scholarly journals Stability analysis of Turing patterns generated by the Schnakenberg model

2004 ◽  
Vol 49 (4) ◽  
pp. 358-390 ◽  
Author(s):  
David Iron ◽  
Juncheng Wei ◽  
Matthias Winter
Keyword(s):  
Stability Analysis ◽  
Turing Patterns ◽  
Schnakenberg Model

Stripe and Spot Patterns for General Gierer–Meinhardt Model with Common Sources

2017 ◽  
Vol 27 (02) ◽  
pp. 1750018 ◽  
Author(s):  
You Li ◽  
Jinliang Wang ◽  
Xiaojie Hou
Keyword(s):  
Periodic Solution ◽  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Theoretical Analysis ◽  
Stable Equilibrium ◽  
Turing Instability ◽  
Turing Patterns ◽  
Stability Conditions ◽  
Ode System ◽  
The Stability

This paper focuses on the Turing patterns in the general Gierer–Meinhardt model of morphogenesis. The stability analysis of the equilibrium for the associated ODE system is carried out and the stability conditions are obtained. Furthermore, we perform a detailed Hopf bifurcation analysis for this system. The results show that the equilibrium undergoes a supercritical Hopf bifurcation in certain parameter range and the bifurcated limit cycle is stable. With added diffusions, we then show that both the stable equilibrium and the Hopf periodic solution experience Turing instability with unequal spatial diffusions and obtain the instability conditions. Numerical simulations are given to illustrate the theoretical analysis, which show that the Turing patterns are of either spot or stripe type.

Pattern formation by fractional cross-diffusion in a predator–prey model with Beddington–DeAngelis type functional response

2019 ◽  
Vol 33 (25) ◽  
pp. 1950296
Author(s):  
Naveed Iqbal ◽  
Ranchao Wu
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Equilibrium Points ◽  
Turing Instability ◽  
Turing Patterns ◽  
Scaling Analysis ◽  
Amplitude Equations ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
The Stability

In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations. Finally, we ensure the existence of Turing patterns such as squares, spots and stripes by using the stability analysis of the amplitude equations. Moreover, with the assistance of numerical simulations, we verify the theoretical results.

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