Pith. sign in

REVIEW 5 major objections 5 minor 95 references

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

T0 review · 5 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read This paper builds a stochastic partial differential equation for river eDNA driven by a delayed, fluctuating fish-migration source, proves it well-posed, and derives its Laplace functional in closed form.

desk verdict Genuinely novel tractable SPDE for eDNA with a closed-form Laplace functional, but the empirical application is a fit, not a validation. read the letter →

arxiv 2608.03364 v1 pith:VFEDZZRX submitted 2026-08-04 q-bio.QM math.PR

Hidekazu Yoshioka This is my paper · ORCID
classification q-bio.QMmath.PR MSC 60H1560J2592D40
keywords environmentalDNAupstreamfishmigrationstochasticpartialdifferentialequationswithdelayaffineprocessesCIRbridgenonnegativity-preservingnumericalschemePlecoglossusaltivelisHiiRivercasestudy
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 proposes and analyzes a stochastic partial differential equation that ties the concentration of environmental DNA (eDNA) in a river to the upstream migration of fish. The eDNA is shed by migrants moving upstream, carried downstream by the current, and decays in place; the model writes those three processes as an advection equation with a delayed source and square-root multiplicative noise. Because the equation is affine, the paper proves the model is well-posed and derives an explicit formula for its Laplace functional, which determines all exponential moments. A matching numerical scheme keeps solutions nonnegative by construction. The framework is demonstrated on weekly eDNA samples of Plecoglossus altivelis in the Hii River system, reproducing the observed concentration peaks. If the model holds up, routine eDNA sampling could be converted into quantitative statements about fish abundance and migration timing.

What carries the argument

The machinery is the affine structure shared by the fish-count SDE and the eDNA SPDE: drift linear in the state and square-root multiplicative noise. With upwind spatial differences, the semidiscrete model becomes a cascade of affine SDEs solvable from the upstream-most grid point downstream, so the Laplace functional is computable through a backward system of ODEs for the coefficients $\alpha_t,\beta_t,\gamma_t,\omega_t$ (Proposition 2, continuum version Proposition 3). The delayed source term $Z_{t-x/v}$ is the coupling: it carries fish-count information into the eDNA equation with the travel-time lag $x/v$. The affine cascade also turns existence into a tightness argument for measure-valu

What would settle it

Install a continuous fish counter at the downstream boundary and collect high-frequency eDNA samples at a fixed midstream point during one migration season. The model predicts the mean eDNA signal at $(t,x)$ is a specified convolution of the expected fish count with the delay $x/v$ and exponential decay $e^{-r(t-s)}$; if the observed eDNA peak time or shape deviates from that prediction beyond sampling error, or if the $g$ and $r$ estimated at one site cannot predict the other site's concentrations, the proportionality-and-delay mechanism is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the SPDE (3), $$dY_t(x) = \left(-u \frac{\partial Y}{\partial x} + g Z_{t-x/v} - r Y_t\right)dt + \$\sigma$ \sqrt{Y_t}\,dW(t,x),$$ with $Z_t$ the unit-time fish count following the affine diffusion bridge (5), is a well-posed stochastic model for the eDNA concentration $Y_t(x)$ in a 1-D river. Proposition 7 establishes existence and uniqueness of weak solutions in law; Proposition 3 gives the Laplace functional (20) in closed form. The paper also shows that the upwind semidiscrete version converges to the SPDE, and that a fully discrete version using an inverse-gamma update stays nonnegative with probability 1 (Proposition 8). The application to the Hii River fits the

Load-bearing premise

The load-bearing premise is that the fish-count process $Z_t$ calibrated on a different river system transfers to the Hii River and that eDNA shedding is strictly proportional to that delayed fish count with one constant $g$; if either fails, the source term is misspecified and the apparent fit comes from freely adjusting $g$, the decay $r$, and the migration start date.

Editorial extensions

If this is right

  • The closed-form Laplace functional makes the mean, variance, and higher exponential moments of the eDNA concentration available without Monte Carlo, so model calibration can be done by direct moment matching.
  • The discrete scheme is nonnegative for any time step, so simulations near zero concentration—a regime where standard Euler-type schemes fail—remain well behaved.
  • The delayed source implies downstream eDNA peaks lag fish-count peaks; matching that lag can constrain the fish ground speed $v$ and the ratio of shedding to decay.
  • The computations indicate that noise intensity $\sigma$ is best estimated from high-frequency eDNA time series at the downstream end, not from spatially averaged concentration profiles.
  • The cascade/upwind structure means eDNA is determined from upstream to downstream, so boundary data at the upstream end are not needed; only the free-outflow condition at $x=0$ enters.

Reading between the lines

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

  • The same affine-delay machinery is a natural candidate for river-network (graph) extensions, which the paper explicitly leaves open; the main mathematical obstacle is well-posedness at confluence nodes.
  • The single proportionality constant $g$ between fish count and eDNA shedding is the most fragile input; independent measurement of the per-fish shedding rate would break the confounding between $g$ and the fitted decay $r$ in sparse weekly data.
  • A direct test of the delay mechanism would use sub-daily eDNA sampling at a midstream site together with fish counts at the river mouth; the model predicts the eDNA peak should trail the fish-count peak by roughly the travel time $x/v$ filtered through exponential decay.
  • The paper's sensitivity experiments suggest the noise intensity $\sigma$ is identifiable from the variability of downstream-end sample paths; this makes high-frequency eDNA monitoring, rather than more spatial sites, the information-rich experiment for parameter estimation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript proposes a one-dimensional SPDE for environmental DNA (eDNA) concentration in a river, with downstream advection, linear decay, multiplicative square-root space-time white noise, and a delayed source term driven by a CIR-type SDE for upstream fish migration. The authors derive a closed-form Laplace functional for the semidiscrete and continuum models, prove weak well-posedness via a martingale-problem/tightness argument, and propose a fully discrete operator-splitting scheme that preserves nonnegativity. The paper concludes with an application to Ayu (Plecoglossus altivelis) eDNA data from two sites in the Hii River system, fitting g, r, and the initial migration time. The theoretical results are plausible and potentially useful, but the proofs are compressed and several key steps are asserted rather than fully demonstrated. The application is presented as support for the model, but it relies on fish-migration parameters borrowed from a different river and on a proportionality assumption that is not tested; the fitted agreement is therefore not yet convincing evidence for the source-term specification.

Significance. If the mathematical claims hold, the paper would make a useful contribution: it introduces a tractable affine SPDE with a delayed stochastic source, obtains a closed-form Laplace functional despite the non-Lipschitz diffusion coefficient, gives a weak well-posedness result for a measure-valued solution, and provides a nonnegativity-preserving numerical scheme. The availability of reproducible code and the explicit closed-form fitting formula (Eq. 42) are strengths. However, the significance of the empirical application is considerably weaker than the theoretical contribution because the source process Z and its parameters are taken from earlier studies on a different river system, and the agreement in Figure 4 is obtained by fitting g, r, and the initial time to the same data. The theoretical results are independent of the specific Z parameter values, so the mathematical contribution can stand even if the application is reframed as an illustration rather than validation.

major comments (5)
  1. [§4.2 / Eq. (42) / Fig. 4] The case-study agreement does not validate the source process. The fitted mean E[Y_t(0)] is proportional to g (Eq. 42), and g is a free parameter; r and the initial time further shift and scale the curve. With only three free parameters fitted to a single weekly time series at each site, Figure 4 only shows that a scaled and time-shifted version of the chosen Z-bridge shape can be matched. The manuscript should either add out-of-sample validation (e.g., hold-out dates or sites), estimate Z parameters from Hii River data, or explicitly present Section 4 as a demonstration rather than empirical support.
  2. [§4.2, source-term transferability] The manuscript states 'There are no data about unit-time fish counts of P. altivelis in the Hii River system' and takes the parameters p_i, q_i, and b from [48], [49], which were estimated on a different river system. The shedding term g Z_{t-x/v} also assumes that eDNA shedding is proportional to unit-time fish count, an assumption with no site-specific support. If the source process is misspecified, the fitted g and r are uninterpretable and the apparent agreement could be an artifact of fitting. At minimum, a sensitivity analysis over the borrowed parameters (or a calibration of Z to the eDNA data themselves) is needed before the application can support the model.
  3. [Proof of Proposition 3 / Eq. (73)] The derivation of the Laplace functional applies Itô's formula to an expression containing ∫_0^L γ_t(x) μ_t(dx), where γ_t is only shown to be an entropy solution of a first-order nonlinear PDE (Proposition 4). The required space-time regularity of γ, and the justification for interchanging stochastic integration with the spatial integral, are not established. As written, Eq. (73) and the matching-coefficient argument are formal. Please either provide a rigorous justification via smooth approximations of γ or state explicitly that the formula is an ansatz verified through the martingale problem.
  4. [Proofs of Propositions 4 and 5-7] The convergence and tightness proofs are not self-contained. In Proposition 4, the total-variation bound (97) is obtained through a long chain of inequalities involving sgn(γ_i) whose validity is difficult to check, and the weak form (80) contains a boundary term at x=L that is not explained. In Proposition 5-6, the bound (112) uses an estimate on sup_t E[Σ h Y_t^i] before that estimate is proven (later in Step 5), and the Burkholder-type inequalities in (134) are asserted without sufficient detail. More importantly, Proposition 7 claims 'existence of density at the boundary x=0' from the L2 estimate (148), which is not a density result; a measure-valued solution need not admit a density. Please either prove the density statement or remove it, since the fitting formula (42) relies on evaluating Y_t(0).
  5. [§3.5 / Prop. 8] The numerical section proves nonnegativity of the fully discrete scheme, but it does not prove convergence of the split scheme to the semidiscrete solution as the time step tends to zero, nor does it quantify the operator-splitting error. Since the paper claims an 'unconditionally stable' discretization, convergence in a suitable sense should be stated or at least discussed. This is not fatal to the main theory, but it is a load-bearing point for the claim that the computed paths approximate the SPDE.
minor comments (5)
  1. [Throughout] There are numerous typos and OCR-like errors, e.g., 'aee' for 'are', 'peobability' for 'probability', 'expeessed' for 'expressed'. These should be corrected.
  2. [§4.1 / Fig. 3] The text says samples were collected weekly from April to October 2025, while Figure 3 is described as data from February 1 to November 27. Please clarify the sampling period and the meaning of zeros before April.
  3. [Table 2] The table lists p0 = 10^6 and q0 = 4×10^6; the text says p0 can be absorbed into G and is set to 10^6. Since g is fitted, the absolute values of p0 and q0 are not identifiable from the eDNA data. A sentence explaining the identifiability limitation would help.
  4. [Eq. (39)] The inverse-gamma distribution notation 'IG(λ, μ)' is used without a definition of the parameterization; please state the density or reference the convention in [70].
  5. [§5 / Conclusions] The conclusion says the affine nature of the model guarantees nonnegativity of its solution, but pathwise nonnegativity is only proven for the semidiscrete and fully discrete systems; the SPDE solution is measure-valued and nonnegativity is part of the definition. Please rephrase to avoid overstating.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the mathematical derivation is self-contained and the application is explicitly a fit, not a prediction.

full rationale

The core theoretical claims (Propositions 2, 3, 4, 7) are genuine derivations from the stated SPDE (3) and the assumed CIR-bridge source process (5). Proposition 3 obtains the closed-form Laplace functional via an Itô-calculus verification: the ansatz (20) is justified by solving the associated ODE/PDE system (21)-(26), and Proposition 4 proves the semidiscrete Laplace functionals converge to the continuous one, so the formula is not equivalent to an input by construction. Proposition 7's existence and uniqueness in law is proved with an Aldous-criterion tightness argument and the Laplace-functional convergence, relying on standard external theorems (Perkins, Li), not on the author's prior results. Proposition 8's nonnegativity follows from the externally established scheme of Abi Jaber. The application section is the only place where fitting occurs: g, r, and the initial time are least-squares fitted to the same eDNA concentration data, and the Z parameters are taken from the author's prior studies on a different river system. The paper explicitly states there are no unit-time fish count data for the Hii River and calls Figure 4 a comparison with 'fitted eDNA concentrations,' not a prediction. The reasonable agreement therefore does not independently validate the source-process specification, but this is a correctness/validation limitation, not a circular reduction of the derivation. No load-bearing claim in the paper reduces by definition to its own inputs.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the model is a mathematical construct over existing quantities. The multiplicative noise and affine structure are modeling choices, not entities. The main external inputs are the fitted Z-process parameters from prior self-cited work and the fitted eDNA parameters g, r, t0.

free parameters (6)
  • g (shedding rate) = 1453.6 copies/ml/day (Kisuki), 6546.5 copies/ml/day (Shin-Mitoya)
    Least-squares fitted to observed eDNA concentrations in Section 4.2.
  • r (eDNA decay rate) = 5.70 /day (Kisuki), 10.3 /day (Shin-Mitoya)
    Least-squares fitted with g.
  • initial migration time t0 = 2025/3/27 (Kisuki), 2025/3/20 (Shin-Mitoya)
    Optimized because the start of migration is uncertain (Section 4.2).
  • sigma (noise intensity) = 0.5 (default), varied 0.1 to 20
    Chosen by hand; Section 4.2 states the choice will be discussed later, and Section 4.3 constrains it to 0.1-1 based on a literature qualitative comparison.
  • v (fish ground speed) = 1 km/day
    Chosen as intermediate value of literature range (0.3-0.5 and 2-3 km/day), Section 4.2.
  • Z-process parameters p0,p1,p2,q0,q1,q2,b = p0=1e6, p1=10, p2=10, q0=4e6, q1=20, q2=20, b=61.9
    Borrowed from prior studies (Yoshioka [48], Yoshioka and Louriki [49]) fitted to fish count data from a different river; used as inputs here.
assumptions (5)
  • domain assumption Coefficients g, r, sigma are positive, bounded, and continuous
    Stated before Section 3.1: required for well-posedness in the affine framework.
  • domain assumption Space-time white noise W with covariance (2) drives the model
    Modeling choice for unresolved fluctuations; justifies the square-root multiplicative noise term.
  • ad hoc to paper Z follows CIR-bridge SDE (5) with Z_0=Z_tau=0, parameters from [48], [49]
    The source of eDNA is driven by this process; it is not derived in this paper but imported from the authors' prior migration models.
  • domain assumption Nonnegativity is required; noise must be multiplicative and the Abi Jaber [70] inverse-gamma scheme is used
    Physical requirement on eDNA concentration; motivates the SDE structure.
  • standard math The finite-volume upwind discretization converges to the SPDE weak solution
    Used for Propositions 4-7; relies on monotone scheme convergence (LeVeque) and tightness (Perkins).

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]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

95 extracted references · 62 canonical work pages

  1. [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]

    work page arXiv 2026

  2. [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

    work page Pith review arXiv 2026

  3. [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

    work page doi:10.1093/tafafs/vnaf032 2025

  4. [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

    work page doi:10.1093/najfmt/vqaf109 2026

  5. [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

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

  6. [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

  7. [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

  8. [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

Show all 95 references
  1. [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....

    2025

  2. [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/1...

    2025 doi

  3. [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

  4. [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

  5. [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

  6. [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

  7. [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

  8. [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

  9. [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

  10. [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

  11. [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

  12. [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

  13. [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

  14. [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

  15. [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

  16. [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

  17. [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

  18. [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

  19. [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

  20. [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

  21. [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

  22. [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

  23. [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

  24. [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

  25. [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

  26. [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

  27. [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

  28. [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

  29. [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

  30. [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

  31. [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

  32. [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

  33. [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

  34. [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

  35. [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

  36. [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 47

    2025 doi

  37. [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

  38. [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

  39. [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

  40. [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

  41. [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

  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. 48

    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 46

    doi

  87. [1931]

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

    doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.