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# Dynamic transitions and stability for the acetabularia whorl formation

• * Corresponding author: Yiqiu Mao
• 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.

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

 Citation: • • Figure 1.  Classification in the parameter space

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

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

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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