Diffusion bridge with randomized initial and terminal times and its application to fish migration

Hidekazu Yoshioka; Mohammed Louriki

REVIEW 4 major objections 3 minor 1 cited by

Diffusion bridge with randomized initial and terminal times and its application to fish migration

T0 review · 4 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A temperature-driven biological clock randomizes the endpoints of a fish-migration bridge, and the resulting model is provably well-posed.

desk verdict The randomized-time CIR bridge is a genuinely new and essentially sound piece of stochastic modeling; the real weakness is the empirical calibration, which fits thresholds to the very data the model then fails to reproduce under stochastic WT. read the letter →

arxiv 2607.04253 v3 pith:NETWABDI submitted 2026-07-05 q-bio.PE math.PR

Hidekazu Yoshioka , Mohammed Louriki This is my paper · ORCID

classification q-bio.PEmath.PR MSC 60H1060G4060J6092D25

keywords CIRbridgerandominitialandterminaltimestimechangefishmigrationPlecoglossusaltivelis(Ayu)watertemperatureenvironmentalDNAstochasticdifferentialequation

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 establish that the Cox-Ingersoll-Ross (CIR) bridge, a diffusion bridge used to model the number of fish migrating past a fixed point, remains well-posed when its initial and terminal times are made random through a time-change driven by an environmental process. The authors define a 'biological clock' as the time integral of a nonnegative factor such as water temperature, and let migration start when that clock crosses a lower threshold and end when it crosses an upper threshold. The central result is a proof that the resulting stochastic differential equation has a pathwise unique, nonnegative solution that vanishes at both random endpoints. The empirical application calibrates the thresholds to observed water temperatures on Ayu migration start and end dates in the Nagara River, and demonstrates that total fish counts are largely robust to temperature noise. If the mathematical claim holds, this offers a tractable way to let environmental variation set migration timing without abandoning the closed-form statistical structure of the bridge.

What carries the argument

The key machinery is the time-change (5): the biological clock τ_t = ∫0^t M_u du, where M is a nonnegative, càdlàg environmental process. The initial and terminal times θ_i are the first times the clock crosses thresholds T_i. Under biological time s = τ_t, the state Y_s = X_t satisfies the original fixed-time CIR bridge (6), so the well-posedness of the original bridge is transferred to the randomized-time model. The load-bearing identity is that the transformed noise is a standard Brownian motion, which requires strict positivity of M almost everywhere; this is what turns the formal time-change into a rigorous equivalence.

What would settle it

Hold out one or more years of Nagara River data; use the fitted temperature thresholds (9.07 °C, 23.23 °C) to predict the start and end dates in the held-out years. If the prediction error is no smaller than using the fixed average calendar dates (start Feb 20, end Jun 30), then the temperature-driven clock adds no predictive power. For the mathematical claim, a counterexample would be a nonnegative càdlàg M that is positive a.e. but for which the solution of (4) fails uniqueness or nonnegativity; checking edge cases where M vanishes on a set of positive measure would test whether the strict-p

Watch

Extended reading notes

Core claim

The central claim, Proposition 1, is that under Assumptions 1–2 the randomized-time CIR bridge has a well-posed solution: the stopping times θ1 < θ2 are almost surely finite and strictly ordered, and the SDE (4) admits a pathwise unique, almost surely nonnegative solution on (θ1, θ2) with X_{θ1} = X_{θ2} = 0. The argument turns on the time-change (5): in biological time s = ∫0^t M_u du, the randomized-time bridge becomes the original fixed-time CIR bridge, provided the environmental process is almost everywhere strictly positive so that the time change is invertible. The proof constructs the driving Brownian motion explicitly and verifies it is a genuine Brownian motion via Lévy's characteri

Load-bearing premise

The empirical application assumes water temperature is the dominant driver of Ayu migration start and end; if calendar date or some other factor actually controls timing, then the temperature-threshold clock is fitted to the data it claims to explain, and the random-time mechanism has no causal content.

Editorial extensions

If this is right

Well-posedness (Proposition 1): the randomized-time CIR bridge has a unique, continuous, nonnegative solution on (θ1, θ2) that vanishes at both random endpoints, justifying simulation and estimation.
A 10% change in the migration-duration parameter T_emp changes the expected total fish count by about 10%; this parameter, not the temperature noise or the clock acceleration ω, is the dominant sensitivity in the Ayu application.
Total fish-count statistics are nearly insensitive to doubling or halving the water-temperature noise (relative differences around 0.5%), because the identified shape parameters m,n,p,q are large and create low-count buffer zones near the endpoints.
The eDNA model obtained by coupling the bridge to a degradation-accumulation ODE is well-posed; a sublinear allometric exponent H≈0.75 fits the Hii River eDNA data about 5% better than H=1 (RMSE), and the attenuation constant ≈0.4/day suggests the eDNA signal retains a memory of the fish count for a few days.
Direct transfer of parameters from the Nagara to the Hii River gives an unphysical attenuation constant (R≈100), so site-specific estimation is required.

Reading between the lines

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

If water temperature is only a correlate of calendar date, then the fitted thresholds might simply be an elaborate clock that re-labels the season: a decisive test is to compare start/end predictions in an anomalous-temperature year (e.g., a warm early March) against the fixed-date baseline – if the temperature clock predicts no better, the random-time machinery is not adding causal content.
The time-change construction is general: any diffusion bridge with an affine drift can be randomized this way by choosing M as a function of a covariate (e.g., discharge, salinity, day length), so the paper's Proposition 1 sets up a template for other environmentally timed animal movements.
The buffer-zone effect suggests a monitoring implication: fish-count sampling in the central migration window matters most for estimating total abundance, since the endpoints are insensitive regions – a testable prediction for survey design.
The eDNA attenuation constant R may serve as a river-specific fingerprint of mixing and degradation; the failure of parameter transfer is itself informative and points to collecting local eDNA decay experiments before applying the model to a new site.

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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

The core contribution here is the randomized initial/terminal time CIR bridge and the well-posedness result. The time-change construction is standard but carefully applied, and Proposition 1's proof checks out in substance. The reader's worry about an L2-vs-pathwise gap at the terminal point is overcautious: the variation-of-constants decomposition has a deterministic term tending to 0 and a martingale term whose variance vanishes, so L2-bounded martingale convergence gives the a.s. limit 0. Pathwise continuity at the endpoint follows by extension. The nonnegativity and the careful treatment of the singular coefficients under Assumption 2 are also solid. That part deserves serious attention.

The soft spot is the application, and it is genuine. In §3.2.2 the thresholds w and w̄ are set to the observed average water temperatures on the migration start and end dates (9.07 and 23.23 °C). With the deterministic linear WT model (21), this reproduces the observed average start/end dates by construction. But the nominal stochastic model (22) gives average start and end dates of 13.5 and 138.4 days, versus the observed 20.7 and 148.5. So the model does not reproduce the very statistics to which the thresholds were calibrated, and no out-of-sample test is offered. The paper honestly cites the literature suggesting calendar date is a dominant driver, but it does not resolve the tension. The eDNA example uses the same thresholds and a chosen endpoint of July 14, which is again circular in spirit.

That said, the sensitivity analysis is competently done and the parameter ranges are clearly reported. The mathematical contribution stands independently of the data application, and the authors do not hide the limitations. This is a paper with a solid theoretical core and a thin, partially circular empirical section.

I would send it to peer review: the stochastic bridge result is new and should be accessible to the community. But the referee should push for a proper calibration or reframing of the applied work as an illustrative case study, not validation. The paper is worth engaging with, and a serious referee can help separate the durable math from the weak application.

Referee Report

4 major / 3 minor

Summary. The paper proposes a Cox–Ingersoll–Ross (CIR) bridge whose initial and terminal times are random, generated by hitting times of the integral of a nonnegative environmental process M. The central theoretical result (Proposition 1, §2.2.2) asserts pathwise existence, uniqueness, nonnegativity, and pinning X_{θ1}=X_{θ2}=0 under Assumptions 1–2, via time-change to a standard CIR bridge with singular coefficients. The authors then apply the model to juvenile Ayu migration in the Nagara River, using water temperature (WT) thresholds to define the migration window and Ornstein–Uhlenbeck WT dynamics, and to eDNA time series in the Hii River through a linear/nonlinear ODE coupled to the bridge.

Significance. The mathematical construction is a genuine contribution: randomizing the bridge endpoints through a time-change preserves the affine structure of the CIR bridge while allowing environmental information to enter, and the proof gives an explicit construction of the time-changed Brownian motion. The use of closed-form moment formulas from the underlying CIR bridge is a practical strength, and the paper is transparent about several limitations of the empirical analysis. However, the application section does not currently validate the environmental-clock mechanism: the WT thresholds are calibrated from the very start/end dates the model is said to explain, and the nominal stochastic model does not reproduce those calibration targets. The eDNA case study is even more clearly exploratory. If the empirical claims are reframed or independently tested, the manuscript could be publishable; as it stands, the application is a calibrated illustration rather than a demonstration that WT is the causal clock.

major comments (4)

[§3.2.2, Table 2, Eqs. (8) and (21), Table B2] The thresholds w=9.07 °C and w̄=23.23 °C are the observed average WTs on the migration start/end dates in Table 2, and the deterministic WT (21) is anchored at the observed mean start date t_start=20.7 d. Thus the deterministic model reproduces the observed mean start/end dates by construction, not by independent mechanism. Under the stochastic WT model used as nominal (Eq. (22), ω=2), Table B2 gives mean start 13.46 d and mean end 138.4 d, against observed means ≈20.7 d and ≈148.5 d. So the nominal model does not even match the calibration targets, and no out-of-sample or cross-validation is provided. Please either identify thresholds independently, fit the full start/end distribution with uncertainty, or explicitly reframe this section as an illustration rather than validation.
[§3.2.3, Fig. 4, Tables B2–B4] The duration and end-date statistics appear to be mixtures with a large degenerate component: the Fig. 4 caption reports a 47% point mass at T_emp=127.8 d, and Table B3 shows the duration standard deviation is 2×10^{-9} d for ω=1. Consequently, the reported means and standard deviations in Tables B2–B4 combine a continuous distribution with a point mass placed exactly at the calibration value. This must be stated explicitly, and its biological interpretation discussed; as presented, the histograms, particularly Fig. 4(c), obscure the structure of the model output.
[§3.3.3, Table 3, Eq. (23)] The eDNA analysis inherits the threshold circularity: the 2025 start (Mar 10) and end (Jul 14) dates are inferred from the same w̄=23 °C threshold, giving T_emp=127 d, and G,H,R are calibrated by least squares to the same weekly eDNA series. The allometric exponent H≈0.75 is reported without confidence bounds and rests on roughly 15 weekly samples; moreover, the paper itself notes that a transferred model produces R≈100, which is unphysical. I recommend presenting this as a proof-of-concept and adding uncertainty quantification, rather than as an estimated allometric relationship.
[Appendix A, proof of Proposition 1, Eqs. (29)–(33)] The exponent bookkeeping in the proof of the two limits in (29) is not transparent and appears garbled. In Eq. (30) the integrand is said to be O((1-u)^{α-1}), but the preceding estimate contains the factor (1-u)^{-r} from the exponential weight; the printed condition “if 1? ... i.e., α>-1” seems to drop the dependence on r. With the application value r=61.9 (Table 1), a bounded â (α=0) would not make ∫_0^1 (1-u)^{-r} a_u du finite, so a more careful argument or a corrected assumption is needed. Please rewrite this part so that the sufficiency of Assumption 2 can be checked.

minor comments (3)

[Eq. (14)–(15)] The inverse gamma density in (15) appears misprinted: the normalization, the exponent of z, and the argument of the exponential need to be checked against the stated parameters (μ, λ).
[Throughout] Typos and wording: “This modal” should be “This model”; “CIR brides” should be “CIR bridges”; the Declaration says “The author used” while the paper has two authors; and in §3.2.3 “approximately 7 (day) and 10 (day)” should specify that these are standard deviations of the start/end dates, not mean shifts.
[Table 1] The value r=6.190E+01 is very large relative to the usual CIR reversion scale; please clarify whether this is the same r as in Proposition 1 and why the moment formulas (37)–(38) remain valid for this value.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mathematical well-posedness result is self-contained, and the applied thresholds are transparently calibrated inputs rather than disguised predictions.

full rationale

Proposition 1 is not circular: the proof explicitly constructs the time-changed Brownian motion in Eq. (25), verifies its quadratic variation in Eq. (27), invokes Lévy's characterization, and proves the terminal pinning through the estimates (30)–(33) under Assumption 2. The proof refers to Yoshioka [35] only for the structure of the argument, not as a substitute for the proof. The baseline CIR parameters (A,V,m,n,p,q,r) are imported from previous work by the same author, but those are empirical fits from earlier data and are not used to derive the new theorem; thus self-citation is not load-bearing for the central mathematical claim. In the application, the thresholds w=9.07 °C and w̄=23.23 °C are taken from the observed averages of start/end-date water temperatures in Table 2, and the deterministic WT model (21) is constructed with t_start=20.7 and T_emp=127.8. As a result, the deterministic limiting case reproduces those calibration targets by construction. However, the paper presents this as calibration and sensitivity analysis, not as an out-of-sample prediction, and it is transparent about the construction. Moreover, the nominal stochastic model in Table B2 gives different averages (start 13.5 d vs 20.7 d observed; end 138.4 d vs 148.5 d observed), so the model output is not forced to equal the calibration inputs. The paper also explicitly acknowledges limitations: the end-date estimates may be biased, the Hii River transfer produced an unphysical attenuation constant, and the model was not constructed to reflect the calendar-date findings of ref. [16]. These admissions further show that calibrated outputs are not being presented as independent predictions. Overall, the derivation chain does not reduce any claimed result to its own inputs.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central mathematical result depends mainly on standard stochastic calculus tools plus explicit regularity assumptions on the environmental process. The applied conclusions depend on many fitted constants: temperature thresholds, OU parameters, imported CIR shape parameters, and eDNA kinetic parameters. No new physical entities are introduced.

free parameters (6)

WT thresholds w, w̄ = 9.07 °C, 23.23 °C
Computed as the average daily WT on observed start/end dates of Ayu migration, Table 2; these thresholds define θ1, θ2 through Eq. (8).
OU parameters a_w, b_w = 0.1884 /day, 0.8533 °C/day^1/2
Averages of yearly least-squares estimates of Eq. (22)/(39) from daily WT; Appendix B.2.
empirical migration duration T_emp = 127.8 day
Average duration in Table 2; used to fix S via Eq. (20) and as baseline for sensitivity.
CIR shape parameters A,V,m,n,p,q,r = Table 1 values
Imported from Yoshioka [39]; fitted to earlier Ayu data by least squares and not re-derived here.
eDNA parameters G,H,R = H=0.748, G=136.9, R=0.395 (nonlinear); G=95.33, R=0.411 (linear)
Least-squares fit to 2025 Hii River eDNA samples via mean-field delta method, Table 3.
biological-clock speed ω = 2 (nominal)
Chosen by hand; sensitivity shows small effect for Ayu.

assumptions (6)

domain assumption M_t > 0 Lebesgue-a.e. and ∫M = ∞ (Assumption 1)
Needed so τ is strictly increasing and the time-change B̂ in Eq. (25) is a Brownian motion; if the environmental process stalls, the bridge construction breaks.
standard math Coefficient blow-up conditions α>-1, β>-1/2-α (Assumption 2)
Regularity bounds that make the terminal-time integrals in Appendix A finite; these are stated assumptions, not derived.
domain assumption M independent of B
Used to define a new Brownian motion B̂ via Lévy's theorem; asserts environment drives fish, not reverse.
domain assumption Water temperature thresholds determine migration start/end
The biological-clock mechanism Eq. (8) with w,w̄; the paper cites [46,47], but the specific thresholds are estimated from the same data.
domain assumption eDNA concentration follows dE=(G X^H - R E)dt
Aggregation/degradation ODE Eq. (23); no transport or spatial mixing modeled.
standard math Martingale convergence and Lévy characterization theorems
Used in Appendix A to identify B̂ and to attempt terminal pinning.

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Cite this review

Pith. "Pith review of Diffusion bridge with randomized initial and terminal times and its application to fish migration." pith.science (2026). https://pith.science/paper/NETWABDI

@misc{pith2026260704253,
  author       = {Pith},
  title        = {Pith review of: Diffusion bridge with randomized initial and terminal times and its application to fish migration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NETWABDI}},
  note         = {Machine review of arXiv:2607.04253}
}

read the original abstract

We mathematically model the dynamics of the number of migratory fish observed at a fixed location along a river in a random environment. Particularly, as a new approach, we construct a stochastic differential equation that incorporates the influence of environmental factors on the fluctuations in the start and end of migration. The model is a diffusion bridge with a non-Lipschitz diffusion coefficient, called the Cox-Ingersoll-Ross bridge, and has random initial and terminal times arising from time-change, so that the influences of environmental factors can be efficiently incorporated. The well-posedness of the model is first established, which is considered novel and significant in applied mathematics. Second, we estimate the parameters of the model based on the latest multiyear daily data set for the upstream migration of Plecoglossus altivelis altivelis (Ayu) by relying on the hypothesis that water temperature affects the migration of the fish, which has been suggested in existing studies. We also explore the application of the proposed model to the challenging task of analyzing environmental DNA data. This study advances the development of a theory of fish migration that is simple yet can take environmental factors into account.

Figures

Figures reproduced from arXiv: 2607.04253 by the authors.

**Figure 1.** Map of the study site in the Nagara River (modified from Chiri-in Chizu Vector: https://github.com/gsi-cyberjapan/gsimaps-vector-experiment) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Daily data of (a) fish count of Ayu at the Nagara River Estuarine Barrage from 2003 to 2025 and (b) daily WT (°C) at Oyabu from 2016 to 2025. Gray circles represent the zero count. Black circles represent the start and end dates of the migration of Ayu each year [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** A set of sample paths of WT w , biological clock  , and unit-time fish count X with normalization for visualization proposals [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Histograms (Hist) of (a) start date (day), (b) end date (day), (c) migration duration (except for the 47% mass at 127.8), and (d) total fish count (ind) for the nominal case [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: Histograms of the total fish count for the nominal case (black), deterministic case (w) a → + (red), less-fluctuating case ( ) ( ) 1/ 2 ww bb → (green), and more fluctuating case ( ) ( ) 2 ww bb → (blue) [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: Histograms of the total fish count for the nominal case (black), emp emp TT →1.1 (red), emp emp TT → 0.9 (blue), and randomized Temp (magenta) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: Map of the study site in the Hii River (modified from Chiri-in Chizu Vector: https://github.com/gsi-cyberjapan/gsimaps-vector-experiment) [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: Observed data of the eDNA concentration (copies/ml) of Ayu (black circles) and daily WT (°C) (blue circles) at Kisuki. The gray line represents the linear regression of the daily WT data, and the green vertical line corresponds to July 14, around which WT exceeds 23 (°…

**Figure 9.** Figure 9: eDNA concentration (black) (copies/ml) For the empirical result (black) and theoretical ones for the linear (red) and nonlinear cases (blue) [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]

**Figure 10.** Figure 10: Sample paths of the (nondimensionalized) unit-time fish count X (-) and eDNA concentration E (copies/ml) [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

**Figure 11.** Figure 11: Comparison between the average (Ave) (copies/ml) and standard deviation (Std) (copies/ml) of the eDNA concentration between the linear (red) and nonlinear models (blue) [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

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Stochastic partial differential equation model for environmental DNA dynamics in river environments
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A stochastic partial differential equation with a delayed source models riverine eDNA from migrating fish, with a closed-form Laplace functional and a nonnegativity-preserving numerical scheme.

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Reviewed August 2, 2026 · model on record in the stance chip above.