Unified framework for localized patterns in reaction-diffusion systems; the Gray-Scott and Gierer-Meinhardt cases.

Area of Science:

• Mathematical modeling
• Chemical kinetics
• Pattern formation

Background:

• Activator-inhibitor models are crucial for understanding pattern formation in biological and chemical systems.
• Canonical models like Schnakenberg-like systems have been studied extensively on infinite domains.
• Extending these models to include bistability, as seen in the Gray-Scott model, presents new theoretical challenges.

Purpose of the Study:

• To investigate pattern transitions in activator-inhibitor models with bistable equilibria.
• To explore the use of homotopy methods to connect different model types, specifically linking Schnakenberg-like and Gray-Scott models.
• To identify and characterize bifurcations governing pattern formation dynamics.

Main Methods:

• Utilized a homotopy approach to continuously deform a Schnakenberg-like glycolysis model into a Gray-Scott model.
• Employed numerical continuation techniques to analyze bifurcations in parameter space.
• Extended the analysis to the Gierer-Meinhardt system, which operates outside the canonical framework.

Main Results:

• Discovered several codimension-two bifurcations, including cusp and quadruple zero points for homogeneous steady states.
• Identified a degenerate heteroclinic connection and alterations in the homoclinic snaking structure.
• Observed similar pattern transition phenomena in the Gierer-Meinhardt system under homotopy, including changes related to active and inactive field feeds.

Conclusions:

• The study demonstrates a robust method for analyzing pattern transitions across different activator-inhibitor models.
• The findings reveal complex bifurcation behaviors and structural changes in pattern formation.
• The results have broader implications for understanding pattern formation in diverse natural phenomena and contribute to Turing's theory of morphogenesis.