Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay

@article{Jiang2018TuringIA,
  title={Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay},
  author={Weihua Jiang and Hongbin Wang and Xun Cao},
  journal={Journal of Dynamics and Differential Equations},
  year={2018},
  volume={31},
  pages={2223-2247}
}
In this paper, we study the delayed reaction–diffusion Schnakenberg systems with Neumann boundary conditions. Sufficient and necessary conditions for the occurrence of Turing instability are obtained, and the existence of Turing, Hopf and Turing–Hopf bifurcation for the model are also established. Furthermore, for Turing–Hopf bifurcation, the explicit formula of the truncated normal form up to third order is derived. With the aid of these formulas, we determine the regions on two parameters… 

Figures and Tables from this paper

Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay
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
Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays
We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain
Turing–Hopf bifurcation and multi-stable spatio-temporal patterns in the Lengyel–Epstein system
Abstract In this paper, we consider the Lengyel–Epstein system of the CIMA reaction with homogeneous Neumann condition. Firstly, we derive conditions for existence of Turing/Turing–Hopf bifurcation
Turing-Hopf bifurcation of reaction-diffusion neural networks with leakage delay
TLDR
The sufficient and necessary conditions of Turing instability are obtained and the existence of Turing, Hopf, and Turing-Hopf bifurcations is established, and the truncated normal form up to third order is derived to understand and classify the spatio-temporal dynamics close to the Turing- Hopf bIfurcation point.
Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations
Abstract Turing-Hopf bifurcation is considered as an important mechanism for generating complex spatio-temporal patterns in dynamical systems. In this work, the normal form up to the third order for
The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system
Abstract The memory-based diffusion systems have wide applications in practice. Hopf bifurcations are observed from such systems. To meet the demand for computing the normal forms of the Hopf
Bifurcation Analysis of a Diffusive Predator-Prey Model with Monod-Haldane Functional Response
TLDR
The Turing instability and Hopf bifurcation of the coexisting equilibriums are studied in a diffusive predator–prey model with Monod–Haldane functional response.
Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System
This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we
The Design Principles of Discrete Turing Patterning Systems.
TLDR
A large-scale study on all possible two-species networks is performed and it is found that Turing pattern topologies found to be substantially more robust to changes in the parameters of the model than in the continuous case.
Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis
In this paper, we deal with the Gause–Kolmogorov-type predator–prey system with indirect prey-taxis, which means that directional movement of predators is stimulated by some chemicals emitted by
...
1
2
...

References

SHOWING 1-10 OF 59 REFERENCES
Turing Instabilities at Hopf Bifurcation
TLDR
A way is suggested for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and these are used to build convenient normal modes for the analysis of Turing instabilities of the limit Cycle.
The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system
A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression
Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system
  • Qi An, W. Jiang
  • Mathematics
    Discrete & Continuous Dynamical Systems - B
  • 2019
We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We
Turing-Hopf bifurcation in the reaction-diffusion equations and its applications
TLDR
The reduced dynamics associated with Turing–Hopf bifurcation is exactly the dynamics of codimension–two ordinary differential equations (ODE), which implies the ODE techniques can be employed to classify the reduced dynamics by the unfolding parameters.
Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference
In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference described by the Crowley–Martin-type functional response, under
Hopf bifurcation and Turing instability in the reaction–diffusion Holling–Tanner predator–prey model
The reaction–diffusion Holling–Tanner predator–prey model with Neumann boundary condition is considered. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for
Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay
The normal forms up to the third order for a Hopf-steady state bifurcation of a general system of partial functional differential equations (PFDEs) is derived based on the center manifold and normal
Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations.
TLDR
A generalized predator-prey system on a spatial domain is studied, where diffusion is considered as the principal process of motion and the conditions for Hopf and Turing instabilities without specifying the predator- prey functional responses are derived.
Global Bifurcation and Structure of Turing Patterns in the 1-D Lengyel–Epstein Model
This work continues the mathematical study started in ([13], to appear) on the analytic aspects of the Lengyel–Epstein reaction diffusion system. This system captures the crucial feature of the CIMA
The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays
TLDR
Exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important forpattern formation.
...
1
2
3
4
5
...