoked in Section 3.2 without proof for the CF operator.
domain assumption The function F in (6) is Lipschitz on L[0,t'] with a uniform constant L.
Section 3.1 proof asserts this, but F is quadratic and only locally Lipschitz on bounded sets; the claimed global Lipschitz bound is not established.
ad hoc to paper The equivalent integral formulation (14) correctly represents the CF initial value problem for 0<α<1.
Eq. (14) contains (n-2)! with n=1 and is not the standard integral form for CF; the numerical scheme is derived from it.

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Cite this review

Pith. "Pith review of Three-species predator-prey model with respect to Caputo and Caputo-Fabrizio fractional operators." pith.science (2026). https://pith.science/paper/PHXLA5CK

@misc{pith2026190803685,
  author       = {Pith},
  title        = {Pith review of: Three-species predator-prey model with respect to Caputo and Caputo-Fabrizio fractional operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHXLA5CK}},
  note         = {Machine review of arXiv:1908.03685}
}

read the original abstract

We study distributed lag effects in three-dimensional Lotka-Volterra systems by applying the concept of fractional calculus. We derive a new numerical method that provides enhanced stability for the Caputo-Fabrizio operator based on Adams-Bashforth method, considering non-singular kernel in the definition of Caputo-Fabrizio operator. We investigate the stability conditions of this system with comparisons to the Caputo fractional derivative. Numerical results show that the type of differential operators and the value of orders significantly influence the stability of the numerical solution, and dynamics of the Lotka-Volterra system.

Figures

Figures reproduced from arXiv: 1908.03685 by the authors.

**Figure 2.** (left) Comparing the behavior of Caputo and CF oper [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** System 4 with the parameters of Example 2 and (x0, y0, z0) = (2, 2, 3) is asymptotically stable for Caputo (left) and unstable for CF (right). λ0, λ1, and λ3 are. As a result, [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: System 4 with the parameters of the Example 3 and (x0, y0, z0) = (0.5, 0.1, 5) is asymptotically stable for both Caputo (left) and CF (right) at ǫ2 with different oscillations. 0 50 100 150 200 250 300 350 400 time 0 2 4 6 8 10 12 x,y,z Caputo α=0.4 x y z 100.5 101 101…

**Figure 5.** Figure 5: System 4 with the parameters of the Example 3 and (x0, y0, z0) = (3, 8, 0) is unstable for Caputo (left) and asymptotically stable for CF (right) [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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