Turing instabilities for three interacting species

TL;DR

This paper establishes precise mathematical conditions for Turing instabilities in systems with three interacting species, extending classical two-species results to more complex multi-species interactions.

Contribution

It provides necessary and sufficient inequality conditions for Turing instabilities in three-species systems, including cases with oscillatory growth (Turing-Hopf instability).

Findings

Derived inequality conditions for Turing instabilities with three species.

Distinguished conditions for steady and oscillatory Turing-Hopf instabilities.

Extended classical two-species Turing criteria to three-species systems.

Abstract

In this paper, I prove necessary and sufficient conditions for the existence of Turing instabilities in a general system with three interacting species. Turing instabilities describe situations when a stable steady state of a reaction system (ordinary differential equation) becomes an unstable homogeneous steady state of the corresponding reaction-diffusion system (partial differential equation). Similarly to a well-known inequality condition for Turing instabilities in a system with two species, I find a set of inequality conditions for a system with three species. Furthermore, I distinguish conditions for the Turing instability when spatial perturbations grow steadily and the Turing-Hopf instability when spatial perturbations grow and oscillate in time simultaneously.