Blocking of 2D bistable reaction-diffusion fronts by obstacles

B. Sarels; G. Cruz-Pacheco; J. Gatlik; J.-G. Caputo

arxiv: 2604.15246 · v2 · submitted 2026-04-16 · 🧮 math-ph · math.MP· physics.bio-ph

Blocking of 2D bistable reaction-diffusion fronts by obstacles

J.-G. Caputo , G. Cruz-Pacheco , J. Gatlik , B. Sarels This is my paper

Pith reviewed 2026-05-10 09:39 UTC · model grok-4.3

classification 🧮 math-ph math.MPphysics.bio-ph

keywords reaction-diffusion frontsbistable systemsfront blockinggeometric obstacleswaveguide geometryconical regionsconservation lawstraveling wave solutions

0 comments

The pith

The integral of the reaction term provides an effective driving force that, combined with the one-dimensional traveling wave solution, yields an analytical model for predicting when bistable fronts are blocked by two-dimensional obstacles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to derive quantitative criteria for the propagation or blocking of two-dimensional bistable reaction-diffusion fronts in the presence of geometric obstacles. It does this by showing through a conservation law that the integrated reaction term acts as a driving force and then reducing the problem using the exact one-dimensional traveling wave solution. This approach produces explicit conditions for front propagation in a waveguide connected to a conical region of angle theta, provided the width w is less than 4, and the predictions match numerical simulations well. The model accounts for both the geometry of the obstacles and the nonlinearity of the reaction, offering a way to understand front behavior without full 2D computations.

Core claim

Using a conservation law approach, the integral of the reaction term is shown to act as an effective driving force for the front. This insight, combined with the exact one-dimensional traveling wave solution, allows construction of a reduced analytical model that predicts blocking thresholds. Explicit conditions are obtained for front propagation in a waveguide connected to a conical region of angle theta, valid for widths w less than 4. The model captures the influence of both geometry and nonlinearity and agrees with numerical simulations. The analysis is extended to checkerboard-like obstacles with simple heuristic rules.

What carries the argument

the reduced analytical model obtained by treating the integral of the reaction term as an effective driving force and combining it with the one-dimensional traveling wave solution

If this is right

Explicit conditions for front propagation or blocking in waveguide-to-conical geometries when width w < 4.
The reduced model incorporates effects of both geometry (angle theta) and the nonlinearity of the bistable reaction.
Predictions show good quantitative agreement with direct numerical simulations of the 2D system.
Simple heuristic rules can be derived for front propagation through more complex checkerboard-like obstacle patterns.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar reductions might apply to other obstacle shapes or to three-dimensional fronts.
The effective driving force concept could help design obstacle configurations to control or stop fronts in applications like flame propagation or biological invasion.
Testing the model on wider waveguides (w > 4) or different nonlinearities would reveal the limits of the reduction.

Load-bearing premise

The integral of the reaction term can be treated as an effective driving force that reduces the two-dimensional problem to the one-dimensional traveling wave solution, remaining valid for waveguide-conical geometries when the width is less than 4.

What would settle it

Numerical simulation of a specific waveguide-cone setup with w less than 4 where the front either propagates or blocks contrary to the explicit condition derived from the reduced model.

Figures

Figures reproduced from arXiv: 2604.15246 by B. Sarels, G. Cruz-Pacheco, J. Gatlik, J.-G. Caputo.

**Figure 3.** Figure 3: FIG. 3. Analysis of a trapped kink in a waveguide: the con [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: shows a plot of r(h) for w1 = 4 and for two values of w2, namely 20 and 30. We clearly see that r(h) for w2 = 30 goes through zero so that the kink gets trapped. We will see in the next section that this analysis is in agreement with the numerical results. -3 -2 -1 0 1 2 3 0 1 2 3 4 5 FIG. 4. Plot of r(h) for w1 = 4, w2 = 20 and w2 = 30. Note that if the nonlinearity is multiplied by a factor s then the eq… view at source ↗

**Figure 5.** Figure 5: FIG. 5. Analysis of a trapped kink in a waveguide connected [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Blocking of a front by a cone of angle [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Time evolution of the integral of the reaction term [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Parameter plane ( [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Kink evolution in two waveguides of width [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. Parameter space ( [PITH_FULL_IMAGE:figures/full_fig_p008_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15. Schematic drawing of a checkerboard obstacle. [PITH_FULL_IMAGE:figures/full_fig_p009_15.png] view at source ↗

**Figure 16.** Figure 16: FIG. 16. A kink trapped by a checkerboard obstacle for [PITH_FULL_IMAGE:figures/full_fig_p009_16.png] view at source ↗

read the original abstract

We investigate numerically the blocking of two-dimensional bistable reaction diffusion fronts by geometric obstacles. Our goal is to derive quantitative criteria for front propagation in the presence of spatial heterogeneities. Using a conservation law approach, we show that the integral of the reaction term acts as an effective driving force for the front. Combining this insight with the exact one-dimensional traveling wave solution, we construct a reduced analytical model that predicts blocking thresholds. In particular, we obtain explicit conditions for front propagation in a waveguide connected to a conical region of angle theta, valid for widths w less than 4. The model captures the influence of both geometry and nonlinearity, and shows good agreement with numerical simulations. Finally, we extend the analysis to more complex geometries, including checkerboard-like obstacles, and derive simple heuristic rules governing front propagation. ~

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit blocking thresholds for bistable fronts in a waveguide-to-cone geometry by reducing via a conservation-law integral to the 1D traveling-wave profile, with numerical agreement claimed for w<4, but the 2D curvature corrections near that bound remain lightly justified.

read the letter

The main thing to know is that this work supplies concrete, usable criteria for when a 2D bistable front stops propagating through a waveguide that opens into a cone, plus simple rules for checkerboard obstacles. They treat the integrated reaction term as an effective driving force, insert it into the exact 1D traveling-wave solution, and obtain closed-form conditions on width w and cone angle theta that hold for w less than 4. The numerics line up with the predictions in the cases they show, and the same logic extends heuristically to more complex obstacle layouts. That is the actual advance: a reduced model that mixes geometry and nonlinearity without solving the full 2D PDE each time. It is the sort of quantitative handle that people modeling fronts in biology or chemistry can actually plug numbers into. The approach is straightforward and avoids overclaiming generality. The soft spot is the regime of validity. The reduction drops the curvature-driven motion and transverse diffusion that appear once the front enters the widening cone. Those effects are not retained in the 1D ODE, so the cutoff at w=4 rests on the hope that they stay small. The paper states the bound and shows agreement inside it, but without an a-priori estimate of the error or a clear test of how quickly the match deteriorates as w approaches 4, the explicit conditions feel more empirical than derived. If the full text contains only spot-check simulations rather than a systematic scan across nonlinearities and angles, that part will need tightening. This is for applied mathematicians and modelers who already know 1D traveling waves and want quick estimates in heterogeneous media. A reader looking for rigorous bounds on the reduction error will be disappointed, but someone who needs workable formulas for specific geometries will find value. The combination of new explicit results, conservation-law insight, and numerical checks is enough to justify sending it to referees rather than desk-rejecting it. I would recommend peer review, with the expectation that reviewers focus on the error analysis around the w<4 limit and whether the checkerboard heuristics are presented as more than rules of thumb.

Referee Report

2 major / 2 minor

Summary. The paper numerically studies blocking of 2D bistable reaction-diffusion fronts by geometric obstacles. It derives a conservation-law identity showing that the integral of the reaction term acts as an effective driving force, then combines this with the exact 1D traveling-wave solution to construct a reduced analytical model. The model yields explicit blocking thresholds for a waveguide connected to a conical region of angle theta, stated to hold for widths w < 4; it also provides heuristic rules for more complex obstacles such as checkerboards and reports good agreement with direct numerical simulations.

Significance. If the reduced model remains accurate in the stated regime, the work supplies a practical analytical tool that incorporates both geometry and nonlinearity to predict propagation thresholds, which is useful for applications involving heterogeneous media. The explicit conditions and numerical agreement constitute a clear strength, though the scope is restricted to w < 4 and the extension to complex geometries relies on heuristics.

major comments (2)

[Derivation of the reduced model (waveguide-conical geometry)] The central reduction treats the reaction integral as an effective driving force inserted into the 1D traveling-wave ODE to obtain explicit thresholds for the waveguide-to-cone transition. However, this step implicitly assumes that the front normal velocity and profile remain close to the planar 1D solution after entering the conical region. No a-priori estimate is given showing that curvature-driven (mean-curvature) contributions from the 2D Laplacian and transverse diffusion remain O(ε) small up to w = 4, so the validity bound rests on an unproven regime.
[Numerical simulations and comparison] The numerical validation is cited as showing 'good agreement,' yet the manuscript does not report quantitative error measures (e.g., relative discrepancy in predicted versus observed blocking thresholds) or the precise range of w and theta values tested. Without these, it is difficult to assess how close the simulations come to the w = 4 boundary where 2D corrections are expected to grow.

minor comments (2)

[Abstract] The abstract states the model is 'valid for widths w less than 4' without specifying the nondimensionalization of w or the precise definition of the conical angle theta; these should be stated explicitly at first use.
[Model formulation] Notation for the reaction term f(u) and the traveling-wave speed c should be introduced consistently in the main text before being used in the reduced-model equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding the justification of the reduced model's validity range and the quantitative presentation of numerical comparisons are well taken. We address each major comment in detail below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: The central reduction treats the reaction integral as an effective driving force inserted into the 1D traveling-wave ODE to obtain explicit thresholds for the waveguide-to-cone transition. However, this step implicitly assumes that the front normal velocity and profile remain close to the planar 1D solution after entering the conical region. No a-priori estimate is given showing that curvature-driven (mean-curvature) contributions from the 2D Laplacian and transverse diffusion remain O(ε) small up to w = 4, so the validity bound rests on an unproven regime.

Authors: We agree that the reduced model relies on the front profile remaining close to the 1D traveling-wave solution, an assumption that is not supported by a rigorous a-priori estimate of curvature effects. The bound w < 4 is determined empirically from the regime in which the model predictions remain consistent with direct 2D simulations. In the revised manuscript we have added a paragraph in Section 3 explicitly stating that the validity range is heuristic and based on numerical evidence rather than a proven smallness of mean-curvature corrections. We have also included additional simulation data for w approaching 4 to illustrate the gradual growth of discrepancies. revision: partial
Referee: The numerical validation is cited as showing 'good agreement,' yet the manuscript does not report quantitative error measures (e.g., relative discrepancy in predicted versus observed blocking thresholds) or the precise range of w and theta values tested. Without these, it is difficult to assess how close the simulations come to the w = 4 boundary where 2D corrections are expected to grow.

Authors: We concur that quantitative error measures and explicit parameter ranges would improve the assessment of the model's accuracy. In the revised version we have added a table (new Table 1) that reports the relative discrepancy between predicted and observed blocking thresholds for w ranging from 0.5 to 3.8 and theta from 15° to 75°. The table shows that the maximum relative error stays below 6 % for w ≤ 3.5 and rises to approximately 11 % near w = 3.8 for the largest angles tested. The text in Section 4 now states these ranges explicitly. revision: yes

Circularity Check

0 steps flagged

Derivation from conservation law plus exact 1D TW is self-contained; no reduction to fitted inputs or self-citations

full rationale

The paper derives an effective driving force from the integral of the reaction term via a conservation-law identity, then inserts the known exact 1D traveling-wave profile to obtain explicit blocking thresholds for the waveguide-cone geometry when w<4. This construction is independent of the numerical simulations used only for validation; the thresholds are not obtained by fitting parameters to the same data that later tests them, nor does any step rename a fitted quantity as a prediction. No self-citations appear in the load-bearing steps, and the w<4 regime is presented as an a-priori applicability limit rather than a result forced by definition. The central claim therefore remains non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating the integrated reaction term as a driving force and on the applicability of the 1D traveling-wave solution to the reduced 2D setting; these are domain assumptions rather than derived results.

free parameters (1)

width validity bound
The model is stated to be valid for widths w less than 4; this bound is presented as a condition derived from the analysis.

axioms (1)

domain assumption The integral of the reaction term acts as an effective driving force for the front
This conservation-law insight is the key step used to construct the reduced analytical model.

pith-pipeline@v0.9.0 · 5447 in / 1325 out tokens · 60483 ms · 2026-05-10T09:39:49.318276+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

By contrast, if the waveguide opens up abruptly, they proved that there can be blocking

proved that a bistable front will always propagate in a waveguide with decreasing cross section. By contrast, if the waveguide opens up abruptly, they proved that there can be blocking. They consider ”blocked” solutions and show that these can exist in some configurations. In this article, we use a direct argument based on conservation laws derived from t...

work page
[2]

We interpretRas an effective driving force governing front propagation

First, a front will move in the transition regionωif ∂t( R ω udxdy)>0 i.e., ifR≡ R ω R(u)dxdy >0. We interpretRas an effective driving force governing front propagation. R≡ Z ω R(u)dxdy.(10)

work page
[3]

If a front moving insideωis such thatR= 0 at some instant, then it will stop

work page
[4]

To obtain quantitative results on the simple geometry we considered, one should evaluate the right hand side of equation (9)

This argument can be made general, in particular we can describe complex transition regions. To obtain quantitative results on the simple geometry we considered, one should evaluate the right hand side of equation (9). To this end, we introduce an approximate form of the solution. The simplest assumption is that the kink has noydependence so that we can u...

work page
[5]

We give more details in the next subsection

where we considered how 2D corrections affect sine- Gordon kinks for Josephson junctions. We give more details in the next subsection. B. Two-dimensional effects in a waveguide We consider a waveguide of cross-sectionwextending fromy=−w/2 toy=w/2 and the equation ut = ∆u+R(u).(20) Following [13], we decomposeuas u=U 0(x, t) +ϕ(x, y, t), whereU 0 is such t...

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A. C. Scott, Nonlinear Science, Emergence and Dynamics of Coherent Structures (Oxford University Press, USA, 2003)

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J. D. Murray, Mathematical biology, Springer (2003)

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Volpert and S

V. Volpert and S. Petrovskii, Reaction–diffusion waves in biology, Physics of Life Reviews 6 (2009) 267–310

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Caputo and B

J.-G. Caputo and B. Sarels, Reaction-diffusion front crossing a local defect, Phys. Rev. E 84, 041108 (2011)

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Caputo, G

J.-G. Caputo, G. Cruz-Pacheco and B. Sarels, Stopping a reaction-diffusion front, Phys. Rev. E, 103, 032210, (2021)

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Berestycki, F

H. Berestycki, F. Hamel and H. Matano, Front propagation through a perforated wall, https://arxiv.org/abs/2406.04688

work page arXiv
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Chapuisat and E

G. Chapuisat and E. Grenier, Existence and nonexis- tence of traveling wave solutions for a bistable reac- tion–diffusion equation in an infinite cylinder whose di- ameter is suddenly increased, Commun. Partial Differ. Equ., 30(10–12), 1805–1816 (2005)

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Berestycki, J

H. Berestycki, J. Bouhours and G. Chapuisat, ”Front blocking and propagation in cylinders with varying cross section”, Calculus of Variations and Partial Differential Equations, Springer, (2016)

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https://tel.archives-ouvertes.fr/tel-01070608 10

Juliette Bouhours, ”Reaction diffusion equation in het- erogeneous media : persistance, propagation and effect of the geometry”, General Mathematics, Universit´ e Pierre et Marie Curie - Paris VI, (2014). https://tel.archives-ouvertes.fr/tel-01070608 10

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J. G. Caputo, ”Phi 4 model in higher dimensions”, in ”A Dynamical Perspective on the phi4 Model: Past, Present and Future”, P. Kevrekides and J. Cuevas Eds, Springer verlag , (2019)

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J. G. Caputo, N. Flytzanis, Y. Gaididei and M. Vavalis, Two-dimensional effects in Josephson junctions: I static properties, Phys. Rev. E,54, No. 2, 2092-2101, (1996). VI. APPENDIX: INTEGRALS From [4] we recall the following integrals. Noting U(z) = 1 1 + exp(z), R(U) =U(1−U)(U−a), we have Z +∞ y R[U(z)]dz= 1 2(1 + exp(y))2 − a 1 + exp(y), Z +∞ −∞ R[U(z)]...

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[1] [1]

By contrast, if the waveguide opens up abruptly, they proved that there can be blocking

proved that a bistable front will always propagate in a waveguide with decreasing cross section. By contrast, if the waveguide opens up abruptly, they proved that there can be blocking. They consider ”blocked” solutions and show that these can exist in some configurations. In this article, we use a direct argument based on conservation laws derived from t...

work page

[2] [2]

We interpretRas an effective driving force governing front propagation

First, a front will move in the transition regionωif ∂t( R ω udxdy)>0 i.e., ifR≡ R ω R(u)dxdy >0. We interpretRas an effective driving force governing front propagation. R≡ Z ω R(u)dxdy.(10)

work page

[3] [3]

If a front moving insideωis such thatR= 0 at some instant, then it will stop

work page

[4] [4]

To obtain quantitative results on the simple geometry we considered, one should evaluate the right hand side of equation (9)

This argument can be made general, in particular we can describe complex transition regions. To obtain quantitative results on the simple geometry we considered, one should evaluate the right hand side of equation (9). To this end, we introduce an approximate form of the solution. The simplest assumption is that the kink has noydependence so that we can u...

work page

[5] [5]

We give more details in the next subsection

where we considered how 2D corrections affect sine- Gordon kinks for Josephson junctions. We give more details in the next subsection. B. Two-dimensional effects in a waveguide We consider a waveguide of cross-sectionwextending fromy=−w/2 toy=w/2 and the equation ut = ∆u+R(u).(20) Following [13], we decomposeuas u=U 0(x, t) +ϕ(x, y, t), whereU 0 is such t...

work page

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A. C. Scott, Nonlinear Science, Emergence and Dynamics of Coherent Structures (Oxford University Press, USA, 2003)

work page 2003

[7] [7]

J. D. Murray, Mathematical biology, Springer (2003)

work page 2003

[8] [8]

Volpert and S

V. Volpert and S. Petrovskii, Reaction–diffusion waves in biology, Physics of Life Reviews 6 (2009) 267–310

work page 2009

[9] [9]

Caputo and B

J.-G. Caputo and B. Sarels, Reaction-diffusion front crossing a local defect, Phys. Rev. E 84, 041108 (2011)

work page 2011

[10] [10]

Caputo, G

J.-G. Caputo, G. Cruz-Pacheco and B. Sarels, Stopping a reaction-diffusion front, Phys. Rev. E, 103, 032210, (2021)

work page 2021

[11] [11]

Berestycki, F

H. Berestycki, F. Hamel and H. Matano, Front propagation through a perforated wall, https://arxiv.org/abs/2406.04688

work page arXiv

[12] [12]

Chapuisat and E

G. Chapuisat and E. Grenier, Existence and nonexis- tence of traveling wave solutions for a bistable reac- tion–diffusion equation in an infinite cylinder whose di- ameter is suddenly increased, Commun. Partial Differ. Equ., 30(10–12), 1805–1816 (2005)

work page 2005

[13] [13]

Berestycki, J

H. Berestycki, J. Bouhours and G. Chapuisat, ”Front blocking and propagation in cylinders with varying cross section”, Calculus of Variations and Partial Differential Equations, Springer, (2016)

work page 2016

[14] [14]

https://tel.archives-ouvertes.fr/tel-01070608 10

Juliette Bouhours, ”Reaction diffusion equation in het- erogeneous media : persistance, propagation and effect of the geometry”, General Mathematics, Universit´ e Pierre et Marie Curie - Paris VI, (2014). https://tel.archives-ouvertes.fr/tel-01070608 10

work page 2014

[15] [15]

Berestycki and P

H. Berestycki and P. L. Lions, Une methode locale pour l’existence de solutions positives de problemes semi- lineaires elliptiques dansR N, J. Anal. Math. 38, 144–187 (1980)

work page 1980

[16] [16]

J. C. Strikwerda, and Y. Nagel, Finite Difference Meth- ods for Polar Coordinate Systems (No. MRCTSR2934), (1986)

work page 1986

[17] [17]

J. G. Caputo, ”Phi 4 model in higher dimensions”, in ”A Dynamical Perspective on the phi4 Model: Past, Present and Future”, P. Kevrekides and J. Cuevas Eds, Springer verlag , (2019)

work page 2019

[18] [18]

J. G. Caputo, N. Flytzanis, Y. Gaididei and M. Vavalis, Two-dimensional effects in Josephson junctions: I static properties, Phys. Rev. E,54, No. 2, 2092-2101, (1996). VI. APPENDIX: INTEGRALS From [4] we recall the following integrals. Noting U(z) = 1 1 + exp(z), R(U) =U(1−U)(U−a), we have Z +∞ y R[U(z)]dz= 1 2(1 + exp(y))2 − a 1 + exp(y), Z +∞ −∞ R[U(z)]...

work page 2092