{"id":"9430ce7c-1a0f-434d-b2d2-f9a003eee415","arxiv_id":"2608.03364","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A stochastic partial differential equation with a delayed source models riverine eDNA from migrating fish, with a closed-form Laplace functional and a nonnegativity-preserving numerical scheme.","lead":"This paper builds a mathematical model of how environmental DNA (eDNA) from migrating fish moves along a river. The model combines random fish movement with a stochastic transport equation, giving a framework for interpreting eDNA samples and for planning monitoring.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Weak-solution definition (9) requires a boundary trace Y_t(0) not determined by the measure-valued state; the tightness proof does not show this trace converges, so Prop. 7's well-posedness may be under-specified.","rationale":"The reader's weakest assumption was the portability of the source-process parameters, which threatens the case study but not the mathematical central claim. I find a more central concern in the well-posedness theorem itself: the martingale problem (9) uses a boundary trace Y_s(0) that is not a functional of the measure-valued state μ_s. The existence proof via tightness of the semidiscrete models controls the empirical measures and their integrals against smooth test functions, but the boundary value Y^1_N(t) contributes to those integrals only at order h, so the Aldous criterion does not imply convergence of this trace. The paper's own admission that full density is open (Section 5) is in tension with a definition that presupposes a boundary trace. The explicit characteristic representation of the solution, conditional on Z, gives a natural candidate with a density and a well-defined boundary value; if that candidate satisfies (9), the gap is likely repairable and the theorem stands. The recommended concrete test settles this directly. Because the reader's CONDITIONAL verdict is unchanged—the mathematical result needs this verification, and the application has the independent parameter-portability problem—I set verdict_should_be UNCHANGED and agreement_with_reader partial.","tokens_in":50217,"tokens_out":25373,"duration_ms":248661,"concrete_test":"Construct the explicit characteristic solution: for each x∈D, Y_t(x) solves dY_t(x)=[-r_t(x)Y_t(x)+g_t(x)Z_{t-x/v}]dt+σ_t(x)√(Y_t(x))dB^x_t with B^x independent across x, and Y_t(L)=0. Check (i) that μ_t(dx)=Y_t(x)dx satisfies the martingale problem (9) with Y_t(0) taken as the x=0 characteristic solution, and (ii) numerically verify that the semidiscrete boundary value Y^1_N(t) converges in law to this Y_t(0) as N→∞ at fixed t. If both hold, the boundary-trace gap is bridgeable and Prop. 7 stands; if not, the weak-solution definition needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weak solution of (3) is defined by requiring M_t(φ) in (9) to be a zero-mean martingale for every φ∈C^∞(D). That definition contains u Y_s(0) φ(0), a boundary trace. For a measure-valued process μ_s, Y_s(0) is not determined by μ_s. The semidiscrete approximations have Y^1_N(t) as a separate component, but the tightness estimates in Props. 5–6 control only the empirical measures hΣ Y^i_N δ_{x_i}; the O(1) boundary term enters ⟨μ^N,φ⟩ only at order h and is not controlled by the Aldous criterion. The proof of Prop. 7 cites (148) for 'existence of density at the boundary', but (148) provides only an L^1-in-time bound on the boundary value, which does not imply convergence to a bounded density. Section 5 lists full density as an open issue, yet the stated definition already presupposes a boundary trace. If a tightness limit μ lacks a well-defined trace, the martingale problem is not closed, and uniqueness in law of μ alone does not imply uniqueness of solutions to (9).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":50632,"tokens_out":10214,"duration_ms":103132,"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":[{"comment":"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.","section":"§3.4.2, Proposition 7; §3.2.2, Eq. (9); Appendix A1, Eq. (148)"},{"comment":"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.","section":"§4.2, Table 2; §4.3, Figures 4 and 9-10"},{"comment":"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.","section":"§4.2, Eq. (42); Appendix A3, Figures A1-A2"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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).","section":"Proposition 3"},{"comment":"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.","section":"§2.2, Eq. (4)"},{"comment":"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.","section":"§4.1 and Table 2"},{"comment":"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.","section":"§4.1"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on the author's own prior work for the Z-process parameters; this is legitimate, but it makes the external validity of the application hard to assess. The data are only 'available upon reasonable request', which weakens reproducibility. The main mathematical gap, the boundary trace in Proposition 7, appears fixable either by adding a convergence proof for the boundary flux or by reformulating the weak-solution definition; I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine first SPDE treatment of eDNA dynamics with a delayed stochastic source, and the affine structure buys real analytic traction. The paper deserves a serious referee, but the well-posedness proof has a gap at the boundary, and the case study is thinner than the theory.\n\nWhat's new: the SPDE (3) with square-root multiplicative space-time white noise and a source delayed by x/v is new in the eDNA literature. The closed-form Laplace functional in Prop. 3 is the real contribution: it gives moments without simulation, and the semidiscrete scheme with unconditionally nonnegative solutions is useful. The author is honest about what is a first step. The proofs in the appendix are substantial, and the code is posted.\n\nSoft spots, in order of importance. First, the weak solution definition itself asks for a boundary trace Y_s(0) in (9), but for a measure-valued mu_s that quantity is not determined by mu_s. Prop. 7's proof cites (148) for 'existence of density at the boundary,' but (148) is an L^1-in-time bound on the boundary value, not a pointwise density. Without a trace, the martingale problem is not closed and the uniqueness-in-law statement is under-specified. This is a real gap, not a nitpick; Section 5 lists full density as open, which is in tension with the definition using Y_s(0). Second, the application imports the whole Z-process from a different river via parameters from Yoshioka 2025/2026, and the paper admits there are no Hii River fish count data. The fit is three parameters per site, in-sample, with hand-set noise and a post-hoc site exclusion. That makes the case study illustrative rather than predictive.\n\nThe reader's circularity concern does not land: the Laplace functional is derived from the model, not from the data. Self-citation here is reliance on prior work, not circularity.\n\nWho this is for: people building mechanistic eDNA transport models, and probabilists interested in affine measure-valued processes with delay. A serious editor should send this to peer review, not desk reject. The boundary trace issue needs to be fixed or re-scoped before publication; the rest is solid enough to build on.","headline":"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.","tokens_in":51013,"tokens_out":1666,"would_cite":false,"duration_ms":16978,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60J68","65C30","92D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A stochastic PDE with a delayed source links migrating fish to river eDNA, with proof of well-posedness and a closed-form Laplace functional.","keywords":["environmental DNA","stochastic partial differential equation","affine process","fish migration","delayed source","Laplace functional","measure-valued process","nonnegativity-preserving scheme"],"falsifier":"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.","tokens_in":50022,"feed_emoji":"🧬","tokens_out":5733,"duration_ms":52813,"temperature":0.7,"pith_summary":"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.","feed_headline":"SPDE model ties fish migration to river eDNA forecasts","feed_subtitle":"Closed-form law for eDNA dynamics computes moments without simulation and fits weekly field samples.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the affine SDE model for unit-time fish counts whose coefficients are used in Eq. (5) and Section 4.2.","marker":"[48]"},{"why":"Provides the migration-duration estimate and parameter values imported into the case study, including $\\tau=127$ days.","marker":"[49]"},{"why":"Gives the inverse-gamma simulation scheme that makes the time discretization unconditionally nonnegative.","marker":"[70]"},{"why":"Defines space-time white noise and the SPDE framework in which Eq. (3) is formulated.","marker":"[54]"},{"why":"Supplies the tightness theorem used to prove existence of a weak solution to the SPDE.","marker":"[68]"},{"why":"Provides the Laplace-functional uniqueness theorem used to conclude uniqueness in law.","marker":"[69]"},{"why":"Establishes the measure-valued affine diffusion framework in which the model is interpreted.","marker":"[64]"},{"why":"Motivates the semidiscrete multiplicative-noise discretization used in Eq. (4).","marker":"[61]"}],"fun_headline_variants":["Closed-form SPDE law for river eDNA without simulation","River eDNA dynamics captured by affine SPDE with delay","SPDE model yields exact eDNA moments in rivers","Well-posed SPDE for eDNA from migrating fish"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Closed-form SPDE law for river eDNA without simulation","River eDNA dynamics captured by affine SPDE with delay","SPDE model yields exact eDNA moments in rivers","Well-posed SPDE for eDNA from migrating fish"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1462,"prompt_tokens":907,"completion_tokens":555,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":501}},"tokens_in":523,"tokens_out":555,"duration_ms":5535,"temperature":1.0,"reasoning_tokens":501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:18:08.826397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the measure-valued affine diffusion framework in which the model is interpreted."}],"review_version":2}