Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 449, Issue 2, Pages 1863-1879Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2017.01.019
Keywords
Turing patterns; Stability; Hopf bifurcation; Supercritical; Subcritical; Saturated
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Funding
- National Natural Science Foundation of China [10971009, 10771196]
- National Scholarship Fund [201303070222]
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The Gierer-Meinhardt model of morphogenesis with saturated activator production is considered. For the unique positive equilibrium of the kinetic equations, the precise parameter conditions of stability, instability and Hopf bifurcation are obtained. It is shown that the equilibrium can either undergo supercritical or subcritical Hopf bifurcation under certain parameter range. Furthermore, it is proved that there exists at least one stable limit cycle besides the periodic solution bifurcating from Hopf bifurcation. In addition, Turing instability conditions on the parameters and diffusion coefficients for the positive equilibrium and the periodic solution bifurcating from Hopf bifurcation are given. The dynamics of the model are illustrated by numerical simulations which exhibit that Turing patterns are of either stripe or spot type. (C) 2017 Elsevier Inc. All rights reserved.
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