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April  2022, 9(2): 185-206. doi: 10.3934/jcd.2021027

Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

1. 

Dipartimento di Ingegneria Elettrica e dell'Informazione, Università di Cassino e del Lazio Meridionale, 03043 Cassino, Italy

2. 

Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, 73100 Lecce, Italy

3. 

CNR-IASI (Istituto di Analisi dei Sistemi ed Informatica), Consiglio Nazionale delle Ricerche, 00185 Roma, Italy

4. 

CNR-IRIB (Istituto per la Ricerca e l'Innovazione Biomedica), Consiglio Nazionale delle Ricerche, 90146 Palermo, Italy

* Corresponding author

Received  April 2021 Revised  September 2021 Published  April 2022 Early access  December 2021

In this study, the formation of the adult sea urchin shape is rationalized within the Turing's theory paradigm. The emergence of protrusions from the expanding underlying surface is described through a reaction-diffusion model with Gray-Scott kinetics on a growing oblate spheroid. The case of slow exponential isotropic growth is considered. The model is first studied in terms of the spatially homogenous equilibria and of the bifurcations involved. Turing diffusion-driven instability is shown to occur and the impact of the slow exponential growth on the resulting Turing regions adequately discussed. Numerical investigations validate the theoretical results showing that the combination between an inhibitor and an activator can result in a distribution of spot concentrations that underlies the development of ambulacral tentacles in the sea urchin's adult stage. Our findings pave the way for a model-driven experimentation that could improve the current biological understanding of the gene control networks involved in patterning.

Citation: Deborah Lacitignola, Massimo Frittelli, Valerio Cusimano, Andrea De Gaetano. Pattern formation on a growing oblate spheroid. an application to adult sea urchin development. Journal of Computational Dynamics, 2022, 9 (2) : 185-206. doi: 10.3934/jcd.2021027
References:
[1]

J. AragonM. TorresD. GilR. Barrio and P. Maini, Turing patterns with pentagonal symmetry, Phys. Rev. E, 65 (2002), 051913.  doi: 10.1103/PhysRevE.65.051913.

[2]

R. A. Barrio, Turing systems: A general model for complex patterns in nature, Physics of Emergence and Organization, 2008,267–296. doi: 10.1142/9789812779953_0011.

[3]

J. A. CastilloF. Sanchez-Garduno and P. Padilla, A Turing-Hopf Bifurcation Scenario for Pattern Formation on Growing Domains, Bulletin of Mathematical Biology, 78 (2016), 1410-1449.  doi: 10.1007/s11538-016-0189-6.

[4]

M. A. Chakra and J. R. Stone, Classifying echinoid skeleton models: Testing ideas about growth and form, Paleobiology, 37 (2011), 686-695.  doi: 10.1666/10012.1.

[5]

J. Claxton, The determination of patterns with special reference to that of the central primary skin follicles in sheep, Journal of Theoretical Biology, 7 (1964), 302-317.  doi: 10.1016/0022-5193(64)90074-8.

[6]

E. J. CrampinE. A. Gaffney and P. K. Maini, Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation, Bulletin of Mathematical Biology, 61 (1999), 1093-1120.  doi: 10.1006/bulm.1999.0131.

[7]

S. Damle and E. Davidson, Precise cis-regulatory control of spatial and temporal expression of the alx-1 gene in the skeletogenic lineage of s. purpuratus, Developmental Biology, 357 (2011), 505-517.  doi: 10.1016/j.ydbio.2011.06.016.

[8]

E. DavidsonJ. P. RastP. OliveriA. RansickC. CalestaniC. YuhT. MinokawaG. AmoreV. HinmanC. Arenas-MenaO. OtimC. BrownC. LiviP. Y. LeeR. RevillaM. J. SchilstraP. J. ClarkeA. G. RustZ. PanM. I. ArnoneL. RowenR. CameronD. R. McClayL. Hood and H. Bolouri, A provisional regulatory gene network for specification of endomesoderm in the sea urchin embryo, Developmental Biology, 246 (2002), 162-190.  doi: 10.1006/dbio.2002.0635.

[9]

O. Ellers, A mechanical model of growth in regular sea urchins: Predictions of shape and a developmental morphospace, Proceedings of the Royal Society of London. Series B: Biological Sciences, 254 (1993), 123-129. 

[10]

S. G. Ernst, A century of sea urchin development, American Zoologist, 37 (1997), 250-259.  doi: 10.1093/icb/37.3.250.

[11]

C. A. Ettensohn, Sea urchins as a model system for studying embryonic development, Reference Module in Biomedical Sciences, 2017. doi: 10.1016/B978-0-12-801238-3.99509-6.

[12]

M. FrittelliA. MadzvamuseI. Sgura and C. Venkataraman, Numerical preservation of velocity induced invariant regions for reaction–diffusion systems on evolving surfaces, J. Sci. Comput., 77 (2018), 971-1000.  doi: 10.1007/s10915-018-0741-7.

[13]

P. Guidetti and M. Mori, Morpho-functional defences of mediterranean sea urchins, paracentrotus lividus and arbacia lixula, against fish predators, Marine Biology, 147 (2005), 797-802.  doi: 10.1007/s00227-005-1611-z.

[14]

L. H. Hyman, The Invertebrates: Echinodermata, McGraw Hill, New York, 1955.

[15]

A. S. JohnsonO. EllersJ. LemireM. Minor and H. A. Leddy, Sutural loosening and skeletal flexibility during growth: Determination of drop-like shapes in sea urchins, Proceedings of the Royal Society of London. Series B: Biological Sciences, 269 (2002), 215-220.  doi: 10.1098/rspb.2001.1881.

[16]

S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0.

[17]

D. Lacitignola, The mathematical beauty of nature and Turing pattern formation, Mat. Cult. Soc. Riv. Unione Mat. Ital. (I), 1 (2016), 93-103. 

[18]

D. LacitignolaB. BozziniM. Frittelli and I. Sgura, Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 484-508.  doi: 10.1016/j.cnsns.2017.01.008.

[19]

D. LacitignolaI. SguraB. BozziniT. Dobrovolska and I. Krastev, Spiral waves on the sphere for an alloy electrodeposition model, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104930.  doi: 10.1016/j.cnsns.2019.104930.

[20]

M. Lamare and P. Mladenov, Modelling somatic growth in the sea urchin evechinus chloroticus (echinoidea: Echinometridae), Journal of Experimental Marine Biology and Ecology, 243 (2000), 17-43.  doi: 10.1016/S0022-0981(99)00107-0.

[21]

B. LefebvreC. D. SumrallR. A. Shroat-LewisM. ReichG. D. WebsterA. W. HunterE. NardinS. V. RozhnovT. E. GuensburgA. TouzeauF. Noailles and J. Sprinkle, Chapter 14 palaeobiogeography of ordovician echinoderms, Geological Society, London, Memoirs, 38 (2013), 173-198.  doi: 10.1144/M38.14.

[22]

S. LiawC. YangR. Liu and J. Hong, Turing model for the patterns of lady beetles, Phys. Rev. E, 64 (2001), 041909.  doi: 10.1103/PhysRevE.64.041909.

[23]

A. Madzvamuse, Stability analysis of reaction-diffusion systems with constant coefficients on growing domains, Int. J. Dyn. Syst. Differ. Equ., 1 (2008), 250-262.  doi: 10.1504/IJDSDE.2008.023002.

[24]

A. Madzvamuse, Turing instability conditions for growing domains with divergence free mesh velocity, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e2250–e2257. doi: 10.1016/j.na.2009.05.027.

[25]

A. Madzvamuse and R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces, Phys. Rev. E, 90 (2014), 043307.  doi: 10.1103/PhysRevE.90.043307.

[26]

A. MadzvamuseE. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains, J. Math. Biol., 61 (2010), 133-164.  doi: 10.1007/s00285-009-0293-4.

[27]

P. MainiE. CrampinA. MadzvamuseA. Wathen and R. D. Thomas, Implications of domain growth in morphogenesis, Mathematical Modelling & Computing in Biology and Medicine, Milan Res. Cent. Ind. Appl. Math. MIRIAM Proj., Esculapio, Bologna, 1 (2003), 67-73. 

[28]

P. MainiT. WoolleyR. BakerE. Gaffney and S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496.  doi: 10.1098/rsfs.2011.0113.

[29]

H. Meinhardt, Pigment patterns on sea shells - A beautiful case of biological pattern formation, Growth, Dissolution and Pattern Formation in Geosystems, (1999), 221–236. doi: 10.1007/978-94-015-9179-9_10.

[30]

H. Meinhardt and M. Klingler, A model for pattern formation on the shells of molluscs, J. Theoret. Biol., 126 (1987), 63-89.  doi: 10.1016/S0022-5193(87)80101-7.

[31]

J. Murray, Mathematical Biology II - Spatial Models and Biomedical Applications, 3$^{rd}$ edition, Springer-Verlag, New York, 2003.

[32]

B. Nagorcka and J. Mooney, The role of a reaction-diffusion system in the formation of hair fibres, J. Theoret. Biol., 98 (1982), 575-607.  doi: 10.1016/0022-5193(82)90139-4.

[33]

P. OliveriQ. Tu and E. H. Davidson, Global regulatory logic for specification of an embryonic cell lineage, Proceedings of the National Academy of Sciences, 105 (2008), 5955-5962.  doi: 10.1073/pnas.0711220105.

[34]

H. G. OthmerK. PainterD. Umulis and C. Xue, The intersection of theory and application in elucidating pattern formation in developmental biology, Math. Model. Nat. Phenom., 4 (2009), 3-82.  doi: 10.1051/mmnp/20094401.

[35]

K. PainterP. Maini and H. Othmer, Stripe formation in juvenile Pomacanthus explained by a generalized turing mechanism with chemotaxis, Proceedings of the National Academy of Sciences of the United States of America, 96 (1999), 5549-5554.  doi: 10.1073/pnas.96.10.5549.

[36]

V. PerriconeT. GrunF. MarmoC. Langella and M. Candia Carnevali, Constructional design of echinoid endoskeleton: Main structural components and their potential for biomimetic applications, Bioinspiration & Biomimetics, 16 (2020), 011001.  doi: 10.1088/1748-3190/abb86b.

[37]

U. Philippi and W. Nachtigall, Functional morphology of regular echinoid tests (echinodermata, echinoida): A finite element study, Zoomorphology, 116 (1996), 35-50.  doi: 10.1007/BF02526927.

[38]

R. PlazaF. Sánchez-GarduñoP. PadillaR. Barrio and P. Maini, The effect of growth and curvature on pattern formation, J. Dynam. Differential Equations, 16 (2004), 1093-1121.  doi: 10.1007/s10884-004-7834-8.

[39]

D. Raup, Theoretical morphology of echinoid growth, Journal of Paleontology, 42 (1968), 50-63.  doi: 10.1017/S0022336000061643.

[40]

R. Revilla-i DomingoP. Oliveri and E. H. Davidson, A missing link in the sea urchin embryo gene regulatory network: Hesc and the double-negative specification of micromeres, Proceedings of the National Academy of Sciences, 104 (2007), 12383-12388. 

[41]

L. Rogers-BennettD. RogersW. Bennett and T. Ebert, Modeling red sea urchin (strongylocentrotus franciscanus) growth using six growth functions, Fish Bull (Wash DC), 101 (2003), 614-626. 

[42]

F. Sanchez-GardunoA. KrauseJ. Castillo and P. Padilla, Turing-Hopf patterns on growing domains: The torus and the sphere, J. Theoret. Biol., 481 (2019), 136-150.  doi: 10.1016/j.jtbi.2018.09.028.

[43]

A. Seilacher and A. Gishlick, Morphodynamics, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2014. doi: 10.1201/b17557.

[44]

S. SickS. ReinkerJ. Timmer and T. Schlake, Wnt and dkk determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450.  doi: 10.1126/science.1130088.

[45]

J. Smith and E. Davidson, Regulative recovery in the sea urchin embryo and the stabilizing role of fail-safe gene network wiring, Proceedings of the National Academy of Sciences, 106 (2009), 18291-18296.  doi: 10.1073/pnas.0910007106.

[46]

M. SunX. Cheng and J. E. Socolar, Regulatory logic and pattern formation in the early sea urchin embryo, Journal of Theoretical Biology, 363 (2014), 80-92.  doi: 10.1016/j.jtbi.2014.07.023.

[47] D. Thompson, On Growth and Form, Cambridge University Press, Cambridge, 1942. 
[48]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[49]

D. M. Umulis and H. G. Othmer, Mechanisms of scaling in pattern formation, Development, 140 (2013), 4830-4843.  doi: 10.1242/dev.100511.

[50]

C. VareaJ. L. Aragón and R. A. Barrio, Turing patterns on a sphere, Phys. Rev. E, 60 (1999), 4588-4592.  doi: 10.1103/PhysRevE.60.4588.

[51]

L. G. Zachos, A new computational growth model for sea urchin skeletons, J. Theoret. Biol., 259 (2009), 646-657.  doi: 10.1016/j.jtbi.2009.04.007.

show all references

References:
[1]

J. AragonM. TorresD. GilR. Barrio and P. Maini, Turing patterns with pentagonal symmetry, Phys. Rev. E, 65 (2002), 051913.  doi: 10.1103/PhysRevE.65.051913.

[2]

R. A. Barrio, Turing systems: A general model for complex patterns in nature, Physics of Emergence and Organization, 2008,267–296. doi: 10.1142/9789812779953_0011.

[3]

J. A. CastilloF. Sanchez-Garduno and P. Padilla, A Turing-Hopf Bifurcation Scenario for Pattern Formation on Growing Domains, Bulletin of Mathematical Biology, 78 (2016), 1410-1449.  doi: 10.1007/s11538-016-0189-6.

[4]

M. A. Chakra and J. R. Stone, Classifying echinoid skeleton models: Testing ideas about growth and form, Paleobiology, 37 (2011), 686-695.  doi: 10.1666/10012.1.

[5]

J. Claxton, The determination of patterns with special reference to that of the central primary skin follicles in sheep, Journal of Theoretical Biology, 7 (1964), 302-317.  doi: 10.1016/0022-5193(64)90074-8.

[6]

E. J. CrampinE. A. Gaffney and P. K. Maini, Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation, Bulletin of Mathematical Biology, 61 (1999), 1093-1120.  doi: 10.1006/bulm.1999.0131.

[7]

S. Damle and E. Davidson, Precise cis-regulatory control of spatial and temporal expression of the alx-1 gene in the skeletogenic lineage of s. purpuratus, Developmental Biology, 357 (2011), 505-517.  doi: 10.1016/j.ydbio.2011.06.016.

[8]

E. DavidsonJ. P. RastP. OliveriA. RansickC. CalestaniC. YuhT. MinokawaG. AmoreV. HinmanC. Arenas-MenaO. OtimC. BrownC. LiviP. Y. LeeR. RevillaM. J. SchilstraP. J. ClarkeA. G. RustZ. PanM. I. ArnoneL. RowenR. CameronD. R. McClayL. Hood and H. Bolouri, A provisional regulatory gene network for specification of endomesoderm in the sea urchin embryo, Developmental Biology, 246 (2002), 162-190.  doi: 10.1006/dbio.2002.0635.

[9]

O. Ellers, A mechanical model of growth in regular sea urchins: Predictions of shape and a developmental morphospace, Proceedings of the Royal Society of London. Series B: Biological Sciences, 254 (1993), 123-129. 

[10]

S. G. Ernst, A century of sea urchin development, American Zoologist, 37 (1997), 250-259.  doi: 10.1093/icb/37.3.250.

[11]

C. A. Ettensohn, Sea urchins as a model system for studying embryonic development, Reference Module in Biomedical Sciences, 2017. doi: 10.1016/B978-0-12-801238-3.99509-6.

[12]

M. FrittelliA. MadzvamuseI. Sgura and C. Venkataraman, Numerical preservation of velocity induced invariant regions for reaction–diffusion systems on evolving surfaces, J. Sci. Comput., 77 (2018), 971-1000.  doi: 10.1007/s10915-018-0741-7.

[13]

P. Guidetti and M. Mori, Morpho-functional defences of mediterranean sea urchins, paracentrotus lividus and arbacia lixula, against fish predators, Marine Biology, 147 (2005), 797-802.  doi: 10.1007/s00227-005-1611-z.

[14]

L. H. Hyman, The Invertebrates: Echinodermata, McGraw Hill, New York, 1955.

[15]

A. S. JohnsonO. EllersJ. LemireM. Minor and H. A. Leddy, Sutural loosening and skeletal flexibility during growth: Determination of drop-like shapes in sea urchins, Proceedings of the Royal Society of London. Series B: Biological Sciences, 269 (2002), 215-220.  doi: 10.1098/rspb.2001.1881.

[16]

S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0.

[17]

D. Lacitignola, The mathematical beauty of nature and Turing pattern formation, Mat. Cult. Soc. Riv. Unione Mat. Ital. (I), 1 (2016), 93-103. 

[18]

D. LacitignolaB. BozziniM. Frittelli and I. Sgura, Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 484-508.  doi: 10.1016/j.cnsns.2017.01.008.

[19]

D. LacitignolaI. SguraB. BozziniT. Dobrovolska and I. Krastev, Spiral waves on the sphere for an alloy electrodeposition model, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104930.  doi: 10.1016/j.cnsns.2019.104930.

[20]

M. Lamare and P. Mladenov, Modelling somatic growth in the sea urchin evechinus chloroticus (echinoidea: Echinometridae), Journal of Experimental Marine Biology and Ecology, 243 (2000), 17-43.  doi: 10.1016/S0022-0981(99)00107-0.

[21]

B. LefebvreC. D. SumrallR. A. Shroat-LewisM. ReichG. D. WebsterA. W. HunterE. NardinS. V. RozhnovT. E. GuensburgA. TouzeauF. Noailles and J. Sprinkle, Chapter 14 palaeobiogeography of ordovician echinoderms, Geological Society, London, Memoirs, 38 (2013), 173-198.  doi: 10.1144/M38.14.

[22]

S. LiawC. YangR. Liu and J. Hong, Turing model for the patterns of lady beetles, Phys. Rev. E, 64 (2001), 041909.  doi: 10.1103/PhysRevE.64.041909.

[23]

A. Madzvamuse, Stability analysis of reaction-diffusion systems with constant coefficients on growing domains, Int. J. Dyn. Syst. Differ. Equ., 1 (2008), 250-262.  doi: 10.1504/IJDSDE.2008.023002.

[24]

A. Madzvamuse, Turing instability conditions for growing domains with divergence free mesh velocity, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e2250–e2257. doi: 10.1016/j.na.2009.05.027.

[25]

A. Madzvamuse and R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces, Phys. Rev. E, 90 (2014), 043307.  doi: 10.1103/PhysRevE.90.043307.

[26]

A. MadzvamuseE. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains, J. Math. Biol., 61 (2010), 133-164.  doi: 10.1007/s00285-009-0293-4.

[27]

P. MainiE. CrampinA. MadzvamuseA. Wathen and R. D. Thomas, Implications of domain growth in morphogenesis, Mathematical Modelling & Computing in Biology and Medicine, Milan Res. Cent. Ind. Appl. Math. MIRIAM Proj., Esculapio, Bologna, 1 (2003), 67-73. 

[28]

P. MainiT. WoolleyR. BakerE. Gaffney and S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496.  doi: 10.1098/rsfs.2011.0113.

[29]

H. Meinhardt, Pigment patterns on sea shells - A beautiful case of biological pattern formation, Growth, Dissolution and Pattern Formation in Geosystems, (1999), 221–236. doi: 10.1007/978-94-015-9179-9_10.

[30]

H. Meinhardt and M. Klingler, A model for pattern formation on the shells of molluscs, J. Theoret. Biol., 126 (1987), 63-89.  doi: 10.1016/S0022-5193(87)80101-7.

[31]

J. Murray, Mathematical Biology II - Spatial Models and Biomedical Applications, 3$^{rd}$ edition, Springer-Verlag, New York, 2003.

[32]

B. Nagorcka and J. Mooney, The role of a reaction-diffusion system in the formation of hair fibres, J. Theoret. Biol., 98 (1982), 575-607.  doi: 10.1016/0022-5193(82)90139-4.

[33]

P. OliveriQ. Tu and E. H. Davidson, Global regulatory logic for specification of an embryonic cell lineage, Proceedings of the National Academy of Sciences, 105 (2008), 5955-5962.  doi: 10.1073/pnas.0711220105.

[34]

H. G. OthmerK. PainterD. Umulis and C. Xue, The intersection of theory and application in elucidating pattern formation in developmental biology, Math. Model. Nat. Phenom., 4 (2009), 3-82.  doi: 10.1051/mmnp/20094401.

[35]

K. PainterP. Maini and H. Othmer, Stripe formation in juvenile Pomacanthus explained by a generalized turing mechanism with chemotaxis, Proceedings of the National Academy of Sciences of the United States of America, 96 (1999), 5549-5554.  doi: 10.1073/pnas.96.10.5549.

[36]

V. PerriconeT. GrunF. MarmoC. Langella and M. Candia Carnevali, Constructional design of echinoid endoskeleton: Main structural components and their potential for biomimetic applications, Bioinspiration & Biomimetics, 16 (2020), 011001.  doi: 10.1088/1748-3190/abb86b.

[37]

U. Philippi and W. Nachtigall, Functional morphology of regular echinoid tests (echinodermata, echinoida): A finite element study, Zoomorphology, 116 (1996), 35-50.  doi: 10.1007/BF02526927.

[38]

R. PlazaF. Sánchez-GarduñoP. PadillaR. Barrio and P. Maini, The effect of growth and curvature on pattern formation, J. Dynam. Differential Equations, 16 (2004), 1093-1121.  doi: 10.1007/s10884-004-7834-8.

[39]

D. Raup, Theoretical morphology of echinoid growth, Journal of Paleontology, 42 (1968), 50-63.  doi: 10.1017/S0022336000061643.

[40]

R. Revilla-i DomingoP. Oliveri and E. H. Davidson, A missing link in the sea urchin embryo gene regulatory network: Hesc and the double-negative specification of micromeres, Proceedings of the National Academy of Sciences, 104 (2007), 12383-12388. 

[41]

L. Rogers-BennettD. RogersW. Bennett and T. Ebert, Modeling red sea urchin (strongylocentrotus franciscanus) growth using six growth functions, Fish Bull (Wash DC), 101 (2003), 614-626. 

[42]

F. Sanchez-GardunoA. KrauseJ. Castillo and P. Padilla, Turing-Hopf patterns on growing domains: The torus and the sphere, J. Theoret. Biol., 481 (2019), 136-150.  doi: 10.1016/j.jtbi.2018.09.028.

[43]

A. Seilacher and A. Gishlick, Morphodynamics, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2014. doi: 10.1201/b17557.

[44]

S. SickS. ReinkerJ. Timmer and T. Schlake, Wnt and dkk determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450.  doi: 10.1126/science.1130088.

[45]

J. Smith and E. Davidson, Regulative recovery in the sea urchin embryo and the stabilizing role of fail-safe gene network wiring, Proceedings of the National Academy of Sciences, 106 (2009), 18291-18296.  doi: 10.1073/pnas.0910007106.

[46]

M. SunX. Cheng and J. E. Socolar, Regulatory logic and pattern formation in the early sea urchin embryo, Journal of Theoretical Biology, 363 (2014), 80-92.  doi: 10.1016/j.jtbi.2014.07.023.

[47] D. Thompson, On Growth and Form, Cambridge University Press, Cambridge, 1942. 
[48]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[49]

D. M. Umulis and H. G. Othmer, Mechanisms of scaling in pattern formation, Development, 140 (2013), 4830-4843.  doi: 10.1242/dev.100511.

[50]

C. VareaJ. L. Aragón and R. A. Barrio, Turing patterns on a sphere, Phys. Rev. E, 60 (1999), 4588-4592.  doi: 10.1103/PhysRevE.60.4588.

[51]

L. G. Zachos, A new computational growth model for sea urchin skeletons, J. Theoret. Biol., 259 (2009), 646-657.  doi: 10.1016/j.jtbi.2009.04.007.

Figure 1.  Case of no growth: $ r = 0 $. Bifurcation diagram in the $ (c_1, v_e) $ plane. The branches of the spatially homogeneous equilibria $ (u_e, v_e) $ for model (16) are shown as functions of the parameter $ c_1 $. The parameter $ c_2 $ is fixed as $ c_2 = 0.04 $. In this specific case, the trivial equilibrium $ P_0 $ is an equilibrium for all the values of $ c_1 $ whereas the two equilibria $ P_1 $ and $ P_2 $ exist only in the range $ 0.01 < c_1 < 0.16 $
Figure 2.  Case of growth: $ r \neq 0 $. In the parameter space $ (c_1, r) $, the region below the curve is where existence of the two non trivial equilibria $ P_1^{(r)} $ and $ P_2^{(r)} $ is ensured. The parameter $ c_2 $ is fixed as $ c_2 = 0.04 $. (left) The case $ \omega = 1 $ (right) The case $ \omega = 10 $
Figure 3.  Case of growth: $ r \neq 0 $. The branches of the spatially homogeneous equilibria $ (u_e, v_e) $ for model (16) as functions of the growth parameter $ r $. The parameters $ c_1 $ and $ c_2 $ are fixed at: $ c_1 = 0.03 $, $ c_2 = 0.04 $. The trivial equilibrium $ P_0 $ is an equilibrium for all the values of $ r $ whereas the non trivial equilibria $ P_1^{(r)} $ and $ P_2^{(r)} $ exist only within a particular range of the parameter $ r $ and disappear because of a saddle-node bifurcation. (Left) The case $ \omega = 1 $, (right) The case $ \omega = 10 $
Figure 4.  Effects of the growth parameter $ r $ on the Turing region of the equilibrium $ P_{2} $. Turing regions in the parameter space $ (c_1, d) $ are shown for different values of the parameter $ r $. The other parameters are fixed as: $ c_2 = 0.04 $ and $ \omega = 1 $. Turing regions are bounded by the solid curve, the dashed vertical line and the dash-dot vertical line. Top row: (left) $ r = 0 $; (right) $ r = 0.001 $. Bottom: $ r = 0.002 $
Figure 5.  The stationary case: $ r = 0 $. LSFEM numerical solutions $ v({\bf{x}}, T) $ of the model (16)-(2) on the oblate spheroid with $ f_0 = 1.9645 $ and $ \xi = 4.1134 $ attained at the corresponding integration times $ T = [0, 20, 100, 500, 1500] $. The model parameters are chosen so that $ (c_1, c_2) = (0.03, 0.04) $, $ d = 0.05 $. The size parameter is $ \omega = 10 $
Figure 6.  The slow exponential evolving case: $ r = 0.001 $. LSFEM numerical solutions $ v({\bf{x}}, T) $ of the model (16)-(2) on the oblate spheroid with $ f_0 = 1.9645 $ and $ \xi = 4.1134 $ attained at the corresponding integration times $ T = [0, 20, 100, 500, 1500] $. The model parameters are chosen so that $ (c_1, c_2) = (0.03, 0.04) $, $ d = 0.05 $. The size parameter is $ \omega = 10 $
Figure 7.  Spatial pattern in the biological plot related to the variable $ v({\bf{x}}, T) $ of model (16)-(2) at $ T = 1500 $. The oblate spheroid parameters and the model parameters are fixed as in Fig. 6. (Left) stationary case, $ r = 0 $. (Right) slow exponential evolving case, $ r = 0.001 $
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