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Keywords = Gierer-Meinhardt system

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18 pages, 1290 KiB  
Article
Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative
by Messaoud Berkal and Mohammed Bakheet Almatrafi
Fractal Fract. 2023, 7(5), 344; https://doi.org/10.3390/fractalfract7050344 - 22 Apr 2023
Cited by 38 | Viewed by 1890
Abstract
In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the [...] Read more.
In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains. Full article
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14 pages, 21978 KiB  
Article
Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System
by Biao Liu and Ranchao Wu
Mathematics 2022, 10(2), 243; https://doi.org/10.3390/math10020243 - 13 Jan 2022
Cited by 5 | Viewed by 2579
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 [...] Read more.
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. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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