Dynamic transitions and stability for the acetabularia whorl formation

Yiqiu Mao; Dongming Yan; ChunHsien Lu

doi:10.3934/dcdsb.2019117

Article Contents

2019, Volume 24, Issue 11: 5989-6004. Doi: 10.3934/dcdsb.2019117

This issue Previous Article Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution Next Article Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model

Dynamic transitions and stability for the acetabularia whorl formation

1.
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
2.
School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou, 310018, China

^* Corresponding author: Yiqiu Mao
^* Corresponding author: Yiqiu Mao

Received: September 30, 2018

Early access: June 2019

Published: August 2019

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Abstract

Dynamical transitions of the Acetabularia whorl formation caused by outside calcium concentration is carefully analyzed using a chemical reaction diffusion model on a thin annulus. Restricting ourselves with Turing instabilities, we found all three types of transition, continuous, catastrophic and random can occur under different parameter regimes. Detailed linear analysis and numerical investigations are also provided. The main tool used in the transition analysis is Ma & Wang's dynamical transition theory including the center manifold reduction.

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Mathematics Subject Classification: Primary: 35Q92, 37L10; Secondary: 35K57, 35G60.

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Figure 1. Classification in the parameter space

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Figure 2. $ n_c $ plot against $ a $ and $ d $, numbers inside indicates critical wave number $ n_c $ when $ \lambda $ crosses $ \lambda_c $

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Figure 3. Critical Eigenvector $ \cos{n_c\theta}R_{n_c, J_c}(r) $ with $ n_c = 6, j_c = 3 $ and $ \delta = 1.2 $, dark or bright regions indicate potential whorl hair growth, in this graph we have around 18 growth regions

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Table 1. Calculation for $ q(\lambda_c) $

$ \delta $	$ a $	$ d $	$ R $	$ (n_c, j_c) $	$ q(\lambda_c)(u_c-v_c) $
$ 1.05 $	$ 0.2 $	$ 13 $	$ 4 $	$ (2, 1) $	$ 73.5557 $
$ 1.05 $	$ 0.2 $	$ 15 $	$ 4 $	$ (2, 1) $	$ 147.17 $
$ 1.05 $	$ 0.2 $	$ 20 $	$ 4 $	$ (2, 1) $	$ 240.462 $
$ 1.05 $	$ 0.2 $	$ 80 $	$ 4 $	$ ( 1, 1) $	$ 82.1464 $
$ 1.05 $	$ 0.4 $	$ 65 $	$ 4 $	$ ( 2, 1) $	$ 79.4266 $
$ 1.05 $	$ 0.2 $	$ 13 $	$ 10 $	$ (5, 1) $	$ 459.715 $
$ 1.05 $	$ 0.4 $	$ 80 $	$ 10 $	$ ( 4, 1) $	$ 527.142 $
$ 1.2 $	$ 0.2 $	$ 15 $	$ 4 $	$ (2, 1 ) $	$ 31.0966 $
$ 1.2 $	$ 0.2 $	$ 15 $	$ 20 $	$ ( 9, 1) $	$ 523.768 $
$ 1.2 $	$ 0.4 $	$ 175 $	$ 4 $	$ ( 1, 1) $	$ -1.97781 $
$ 2 $	$ 0.2 $	$ 30 $	$ 4 $	$ ( 2, 1) $	$ 5.66053 $
$ 2 $	$ 0.4 $	$ 60 $	$ 4 $	$ ( 3, 1) $	$ 2.0822 $
$ 2 $	$ 0.4 $	$ 100 $	$ 4 $	$ ( 2, 1) $	$ 2.479171 $
$ 8 $	$ 0.4 $	$ 80 $	$ 10 $	$ ( 10, 5) $	$ 6.07011 $

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