Turing patterns of Gierer–Meinhardt model on complex networks

Guo, Luyao; Shi, Xinli; Cao, Jinde

doi:10.1007/s11071-021-06618-6

Cited by 13 publications

(9 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Pasemann et al [5] developed a theory for diffusivity estimation for the activator-inhibitor model. Guo et al [6] discussed the Turing patterns of the activator-inhibitor system on regular lattice networks. In addition, several complex networks were also examined in [6].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

Berkal

Almatrafi

2023

Fractal Fract

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Guo et al [6] discussed the Turing patterns of the activator-inhibitor system on regular lattice networks. In addition, several complex networks were also examined in [6]. Song et al [7] investigated the stability and the bifurcation of the activator-inhibitor system with a saturating term.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

Berkal

Almatrafi

2023

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…In the past decade, there appeared a large number of achievements which has broadened our understanding of patterns in various networks. For example, Mimar et al [16] reported that the Turning instability in undirected (symmetry) networks can be induced by mediating network topology such as network connectivity, and Guo et al [10] presented an exponential decay of Turning patterns on undirected networks, providing a way to quantitatively characterize the influence of network topology on pattern formations. Generally, the topology of undirected networks might determine the relevant directions for the spreading of the perturbation, but the onset of Turning instability will emerge under the very same conditions that apply when the system is defined on a continuous support.…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation and Its Normal Form of Reaction Diffusion Systems Defined on Directed Networks

Cui¹,

Jin²

2023

CSIAM-AM

View full text Add to dashboard Cite

Compared with the real Laplacian eigenvalues of undirected networks, the ones of asymmetrical directed networks might be complex, which is able to trigger additional collective dynamics, including the oscillatory behaviors. However, the high dimensionality of the reaction-diffusion systems defined on directed networks makes it difficult to do in-depth dynamic analysis. In this paper, we strictly derive the Hopf normal form of the general two-species reaction-diffusion systems defined on directed networks, with revealing some noteworthy differences in the derivation process from the corresponding on undirected networks. Applying the obtained theoretical framework, we conduct a rigorous Hopf bifurcation analysis for an SI reaction-diffusion system defined on directed networks, where numerical simulations are well consistent with theoretical analysis. Undoubtedly, our work will provide an important way to study the oscillations in directed networks.

show abstract

“…The statistical properties of the network structure have been studied [2][3][4][5][6][7][8] that each of real-world networks has a different physical form and is described in its respective systems, which has made great progress and revealed the inherent relationship between the real systems. By the rapid development of network theory and the application of computer technology, figures are collected more true and analyzed more accurate.…”

Section: Introductionmentioning

confidence: 99%

“…The degrees at the both end points of any given edge are not usually independent. It has significant effect on the dynamical evolution processes of the real life complex networks, such as liquid percolation [8] , games [18] , synchronization [17] and epidemic spreading [7] .…”

Section: Introductionmentioning

confidence: 99%

Scale-free and small-world properties in an incompatibility hierarchical network

Wang

Tian

2022

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

In accordance with a particular iterative method, an incompatibility hierarchical network is introduced and the topological features of the network is investigated, It contains the degree distribution, diameter, clustering coefficient, second moment of distribution and degree correlations. The incompatibility hierarchical network is a simultaneous small-world, scale-free, assortative and divergent, which makes it possible to research the complexity of some real networks.

show abstract

Turing patterns of Gierer–Meinhardt model on complex networks

Cited by 13 publications

References 31 publications

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

Hopf Bifurcation and Its Normal Form of Reaction Diffusion Systems Defined on Directed Networks

Scale-free and small-world properties in an incompatibility hierarchical network

Contact Info

Product

Resources

About