Pattern Formations
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Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 243
Author(s):  
Biao Liu ◽  
Ranchao Wu

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.


2022 ◽  
Author(s):  
Sakura Takada ◽  
Natsuhiko Yoshinaga ◽  
Nobuhide Doi ◽  
Kei Fujiwara

Reaction-diffusion coupling (RDc) generates spatiotemporal patterns, including two dynamic wave modes: traveling and standing waves. Although mode selection plays a significant role in the spatiotemporal organization of living cell molecules, the mechanism for selecting each wave mode remains elusive. Here, we investigated a wave mode selection mechanism using Min waves reconstituted in artificial cells, emerged by the RDc of MinD and MinE. Our experiments and theoretical analysis revealed that the balance of membrane binding and dissociation from the membrane of MinD determines the mode selection of the Min wave. We successfully demonstrated that the transition of the wave modes can be regulated by controlling this balance and found hysteresis characteristics in the wave mode transition. These findings highlight a novel role of the balance between activators and inhibitors as a determinant of the mode selection of waves by RDc and depict a novel mechanism in intracellular spatiotemporal pattern formations.


Author(s):  
Rui-Cheng Liu ◽  
Yang Liu ◽  
Zongxi Cai

Growth-induced pattern formations in curved film-substrate structures have attracted extensive attention recently. In most existing literature, the growth tensor is assumed to be homogeneous or piecewise homogeneous. In this paper, we aim at clarifying the influence of a growth gradient on pattern formation and pattern evolution in bilayered tubular tissues under plane-strain deformation. In the framework of finite elasticity, a bifurcation condition is derived for a general material model and a generic growth function. Then we suppose that both layers are composed of neo-Hookean materials. In particular, the growth function is assumed to decay linearly either from the inner surface or from the outer surface. It is found that a gradient in the growth has a weak effect on the critical state, compared with the homogeneous growth type where both layers share the same growth factor. Furthermore, a finite-element model is built to validate the theoretical model and to investigate the post-buckling behaviours. It is found that the associated pattern transition is not controlled by the growth gradient but by the ratio of the shear modulus between two layers. Different morphologies can occur when the modulus ratio is varied. The current analysis could provide useful insight into the influence of a growth gradient on surface instabilities and suggests that a homogeneous growth field may provide a good approximation on interpreting complicated morphological formations in multiple systems.


2021 ◽  
Author(s):  
Meng Yan ◽  
Qingshan Zhang

Abstract In this paper, we are concerned with the spatial epidemic model with infected-taxis in which the susceptible individuals could avoid the infected ones. The spatial pattern for the resulted model is investigated under homogeneous Neumann boundary condition. We gain the condition for spatial pattern induced by diffusion term and infected-taxis term. Moreover, we obtain the condition for the occurrence of pattern formations induced by infected-taxis, in which the diffusion-driven Turing instability case is excluded. We give numerical examples to support the theoretical scheme.


2021 ◽  
Author(s):  
Akiko Nakamasu

Abstract Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.


2021 ◽  
Vol 31 (09) ◽  
pp. 2150129
Author(s):  
Shihong Zhong ◽  
Jinliang Wang ◽  
Juandi Xia ◽  
You Li

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.


2021 ◽  
Author(s):  
Akiko Nakamasu

Abstract Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.


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