Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model

Abstract

For a model glycolytic diffusion-reaction system, an amplitude equation has been derived in the framework of a weakly nonlinear theory. The linear stability analysis of this amplitude equation interprets the structural transitions and stability of various forms of Turing structures. This amplitude equation also conforms to the expectation that time-invariant amplitudes in Turing structures are independent of complexing reaction with the activator species, whereas complexing reaction strongly influences Hopf-wave bifurcation.

Pattern formation in the FitzHugh–Nagumo model

“…The proof of Lemma 1 and Theorem 1 can be obtained easily by using the theorem in [24], we ignore it here.…”

“…However, the amplitude equation is a lengthy process, and only some systems have been chosen in the past for amplitude equation [24]. In this paper, we would make research on pattern selection of FitzHugh-Nagumo model (FN model) by using the standard multiple scale analysis [25,26].…”

Dynamics and pattern formation in a cancer network with diffusion

“…But analysis of the amplitude equation is a time-consuming process. Only a few systems have been previously analyzed in this way [23]. Pattern selection can be studied with standard multiple scale analysis [24,25], but previous work did not take into account the effect of space, or their results were obtained by choosing particular initial conditions [26].…”

Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

“…In general, Turing model contains two reactants: activator and inhibitor, which engage in diffusion. Recently, the study of Turing bifurcation, amplitude equation, and secondary bifurcation have paid more attention on the pattern formation [2][3][4], and Lee and Cho found that dynamical parameters and external periodic forcing play an important role in the shape and type of pattern formation [5]. And the robustness problem is also investigated [6].…”

“…Recently, the pattern formation with three or four variables has been investigated, and it obtains promising results [19,20], and Xu et al made a concrete analysis with three variables [21]. As we all know that amplitude equation is a promising tool to investigate the pattern dynamics of the reaction-diffusion system [2,22], however, the amplitude equation is a complex process [3], and the researcher often chose the amplitude equation [23][24][25][26] to investigate the reaction-diffusion system. In conclusion, spatial patterns in reaction-diffusion systems have attracted the interest of experimentalists and theorists during the last few decades.…”