Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay
Shuangrui Zhao
Harbin Institute of Technology
Hongbin Wang
Harbin Institute of Technology
Weihua Jiang
Harbin Institute of Technology
Published 2021-05-01


Gierer–Meinhardt system
Turing–Hopf bifurcation
normal form
spatiotemporal patterns

How to Cite

Zhao S., Wang H. and Jiang W. (2021) “Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay”, Nonlinear Analysis: Modelling and Control, 26(3), pp. 461-481. doi: 10.15388/namc.2021.26.23054.


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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