Ricci Flow and Nonlinear Reaction--Diffusion Systems in Biology, Chemistry, and Physics

TL;DR

This paper introduces a geometric framework based on Ricci flow to model nonlinear reaction-diffusion systems across biology, chemistry, and physics, aiming to unify various models under a common geometric approach.

Contribution

It proposes a novel conjecture that Ricci flow can serve as a universal geometric model for reaction-diffusion processes in multiple scientific fields.

Findings

Demonstrates that several popular reaction-diffusion systems can be embedded in the Ricci flow framework.

Provides a review linking geometric Ricci flow to biological, chemical, and physical models.

Suggests a unifying geometric perspective for nonlinear dissipative systems.

Abstract

This paper proposes the Ricci-flow equation from Riemannian geometry as a general geometric framework for various nonlinear reaction-diffusion systems (and related dissipative solitons) in mathematical biology. More precisely, we propose a conjecture that any kind of reaction-diffusion processes in biology, chemistry and physics can be modelled by the combined geometric-diffusion system. In order to demonstrate the validity of this hypothesis, we review a number of popular nonlinear reaction-diffusion systems and try to show that they can all be subsumed by the presented geometric framework of the Ricci flow. Keywords: geometrical Ricci flow, nonlinear reaction-diffusion, dissipative solitons and breathers