Pith. sign in

REVIEW 3 major objections 6 minor 95 references

Stochastic partial differential equation model for environmental DNA dynamics in river environments

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A stochastic PDE with a delayed source links migrating fish to river eDNA, with proof of well-posedness and a closed-form Laplace functional.

desk verdict A genuine first SPDE for eDNA with delayed stochastic source and closed-form Laplace functional, but the boundary trace gap in the well-posedness proof needs fixing and the case study is illustrative rather than predictive. read the letter →

arxiv 2608.03364 v2 pith:VFEDZZRX submitted 2026-08-04 q-bio.QM math.PR

Hidekazu Yoshioka This is my paper · ORCID
classification q-bio.QMmath.PR MSC 60H1560J6865C3092D40
keywords environmentalDNAstochasticpartialdifferentialequationaffineprocessfishmigrationdelayedsourceLaplacefunctionalmeasure-valuednonnegativity-preservingscheme
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 spatiotemporal spread of environmental DNA (eDNA) from migratory fish can be described by a stochastic partial differential equation whose noise term is non-Lipschitz yet still leaves the model exactly solvable in distribution. The proposed model tracks eDNA concentration along a 1-D river with advection, decay, and a source that is delayed because it comes from fish that passed the river mouth a travel time earlier; fish counts themselves follow a stochastic differential equation. Because the whole system is affine, a closed-form Laplace functional is derived, a weak solution is shown to exist and be unique in law, and a fully discrete scheme is given whose numerical solutions stay nonnegative. The model is fitted to weekly eDNA measurements of ayu in the Hii River system and captures the observed migration peaks. The appeal is that ecologically relevant quantities such as means, variances, and extinction times can be computed directly from the formula rather than by expensive simulation.

What carries the argument

The load-bearing object is the pair consisting of the square-root multiplicative noise $\sigma_t(x)\sqrt{Y_t(x)}W(dt,dx)$ and the delayed source $g_t(x)Z_{t-x/v}$. Square-root noise is the standard affine-process ingredient: it keeps concentrations nonnegative while making the Laplace functional exponential-affine. The delay ties the source at location $x$ to the fish-count SDE at the river mouth $x/v$ time units earlier, which is what lets the model convert migration measurements into eDNA predictions. That conversion is carried by the closed-form Laplace functional (20), whose coefficient functions solve the Riccati-type system (21)-(24); the same structure makes the semidiscrete model a finite-dimensional affine process and the numerical scheme unconditionally nonnegative.

What would settle it

Collect independent daily fish counts at the Hii River mouth together with daily eDNA samples at a few fixed stations along the reach during the same migration season, then compare the observed eDNA peak timing and spatial profile with the predictions from the closed-form Laplace functional (20) using the paper's fitted parameters; a systematic separation beyond sampling noise would indicate that the delayed-source assumption or the imported fish-count parameters are wrong.

Watch

Extended reading notes

Core claim

The central claim is that the SPDE $dY_t(x) = \left(-u\partial_x Y_t(x) + g_t(x)Z_{t-x/v} - r_t(x)Y_t(x)\right)dt + \sigma_t(x)\sqrt{Y_t(x)}\,W(dt,dx)$, with the fish count $Z$ following the affine SDE (5), is a well-posed stochastic evolution equation for the measure-valued eDNA concentration. In particular, Proposition 7 states that a weak solution exists and that weak solutions are unique in law, and Proposition 3 gives the Laplace functional in the closed form (20), with coefficients determined by the coupled differential system (21)-(24). This makes the model an infinite-dimensional affine process with a distributed delay, so moments and extinction-related statistics can be evaluated without Monte Carlo simulation. The paper also proves that its upwind semidiscretization converges in law to the continuum model, and that a time-stepping scheme based on inverse-gamma draws keeps computed concentrations nonnegative.

Load-bearing premise

The load-bearing premise is that the fish-count process follows the specific stochastic equation with parameter values estimated from fish migration in a different river, because the paper has no unit-time fish counts for the Hii River system; if the migration timing, speed, or count distribution differs there, the delayed source term is misspecified.

Editorial extensions

If this is right

  • If the central claim is right, the full distribution of eDNA concentration is accessible through the Laplace functional (20); means, variances, and extinction probabilities follow by differentiation, with no stochastic simulation needed.
  • Spatially resolved predictions inherit the upstream-to-downstream causal structure of the upwind discretization, so eDNA fronts and their timing can be computed consistently along a 1-D river reach.
  • The nonnegativity-preserving time-stepping scheme using inverse-gamma draws lets the model be run with large noise intensities without producing negative concentrations.
  • In the case study, the fitted model reproduces the observed weekly eDNA peaks at both sampling sites and indicates that eDNA extinction is driven mainly by decay and outflow once the migrating source vanishes.

Reading between the lines

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

  • The same affine-delay architecture might transfer to river networks, but the graph case requires internal boundary conditions at each node and a new well-posedness proof; the paper leaves this open.
  • Because the closed form separates fish-count parameters from transport parameters, weekly eDNA samples from several sites along a reach could in principle be used to identify migration speed $v$ and shedding rate $g$ without high-frequency fish counts.
  • The model's predictive value in the Hii River rests on parameters estimated in other rivers; direct fish-count data from the Hii system would be the natural next test.
  • The finding that spatially averaged noise statistics are insensitive to the noise intensity while pointwise sample paths are sensitive suggests that future field protocols should favor high-frequency sampling at fixed stations over spatial averaging.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a stochastic partial differential equation (SPDE) model for spatiotemporal environmental DNA (eDNA) concentration during upstream fish migration in a one-dimensional river. The source term is a delayed functional of an auxiliary stochastic differential equation for fish counts, the decay is linear, and the noise is multiplicative square-root type. The main theoretical claims are: well-posedness of the semidiscrete affine SDE system, existence and uniqueness in law of a measure-valued weak solution to the SPDE, a closed-form Laplace functional for the SPDE, convergence of the semidiscrete Laplace functional to the continuum one, and unconditional nonnegativity of a fully discrete scheme. The paper also reports an application to eDNA data from the Hii River system, fitting a few parameters by least squares and performing a sensitivity analysis of the noise intensity.

Significance. If the well-posedness proof is completed, the paper makes a useful contribution: it connects an affine SDE model of fish migration to a measure-valued SPDE with delay, and the closed-form Laplace functional is a nontrivial and practically valuable result that avoids direct simulation for moment computations. The semidiscrete cascade structure and the nonnegativity-preserving time-stepping scheme are also clear strengths, and the paper is explicit about the non-Lipschitz nature of the diffusion coefficient. The application, however, is currently an in-sample illustration rather than a validation: the fish-count process is imported from another river, only three parameters per site are fitted, and no uncertainty quantification or out-of-sample check is provided. The central mathematical gap concerns the boundary trace in the weak-solution definition and in the uniqueness proof; this is fixable but must be addressed before the paper's main claim can be accepted.

major comments (3)
  1. [§3.4.2, Proposition 7; §3.2.2, Eq. (9); Appendix A1, Eq. (148)] The proof that a tightness limit satisfies the martingale problem is incomplete at the boundary. The weak-solution definition (9) contains the term u∫_0^t Y_s(0) φ(0) ds, i.e., a boundary trace, which is not a functional of the measure-valued process μ_s. The semidiscrete approximations carry Y^1_N(t) as a separate component, but Propositions 5 and 6 establish tightness only for the empirical measures μ^N = h Σ_i Y^i_N δ_{x_i}. The estimate (148) controls ∫_0^T (u Y^1_N(s))^2 ds uniformly in N, which is an L^2 bound and does not by itself provide convergence to a boundary density; Section 5 also lists full density as an open issue. Consequently, for a general tightness limit μ, the boundary term in (9) is not determined by μ, the martingale problem is not closed, and uniqueness in law of μ does not imply uniqueness of weak solutions as defined. Please either prove convergence of the boundary flux and uniqueness for the pair (μ, Y(0)), or reformulate the definition so that the boundary flux is part of the state process.
  2. [§4.2, Table 2; §4.3, Figures 4 and 9-10] The application section treats an in-sample fit as empirical support for the model, but the source-process parameters are imported from a different river. The paper states in §4.2 that 'There are no data about unit-time fish counts of P. altivelis in the Hii River system,' yet the SDE (5) and its parameters (τ=127 d, p0=1e6, p1=p2=10, q0=4e6, q1=q2=20, b=61.9, and a_t, c_t from (40)-(41)) are taken from Yoshioka (2025, 2026) and Yoshioka & Louriki (2026). Only g, r, and the initial time are fitted per site, with no uncertainty quantification, no out-of-sample check, and no comparison with a simpler benchmark model. If the migration timing, count distribution, or ground speed in the Hii River differs from the imported values, the delayed source in (3) is misspecified and the fitted curves in Figure 4 carry no predictive information. Given the paper's stated goal of demonstrating model operation, the conclusions in §4.3 should be explicitly restricted to an illustration, or the analysis should be supplemented with validation or a sensitivity analysis over the imported Z-parameters.
  3. [§4.2, Eq. (42); Appendix A3, Figures A1-A2] The parameter estimation uses the infinite-domain closed-form formula (42) for E[Y_t(0)], while the reported simulations use finite domains L=15 km (Kisuki) and L=30 km (Shin-Mitoya). Figure A1 shows that the finiteness of L noticeably affects the downstream concentration tail for the Kisuki case, yet Table 2 reports g and r obtained from the infinite-domain formula. This inconsistency can bias the fitted shedding and decay rates, especially for Kisuki. Please either estimate parameters with the same finite-domain solution used in the simulations, or provide a quantitative check that the finite-domain correction is negligible for the fitted values.
minor comments (6)
  1. [Throughout] There are many typographical errors in the propositions and section headings, including 'aee', 'wheee', 'expeessed', 'boundaey', 'Moeeovee', 'peobability', 'deteemined', and 'teeminal'. The manuscript needs a careful proofreading pass.
  2. [Proposition 3] In Proposition 3, the sentence introducing the coefficients says that α, β, γ, ω 'in (12)' are determined from the system; the reference should be to (20), not (12).
  3. [§2.2, Eq. (4)] Equation (4) is typeset in a way that makes the advection term, the indicator (i<N), and the noise scaling hard to read; the formula should be rewritten with explicit indices and brackets.
  4. [§4.1 and Table 2] Table 3 reports the 'Average of τ_ext/τ (day)' with units of day, but τ_ext/τ is dimensionless; the column headings should be corrected.
  5. [§4.1] The statement that the higher eDNA concentration in the tributary is 'due to the higher attraction of the former having a higher flow speed' is a causal interpretation of two observational time series and is confounded by other site differences; it should be phrased as a hypothesis.
  6. [References] Reference [48] has a stray closing bracket in the URL field, and the reference to 'Yoshioka (2025)' in §3.5 should use the numbered citation format consistently.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity; affine Laplace functional and uniqueness-in-law are derived from model equations, while the case study is an in-sample fit with parameters imported from prior work.

full rationale

The paper's central theoretical chain is not circular. Proposition 2 and Proposition 3 derive the Laplace functional (20) by postulating an exponential-affine form, applying Itô's formula to the conditional expectation C(t,T), and matching drift terms; the coefficients are then solved from Riccati-type ODEs/PDEs (13)-(17) and (21)-(24). Proposition 4 obtains the continuum Laplace functional as the pointwise limit of the semidiscrete one, and Proposition 7 couples this with tightness (Propositions 5-6) and external uniqueness theorems (Perkins; Li) to give existence and uniqueness in law. None of these steps uses the observed eDNA data or the fitted parameters. The application section fits g, r, and the initial time to the same observed eDNA concentrations and explicitly presents Figure 4 as a comparison of observed and fitted eDNA concentrations rather than as an out-of-sample prediction; the mean formula (42) is derived from the model and then used for least-squares fitting, which is honest in-sample calibration, not a predicted quantity reduced to its own input. The Z-process parameters are imported from Yoshioka [48] and Yoshioka and Louriki [49]; those self-citations supply empirical inputs and the SDE form, but they do not reappear as conclusions of this paper's derivation, and the paper explicitly flags the lack of local unit-time fish-count data in Section 4.2. Section 5's caveats about full density and strong convergence are limitations, not circular moves. The skeptical concern about the boundary trace in the martingale problem (9) is a possible mathematical gap, but it is not a definitional equivalence or a fitted-parameter renaming, so it does not raise the circularity score.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The mathematical core rests on standard SPDE and affine-process machinery. The applied part depends on six externally or hand-set parameters (three fitted here, sigma chosen, tau and v imported), and on the domain simplifications listed above. No new physical entities are introduced.

free parameters (6)
  • initial_time (start of upstream migration) = 2025/3/27 (Kisuki), 2025/3/20 (Shin-Mitoya)
    Estimated by least-squares fit to the observed eDNA concentration time series (Section 4.2, Table 2).
  • g (eDNA shedding rate) = 1453.6 copies/ml/day (Kisuki), 6546.5 copies/ml/day (Shin-Mitoya)
    Fitted with r and initial time against observed eDNA concentrations (Section 4.2, Table 2).
  • r (eDNA decay rate) = 5.70 1/day (Kisuki), 10.3 1/day (Shin-Mitoya)
    Fitted with g and initial time against observed eDNA concentrations (Section 4.2, Table 2).
  • sigma (noise intensity) = 0.5 (baseline), varied 0.1 to 20
    Chosen by hand rather than estimated; sensitivity analysis varies it (Sections 4.2, 4.3).
  • tau (migration duration) = 127 days
    Taken from the threshold water temperature analysis of Yoshioka and Louriki [49]; not re-estimated in this paper (Section 4.2).
  • v (fish ground speed) = 1 km/day
    Intermediate value chosen from ranges in the literature for P. altivelis; not fitted (Section 4.2).
assumptions (4)
  • standard math Space-time white noise W with formal covariance (2) and the martingale-problem formulation (6)-(10) for measure-valued solutions.
    The weak formulation is standard for SPDEs driven by white noise (Dalang and Sanz-Solé [54]).
  • domain assumption Coefficients g, r, sigma are positive, bounded, and continuous on [0,T] x D (Section 3).
    Stated at the start of Section 3; these regularity conditions are needed for the affine analysis and for the numerical scheme.
  • domain assumption The unit-time fish count Z follows the affine SDE (5) with Z_0 = Z_tau = 0 and has a unique nonnegative square-integrable solution (Section 3.1).
    This is the driver of the delayed source; well-posedness is cited to Proof of Proposition 1 and prior work [48].
  • domain assumption eDNA transport is purely advective at constant speed u with linear decay; dispersion, deposition, resuspension, and settling are represented only through multiplicative noise (Section 1.2.1, Eq. (3)).
    The model explicitly excludes second-order spatial terms and variable flow; these are acknowledged simplifications (Section 5).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stochastic partial differential equation model for environmental DNA dynamics in river environments." pith.science (2026). https://pith.science/paper/VFEDZZRX

@misc{pith2026260803364,
  author       = {Pith},
  title        = {Pith review of: Stochastic partial differential equation model for environmental DNA dynamics in river environments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFEDZZRX}},
  note         = {Machine review of arXiv:2608.03364}
}
read the original abstract

Environmental DNA (eDNA) has emerged as a novel tool for quantifying the seasonal abundance of aquatic species in water bodies; however, its mathematical modeling is still at a germinating stage because of its mechanistic uncertainties. We propose a first-step mathematical and computational framework for the eDNA dynamics of migratory fish based on a novel stochastic partial differential equation model with a delayed source input. The model governs spatiotemporal eDNA concentration in rivers where the source comes from a stochastic differential equation for the migration dynamics of the fish. The affine nature of the model facilitates its theoretical analysis, including the guarantee of well-posedness and the closed-form derivation of the Laplace functional despite the proportionality coefficient of the multiplicative noise term being non-Lipschitz. We also propose a discretization scheme for the model that theoretically generates nonnegative numerical solutions. We finally apply the proposed model to eDNA concentration data sampled from midstream reaches of a river system and perform sensitivity analysis.

Figures

Figures reproduced from arXiv: 2608.03364 by the authors.

Figure 1
Figure 1. Conceptual diagram for the modeling in this paper. 1.2.2 Mathematical analysis and numerical discretization The square-root multiplicative noise in (1) combined with the specific form of the SDE model [48] allows for the closed-form derivation of the Laplace functional of the eDNA concentration. In this case, the proposed SPDE model is an infinite-dimensional (measure-valued) affine process [64]; our model additiona… view at source ↗
Figure 2
Figure 2. Map of the study site [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Collected eDNA data from February 1 to November 27: Kisuki (red) and Shin-Mitoya (blue) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Comparison between observed (circles) and fitted eDNA concentrations (curves) in units of copies/ml from March 10 to July 14 in 2025: Kisuki (red) and Shin-Mitoya (blue) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 6
Figure 6. Figure 6: Computed sample path of the eDNA concentration Y (copies/ml) for the Shin-Mitoya case at selected time instances [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Computed average (Ave) (copies/ml) of the computed eDNA concentration [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Computed standard deviation (Std) (copies/ml) of the computed eDNA concentration [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Computed average (sAve, blue) and standard deviation (sStd, red) of the spatially averaged eDNA concentration Y (copies/ml): (a)  = 0.5 , (b)  =1 , (c)  = 5 , (d)  =10 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

95 extracted references · 2 linked inside Pith

  1. [1]

    L., Wilson, B

    Piczak, M. L., Wilson, B. M., Bergeron, H., Black, M., Hawkes, J. P., Hardie, D. C., ... & Trudel, M. (2025). Movement ecology of postspawning Alewife. Transactions of the American Fisheries Society, 154(5), 572-584. https://doi.org/10.1093/tafafs/vnaf032

    doi:10.1093/tafafs/vnaf032 2025

  2. [2]

    M., Wulff, M

    Buxton, J. M., Wulff, M. L., Huntsman, B. M., Palenscar, K., Mills, B., Russell, K., ... & Bui, T. (2026). Effect of passive integrated transponder tag size on survival, tag loss, and growth of Santa Ana Sucker. North American Journal of Fisheries Management, 46(1), 269-275. https://doi.org/10.1093/najfmt/vqaf109

    doi:10.1093/najfmt/vqaf109 2026

  3. [3]

    B., Raby, G

    Shorgan, M. B., Raby, G. D., Fedus, A. L., Howell, B. E., Haniford, L. S., Howitt, L. C., ... & Fisk, A. T. (2026). Effects of surgical implantation of electronic tags in fishes: a review and meta-analysis. Reviews in Fish Biology and Fisheries, 36(1), 14. https://doi.org/10.1007/s11160-025-10020-5

    doi:10.1007/s11160-025-10020-5 2026

  4. [4]

    Doi, H., Inui, R., Akamatsu, Y., Kanno, K., Yamanaka, H., Takahara, T., & Minamoto, T. (2017). Environmental DNA analysis for estimating the abundance and biomass of stream fish. Freshwater Biology, 62(1), 30 -39. https://doi.org/10.1111/fwb.12846

    doi:10.1111/fwb.12846 2017

  5. [5]

    & Dejean, T

    Pont, D., Rocle, M., Valentini, A., Civade, R., Jean, P ., Maire, A., ... & Dejean, T. (2018). Environmental DNA reveals quantitative patterns of fish biodiversity in large rivers despite its downstream transportation. Scientific Reports, 8(1), 10361. https://doi.org/10.1038/s41598-018-28424-8

    doi:10.1038/s41598-018-28424-8 2018

  6. [6]

    L., Fowler, A

    Rourke, M. L., Fowler, A. M., Hughes, J. M., Broadhurst, M. K., DiBattista, J. D., Fielder, S., ... & Furlan, E. M. (2022). Environmental DNA (eDNA) as a tool for assessing fish biomass: A review of approaches and future considerations for resource surveys. Environmental DNA, 4(1), 9-33. https://doi.org/10.1002/edn3.185

    doi:10.1002/edn3.185 2022

  7. [7]

    Kasmi, Y., Nunez-Riboni, I., Blancke, T., Möckel, B., Bernreuther, M., Stransky, C., & Hanel, R. (2025). Fish diversity assessment and semiquantitative biomass estimation through metabarcoding of environmental DNA. Ecological Indicators, 173, 113406. https://doi.org/10.1016/j.ecolind.2025.113406

    arXiv 2025

  8. [8]

    D., & Pellissier, L

    Vance, G., Kirschner, D., Willett, S. D., & Pellissier, L. (2025). Drainage reorganization and intraspecific genetic diversity of riverine fish in the Ligurian Alps and northern Apennines. Journal of Geophysical Research: Earth Surface, 130(8), e2024JF008028. https://doi.org/10.1029/2024JF008028

    doi:10.1029/2024jf008028 2025

Show all 95 references
  1. [9]

    M., Goldenberg‐Vilar, A., Morán‐Luis, M., Vieites, D

    Losa, A. M., Goldenberg‐Vilar, A., Morán‐Luis, M., Vieites, D. R., Barquín, J., & Monteoliva, A. P . (2025). Metabarcoding Versus Species‐Specific Primers to Estimate Salmo teutta Biomass and Density in Mountain Streams. Environmental DNA, 7(2), e70090. https://doi.org/10.1002...

    2025 doi

  2. [10]

    A., & Price, S

    Tomke, S. A., & Price, S. J. (2025). Drivers Behind Spatiotemporal Variation in Environmental DNA: An Assessment Using a Rare Aquatic Salamander, the Eastern Hellbender ( Ceyptobeanchus alleganiensis alleganiensis). Environmental DNA, 7(6), e70217. https://doi.org/10.1002/edn3.70217

    2025 doi

  3. [11]

    Tan, W . J. D., Chang, J. J. M., Kwan, V ., & Huang, D. (2025). Characterizing the diversity and distribution of tropical coastal blue carbon using environmental DNA. iScience, 28(11). https://doi.org/10.1016/j.isci.2025.113837

    2025

  4. [12]

    X., Zheng, R., Zhang, C., Xu, Y ., Nasreen, Z., & Mu, D

    ur Rahman, M., Yu, W. X., Zheng, R., Zhang, C., Xu, Y ., Nasreen, Z., & Mu, D. S. (2026). A source-state-space (S3) framework for quantifying environmental DNA fate: bridging mechanism to predictive monitoring. Ecological Indicators, 189, 115185. https://doi.org/10.1016/j.ecol...

    2026

  5. [13]

    N., Tiemann, J

    Curtis, A. N., Tiemann, J. S., Douglass, S. A., Davis, M. A., & Larson, E. R. (2021). High stream flows dilute environmental DNA (eDNA) concentrations and reduce detectability. Diversity and Distributions, 27(10), 1918-

    2021

  6. [14]

    D., Kelly, R

    Tillotson, M. D., Kelly, R. P., Duda, J. J., Hoy, M., Kralj, J., & Quinn, T. P. (2018). Concentrations of environmental DNA (eDNA) reflect spawning salmon abundance at fine spatial and temporal scales. Biological conservation, 220, 1-11. https://doi.org/10.1016/j.biocon.2018.01.030

    2018 doi

  7. [15]

    H., Biernaski, V ., & Val, A

    dos Anjos dos Santos, C. H., Biernaski, V ., & Val, A. L. (2025). Effect of fish biomass on environmental DNA shedding and degradation. Molecular Biology Reports, 52(1), 1017. https://doi.org/10.1007/s11033-025-10991- 5

    2025 doi

  8. [16]

    Y ., Trenkel, V

    Zanni, M. Y ., Trenkel, V. M., Albouy, C., Vaz, A. C., Paris, C. B., & Faillettaz, R. (2026). Beyond Exponential Decay: How Biphasic and Delayed Decay Dynamics Shape Marine eDNA Dispersal. Ecology and evolution, 16(1), e72987. https://doi.org/10.1002/ece3.72987

    2026 doi

  9. [17]

    Jo, T. S. (2025). Integrating temperature-dependent production of environmental DNA into its relationship with organism abundance. Journal of Thermal Biology, 129, 104120. https://doi.org/10.1016/j.jtherbio.2025.104120

    2025

  10. [18]

    D., Tank, J

    Snyder, E. D., Tank, J. L., Pruitt, A. N., Peters, B., Brandão‐Dias, P. F., Curtis, E. M., ... & Lamberti, G. A. (2025). Warming Increases Environmental DNA (eDNA) Removal Rates in Flowing Waters. Environmental DNA, 7(3), e70094. https://doi.org/10.1002/edn3.70094

    2025 doi

  11. [19]

    Jo, T., & Yamanaka, H. (2022). Meta‐analyses of environmental DNA downstream transport and deposition in relation to hydrogeography in riverine environments. Freshwater Biology, 67(8), 1333 -1343. https://doi.org/10.1111/fwb.13920

    2022 doi

  12. [20]

    Pont, D. (2024). Predicting downstream transport distance of fish eDNA in lotic environments. Molecular ecology resources, 24(4), e13934. https://doi.org/10.1111/1755-0998.13934

    2024

  13. [21]

    Coston‐Guarini, J., Hinz, S., Mirimin, L., & Guarini, J. M. (2023). A new simulation framework to evaluate the suitability of eDNA for marine and aquatic environmental impact assessments. Environmental DNA, 5(5), 1116-

    2023

  14. [22]

    Wu, Y., Zhang, Y ., Guo, F., Li, B., Du, Q., Gao, W ., ... & Li, F. (2025). Integrating Dam-Induced Effects into a Bayesian eDNA-Hydrodynamic Model to Improve Fish Monitoring in Regulated Rivers. Environmental Science & Technology, 59(40), 21670-21681. https://doi.org/10.1021/...

    2025 doi

  15. [23]

    Y ., Trenkel, V

    Zanni, M. Y ., Trenkel, V. M., & Faillettaz, R. (2025). Decaying Uncertainties: Exploring the Role of Decay Rate Variability in Marine eDNA Dispersal Using Lagrangian Transport Modeling. Environmental DNA, 7(3), e70140. https://doi.org/10.1002/edn3.70140

    2025 doi

  16. [24]

    Carraro, L., Hartikainen, H., Jokela, J., Bertuzzo, E., & Rinaldo, A. (2018). Estimating species distribution and abundance in river networks using environmental DNA. Proceedings of the National Academy of Sciences, 115(46), 11724-11729. https://doi.org/10.1073/pnas.1813843115

    2018 doi

  17. [25]

    D., & Ellner, S

    Lambert, T. D., & Ellner, S. P . (2025). SDM meets eDNA: optimal sampling of environmental DNA to estimate species–environment relationships in stream networks. Ecography, 2025(5), e07644. https://doi.org/10.1111/ecog.07644

    2025 doi

  18. [26]

    Wang, Y ., Fan, J., Lu, J., Hu, Y., & Guo, F. (2026). eDNA and AI Identification Reveal Complementary Signals in Phytoplankton Monitoring. Environmental Science & Technology, 60(17), 12894 -12905. https://doi.org/10.1021/acs.est.5c17985

    2026 doi

  19. [27]

    H., Goodman, A., Jorgensen, J

    Fullerton, A. H., Goodman, A., Jorgensen, J. C., Bond, M. H., Bowerman, T. E., Siegel, J. E., ... & Jordan, C. E. (2026). Predicted Stream Temperatures Suggest Challenges for Pacific Salmon in Coming Decades. JAWRA Journal of the American Water Resources Association, 62(3), e7...

    2026

  20. [28]

    S., & Enright, D

    Rosenfeld, J. S., & Enright, D. (2025). Developing generalized flow ecology relationships for stream salmonids: Providing a clearer empirical basis for minimum flow regulations. Transactions of the American Fisheries Society, 154(2), 162-178. https://doi.org/10.1093/tafafs/vnaf001

    2025 doi

  21. [29]

    D., Page, K

    Shane, K. D., Page, K. S., Pritt, J. J., Conroy, J. D., & Marschall, E. A. (2021). Season and discharge predict downstream emigration rates for reservoir sport fish populations. North American Journal of Fisheries Management, 41(6), 1798-1811. https://doi.org/10.1002/nafm.10700

    2021 doi

  22. [30]

    M., & Næsje, T

    Foldvik, A., Ulvan, E. M., & Næsje, T. (2024). Optimal timing of return migration in Atlantic salmon. Fish and Fisheries, 25(3), 429-440. https://doi.org/10.1111/faf.12816

    2024 doi

  23. [31]

    & Swanson, H

    Smith, R., Hitkolok, E., Dumond, A., DePasquale, S., Thibault, H., Weinstein, S., ... & Swanson, H. (2025). Environment influences migration timing of cold-adapted salmonids (Salvelinus alpinus and S. malma malma) in the Canadian Arctic. Canadian Journal of Fisheries and Aquat...

    2025 doi

  24. [32]

    E., Fadlovich, R., Fonken, D., Heinle, K

    Walsworth, T. E., Fadlovich, R., Fonken, D., Heinle, K. B., May, E., Rousseau, S., ... & Landom, K. (2024). Interactions between runoff volume, timing, and annual temperatures shape migration phenology of a threatened adfluvial sucker. Ecology of Freshwater Fish, 33(4), e12791...

    2024 doi

  25. [33]

    P ., Anderson, J

    Kuruvilla, M., Quinn, T. P ., Anderson, J. H., Scheuerell, M. D., Miller, E. M., Berger, A. G., ... & Berdahl, A. M. (2026). Social influences complement environmental cues to stimulate migrating juvenile salmon. Movement Ecology, published online. https://doi.org/10.1186/s404...

    2026 doi

  26. [34]

    R., & Gaeta, J

    Rigby, C. R., & Gaeta, J. W . (2026). Evaluating Patterns of Juvenile Chinook Salmon Out‐Migration During Pulse Flow Events in California's Sacramento River to Inform Operations of a Newly Proposed Water Infrastructure Project. River Research and Applications, published online...

    2026 doi

  27. [35]

    G., Tarkan, A

    Roberts, C. G., Tarkan, A. S., Hanley, M. E., & Britton, J. R. (2025). Angler catch data as a monitoring tool for European barbel Baebus baebus in a data limited recreational fishery. Fisheries Research, 281, 107224. https://doi.org/10.1016/j.fishres.2024.107224

    2025

  28. [36]

    Trivedi, Y ., & Mohapatra, A. (2025). Stochastic environments and migrating population dynamics. Mathematical Biosciences, 109589. https://doi.org/10.1016/j.mbs.2025.109589

    2025

  29. [37]

    Talapatra, C. (2025). Stochastic Dynamics of Dual -Prey–Predator Interactions under Harvesting Pressure: Insights from the California Current Ecosystem. Earthline Journal of Mathematical Sciences, 15(6), 989-1020. https://doi.org/10.34198/ejms.15625.9891020

    2025

  30. [38]

    D., Phillips, J

    Nooteboom, P. D., Phillips, J. S., Kehl, C., Nicol, S., & van Sebille, E. (2023). Modeling of tuna around fish aggregating devices: The importance of ocean flow and prey. Ecological Modeling, 475, 110188. https://doi.org/10.1016/j.ecolmodel.2022.110188

    2023

  31. [39]

    K., Abbas, S., & Debbouche, A

    Mishra, S. K., Abbas, S., & Debbouche, A. (2026). Ergodic stationary distribution and extinction of a stochastic eco-epidemiological model with disease in prey. Chaos, Solitons & Fractals, 203, 117661. https://doi.org/10.1016/j.chaos.2025.117661

    2026

  32. [40]

    Buyse, J., Reubens, J., Hostens, K., Degraer, S., Goossens, J., & De Backer, A. (2025). European plaice movements show evidence of high residency, site fidelity, and feeding around hard substrates within an offshore wind farm. ICES Journal of Marine Science, 82(4), fsad179. ht...

    2025 doi

  33. [41]

    P., Friston, K

    Heins, C., Millidge, B., Da Costa, L., Mann, R. P., Friston, K. J., & Couzin, I. D. (2024). Collective behavior from surprise minimization. Proceedings of the National Academy of Sciences, 121(17), e2320239121. https://doi.org/10.1073/pnas.2320239121

    2024 doi

  34. [42]

    Liu, D., & Burbano, D. (2025). Collective intermittent exploration in fish schools is mediated by visual cues. Royal Society Open Science, 12(6), 250033. https://doi.org/10.1098/rsos.250033 48

    2025 doi

  35. [43]

    S., Kamrujjaman, M., Mohebujjaman, M., & Khan, T

    Tisha, M. S., Kamrujjaman, M., Mohebujjaman, M., & Khan, T. (2025). Decoupled algorithms and analyses for an advection -reaction-diffusion model with stocking and harvesting. Computers & Mathematics with Applications, 189, 24-47. https://doi.org/10.1016/j.camwa.2025.03.024

    2025 doi

  36. [44]

    A., Buchanan, R

    Min, M. A., Buchanan, R. A., & Scheuerell, M. D. (2025). Modeling Climate and Hydropower Influences on the Movement Decisions of an Anadromous Species. Global Change Biology, 31(10), e70533. https://doi.org/10.1111/gcb.70533

    2025 doi

  37. [45]

    Matsuzaki, S. I. S., Fukaya, K., Mabuchi, K., Kikko, T., & Takamura, N. (2025). Changes of cyprinid fishery resources in Lake Biwa over 57 years: association with multiple stressors and restoration measures. Oecologia, 207(7), 121. https://doi.org/10.1007/s00442-025-05762-9

    2025 doi

  38. [46]

    Yoshioka, H. (2025). CIR bridge for modeling of fish migration on subhourly scale. Chaos, Solitons & Fractals, 199, 116874. https://doi.org/10.1016/j.chaos.2025.116874

    2025

  39. [47]

    Yoshioka, H. (2026a). A minimization principle behind the diffusion bridge of diurnal fish migration. Mathematical Biosciences, 109684. https://doi.org/10.1016/j.mbs.2026.109684

    2026

  40. [48]

    Yoshioka, H. (2026b). Multiple timescales in collective motion daily and intraday upstream fish migration focusing on Feller condition. Physica A: Statistical Mechanics and its Applications, 131695. https://doi.org/10.1016/j.physa.2026.131695]

    2026

  41. [49]

    Yoshioka, H., & Louriki, M. (2026). Diffusion bridge with randomized initial and terminal times and its application to fish migration. Preprint. https://arxiv.org/abs/2607.04253

    2026 arXiv

  42. [50]

    Hundermark, E., Stoudt, S., & Takahashi, M. (2026). Fine -scale aquatic eDNA sampling reveals significant within-and across -site variation during the expected breeding season of Cryptobranchus alleganiensis alleganiensis. Hydrobiologia, published online. https://doi.org/10.10...

    2026 doi

  43. [51]

    Thalinger, B., Wolf, E., Traugott, M., & Wanzenböck, J. (2019). Monitoring spawning migrations of potamodromous fish species via eDNA. Scientific reports, 9(1), 15388. https://doi.org/10.1038/s41598-019- 51398-0

    2019 doi

  44. [52]

    S., Takeuchi, A., & Itakura, H

    Jo, T. S., Takeuchi, A., & Itakura, H. (2025). Allometric Scaling in Environmental DNA Concentration of Japanese Eel Anguilla japonica Confirmed Under Laboratory and Natural Conditions. Integrative Conservation, 4(4), 675-685. https://doi.org/10.1002/inc3.70050

    2025 doi

  45. [53]

    S., & Doi, H

    Jo, T. S., & Doi, H. (2026). Does Allometric Scaling Improve Estimates of Population Abundance Based on Environmental DNA?. Molecular Ecology, 35(6), e70303. https://doi.org/10.1111/mec.70303

    2026 doi

  46. [54]

    C., & Sanz -Solé, M

    Dalang, R. C., & Sanz -Solé, M. (2026). Stochastic partial differential equations, space -time white noise and random fields. Springer, Cham

    2026

  47. [55]

    Mena, H., & Pfurtscheller, L. (2019). An efficient SPDE approach for El Niño. Applied Mathematics and Computation, 352, 146-156. https://doi.org/10.1016/j.amc.2019.01.071

    2019 doi

  48. [56]

    A., Babaei, A., & Moghaddam, B

    Moniri, Z., Zaky, M. A., Babaei, A., & Moghaddam, B. P . (2026). Stochastic approaches to uncertainty quantification in fractional drift -flux models with concentration -dependent sources for pollutant dispersion. Stochastic Environmental Research and Risk Assessment, 40(2), 4...

    2026 doi

  49. [57]

    W., Ahmed, N., Saeed, J., Baber, M

    Yasin, M. W., Ahmed, N., Saeed, J., Baber, M. Z., Ali, S. M., Akgül, A., ... & Ali, M. (2024). Numerical study of diffusive fish farm system under time noise. Scientific Reports, 14(1), 14711. https://doi.org/10.1038/s41598- 024-62304-8

    2024 doi

  50. [58]

    Cuchiero, C., Di Persio, L., Guida, F., & Svaluto‐Ferro, S. (2025). Measure‐valued processes for energy markets. Mathematical Finance, 35(2), 520-566. https://doi.org/10.1111/mafi.12452

    2025 doi

  51. [59]

    Stein, A., & Barth, A. (2025). Stochastic transport with Lévy noise fully discrete numerical approximation. Mathematics and Computers in Simulation, 227, 347-370. https://doi.org/10.1016/j.matcom.2024.07.036

    2025 doi

  52. [60]

    El Saadi, N., & Arino, O. (2006). A stochastic modeling of phytoplankton aggregation. Revue Africaine de Recherche en Informatique et Mathématiques Appliquées, 5, 1856. https://doi.org/10.46298/arima.1856

    2006 doi

  53. [61]

    El Saadi, N., & Bah, A. (2015). Numerical simulations of a nonlinear stochastic partial differential equation modeling phytoplankton aggregation. Journal of Biological Systems, 23(04), 1550032. https://doi.org/10.1142/S0218339015500321

    2015 doi

  54. [62]

    Kanamori, Y ., Yano, T., Okamura, H., & Yagi, Y . (2024). Spatio‐temporal model and machine learning method reveal patterns and processes of migration under climate change. Journal of Biogeography, 51(4), 522 -532. https://doi.org/10.1111/jbi.14595

    2024 doi

  55. [63]

    B., & Berdahl, A

    Okasaki, C., Hooten, M. B., & Berdahl, A. M. (2022). Source reconstruction for spatiotemporal physical statistical models. Spatial Statistics, 52, 100707. https://doi.org/10.1016/j.spasta.2022.100707

    2022

  56. [64]

    Cuchiero, C., Di Persio, L., Guida, F., & Svaluto -Ferro, S. (2024b). Measure -valued affine and polynomial diffusions. Stochastic Processes and their Applications, 175, 104392. https://doi.org/10.1016/j.spa.2024.104392

    2024

  57. [65]

    Flore, F., & Nappo, G. (2019). A Feynman-Kac type formula for a fixed delay CIR model. Stochastic Analysis and Applications, 37(4), 550-573. https://doi.org/10.1080/07362994.2019.1592691

    2019

  58. [66]

    Guinea Juliá, Á., & Caro‐Carretero, R. (2025). Option pricing in a stochastic delay volatility model. Mathematical methods in the Applied Sciences, 48(2), 1927-1951. https://doi.org/10.1002/mma.10417

    2025 doi

  59. [67]

    Rudnicki, R., & Wieczorek, R. (2024). Individual-Based Models and Their Limits. Springer, Cham. 49

    2024

  60. [68]

    Perkins, E. (2002). Dawson -Watanabe superprocesses and measure -valued diffusions. École D’étÉ de Probabilités de Saint -Flour XXIX -1999, Springer, Berlin, pp. 125 -329. https://personal.math.ubc.ca/~perkins/dawsonwatanabesuperprocesses.pdf (last accessed on July 15, 2026)

    2002

  61. [69]

    Li, Z. (2023). Measure-Valued Branching Markov Processes. Springer, Berlin, Heidelberg

    2023

  62. [70]

    Abi Jaber, E. (2025). Simulation of square-root processes made simple: applications to the Heston model. Preprint. https://doi.org/10.48550/arXiv.2412.11264

    2025 doi arXiv

  63. [71]

    Abi Jaber, E., Larsson, M., & Pulido, S. (2019). Affine Volterra processes. The Annals of Applied Probability, 29(5), 3155-3200. https://doi.org/10.1214/19-AAP1477

    2019 doi

  64. [72]

    Alfonsi, A. (2025). Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation. Stochastic Processes and their Applications, 181, 104535. https://doi.org/10.1016/j.spa.2024.104535

    2025

  65. [73]

    C., Wang, L., & Wong, H

    Chiu, M. C., Wang, L., & Wong, H. Y. (2026). Long -range dependent mortality modeling with cointegration. Scandinavian Actuarial Journal, 2026(1), 60-93. https://doi.org/10.1080/03461238.2025.2503290

    2026

  66. [74]

    Kang, J., Jin, Z., Qian, L., & Zhang, N. (2025). Fairness and risk sharing in integrated LRD -tontine schemes under Volterra mortality risk. ASTIN Bulletin: The Journal of the IAA, 55(3), 644 -667. https://doi.org/10.1017/asb.2025.10057

    2025

  67. [75]

    Moro, E., & Schurz, H. (2007). Boundary preserving semianalytic numerical algorithms for stochastic differential equations. SIAM Journal on Scientific Computing, 29(4), 1525-1549. https://doi.org/10.1137/05063725X

    2007 doi

  68. [76]

    Kelly, C., & Lord, G. J. (2023). An adaptive splitting method for the Cox -Ingersoll-Ross process. Applied Numerical Mathematics, 186, 252-273. https://doi.org/10.1016/j.apnum.2023.01.014

    2023 doi

  69. [77]

    Grün, G., Mecke, K., & Rauscher, M. (2006). Thin-film flow influenced by thermal noise. Journal of Statistical Physics, 122(6), 1261-1291. https://doi.org/10.1007/s10955-006-9028-8

    2006 doi

  70. [78]

    Feppon, F., & Lermusiaux, P . F. (2018). Dynamically orthogonal numerical schemes for efficient stochastic advection and Lagrangian transport. Siam Review, 60(3), 595-625. https://doi.org/10.1137/16M1109394

    2018 doi

  71. [79]

    Tsukamoto, K., & Uchida, K. (1992). Migration Mechanism of the ayu, in Oceanic and Anthropogenic Controls of Life in the Pacific Ocean, Ilyichev, V .I. and Anikiev, V .V ., eds., Springer, Dordrecht, 145 -172. https://doi.org/10.1007/978-94-011-2773-8_12

    1992 doi

  72. [80]

    Yamanaka, H., & Minamoto, T. (2016). The use of environmental DNA of fishes as an efficient method of determining habitat connectivity. Ecological Indicators, 62, 147 -153. https://doi.org/10.1016/j.ecolind.2015.11.022

    2016 doi

  73. [81]

    Environmental DNA Sampling and Experiment Manual , V ersion 3.0

    The eDNA Society (2024). Environmental DNA Sampling and Experiment Manual , V ersion 3.0. https://ednasociety.org/wp- content/uploads/2024/08/%E7%92%B0%E5%A2%83DNA%E8%AA%BF%E6%9F%BB%E3%83%BB%E5 %AE%9F%E9%A8%93%E3%83%9E%E3%83%8B%E3%83%A5%E3%82%A2%E3%83%AB_ver3_0.pdf (Last acces...

    2024

  74. [82]

    Yoshioka, H. (2016). Mathematical analysis and validation of an exactly solvable model for upstream migration of fish schools in one -dimensional rivers. Mathematical biosciences, 281, 139 -148. https://doi.org/10.1016/j.mbs.2016.09.014

    2016 doi

  75. [83]

    Metsäniemi, T., Orell, P ., Foldvik, A., Kuusela, J., Kurkilahti, M., & Erkinaro, J. (2025). Mesohabitat Evaluation Reveals Variable Abundances and Habitat Choice in Juvenile Atlantic Salmon Across Diverse Habitats in the Main Stem of a Large Sub‐Arctic Riv er. Ecology of Fres...

    2025 doi

  76. [84]

    A., Boehm, A

    Searcy, C. A., Boehm, A. E., Weinstock, C., Preston, C. M., …, Yamashita, K. M. (2022). High‐frequency and long‐term observations of eDNA from imperiled salmonids in a coastal stream: Temporal dynamics, relationships with environmental factors, and comparisons with conventiona...

    2022

  77. [85]

    Alfonsi, A. (2015). Affine diffusions and related processes: simulation, theory and applications. Springer, Cham

    2015

  78. [86]

    Yoshioka, H., Yoshioka, Y., Tsujimura, M., & Hashiguchi, A. (2026). Mathematical model for sustainable fisheries resource management accounting for size spectrum. Preprint. https://doi.org/10.48550/arXiv.2602.07511

    2026 doi

  79. [87]

    Klenke, A. (2020). Probability Theory. Springer, Cham

    2020

  80. [88]

    Karatzas, I., & Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York

    1991

  81. [89]

    Martin, S. (2007). First order quasilinear equations with boundary conditions in the L∞ framework. Journal of Differential Equations, 236(2), 375-406. https://doi.org/10.1016/j.jde.2007.02.007

    2007 doi

  82. [90]

    J., & Leveque, R

    LeVeque, R. J., & Leveque, R. J. (1992). Numerical methods for conservation laws. Birkhäuser, Basel

    1992

  83. [91]

    Mao, X. (2007). Stochastic differential equations and applications. Elsevier

    2007

  84. [92]

    Bansaye, V., & Méléard, S. (2015). Stochastic models for structured populations. Springer, Berlin

    2015

  85. [849]

    https://doi.org/10.1002/edn3.293

    doi

  86. [1130]

    https://doi.org/10.1002/edn3.429 47

    doi

  87. [1931]

    https://doi.org/10.1111/ddi.13196

    doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.