# Classification of parameter spaces for reaction-diffusion systems on stationary domains

## TL;DR

This paper classifies parameter spaces for reaction-diffusion systems on stationary domains, analyzing how domain size, reaction, and diffusion rates influence system stability and bifurcations, supported by theoretical and numerical results.

## Contribution

It provides a comprehensive classification of parameter spaces for reaction-diffusion systems, linking domain size and parameters to different bifurcation types, with new theoretical bounds and conditions.

## Key findings

- Domain size bounds determine bifurcation types.
- Turing instability occurs within specific domain size conditions.
- Numerical simulations support theoretical predictions.

## Abstract

This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. In the absence of diffusion the results on the classification of parameter space are supported by simulations of the corresponding vector-field and some trajectories around the uniform steady state. In the presence of diffusion, the main findings are the quantitative analysis relating the domain-size with the reaction and diffusion rates and their corresponding influence on the dynamics of the reaction-diffusion system when perturbed in the neighbourhood of the uniform steady state. Theoretical predictions are supported by numerical simulations both in the presence as well as in the absence of diffusion. Conditions on the domain size with respect to the diffusion and reaction rates are related to the types of diffusion-driven instabilities namely Turing, Hopf and Transcritical types of bifurcations. The first condition is an upper bound on the area of a rectangular domain in terms of the diffusion and reaction rates, which forbids the existence of Hopf and Transcritical types of bifurcations, yet allowing Turing type instability to occur. The second condition (necessary) is a lower bound on the domain size in terms of the reaction and diffusion rates to give rise to Hopf and Transcritical types of bifurcations as well as Turing instability.