{"id":"5a16d466-02e8-4704-8b22-c8869e277681","arxiv_id":"1908.05026","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact spreading speeds are derived for the two-species competition-diffusion system with exponentially decaying initial data, showing the slower species' speed is sometimes nonlocally determined.","lead":"This paper derives exact formulas for how fast two competing species spread into empty territory when the initial populations decay exponentially in space. It shows the slower species' speed can depend on the faster species' speed and on the initial decay rate, a nonlocal effect central to predicting biological invasions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Upper bound for c2 rests on unproved Lemma B.2(a) imported from companion [41]; the comparison-principle assumption (A3) is not the deciding gap.","rationale":"I read the central proof chain carefully: Theorem 1.3 is established through uniform WKB bounds, half-relaxed limits, the new comparison principle, explicit piecewise-linear super/sub-solutions, and the imported Lemmas B.1-B.2. The reader's weakest-assumption pick, A3, does not appear to land: for a step-function R(x/t), the directional monotonicity can be satisfied by choosing the unit vector pointing from the low side to the high side of the jump. At points on the discontinuity, the cone condition in A3 is strict enough to exclude the tangent construction that would produce a positive jump; with the normal choice, H^* at the first point is no larger than H_* at the second, giving RHS >= LHS with room to absorb the omega term. The super-solution verifications in Lemma 3.8 are consistent with the standard convention that super-solutions use the upper envelope H^* (and subsolutions use H_*). The truly load-bearing gap is Lemma B.2(a): it is stated in full but not proved, and it is the mechanism that converts the Hamilton-Jacobi exponential estimate into the upper bound on c2. Without an independent derivation of its threshold formula, the main theorem's c2 formula is conditional. The omitted proofs of Lemmas 3.7, 4.1, 4.4 and of Lemma 3.8(b), and the unproved Theorem 6.1, are less central because they appear to be routine analogues, whereas Lemma B.2 is the one whose failure would directly change the headline formula. I therefore keep the reader's CONDITIONAL verdict, with no change.","tokens_in":1076,"tokens_out":1316,"duration_ms":465886,"concrete_test":"Write out a complete proof of Lemma B.2(a) in the notation of this paper, starting from the single-species Fisher-KPP comparison used in [41, Lemma 2.4], and independently re-derive the closed form c_{c,mu} in (4.10) by solving the linearized eigenvalue problem at the trailing edge x = c t. In particular, check the boundary case mu = lambda_LLW (c - cLLW) and confirm that the two branches of c_{c,mu} agree there. If the re-derivation requires an additional condition (for instance, a strict inequality or a constraint on the boundary-layer width), then Proposition 4.2's application in Theorem 1.3 has a hidden assumption and the upper bound on c2 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main formula c2 = max{cLLW, c_nlp} is only as secure as the black-box Lemma B.2(a), imported without proof from the companion preprint [41] and used in Proposition 4.2 to convert the exponential estimate u(t, c t) = O(e^{-(w2(1,c)+o(1))t}) into a hard upper bound on c2. The lemma's threshold formula for c_{c,mu} in (4.10) fixes the exact boundary between the LLW-controlled branch and the nonlinear-decay branch, and a misstatement of that threshold would directly corrupt the algebraic identity c_{c,mu} = max{cLLW, c_nlp} verified in Step 3 of Proposition 4.2. The paper gives no proof of Lemma B.2 and only cites [41], an arXiv preprint; the boundary and initial data produced by the Hamilton-Jacobi construction must be checked against the lemma's hypotheses (i) and (ii). The comparison-principle assumption singled out by the reader, condition (A3), is less concerning: for Hamiltonians of the form H(t,x,p) = |p|^2 + R(x/t) with R a step function, the directional local monotonicity A3 can be verified by choosing the unit normal pointing into the side where H is larger; the apparent counterexamples require tangent separation on the discontinuity line, which is excluded by the strict cone condition in A3. Thus the surviving load-bearing gap is the unproved transfer lemma from [41], not the comparison principle itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the two-species Lotka–Volterra competition–diffusion system (1.1) with 0<a,b<1, for initial data satisfying the exponential-decay hypothesis (Hλ). The main result (Theorem 1.3) states that, when σ1>σ2, the solution has four asymptotic zones with speeds c1=σ1, c2=max{cLLW,ĉnlp}, and c3=−max{c̃LLW,σ3}, where ĉnlp is given explicitly by (1.6) and depends on σ1 (the speed of the faster species v) and λu (the decay rate of u). In particular, the speed of the slower species is nonlocally determined by the leading front. The proof uses the WKB transform, half-relaxed limits, a comparison principle for discontinuous Hamilton–Jacobi equations (Appendix A), and explicit piecewise-linear super- and sub-solutions. Theorem 1.5 treats the degenerate case σ1=σ2, recovering the direct invasion of (0,0) by the coexistence state (Tang–Fife wave as two coincident fronts). Section 6 extends the formula to 0<a<1<b, and Section 7 gives an extension with forcing terms.","tokens_in":40607,"tokens_out":21312,"duration_ms":199674,"significance":"If correct, the paper resolves the open question of the second spreading speed for exponentially decaying data and makes precise the nonlocal dependence of c2 on σ1 and λu. The proof is largely self-contained: Appendix A contains a full proof of the comparison principle; the piecewise-linear super-solutions in Lemma 3.8 are explicit; the algebraic identity in Proposition 4.2, Step 3, is verifiable and correct. The formulas contain no fitted parameters. The main weakness is not an internal inconsistency but the dependence on the companion preprint [41] for two lemmas (B.1 and B.2) that are load-bearing in the proof of the upper bound on c2 and the lower bound on c3.","major_comments":[{"comment":"Lemmas B.1(a),(b),(d) and B.2(a),(b) are stated without proofs and cited to [41]. Lemma B.2(a) is the decisive instrument: in Proposition 4.2, Step 3 it converts the exponential estimate (4.4) into the upper bound c2 ≤ c_{σ1,μ} (Eq. (4.11)), and Lemma B.2(b) plays the same role in Proposition 4.5, Step 3 for c3. Since [41] is a preprint, the central proof is not self-contained at this point. I do not see a mathematical error in the way the lemma is applied — the hypotheses (i) and (ii) appear to be met, and the final identity c_{σ1,μ} = max{cLLW, ĉnlp} checks out — but the missing proof or a published reference must be supplied.","section":"Appendix B; Proposition 4.2, Step 3; Proposition 4.5, Step 3"}],"minor_comments":[{"comment":"Remark A.2 asserts that (A1)–(A4) are easy to verify for H(p)+R(x/t) when R has bounded variation. For the step-function R used here, condition (A3) is the key point; the orientation in (A3) makes it plausible by choosing the normal pointing into the side where H is larger, but the verification should be written out because Theorem A.1 is the engine of all bounds.","section":"Remark A.2"},{"comment":"The super-solution check for w2 in case (b) at the boundaries x/t=σ1 and x/t=˜cnlp is only asserted; please provide the omitted details or state explicitly that they are analogous to case (a).","section":"Lemma 3.8, case (b)"},{"comment":"The verification that the function in (4.2) is a viscosity sub-solution of (4.1) is delegated to 'straightforward'. Since this function is used to produce the important estimate (4.4), the verification at the corner x/t=2 and at the kink for λu≤1 should be included.","section":"Proposition 4.2, Step 1"},{"comment":"In the displayed formula for σ^δ_3 after (5.5), the denominator contains λ_v^+; by analogy with (1.3) it should be λ_v^-.","section":"Section 5, formula for σ3^δ"},{"comment":"Theorem 6.1 and Proposition 6.2 are stated without proofs; the text only says 'by arguing similarly as in Theorem 1.3'. If these are intended as original results, proofs are needed; otherwise they should be explicitly labeled as remarks or conjectures.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong, but the reliance on the companion preprint [41] for Lemmas B.1 and B.2 should be resolved before publication. I suggest asking the authors to include proofs of Lemma B.2 or to cite a published or accepted version of [41]. The Section 6 claims also need either proof or explicit downgrading. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper is worth your time. It settles a real open question: for the weak competition Lotka-Volterra system with exponentially decaying initial data, it gives exact formulas for all three spreading speeds, and shows the slower species' speed c2 depends on the faster species' speed σ1 and on the initial decay rate λu. That nonlocal dependence is genuinely new, and the explicit formula (1.6) is the headline. The identification of the Tang–Fife invasion front as the merged limit σ1=σ2 is a nice bonus and makes sense.\n\nThe paper also does something technically substantial: it proves a comparison principle for discontinuous viscosity solutions with piecewise Lipschitz Hamiltonians (Appendix A). That is new, and the proof is presented in full. The overall chain—WKB bounds, half-relaxed limits, explicit piecewise-linear super- and sub-solutions, then an algebraic reduction of the upper-bound constant—is coherent and largely convincing.\n\nWhere I would push back: the weakest point is not the comparison principle, despite what one might worry about with (A3). For the step-function Hamiltonians used here, (A3) should be verifiable. The real gap is Lemma B.2, imported from the companion paper [41] without proof, and used in Proposition 4.2 to go from exponential decay estimates to hard speed bounds. The threshold formula in (4.10) is load-bearing, and a wrong threshold would corrupt c2. Also Lemmas 3.7, 4.1, 4.4 and Lemma 3.8(b) are asserted or sketched, Theorem 6.1 (the Girardin–Lam case) is stated with no proof, and Theorem 7.1 is a sketch. A referee should ask the authors to prove all of these or at least make the companion proofs available.\n\nAll that said, the result is important, the derivation shown is sound, and there are no fitted parameters or circular definitions. It deserves a serious referee. I would send it out.","headline":"Resolves the second spreading speed for exponentially decaying data; proof is mostly coherent but leans on unproved companion lemmas.","tokens_in":41334,"tokens_out":2567,"would_cite":true,"duration_ms":25169,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K57","35B40","35D40","92D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves exact spreading speeds for two competing species with exponentially decaying initial data and shows the slower species' speed depends on the faster species' front.","keywords":["Lotka-Volterra competition-diffusion","spreading speeds","exponential decaying initial data","Hamilton-Jacobi equations","viscosity solutions","comparison principle","nonlocal determination","stacked fronts"],"falsifier":"Run a numerical simulation of (1.1) with exponentially decaying initial data in a regime with $σ_1 > σ_2$, e.g. $\\lambda_u<1<\\lambda_v^+$, measure the large-time rightward front speed of $u$, and compare it with $\\max\\{c_{LLW},\\hat{c}_{nlp}\\}$ from (1.6). Varying $\\lambda_v^+$ while holding $\\lambda_u$ fixed should change $c_2$ through $\\hat{c}_{nlp}$; a measured $c_2$ that is insensitive to $\\lambda_v^+$ would falsify the central claim. A more direct test targets the comparison principle: check whether condition (A3) admits a counterexample for $H(t,x,p)=|p|^2+1-a\\chi_{\\{x<σ_1 t\\}}$, since that would break both bounds on $c_2$.","tokens_in":40100,"feed_emoji":"🌱","tokens_out":11622,"duration_ms":101947,"temperature":0.7,"pith_summary":"This paper establishes exact spreading speeds for the Lotka–Volterra competition–diffusion system in the weak-competition regime ($0<a,b<1$), when the initial populations decay exponentially in space. For initial data satisfying (Hλ) with $σ_1 > σ_2$, the habitat splits into four moving regions: empty, only the faster species, a coexistence zone, and only the slower species. The paper's central result is an explicit formula for the three front speeds, and the striking consequence is that the speed $c_2$ of the slower species is not determined locally by its own growth and diffusion alone; it also depends on $σ_1$, the speed of the faster species' leading front, through formula (1.6). This provides a proof for the stacked-front picture and shows when the slower species' invasion is pulled by the faster species' front.","feed_headline":"Slower species' speed depends on the faster front","feed_subtitle":"New Hamilton–Jacobi analysis ties the slower species' front speed to the faster species' front.","key_machinery":"The central object is the limiting Hamilton–Jacobi equation obtained from the WKB ansatz $w^\\epsilon_2(t,x) = -\\epsilon \\log u(t/\\epsilon,x/\\epsilon)$, namely $\\min\\{\\partial_t w + |\\partial_x w|^2 + 1 - a\\chi_{\\{x<σ_1 t\\}}, w\\} = 0$, with initial data $w(0,x) = \\lambda_u \\max\\{x,0\\}$. The zero level set of its viscosity solution gives the boundary of the region where species $u$ survives, and the speed $\\hat{c}_{nlp}$ is read off from the slope of that level set. The argument is carried by the comparison principle (Theorem A.1) for viscosity super- and sub-solutions of Hamilton–Jacobi equations with piecewise Lipschitz, possibly discontinuous Hamiltonians of the form $H(t,x,p)=|p|^2 + R(x/t)$ with $R$ of bounded variation; the paper constructs explicit piecewise linear super-solutions and uses the comparison principle to turn large-deviation estimates into matching lower and upper bounds on $c_2$.","core_discovery":"For initial data with prescribed exponential decay rates $u_0(x) \\sim e^{-\\lambda_u x}$ and $v_0(x) \\sim e^{-\\lambda_v^+ x}$ as $x \\to +\\infty$, and $v_0(x) \\sim e^{\\lambda_v^- x}$ as $x \\to -\\infty$, the paper proves that if $σ_1 > σ_2$, the four spreading limits in (1.4) hold with speeds $c_1 = σ_1$, $c_2 = \\max\\{c_{LLW}, \\hat{c}_{nlp}\\}$, and $c_3 = -\\max\\{\\tilde{c}_{LLW}, σ_3\\}$. The quantity $\\hat{c}_{nlp}$ is given by the three-branch formula (1.6), in which $σ_1$ and $\\lambda_u$ enter through $\\tilde{\\lambda}_{nlp} = \\tfrac12[\\sigma_1 - \\sqrt{(\\sigma_1-2\\lambda_u)^2+4a}]$. This is the precise sense in which the slower species' rightward spreading speed $c_2$ is nonlocally determined by the faster species' speed $σ_1$. When $σ_1 = σ_2$, the two fronts coincide and (1.9) shows a single front connecting the coexistence state $(k_1,k_2)$ to the empty state $(0,0)$, confirming the coexistence-to-empty invasion mode as a degenerate case.","pith_inferences":["The regime boundaries in (1.6) predict a measurable threshold: continuously varying the faster species' decay rate should cause the slower species' speed $c_2$ to switch between the three functional forms of $\\hat{c}_{nlp}$, a feature that could be tested experimentally or numerically.","The same mechanism likely produces a chain of nonlocal dependences in three-species competition, where a middle species' speed may be set by the fastest front ahead of it; Section 7's forcing-term extension appears intended as a step in that direction.","The results suggest a management lever: altering the shape of the faster species' leading tail (not just its growth rate) can change whether and how fast the slower species follows into open habitat."],"forward_implications":["If the initial data decay fast enough ($\\lambda_v^+ \\ge \\sqrt{r/d}$ and $\\lambda_u > σ_1/2$), the speeds $c_1$ and $c_2$ reduce to the compactly supported initial-data values from the companion paper.","The speed $c_2$ is non-increasing in both $σ_1$ and $\\lambda_u$: a faster leading front or a sparser tail of $u$ slows the invasion of $u$.","As $|σ_1-σ_2|\\to 0$ the spacing between the two fronts shrinks to zero, and at $σ_1=σ_2$ the solution exhibits a single front connecting the coexistence state to the empty state, exactly the coexistence-to-empty invasion mode.","The same Hamilton–Jacobi comparison machinery recovers and extends the spreading formula for the case $0<a<1<b$ previously obtained by other methods, now with full dependence on the initial exponential decay rates.","The approach extends to higher space dimensions under minor modifications, so the nonlocal-determination phenomenon is not an artifact of the one-dimensional setting."],"supporting_citations":[{"why":"Companion paper that supplies Lemmas B.1 and B.2, which convert large-deviation estimates into upper and lower bounds on spreading speeds.","marker":"[41]"},{"why":"Defines $c_{LLW}$, the compact-support spreading baseline that the slower species always retains.","marker":"[36]"},{"why":"Supplies the large-deviation PDE method for scalar KPP fronts that this paper adapts to the system.","marker":"[17]"},{"why":"Origin of the Hamilton–Jacobi large-deviation approach for reaction–diffusion fronts.","marker":"[22]"},{"why":"Provides a comparison principle for discontinuous Hamilton–Jacobi equations, a key ingredient in Theorem A.1.","marker":"[33]"},{"why":"Provides a comparison theorem for piecewise Lipschitz Hamiltonians, used to handle the discontinuous indicator terms.","marker":"[51]"},{"why":"The traveling wave connecting the coexistence state to the empty state, shown to be the degenerate two-front case.","marker":"[50]"},{"why":"The previously studied case $0<a<1<b$ whose $c_2$ formula is rederived and extended here with exponential-decay data.","marker":"[26]"},{"why":"Scalar Fisher–KPP spreading speed for exponentially decaying initial data, giving the formulas for $σ_1,σ_2,σ_3$.","marker":"[44]"}],"fun_headline_variants":["Fast front sets slow species' speed","Slow species' speed: fast front decides","Exponential decay links two front speeds","Hamilton-Jacobi reveals nonlocal front speed","Nonlocal spreading: fast front rules slow species"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the comparison principle stated in Theorem A.1 for Hamilton–Jacobi equations whose Hamiltonian may be discontinuous along a set $Γ$; the paper's Remark A.2 asserts that the required hypotheses, especially the directional local monotonicity condition (A3), hold for the indicator-type coefficients $1-a\\chi_{\\{x<σ_1 t\\}}$ and $1-a\\chi_{\\{σ_2 t<x<σ_1 t\\}}$ because the profile $R(x/t)$ has bounded variation. If (A3) fails for these coefficients, the lower and upper bounds on $c_2$ collapse even though all other estimates stand.","fun_headline_variants_meta":{"raw":{"variants":["Fast front sets slow species' speed","Slow species' speed: fast front decides","Exponential decay links two front speeds","Hamilton-Jacobi reveals nonlocal front speed","Nonlocal spreading: fast front rules slow species"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001459,"raw_usage":{"total_tokens":5887,"prompt_tokens":974,"completion_tokens":4913,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":4848}},"tokens_in":590,"tokens_out":4913,"duration_ms":39043,"temperature":1.0,"reasoning_tokens":4848,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:26:50.727463+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a numerical simulation of (1.1) with exponentially decaying initial data in a regime with $σ_1 > σ_2$, e.g. $\\lambda_u<1<\\lambda_v^+$, measure the large-time rightward front speed of $u$, and compare it with $\\max\\{c_{LLW},\\hat{c}_{nlp}\\}$ from (1.6). Varying $\\lambda_v^+$ while holding $\\lambda_u$ fixed should change $c_2$ through $\\hat{c}_{nlp}$; a measured $c_2$ that is insensitive to $\\lambda_v^+$ would falsify the central claim. A more direct test targets the comparison principle: check whether condition (A3) admits a counterexample for $H(t,x,p)=|p|^2+1-a\\chi_{\\{x<σ_1 t\\}}$, since that would break both bounds on $c_2$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Companion paper that supplies Lemmas B.1 and B.2, which convert large-deviation estimates into upper and lower bounds on spreading speeds."},{"cited_title":"Lewis, B.T","cited_arxiv_id":null,"evidence_quote":"Defines $c_{LLW}$, the compact-support spreading baseline that the slower species always retains."},{"cited_title":"Evans, P.E","cited_arxiv_id":null,"evidence_quote":"Supplies the large-deviation PDE method for scalar KPP fronts that this paper adapts to the system."},{"cited_title":"Ishii, Comparison results for hamilton-jacobi equa tions without growth condition on solutions from above, Appl","cited_arxiv_id":null,"evidence_quote":"Provides a comparison principle for discontinuous Hamilton–Jacobi equations, a key ingredient in Theorem A.1."},{"cited_title":"Tourin, A comparison theorem for a piecewise Lipschi tz continuous Hamilto- nian and application to Shape-from-Shading problems, Nume r","cited_arxiv_id":null,"evidence_quote":"Provides a comparison theorem for piecewise Lipschitz Hamiltonians, used to handle the discontinuous indicator terms."},{"cited_title":"Tang, P.C","cited_arxiv_id":null,"evidence_quote":"The traveling wave connecting the coexistence state to the empty state, shown to be the degenerate two-front case."},{"cited_title":"Girardin, K.Y","cited_arxiv_id":null,"evidence_quote":"The previously studied case $0<a<1<b$ whose $c_2$ formula is rederived and extended here with exponential-decay data."}],"review_version":1}