{"id":"e75e2e17-526c-4790-b0a3-11e11990c745","arxiv_id":"1908.10287","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A rigorous limit procedure shows that solutions of modified nonlocal adhesion and chemotaxis systems converge, along a subsequence, to solutions of the local haptotaxis and chemotaxis models as the sensing radius tends to zero.","lead":"This paper proves that, under certain conditions, solutions of nonlocal cell-adhesion and chemotaxis models converge to solutions of the corresponding local models as the sensing radius shrinks to zero. The proof works by rewriting the nonlocal operators as averages of gradients, then passing to the limit in the PDE systems; 1D simulations illustrate the theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence proof in Theorem 5.8 relies on an incorrect uniform bound: replacing ||R_r|| by C11 underestimates norm-dependent constants; the valid bound is sup ||R_{r_m}|| < 1/C11.","rationale":"I agree with the reader that the paper makes a substantial contribution but is conditional. My primary concern differs slightly: the most load-bearing point is not only the boundary-layer difference between A_r and T_r, but the technical uniformity step in the proof of Theorem 5.8. The manuscript itself is transparent about the modified-operator scope, and Example 3.3 explicitly motivates why the original operators are not treated. The reader also flagged the C11 versus 1/C11 issue; I confirm it is real, although it appears to be a typo rather than a conceptual flaw. The second issue, that Theorem 5.8 cites Theorem 5.13 for both branches of Assumption 5.3 while Theorem 5.13 only proves case (b), is an omitted case distinction. Both problems are fixable, so I would not reject or declare the paper unverdictable; I would require a corrected uniformity argument and a treatment of the 5.3(a) case before accepting the main theorem as fully proved.","tokens_in":33909,"tokens_out":10191,"duration_ms":100354,"concrete_test":"Re-read the uniformity step in the proof of Theorem 5.8 and replace the phrase 'Replacing ||R_r|| by C11' with M := sup_m ||R_{r_m}||_{L(L^2(Omega)^n)} < 1/C11. Then trace every occurrence of C22(T, ||R_r||) in estimates (5.40)-(5.47) and verify that C22(T, M) is finite and independent of m. Separately, insert a short argument for Assumption 5.3(a) using the estimates in the proof of Theorem 5.10 with M in place of ||R_r||. If both replacements go through without additional conditions, the theorem can be accepted after a correction; if either step fails, the convergence proof as written is incomplete.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central convergence claim is Theorem 5.8, whose proof obtains uniform-in-m estimates from Theorem 5.13 and then passes to the limit. The proof contains a concrete gap at the uniformity step. It first notes that, because C11 = C12 C13 / C5 < 1 and ||R_{r_m}|| -> 1, one may choose r_m so that sup_m ||R_{r_m}|| < 1/C11. But two sentences later it says 'Replacing ||R_r|| by C11 in C22(T, ||R_r||)' makes the constants independent of m. This is backwards: C11 < 1, while ||R_{r_m}|| is near 1, so replacing the norm by C11 shrinks the constants artificially. The correct uniform replacement is sup_m ||R_{r_m}||, bounded by 1/C11, not by C11. Without that correction, the r-uniform estimates (5.40)-(5.47) are not established as written, and the compactness argument lacks a foundation. A second, related gap is that Theorem 5.8 assumes Assumption 5.3 with either (a) or (b), but the proof invokes Theorem 5.13, which is stated and proved only under 5.3(b). The 5.3(a) Lipschitz case would need r-uniform estimates from Theorem 5.10, which are not stated there. Both issues are repairable, but they affect the proof of the paper's main theorem. The paper's scope caveat about modified operators T_r/S_r versus original A_r/grad-tilde_r is acknowledged in the text and is not, by itself, a correctness defect; the abstract should still be read carefully.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a rigorous limit procedure connecting nonlocal taxis/adhesion models to local haptotaxis/chemotaxis models as the sensing radius tends to zero. The key technical idea is to reformulate the classical nonlocal operators A_r and tilde-grad_r as integral operators T_r and S_r acting directly on gradients of signal-dependent quantities, proving L^p approximation of the identity for these operators. The authors prove global existence for the modified nonlocal system (5.1) under either Lipschitz or dissipative growth of the source terms, and establish existence of solutions and global weak-strong solutions. The main theorem, Theorem 5.8, asserts that under Assumptions 1.1, 5.1, 5.3, and 5.4(b), there exists a sequence r_m -> 0 such that solutions (c_{r_m}, v_{r_m}) of the nonlocal system converge in L^2 to a solution (c,v) of the local system (5.2). The paper also contains one-dimensional numerical simulations illustrating boundary-layer differences between the original and modified nonlocal operators, and convergence of nonlocal to local dynamics as r -> 0.","tokens_in":34324,"tokens_out":6635,"duration_ms":65476,"significance":"If the proof is completed, this is a valuable contribution: it provides the first rigorous convergence result for a broad class of nonlocal adhesion and nonlocal chemotaxis models with solution-dependent coefficients, and it gives a clean operator-theoretic explanation of why the modified operators are the right objects for the limit. The operator estimates in Section 3 (Lemmas 3.5 and 3.7) are carefully proved and include useful adjointness, norm-convergence, and Fourier-multiplier computations. The paper is also honest about the limitation that the convergence concerns the modified operators T_r and S_r, not the original boundary-layer-sensitive operators A_r and tilde-grad_r, and Example 3.3 demonstrates a genuine boundary-layer pathology of A_r. The numerical simulations are a useful illustration. The two gaps identified below affect the proof of the main theorem but appear repairable, so the central claim is not called into question beyond the need for revision.","major_comments":[{"comment":"The uniformity step in the proof of Theorem 5.8 is not valid as written. After choosing r_m so that sup_m ||R_{r_m}|| < 1/C11, the text says that 'Replacing ||R_r|| by C11 in C22(T, ||R_r||)' makes the constants independent of m. This replacement is backwards: C11 = C12C13/C5 satisfies C11 < 1, while ||R_{r_m}|| -> 1, so C11 is strictly smaller than ||R_{r_m}|| for all sufficiently large m. Unless C22 is known to be decreasing in its second argument—which the Gronwall-based estimates in Theorem 5.13 do not suggest—substituting C11 does not provide an upper bound for the uniform constants. The correct uniform substitution is sup_m ||R_{r_m}||, which is admitted to be < 1/C11. As written, the r-uniform estimates (5.40)-(5.47) are not established, and the compactness argument lacks a foundation. This is repairable by replacing C11 with sup_m ||R_{r_m}|| in C22 and making the monotonicity of C22 explicit.","section":"Section 5.4, proof of Theorem 5.8"},{"comment":"Theorem 5.8 is stated under Assumption 5.3, which allows either alternative (a) or (b), but the proof invokes Theorem 5.13, which is stated and proved only under Assumption 5.3(b). In the case of Assumption 5.3(a), Theorem 5.10 provides existence of weak-strong solutions for each fixed r satisfying Assumption 5.4(a), but it does not provide r-uniform a priori estimates of the type (5.40)-(5.47). Therefore the convergence proof as written does not cover the (a) branch of Assumption 5.3. The statement should either be restricted to Assumption 5.3(b), or a proof of r-uniform estimates should be supplied for case (a).","section":"Theorem 5.8 and Section 5.4"}],"minor_comments":[{"comment":"The abstract and introduction present the limit procedure as linking the nonlocal models (1.1) and (1.4) to local models, while Theorem 5.8 concerns the modified systems (5.1) with operators T_r and S_r. Although Sections 3 and 5 are explicit about this modification and Example 3.3 justifies it, the abstract should state this qualification to avoid overstating the scope.","section":"Abstract and Introduction"},{"comment":"The proof of Theorem 5.8 uses 'standard arguments' and 'we omit these details' for several limit passages, including the convergence of the remaining nonlinear terms and boundary conditions. Given the otherwise detailed style, a brief summary of the compactness and limit argument for those terms would improve verifiability.","section":"Proof of Theorem 5.8"},{"comment":"The Fourier-multiplier argument in Lemma 3.5(iv) uses the convention that functions are extended by zero outside Omega; this convention is stated in Section 2, but it would help the reader if it were recalled explicitly at the point where the hat symbol is introduced in the proof.","section":"Lemma 3.5(iv)"},{"comment":"In Figures 3 and 4, the 'nonlocal model' simulations use the original formulation (1.1) rather than the modified formulation (5.1) with T_r. This is correct for the numerical comparison but should be stated explicitly in the captions to avoid confusion with the theoretical convergence result.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The two major comments concern load-bearing steps in the proof of the main theorem. Both are local and repairable: the uniformity step can be fixed by using sup_m ||R_{r_m}|| instead of C11, and the Assumption 5.3(a) branch can be handled by restricting the theorem or supplying uniform estimates. The paper's self-citations and adopted assumptions are contextual and not circular. I see no reason to doubt the soundness of the overall strategy, but the proof as written does not yet fully support the stated theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is real: this is the first rigorous convergence result linking nonlocal adhesion/chemotaxis models to local haptotaxis/chemotaxis, and the T_r/S_r reformulation is a neat idea that deserves to be reused. I'd send it to a referee.\n\nWhat's genuinely new: writing A_r u = T_r(∇u) and ˚∇_r u = S_r(∇u) turns the old formal Taylor expansions into honest L^p operator convergence, and it handles solution-dependent coefficients in one framework. The Section 3 estimates are careful, the existence theory in Sections 4–5 is substantial, and the 1D simulations honestly show where the modified operators differ from the original ones near boundaries. The paper also credits the open question from [29] and answers it, so the citation pattern looks fine.\n\nNow the soft spots, both in the proof of Theorem 5.8. First, the uniformity step says \"Replacing ||R_r|| by C11\" after choosing r_m with sup ||R_{r_m}|| < 1/C11. That is backwards: C11 < 1 while the operator norm is near 1, so shrinking the constant artificially underestimates the dependence. The correct replacement is the sup bound 1/C11, and as written the r-uniform estimates (5.40)–(5.47) are not established. This looks like a typo, not a fatal flaw, but it has to be fixed.\n\nSecond, Theorem 5.8 is stated under Assumptions 5.3 with either (a) or (b), but the proof invokes Theorem 5.13, which is proved only under (b). The Lipschitz case (a) would need analogous uniform-in-r estimates, and those are not stated or proved. This is a genuine gap in the manuscript, though likely patchable by reworking the proof of Theorem 5.10 or 5.13.\n\nMinor framing issue: the abstract says it links nonlocal models to local counterparts without saying the rigorous result is for the modified T_r/S_r operators, not the original A_r/˚∇_r near boundaries. Example 3.3 makes the distinction clear, and the text acknowledges it, so this is a wording problem rather than a correctness problem.\n\nBottom line: the strategy is sound, the novelty is real, and the flaws are repairable. This deserves peer review with a request to fix the uniformity step and clarify the scope of Theorem 5.8. I'd bring it to a reading group for the operator trick alone.","headline":"Rigorous bridge between nonlocal and local taxis models with a genuinely new operator trick; the main theorem is plausible but the proof has two concrete holes that need patching.","tokens_in":34768,"tokens_out":2736,"would_cite":true,"duration_ms":27709,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q92","92C17","35K55","35R09","47G20","35B45","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, as the sensing radius in nonlocal adhesion and chemotaxis models tends to zero, solutions of the nonlocal systems converge to solutions of the classical local haptotaxis and chemotaxis systems, provided the…","keywords":["cell-cell and cell-tissue adhesion","nonlocal and local chemotaxis","haptotaxis","integro-differential equations","unified approach","global existence","rigorous limit behaviour","weak solutions"],"falsifier":"Take the one-dimensional setting of Example 3.3 ($\\Omega=(-1,1)$, $F_r\\equiv 2$, $u\\equiv 1$) and compute numerically the $L^1$ norms of $A_r u$ and $T_r(\\nabla u)$ on $\\Omega$ as $r \\to 0$: the example predicts $\\|A_r u\\|_{L^1}$ stays about $1$ while $\\|T_r(\\nabla u)\\|_{L^1} \\to 0$, which directly checks the boundary-layer gap. Independently of the theorem, one could also run the nonlocal system (5.1) with coefficients violating (5.7) and measure whether the $L^2$ distance between nonlocal and local solutions still tends to zero as $r \\to 0$.","tokens_in":33692,"feed_emoji":"🧫","tokens_out":8679,"duration_ms":82329,"temperature":0.7,"pith_summary":"The paper is trying to establish a rigorous bridge between two modelling traditions: nonlocal equations in which cells sense adhesive or chemical signals over a finite radius, and local equations in which the signal gradient acts pointwise. Its central result, Theorem 5.8, states that under suitable assumptions there are nonlocal solutions that converge in $L^2(0,T;L^2(\\Omega))$ to a solution of the local model as the sensing radius $r$ goes to zero. The key move is to rewrite the nonlocal operators as averages applied to $\\nabla u$, not to $u$ itself; this removes boundary-layer artefacts and makes a compactness argument possible with only weak regularity. A sympathetic reader would care because the result justifies when simpler local models are the appropriate limit of more realistic finite-sensing models, and it exposes exactly where the two families disagree.","feed_headline":"Nonlocal cell-sensing models converge to local ones as radius shrinks","feed_subtitle":"Gradient averaging lets finite-sensing adhesion and chemotaxis equations converge to classical local models.","key_machinery":"The machinery is a pair of integral averaging operators. For a vector field $w$, $$T_r w(x) = \\$int_0^{1}$ \\frac{1}{|B_1|}\\int_{B_1} (w(x+rsy)\\cdot y)\\,\\frac{y}{|y|}\\,F_r(r|y|)\\,dy\\,ds$$ and $$S_r w(x) = n\\$int_0^{1}$ \\frac{1}{|S_1|}\\int_{S_1} (w(x+rsy)\\cdot y)\\,y\\,dS_1(y)\\,ds,$$ where $F_r$ is a smooth positive interaction kernel normalized so that $F_0(0)=n+1$. They carry the argument because they turn the adhesion velocity and the nonlocal gradient into averages of $\\nabla u$, are self-adjoint on $L^2$, have norm bounds uniform in $r$, and converge strongly to the identity in $L^p$ as $r \\to 0$. Those three facts, reformulation, boundedness, and strong convergence, produce the uniform a priori estimates and the passage to the limit in the nonlocal flux, which is the heart of Theorem 5.8.","core_discovery":"The central claim is that the limit procedure works for a family of nonlocal models whose nonlocal terms are the averaging operators $T_r$ and $S_r$ applied to gradients, rather than the original operators $A_r$ and $\\mathring{\\nabla}_r$ applied to signal values. Inside the domain these are the same objects, $A_r u = T_r(\\nabla u)$ and $\\mathring{\\nabla}_r u = S_r(\\nabla u)$, but in a boundary layer they differ, and Example 3.3 shows that the original form can fail to converge in $L^1$ even when the signal is constant. Theorem 5.8 then asserts that, under a smallness condition tying cell sensitivity to diffusion, a sequence of nonlocal solutions with radii $r_m \\to 0$ has a subsequence converging in $L^2(0,T;L^2(\\Omega))$ to a weak-strong solution of the corresponding local haptotaxis or chemotaxis system. The proof obtains uniform-in-$r$ a priori estimates, uses the $L^p$ convergence of $T_r$ and $S_r$ to the identity to pass to the limit in the tactic flux, and closes with a compactness argument. On the paper's own terms, the discovery is that the correct object to average is the gradient of the signal-dependent quantity, not the quantity itself.","pith_inferences":["The line-segment averaging inside $T_r$ and $S_r$ suggests that cells in these models sense the gradient accumulated along a protrusion path rather than at its endpoint; if taken seriously as a modelling principle, it favours gradient-averaging formulations over endpoint-sampling ones in future data-driven models.","Example 3.3 implies that the original nonlocal operators, when naively extended by zero outside the domain, can inject an artificial boundary tendency even for constant signals; a testable prediction is that boundary-confined cell populations should behave differently under the two formulations, as the 1D boundary simulations already show.","The smallness condition $C_{11}<1$ links the convergence to a competition between tactic strength and diffusion; one could test numerically whether the $L^2$ distance to the local solution fails to vanish, or converges more slowly, as that condition is approached.","The same averaging construction is likely to transfer to other integro-differential systems with comparable structure, such as multi-species or structured-population migration models, giving a recipe for proving local limits without high-order regularity."],"forward_implications":["For any sequence of sensing radii satisfying the smallness condition, the nonlocal systems admit global weak-strong solutions whose cell and tissue densities converge, up to a subsequence, to a solution of the local haptotaxis or chemotaxis system.","Away from the boundary the new nonlocal operators coincide with the original adhesion velocity and nonlocal gradient, so the convergence result legitimizes the local model as the $r \\to 0$ limit of finite-sensing models in the interior.","In the numerical experiments the nonlocal formulation remains computable in parameter regimes where the local model's effective diffusion becomes negative and the local problem turns ill-posed, with the nonlocal solution destabilizing into aggregates whose wavelength shrinks as $r \\to 0$.","The same gradient-averaging reformulation treats adhesion and nonlocal chemotaxis in one framework and covers coefficients that depend on the solution itself, extending earlier analyses restricted to simpler settings."],"supporting_citations":[{"why":"Introduces the continuum cell-cell adhesion model whose adhesion velocity $A_r$ is the starting point for system (1.1).","marker":"[2]"},{"why":"Gives the local and nonlocal adhesion models and the heuristic Taylor expansion that the paper upgrades to a rigorous limit.","marker":"[24]"},{"why":"Introduces the finite-sampling-radius chemotaxis model and poses the convergence question answered here.","marker":"[29]"},{"why":"Supplies the nonlocal chemotaxis model with finite sensing radius that underlies system (1.4).","marker":"[39]"},{"why":"Provides the finite-volume numerical scheme used to simulate and compare the nonlocal and local formulations in 1D.","marker":"[23]"},{"why":"Supplies the monotone-operator theory for evolution equations that yields the existence and estimates for the single-equation problem (4.1).","marker":"[47]"},{"why":"Supplies the Leray-Schauder fixed-point principle used to obtain existence for the approximating problems (5.9).","marker":"[52]"},{"why":"Supplies the semilinear parabolic regularity theory used to solve the signal equation for fixed coefficients.","marker":"[33]"},{"why":"Provides the Lions lemma used in the compactness arguments when passing to the limit as $r \\to 0$.","marker":"[35]"}],"fun_headline_variants":["Gradient-based averaging links nonlocal and local taxis models","Cell migration: nonlocal models converge to local via gradient averaging","Averaging gradients, not signals, yields local limits for cell sensing","Nonlocal adhesion and chemotaxis become local via gradient averaging"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem depends on replacing the original nonlocal operators by gradient-averaging versions that agree with them only away from a boundary layer, and on the smallness condition $C_{11}<1$; if the boundary-layer modification is judged illegitimate, or if that smallness condition fails, the stated convergence is not proved.","fun_headline_variants_meta":{"raw":{"variants":["Gradient-based averaging links nonlocal and local taxis models","Cell migration: nonlocal models converge to local via gradient averaging","Averaging gradients, not signals, yields local limits for cell sensing","Nonlocal adhesion and chemotaxis become local via gradient averaging"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000914,"raw_usage":{"total_tokens":3924,"prompt_tokens":940,"completion_tokens":2984,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":2911}},"tokens_in":556,"tokens_out":2984,"duration_ms":21273,"temperature":1.0,"reasoning_tokens":2911,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:48:34.975571+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the one-dimensional setting of Example 3.3 ($\\Omega=(-1,1)$, $F_r\\equiv 2$, $u\\equiv 1$) and compute numerically the $L^1$ norms of $A_r u$ and $T_r(\\nabla u)$ on $\\Omega$ as $r \\to 0$: the example predicts $\\|A_r u\\|_{L^1}$ stays about $1$ while $\\|T_r(\\nabla u)\\|_{L^1} \\to 0$, which directly checks the boundary-layer gap. Independently of the theorem, one could also run the nonlocal system (5.1) with coefficients violating (5.7) and measure whether the $L^2$ distance between nonlocal and local solutions still tends to zero as $r \\to 0$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the continuum cell-cell adhesion model whose adhesion velocity $A_r$ is the starting point for system (1.1)."},{"cited_title":"Gerisch and M","cited_arxiv_id":null,"evidence_quote":"Gives the local and nonlocal adhesion models and the heuristic Taylor expansion that the paper upgrades to a rigorous limit."},{"cited_title":"Hillen, K","cited_arxiv_id":null,"evidence_quote":"Introduces the finite-sampling-radius chemotaxis model and poses the convergence question answered here."},{"cited_title":"Othmer and T","cited_arxiv_id":null,"evidence_quote":"Supplies the nonlocal chemotaxis model with finite sensing radius that underlies system (1.4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the finite-volume numerical scheme used to simulate and compare the nonlocal and local formulations in 1D."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the monotone-operator theory for evolution equations that yields the existence and estimates for the single-equation problem (4.1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Leray-Schauder fixed-point principle used to obtain existence for the approximating problems (5.9)."},{"cited_title":"Ladyzhenskaya, V","cited_arxiv_id":null,"evidence_quote":"Supplies the semilinear parabolic regularity theory used to solve the signal equation for fixed coefficients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Lions lemma used in the compactness arguments when passing to the limit as $r \\to 0$."}],"review_version":1}