Turing structures and stability for the 1-D Lengyel–Epstein system

On the global dynamics of the Lengyel–Epstein system

“…A bifurcation analysis from the constant steady state solution is developed in [8,11,10] for 1-dimensional and 2-dimensional domains, respectively. On the other hand, the presence of Turing patterns can be excluded if ðu à ; v Ã Þ is globally attractive for the Lengyel-Epstein system.…”

Turing Instability and Pattern Formation for the Lengyel–Epstein System with Nonlinear Diffusion

“…[10,11]. Also the analytical properties of the system have been widely studied: the Hopf bifurcation analysis has been performed in [28]; Turing instability and the pattern formation driven by linear diffusion have been investigated in different geometries [13,35,27,9]. The existence and non-existence for the steady states of the system have been proved in [31].…”

“…In this paper we show that the nonlinear diffusion facilitate the Turing instability and the formation of the Turing structures as compared to the case of linear diffusion: in particular, increasing the value of the parameter n in (1), the Turing instability arises even when the diffusion of the inhibitor is significantly slower than that of the activator (which is not the case when the diffusion is linear, i.e. when n = 0, see [31,35]). Moreover, as the Lengyel-Epstein model also supports the Hopf bifurcation, the formation of the Turing structure depends on the mutual location of the Hopf and Turing instability boundaries.…”

Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

“…The diffusion-driven instability of the equilibrium leads to a spatially inhomogeneous distribution of species concentration, which is the so-called Turing instability. Although Turing instability was first investigated in a morphogenesis, it has quickly spread to ecological systems [3,4,6,[13][14][15][16][17][18][19][20][21][22][23][24], chemical reaction system [25][26][27][28][29][30] and other reaction-diffusion system [31][32][33][34][35][36][37][38]. From [39], we know that the phenomenon of spatial pattern formation in (1) with diffusion can not occur under all possible diffusion rates.…”