scholarly journals Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

2010 ◽  
Vol 72 (8) ◽  
pp. 2161-2179 ◽  
Author(s):  
S. Seirin Lee ◽  
E. A. Gaffney
Keyword(s):  
Gene Expression ◽  
Time Delays ◽  
Reaction Diffusion ◽  
Self Organisation ◽  
Expression Time

scholarly journals Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay

2021 ◽  
Vol 26 (3) ◽  
pp. 461-481
Author(s):  
Shuangrui Zhao ◽  
Hongbin Wang ◽  
Weihua Jiang
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Time Delays ◽  
Spatiotemporal Patterns ◽  
Third Order ◽  
Spatially Inhomogeneous ◽  
Math Biol ◽  
Formation Systems ◽  
Expression Time

In this paper, we consider the dynamics of delayed Gierer–Meinhardt system, which is used as a classic example to explain the mechanism of pattern formation. The conditions for the occurrence of Turing, Hopf and Turing–Hopf bifurcation are established by analyzing the characteristic equation. For Turing–Hopf bifurcation, we derive the truncated third-order normal form based on the work of Jiang et al. [11], which is topologically equivalent to the original equation, and theoretically reveal system exhibits abundant spatial, temporal and spatiotemporal patterns, such as semistable spatially inhomogeneous periodic solutions, as well as tristable patterns of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. Especially, we theoretically explain the phenomenon that time delay inhibits the formation of heterogeneous steady patterns, found by S. Lee, E. Gaffney and N. Monk [The influence of gene expression time delays on Gierer–Meinhardt pattern formation systems, Bull. Math. Biol., 72(8):2139–2160, 2010.]

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