Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

Abstract

In this article, Turing instability and the formations of spatial patterns for a general two-component reaction-diffusion system defined on 2D bounded domain, are investigated. By analyzing characteristic equation at positive constant steady states and further selecting diffusion rate d and diffusion ratio ε as bifurcation parameters, sufficient and necessary conditions for the occurrence of Turing instability are established, which is called the first Turing bifurcation curve. Furthermore, parameter regions in which single-mode Turing patterns arise and multiple-mode (or superposition) Turing patterns coexist when bifurcations parameters are chosen, are described. Especially, the boundary of parameter region for the emergence of single-mode Turing patterns, consists of the first and the second Turing bifurcation curves which are given in explicit formulas. Finally, by taking diffusive Schnakenberg system as an example, parameter regions for the emergence of various kinds of spatially inhomogeneous patterns with different spatial frequencies and superposition Turing patterns, are estimated theoretically and shown numerically.

Steady-state bifurcation and spike pattern in the Klausmeier-Gray-Scott model with non-diffusive plants

“…There exists mode-n Turing pattern below the Turing bifurcation curve ε * (n, d) if c ′′ n (0) < 0. Moreover, we obtain from Corollary 2 in [18] that multiple-mode superimposed patterns appear in some regions of the parameter plane. For example, system (1) may exhibit a superposition of mode-2 Turing patterns, mode-3 Turing patterns, and mode-4 Turing patterns when the parameters are taken within the region D in Figure 2.…”

“…For example, system (1) may exhibit a superposition of mode-2 Turing patterns, mode-3 Turing patterns, and mode-4 Turing patterns when the parameters are taken within the region D in Figure 2. For research on the superposition Turing pattern, please refer to [18,63].…”