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

You Li

School of Mathematics and System Science, Beihang University, Beijing 100191, P. R. China

E-mail Address: liyou1265@163.com

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Jinliang Wang

LMIB & School of Mathematics and System Science, Beihang University, Beijing 100191, P. R. China

E-mail Address: jlwang@buaa.edu.cn

Corresponding author.

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, and

Xiaojie Hou

Department of Mathematics and Statistics, University of North Carolina Wilmington, Wilmington, NC 28403, USA

E-mail Address: houx@uncw.edu

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https://doi.org/10.1142/S0218127417500183Cited by:10 (Source: Crossref)

Abstract

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.

This work was supported by the National Natural Science Foundation of China (Nos. 10971009, 10771196), the National Scholarship Fund (No. 201303070222) and the Fundamental Research Funds for the Central Universities.

Keywords:

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