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November  2019, 24(11): 5989-6004. doi: 10.3934/dcdsb.2019117

## 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

Received  October 2018 Published  November 2019 Early access  June 2019

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.

Citation: Yiqiu Mao, Dongming Yan, ChunHsien Lu. Dynamic transitions and stability for the acetabularia whorl formation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5989-6004. doi: 10.3934/dcdsb.2019117
##### References:
 [1] A. M. Ashu, Some Properties of Bessel Functions with Applications to Neumann Eigenvalues in the Unit Disc, Bachelor's Theses in Mathematical Sciences, Lund University, 2013. [2] J. Dumais and L. G. Harrison, Whorl morphogenesis in the dasycladalean algae: The pattern formation viewpoint, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 355 (2000), 281-305.  doi: 10.1098/rstb.2000.0565. [3] J. Dumais, K. Serikawa and D. F. Mandoli, Acetabularia: A unicellular model for understanding subcellular localization and morphogenesis during development, Journal of Plant Growth Regulation, 19 (2000), 253-264.  doi: 10.1007/s003440000035. [4] B. Goodwin, How the Leopard Changed Its Spots: The Evolution of Complexity, Princeton University Press, 2001. [5] B. Goodwin, J. Murray and D. Baldwin, Calcium: The elusive morphogen in acetabularia, in Proc. 6th Intern. Symp. on Acetabularia. Belgian Nuclear Center, CEN-SCK Mol, Belgium, (1984), 101–108. [6] B. C. Goodwin and L. Trainor, Tip and whorl morphogenesis in acetabularia by calcium-regulated strain fields, Journal of theoretical biology, 117 (1985), 79-106.  doi: 10.1016/S0022-5193(85)80165-X. [7] L. G. Harrison, Reaction-diffusion theory and intracellular differentiation, International Journal of Plant Sciences, 153 (1992), S76–S85. doi: 10.1086/297065. [8] L. G. Harrison, J. Snell, R. Verdi, D. Vogt, G. Zeiss and B. R. Green, Hair morphogenesis inacetabularia mediterranea: Temperature-dependent spacing and models of morphogen waves, Protoplasma, 106 (1981), 211-221.  doi: 10.1007/BF01275553. [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [10] T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152. [11] T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4. [12] L. Martynov, A morphogenetic mechanism involving instability of initial forth, Journal of Theoretical Biology, 52 (1975), 471-480.  doi: 10.1016/0022-5193(75)90013-2. [13] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. [14] C. L. Siegel, Über einige anwendungen diophantischer approximationen, in On Some Applications of Diophantine Approximations, Springer, 2 (2014), 81–138. [15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge university press, 1995. [16] Y. You, Global dynamics of the brusselator equations, Dynamics of Partial Differential Equations, 4 (2007), 167-196.  doi: 10.4310/DPDE.2007.v4.n2.a4.

show all references

##### References:
 [1] A. M. Ashu, Some Properties of Bessel Functions with Applications to Neumann Eigenvalues in the Unit Disc, Bachelor's Theses in Mathematical Sciences, Lund University, 2013. [2] J. Dumais and L. G. Harrison, Whorl morphogenesis in the dasycladalean algae: The pattern formation viewpoint, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 355 (2000), 281-305.  doi: 10.1098/rstb.2000.0565. [3] J. Dumais, K. Serikawa and D. F. Mandoli, Acetabularia: A unicellular model for understanding subcellular localization and morphogenesis during development, Journal of Plant Growth Regulation, 19 (2000), 253-264.  doi: 10.1007/s003440000035. [4] B. Goodwin, How the Leopard Changed Its Spots: The Evolution of Complexity, Princeton University Press, 2001. [5] B. Goodwin, J. Murray and D. Baldwin, Calcium: The elusive morphogen in acetabularia, in Proc. 6th Intern. Symp. on Acetabularia. Belgian Nuclear Center, CEN-SCK Mol, Belgium, (1984), 101–108. [6] B. C. Goodwin and L. Trainor, Tip and whorl morphogenesis in acetabularia by calcium-regulated strain fields, Journal of theoretical biology, 117 (1985), 79-106.  doi: 10.1016/S0022-5193(85)80165-X. [7] L. G. Harrison, Reaction-diffusion theory and intracellular differentiation, International Journal of Plant Sciences, 153 (1992), S76–S85. doi: 10.1086/297065. [8] L. G. Harrison, J. Snell, R. Verdi, D. Vogt, G. Zeiss and B. R. Green, Hair morphogenesis inacetabularia mediterranea: Temperature-dependent spacing and models of morphogen waves, Protoplasma, 106 (1981), 211-221.  doi: 10.1007/BF01275553. [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [10] T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152. [11] T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4. [12] L. Martynov, A morphogenetic mechanism involving instability of initial forth, Journal of Theoretical Biology, 52 (1975), 471-480.  doi: 10.1016/0022-5193(75)90013-2. [13] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. [14] C. L. Siegel, Über einige anwendungen diophantischer approximationen, in On Some Applications of Diophantine Approximations, Springer, 2 (2014), 81–138. [15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge university press, 1995. [16] Y. You, Global dynamics of the brusselator equations, Dynamics of Partial Differential Equations, 4 (2007), 167-196.  doi: 10.4310/DPDE.2007.v4.n2.a4.
Classification in the parameter space
$n_c$ plot against $a$ and $d$, numbers inside indicates critical wave number $n_c$ when $\lambda$ crosses $\lambda_c$
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
Calculation for $q(\lambda_c)$
 $\delta$ $a$ $d$ $R$ $(n_c, j_c)$ $q(\lambda_c)(u_c-v_c)$ $1.05$ $0.2$ $13$ $4$ $(2, 1)$ $73.5557$ $1.05$ $0.2$ $15$ $4$ $(2, 1)$ $147.17$ $1.05$ $0.2$ $20$ $4$ $(2, 1)$ $240.462$ $1.05$ $0.2$ $80$ $4$ $( 1, 1)$ $82.1464$ $1.05$ $0.4$ $65$ $4$ $( 2, 1)$ $79.4266$ $1.05$ $0.2$ $13$ $10$ $(5, 1)$ $459.715$ $1.05$ $0.4$ $80$ $10$ $( 4, 1)$ $527.142$ $1.2$ $0.2$ $15$ $4$ $(2, 1 )$ $31.0966$ $1.2$ $0.2$ $15$ $20$ $( 9, 1)$ $523.768$ $1.2$ $0.4$ $175$ $4$ $( 1, 1)$ $-1.97781$ $2$ $0.2$ $30$ $4$ $( 2, 1)$ $5.66053$ $2$ $0.4$ $60$ $4$ $( 3, 1)$ $2.0822$ $2$ $0.4$ $100$ $4$ $( 2, 1)$ $2.479171$ $8$ $0.4$ $80$ $10$ $( 10, 5)$ $6.07011$
 $\delta$ $a$ $d$ $R$ $(n_c, j_c)$ $q(\lambda_c)(u_c-v_c)$ $1.05$ $0.2$ $13$ $4$ $(2, 1)$ $73.5557$ $1.05$ $0.2$ $15$ $4$ $(2, 1)$ $147.17$ $1.05$ $0.2$ $20$ $4$ $(2, 1)$ $240.462$ $1.05$ $0.2$ $80$ $4$ $( 1, 1)$ $82.1464$ $1.05$ $0.4$ $65$ $4$ $( 2, 1)$ $79.4266$ $1.05$ $0.2$ $13$ $10$ $(5, 1)$ $459.715$ $1.05$ $0.4$ $80$ $10$ $( 4, 1)$ $527.142$ $1.2$ $0.2$ $15$ $4$ $(2, 1 )$ $31.0966$ $1.2$ $0.2$ $15$ $20$ $( 9, 1)$ $523.768$ $1.2$ $0.4$ $175$ $4$ $( 1, 1)$ $-1.97781$ $2$ $0.2$ $30$ $4$ $( 2, 1)$ $5.66053$ $2$ $0.4$ $60$ $4$ $( 3, 1)$ $2.0822$ $2$ $0.4$ $100$ $4$ $( 2, 1)$ $2.479171$ $8$ $0.4$ $80$ $10$ $( 10, 5)$ $6.07011$
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