Spot Patterns in the 2-D Schnakenberg Model with Localized Heterogeneities

TL;DR

This paper develops a hybrid asymptotic-numerical theory to analyze how localized heterogeneities affect pattern formation, stability, and dynamics in the 2-D Schnakenberg reaction-diffusion model, revealing novel phenomena like bifurcations and spot self-replication.

Contribution

The paper introduces a hybrid asymptotic-numerical approach to study localized heterogeneities' impact on spot patterns in the Schnakenberg model, uncovering new bifurcation and pattern behaviors.

Findings

Discovery of saddle-node bifurcations in spot patterns.

Identification of spot self-replication phenomena.

Prediction of spot attraction or repulsion due to heterogeneities.

Abstract

A hybrid asymptotic-numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two-component Schnakenberg reaction-diffusion model in a 2-D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed rate and the effect of removing a small hole in the domain, through which the chemical species can leak out. Our hybrid theory reveals a wide range of novel phenomena such as, saddle-node bifurcations for quasi-equilibrium spot patterns that otherwise would not occur for a homogeneous medium, a new type of spot solution pinned at the concentration point of the feed rate, spot self-replication behavior leading to the creation of more than two new spots, and the existence of a creation-annihilation attractor with at most three spots. Depending on the type of localized heterogeneity introduced, localized spots are either repelled or attracted towards the localized defect on asymptotically long time scales. Results for slow spot dynamics and detailed predictions of various instabilities of quasi-equilibrium spot patterns, all based on our hybrid asymptotic-numerical theory, are illustrated and confirmed through extensive full PDE numerical simulations.