Abstract
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.]
Highlights
In developmental biology, embryonic development is mediated by morphogens
By employing these formulas we theoretically prove the existence of various spatiotemporal patterns instead of computational simulations [14, 15], such as semistable spatially inhomogeneous periodic solutions, as well as tristability of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting, in addition, quantitatively give the specific existence region of various forms of solutions near the Turing–Hopf singularity
In order to explore the joint effect of diffusion and time delay, we further investigate the Turing–Hopf bifurcation
Summary
Embryonic development is mediated by morphogens. It is a signal molecule that determines the location, differentiation and fate of many surrounding cells [9]. Jiang et al [11] have derived the formulas of calculating normal forms for a general delayed reaction diffusion equation with Neumann boundary condition, which can greatly simplify the complexity of calculation By employing these formulas we theoretically prove the existence of various spatiotemporal patterns instead of computational simulations [14, 15], such as semistable spatially inhomogeneous periodic solutions, as well as tristability of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting, in addition, quantitatively give the specific existence region of various forms of solutions near the Turing–Hopf singularity.
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