Dynamics and patterns of an activator-inhibitor model with cubic polynomial source

Yanqiu Li; Juncheng Jiang

Applications of Mathematics (2019)

  • Volume: 64, Issue: 1, page 61-73
  • ISSN: 0862-7940

Abstract

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The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns.

How to cite

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Li, Yanqiu, and Jiang, Juncheng. "Dynamics and patterns of an activator-inhibitor model with cubic polynomial source." Applications of Mathematics 64.1 (2019): 61-73. <http://eudml.org/doc/294770>.

@article{Li2019,
abstract = {The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns.},
author = {Li, Yanqiu, Jiang, Juncheng},
journal = {Applications of Mathematics},
keywords = {activator-inhibitor model; cubic polynomial source; Turing pattern; global stability; weakly linear coupling},
language = {eng},
number = {1},
pages = {61-73},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Dynamics and patterns of an activator-inhibitor model with cubic polynomial source},
url = {http://eudml.org/doc/294770},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Li, Yanqiu
AU - Jiang, Juncheng
TI - Dynamics and patterns of an activator-inhibitor model with cubic polynomial source
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 61
EP - 73
AB - The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns.
LA - eng
KW - activator-inhibitor model; cubic polynomial source; Turing pattern; global stability; weakly linear coupling
UR - http://eudml.org/doc/294770
ER -

References

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