Blocking of 2D bistable reaction-diffusion fronts by obstacles

TL;DR

This paper develops a quantitative model to predict when two-dimensional bistable reaction-diffusion fronts are blocked by obstacles, considering geometry and nonlinearity, validated by numerical simulations.

Contribution

It introduces an analytical model combining conservation laws and exact solutions to determine front blocking thresholds in complex geometries.

Findings

Explicit conditions for front propagation in waveguides with conical obstacles.

The model accurately predicts blocking thresholds and agrees with simulations.

Heuristic rules are derived for front behavior in complex obstacle arrangements.

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. ~