{"id":"62b5f58f-8ced-4e0e-af20-665e967f1713","arxiv_id":"1908.05539","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a two-species strong-competition Lotka-Volterra system, the paper establishes the exact asymptotic spreading speed and front profile in both the native-invasive and two-invasive scenarios.","lead":"This paper proves sharp formulas for how fast two competing species with strong competition spread in space, and in what shape their invasion fronts move. The results cover the native-invasive case and the two-invasive-species case, including logarithmic corrections to the front position.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Estimate (2.9)/Lemma 2.6 is the load-bearing foundation; its Appendix proof is incomplete and Lemma 3.11's subsolution signs are inconsistent — both need correction before the sharp convergence claims are certifiable.","rationale":"A careful reading confirms that the central two-species claims are credible and the proof structure is plausible: the super/subsolution constructions in Sections 3-4 are detailed, and the Bramson-shift arguments adapt standard KPP results. However, the chain depends on estimate (2.9) in Lemma 2.6, and the Appendix's proof is not self-contained: it assumes a perturbed-front existence result and tail asymptotics that are only invoked, not derived, and the final comparison on the whole line is summarized rather than verified. This matches the reader's weakest-assumption analysis. An additional verifiability obstacle appears in Lemma 3.11: the stated differential inequalities are those of a supersolution, while the proof computes those of a subsolution and even concludes N2≤0 after establishing a lower bound. Since Lemma 3.12 and Theorem 1 use this as a subsolution, the text as printed is internally inconsistent. This is probably a sign typo, but it is precisely the kind of unresolved detail that prevents certification. The n-species Corollary 4.6 is also stated without proof, though it is not needed for the main two-species theorems. The concern is about completeness and internal consistency, not about a demonstrated mathematical falsehood; therefore the appropriate verdict remains conditional, with no change from the reader's assessment.","tokens_in":45239,"tokens_out":27957,"duration_ms":255728,"concrete_test":"Write out a complete proof of (2.9) from the Appendix: (i) prove existence of a front (U,V) for (4.17) with speed c_ε in (c,c_uv) and justify the analogue of Lemma 2.2 for its tails; (ii) verify the three-case inequalities N1≤0 and N2≥0 for the subsolution; (iii) justify the boundary comparison at |x|≥L and the resulting liminf/limsup limits. Independently, correct the sign convention in Lemma 3.11 and check that its Cases 1-3 yield exactly the subsolution inequalities needed by Lemma 3.12. If these checks pass, the conditional can be lifted; if any step fails, the sharp convergence claims are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"All three main theorems rest on the estimate (2.9) in Lemma 2.6: it turns the local convergence (H3) into exponential decay of (u-1,v) on [-ct,ct], without which Lemma 2.8 and the front-convergence arguments collapse. The proof of (2.9) is deferred to an Appendix that asserts an ε-perturbed front (4.17) exists with c_ε arbitrarily close to c_uv, applies Lemma 2.2 tail estimates to that perturbed front without deriving them, and compresses the final global comparison into a few lines. This is the least secure step in the common preliminaries. In addition, the subsolution lemma used in Theorem 1 has inconsistent signs: Lemma 3.11 is stated as N1≥0, N2≤0, while its proof derives upper bounds for N1 and lower bounds for N2 and concludes N2≤0 only after proving a lower bound. Taken literally, the construction is a supersolution, not the subsolution needed in Lemma 3.12 and the comparison proof of Theorem 1. These issues do not disprove the theorems, but the printed argument cannot be certified as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the two-species Lotka-Volterra competition-diffusion system (1.1) in the strong-competition regime a,b>1, assuming (H3) that the u species invades successfully. It proves three sharp long-time results: in scenario (A1), where u is invasive and v is native, the solution converges on x≥0 to the bistable traveling front (U,V) with speed c_uv and an explicit shift; in scenario (A2) with c_u>c_v, u converges to a shifted Fisher-KPP front with the Bramson logarithmic correction (3d/c_u) ln t while v decays to zero; and in scenario (A2) with c_u 0 (so that ζ′(t) > 0), and the proof of Lemma 3.12 also selects ζ1 < 0, contradicting the hypothesis of Lemma 3.11. As written, the subsolution construction is internally inconsistent in both the sign of the differential inequalities and the sign of the shift parameter; the text must be corrected so that the lemma statements match the proofs and the comparison principle.","section":"§3.3.2, Lemma 3.11 and Lemma 3.12"},{"comment":"The estimate (2.9) is the load-bearing foundation for Lemma 2.6, Lemma 2.8, and ultimately all three theorems, but its proof is deferred to an appendix that leaves several essential points unjustified. In particular, the appendix introduces an ε-perturbed front (U,V) satisfying (4.17) and then applies Lemma 2.2 tail estimates directly to this perturbed front (see the lines after (4.19)), although Lemma 2.2 is only stated for the unperturbed front of (1.4); one needs to show that the analogous exponential estimates hold for (U,V) with constants uniform as ε→0. Furthermore, the final comparison step after (4.18) is compressed to a few lines: the existence of L, the choice of T-hat, and the verification that u(T+T-hat,x) ≥ u(T,x) and v(T+T-hat,x) ≤ v(T,x) on all of R are not fully detailed. Because a failure of (2.9) would invalidate the exponential-decay arguments in the rest of the paper, this gap must be closed or the estimate must be proved directly.","section":"Appendix, proof of (2.9)"},{"comment":"Lemma 3.3 is used to prove Lemma 3.5, which is essential for Proposition 1 and hence Theorem 1, but its proof omits the details for Case (ii) and Case (iii), stating only that they are handled similarly to the corresponding cases in Lemma 3.1. Since the supersolution signs in (3.16) require reversing several inequalities compared with Lemma 3.1, the reader cannot verify the claimed inequality N1 ≥ 0 and N2 ≤ 0 in the middle and trailing regions without a written check. The omitted cases should be supplied.","section":"§3.1.2, Lemma 3.3"},{"comment":"Lemma 4.9 is the key convergence statement for Theorem 3, but the final step of its proof is omitted with the remark that one follows the proof of Proposition 1. This is not a routine repetition: the convergence here must be established on the moving interval [0,ct) with a boundary at ct, and it must be combined with the exponential decay of u on [ct,∞) from Lemma 4.7 and the lower bound for v from Lemma 4.8. The details of the limiting argument and the passage to the sup-norm on [0,ct) need to be written out explicitly for the terrace result to be certified.","section":"§4.2, Lemma 4.9"}],"minor_comments":[{"comment":"In the proof of Lemma 2.8, the display after the comparison step reads 'u(t,x) ≥ 1 −Me δ2t', which should be '1 − M e^{−δ2 t}' with a negative exponent; otherwise the claimed exponential convergence is misstated.","section":"Lemma 2.8"},{"comment":"The parameter condition in the statement of Lemma 3.12 is 'ζ1 < 0', but the proof and the hypotheses of Lemma 3.11 require ζ1 > 0; the sign should be corrected for consistency.","section":"§3.3.2, Lemma 3.12"},{"comment":"Corollary 4.6 states an n-species generalization without proof and without any indication of the additional hypotheses (such as strong competition among all pairs) needed for the arguments of the paper to apply. It would be appropriate to state it as a remark or conjecture, or to provide a proof sketch.","section":"Corollary 4.6"},{"comment":"In the appendix, the ε-perturbed front satisfying (4.17) is denoted by the same symbols (U,V) as the original front of (1.4). This creates confusion in expressions such as 'by Lemma 2.2' where the lemma refers to the original front. It would be clearer to write (U_ε,V_ε) and to restate the required tail estimates for the perturbed front.","section":"Appendix, notation"}],"recommendation":"major_revision","confidential_remarks":"The central results appear plausible and important, but the manuscript is not yet certifiable as written because of the sign inconsistency in Lemma 3.11/Lemma 3.12, the incomplete proof of (2.9) in the appendix, and the omitted details in Lemmas 3.3 and 4.9. These are substantive but local issues that a careful revision can fix within the scope of the paper. I recommend requiring the authors to provide complete proofs for these points, and to correct the sign typos, before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The three main theorems are likely correct and are a real advance: they give the first sharp front-convergence statements for the strong-competition Lotka-Volterra system under two natural initial-data scenarios, including a Bramson log correction for the faster species in the compact-support case. This substantially sharpens Carrere's earlier speed-only results. The proof architecture is sensible: first establish exponential convergence of the solution to (1,0) on expanding boxes, then construct tailored super/subsolutions, then apply the Bramson machinery to the faster species.\n\nWhat the paper does well: the super/subsolution constructions are intricate and mostly carefully checked; Theorem 1's stability-plus-sliding-method proof is detailed; the reduction for Theorem 2 to the Hamel-Nolen-Roquejoffre-Ryzhik argument is clean. The external inputs (Gardner/Kan-On bistable fronts, Bramson correction) are independent of the paper, and the self-citations are not load-bearing.\n\nNow the soft spots, in proportion.\n\nThe main one is the proof of the key estimate (2.9) in Lemma 2.6. All three theorems rest on it. The appendix compresses the argument: it asserts the existence of an ε-perturbed bistable front (4.17) with speed arbitrarily close to c_uv, then applies the tail estimates of Lemma 2.2 to that perturbed front, then compresses the final global comparison. The tail estimates for the perturbed front are not derived (Lemma 2.2 was stated only for the unperturbed front), and continuity of the front family in ε is asserted rather than proved. This is a genuine gap in the printed proof. It is very likely fixable by standard implicit-function/continuation arguments, but as written the paper cannot be certified on this point.\n\nSecond, Lemma 3.11's statement has the inequality signs reversed (N1≥0/N2≤0) relative to what the subsection needs and what the proof actually shows (N1≤0/N2≥0). The proof and Lemma 3.12 agree on the intended subsolution inequalities, so it is a typo, but it should be fixed.\n\nThird, the n-species Corollary 4.6 is stated without proof and under weaker hypotheses than the rest of the paper. For n species, the interaction of the slower species can affect the fastest one; \"the argument can provide\" is not a proof. I would remove it or prove it properly.\n\nMinor: several lemmas say \"we omit the details\"; these are mostly standard, but combined with the appendix gap they make the proof harder to verify.\n\nWho should read this: researchers in reaction-diffusion spreading speeds and spatial ecology. It deserves a serious referee. My recommendation: send to peer review, but require the appendix to be expanded with a rigorous treatment of the perturbed front, the sign typo fixed, and Corollary 4.6 either proved or cut.","headline":"Genuine sharp spreading-speed results for the strong-competition system, but the load-bearing estimate (2.9) is not fully proved and the n-species corollary is unproved; the main theorems are likely correct but the paper needs repair.","tokens_in":45998,"tokens_out":4937,"would_cite":true,"duration_ms":45867,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K57","35K45","92D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves exact spreading speeds and front profiles for the two-species strong-competition Lotka-Volterra diffusion system in three regimes, including a logarithmic slowdown and a two-front terrace.","keywords":["Lotka-Volterra diffusion system","strong competition","bistable traveling waves","spreading speed","front profile","logarithmic slowdown","propagating terraces"],"falsifier":"Run a numerical simulation of (1.1) in the strong-competition regime, for example with $d=r=1$ and $b>a>1$ chosen so that $c_{uv}>0$, with compactly supported $u_0$ and $v_0$ bounded below; measure $\\max_{|x|\\le ct}|u(t,x)-1|$ and $\\max_{|x|\\le ct}v(t,x)$ for some fixed $c1$. Under the assumption that species $u$ invades successfully, the claimed result is that the solution converges to explicit traveling-front profiles with exact speeds: a single bistable front when $v$ is a native resident (Theorem 1), a Fisher-KPP front with the logarithmic shift $(3d/c_u)\\ln t$ when both species invade and $u$'s single-species speed is larger (Theorem 2), and a two-front terrace when $u$ is the slower invader (Theorem 3). The authors identify these as the first precise speed-and-profile results for the strong-competition system. If correct, the results turn a qualitative statement about invasion into quantitative predictions of where each front is at large times and what shape it has.","feed_headline":"Strong-competition invaders get exact front speeds","feed_subtitle":"Native-vs-invader and two-invader cases settle into bistable fronts, KPP fronts with a log slowdown, or a terrace.","key_machinery":"The central object is the unique bistable traveling front $(c_{uv},U,V)$ solving (1.4), which connects the state $(1,0)$ at $-\\infty$ to $(0,1)$ at $+\\infty$; the proof leans on its exponential tails (Lemmas 2.1-2.2) and its local asymptotic stability, so the solution can be trapped between shifted copies of this wave. The carrying mechanism is the comparison principle for the cooperative system, applied to new supersolutions and subsolutions of the form $U(x-c_{uv}t+\\zeta(t))$ and $V(x-c_{uv}t+\\zeta(t))$ with small exponentially decaying corrections $p(t),q(t)$ and a slowly relaxing shift $\\zeta(t)$, sometimes symmetrized as sums $U(x-c_{uv}t+\\zeta(t))+U(-x-c_{uv}t+\\zeta(t))-1$ to handle a resident species occupying the whole line. For the faster-species results, the system is reduced to a perturbed Fisher-KPP equation, and the level-set method of [27] produces the logarithmic time shifts.","core_discovery":"The paper's central claim is that, under assumptions (H) and (A1), the solution converges on $x\\ge 0$ to the unique bistable front $(c_{uv},U,V)$ of (1.4): there is a shift $\\hat h$ such that\n$$\\sup_{x\\ge 0}|u(t,x)-U(x-c_{uv}t-\\hat h)|+\\sup_{x\\ge 0}|v(t,x)-V(x-c_{uv}t-\\hat h)|\\to 0.$$\nUnder (A2), the regime is governed by the ordering of the single-species Fisher speeds $c_u=2\\sqrt{rd}$ and $c_v=2$. If $c_u>c_v$, then $v$ is driven to zero while $u$ approaches $U_{\\mathrm{KPP}}(x-c_u t+(3d/c_u)\\ln t+\\omega(t))$ with bounded $\\omega$. If $c_ua>1$ chosen so that $c_{uv}>0$, with compactly supported $u_0$ and $v_0$ bounded below; measure $\\max_{|x|\\le ct}|u(t,x)-1|$ and $\\max_{|x|\\le ct}v(t,x)$ for some fixed $c