Hopf Bifurcation in an Oscillatory-Excitable Reaction–Diffusion Model with Spatial Heterogeneity

We focus on the qualitative analysis of a reaction–diffusion model with spatial heterogeneity. The system is a generalization of the well-known FitzHugh–Nagumo system, in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of a Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.

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Hopf bifurcation of a delayed reaction–diffusion model with advection term

Nonlinear Analysis ◽

10.1016/j.na.2021.112455 ◽

2021 ◽

Vol 212 ◽

pp. 112455

Author(s):

Li Ma ◽

Dan Wei

Keyword(s):

Hopf Bifurcation ◽

Diffusion Model ◽

Reaction Diffusion ◽

Delayed Reaction ◽

Reaction Diffusion Model ◽

Advection Term

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Numerical simulations of Turing patterns in a reaction-diffusion model with the Chebyshev spectral method

The European Physical Journal Plus ◽

10.1140/epjp/i2018-12265-9 ◽

2018 ◽

Vol 133 (10) ◽

Cited By ~ 1

Author(s):

Maliha Tehseen Saleem ◽

Ishtiaq Ali

Keyword(s):

Numerical Simulations ◽

Spectral Method ◽

Diffusion Model ◽

Reaction Diffusion ◽

Turing Patterns ◽

Reaction Diffusion Model ◽

Chebyshev Spectral Method

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HIERARCHICAL MODELS FOR SPATIAL PATTERN FORMATION IN BIOLOGY

Journal of Biological System ◽

10.1142/s0218339095000885 ◽

1995 ◽

Vol 03 (04) ◽

pp. 987-997 ◽

Cited By ~ 2

Author(s):

P. K. MAINI

Keyword(s):

Pattern Formation ◽

Spatial Heterogeneity ◽

Spatial Pattern ◽

Diffusion Model ◽

Hierarchical Models ◽

Reaction Diffusion ◽

Skeletal Patterning ◽

Reaction Diffusion Model ◽

Spatial Pattern Formation ◽

Set Up

We review some recent work investigating a hierarchy of patterning processes in which a reaction-diffusion model forms the top level. In one such hierarchy, it is assumed that the boundary is differentiated, and it is shown that this can greatly enhance the robustness of the patterns subsequently formed by the reaction-diffusion model. In the second, a spatial heterogeneity in background environment is first set-up by a simple gradient model. The resulting patterns produced by the reaction-diffusion system may be isolated to specific parts of the domain. The application of such hierarchical models to skeletal patterning in the tetrapod limb is considered.

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Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction–diffusion model

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2012.01.001 ◽

2012 ◽

Vol 13 (4) ◽

pp. 1961-1977 ◽

Cited By ~ 22

Author(s):

Chuang Xu ◽

Junjie Wei

Keyword(s):

Hopf Bifurcation ◽

Diffusion Model ◽

Bifurcation Analysis ◽

Reaction Diffusion ◽

One Dimensional ◽

Reaction Diffusion Model

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Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

Discrete Dynamics in Nature and Society ◽

10.1155/2014/139375 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7

Author(s):

Qingsong Liu ◽

Yiping Lin ◽

Jingnan Cao ◽

Jinde Cao

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Numerical Simulations ◽

Bifurcation Analysis ◽

Center Manifold ◽

Local Reaction ◽

Reaction Diffusion ◽

Critical Value ◽

Hopf Bifurcations ◽

The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

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