Abstract
The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.
Highlights
When discrete step increases from t to t + 1, the couple map lattice model (CML) dynamics of the activator and inhibitor consists of two stages; one is the “reaction" stage and the other is “dispersal” stage
We show the spatiotemporally dynamics of Turing instability for flip bifurcation and Neimark–Sacker bifurcation
The flip, Neimark–Sacker and Turing bifurcations of a spatiotemporal discrete GiererMeinhardt system are investigated in this paper
Summary
Ward and Wei [7] studied the stability and oscillatory instability of symmetric k-spike equilibrium solutions to the Gierer-Meinhardt reaction-diffusion system in a one-dimensional spatial domain for various ranges of the reaction-time constant and the diffusivity of the inhibitor field dynamics. Some existing methods and theoretical results can be applied to study the dynamics of the CMLs model [23,24,25,26,27,28]. We will investigate the model theoretically to determine the conditions for such bifurcations to a modified Gierer-Meinhardt system based on CMLs. the influence of parameters on the patterns formation can be illustrated quantitatively. In order to obtain the spatiotemporal discrete Gierer-Meinhardt system, the CMLs method apply to a modified Gierer-Meinhardt model in this paper.
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