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Pattern formation of a Schnakenberg-type plant root hair initiation model
2018
Electronic Journal of Qualitative Theory of Differential Equations
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By means of investigating Turing, steady state and Hopf bifurcations, pattern formation, including Turing patterns, nonconstant spatial patterns or time periodic orbits, is shown. ...
This paper concentrates on the diversity of patterns in a quite general Schnakenberg-type model. ...
Bifurcation analysis In order to better understand patterns of system (1.1), we consider bifurcations from the positive constant equilibrium, such as Turing, steady state and Hopf bifurcations. ...
doi:10.14232/ejqtde.2018.1.88
fatcat:tzh4x4op2bc5jh4hd4u2fx3z2i
Reaction-diffusion models for biological pattern formation
2001
Methods and Applications of Analysis
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After a brief discussion of the Turing instability in such systems we extend the model formulation to incorporate domain growth. ...
We consider the use of reaction-diffusion equations to model biological pattern formation and describe the derivation of the reaction terms for several illustrative examples. ...
EJC acknowledges an earmarked studentship from the BBSRC. ...
doi:10.4310/maa.2001.v8.n3.a3
fatcat:l6w3wlwivzgvtjcnxykbuduvsi
An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion
2016
Journal of nonlinear science
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Finally, the linear stability analysis for these cell-bulk models is shown to be qualitatively rather similar to that of localized spot patterns for activator-inhibitor reaction-diffusion systems in the ...
In the limit of very large bulk diffusion, it is shown that the ODE-PDE system can be approximated by a finite-dimensional dynamical system, which is studied both analytically and numerically. ...
number criterion of complex analysis on (6.9) to count the number of eigenvalues of the linearization when the parameters are off any Hopf bifurcation boundary. ...
doi:10.1007/s00332-016-9296-7
fatcat:tvzgptagufck3liwcxxnzwtlzq
Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme
1995
Journal of Engineering Mathematics
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The picture is similar to the previous case, with the stability of the pattern being lost via a Hopf bifurcation. ...
This graph shows typical Hopf bifurcation behaviour for values of jz close to 47,1. ...
doi:10.1007/bf00043976
fatcat:sjnwrff33rhodcrt3numqvsoau
Pattern formation in a general glycolysis reaction-diffusion system
2015
IMA Journal of Applied Mathematics
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by the bifurcation method and Leray-Schauder degree theory. ...
The effect of various parameters on the existence and non-existence of spatiotemporal patterns is analysed. ...
Acknowledgement We thank the two anonymous reviewers and the associate editor for helpful comments and suggestions. ...
doi:10.1093/imamat/hxv013
fatcat:h2oycybovrh7fn6qulk2fim6yi
The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system
2018
Nonlinear Analysis: Modelling and Control
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We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. ...
Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the ...
Their critical comments and helpful suggestions greatly improve the presentation of the manuscript. ...
doi:10.15388/na.2018.5.7
fatcat:wdu772ywqjfrtkhxuqygd3xjmm
Existence and stability analysis of spiky solutions for the Gierer–Meinhardt system with large reaction rates
2009
Physica D : Non-linear phenomena
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We study the Gierer-Meinhardt system in one dimension in the limit of large reaction rates. ...
First we construct three types of solutions: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. ...
The work of JW is supported by an Earmarked Grant of RGC of Hong Kong. TK is supported by an NSERC discovery grant, Canada. ...
doi:10.1016/j.physd.2009.05.009
fatcat:ag2udhhfnnfchlnqj7svqu7dhq