Internal composite bound states in deterministic reaction diffusion models

TL;DR

This paper demonstrates how composite states in the Sel'kov-Gray-Scott model emerge from a fundamental theory at small scales and evolve into the effective large-scale patterns, illustrating dynamical decoupling.

Contribution

It introduces a fundamental theory with composite states obeying diffusion equations, linking it to the effective GS model at large scales, and highlights scale separation in reaction diffusion systems.

Findings

Composite states act as fundamental species at short scales.

Large-scale patterns emerge from the fundamental theory as parameters vary.

Scale separation exemplifies dynamical decoupling in reaction diffusion models.

Abstract

By identifying potential composite states that occur in the Sel'kov-Gray-Scott (GS) model, we show that it can be considered as an effective theory at large spatio-temporal scales, arising from a more \textit{fundamental} theory (which treats these composite states as fundamental chemical species obeying the diffusion equation) relevant at shorter spatio-temporal scales. When simulations in the latter model are performed as a function of a parameter $M = \lambda^{-1}$, the generated spatial patterns evolve at late times into those of the GS model at large $M$, implying that the composites follow their own unique dynamics at short scales. This separation of scales is an example of \textit{dynamical} decoupling in reaction diffusion systems.