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Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis
1. | Center for Partial Differential Equations, East China Normal University, Minhang, 200241, Shanghai, China |
2. | Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, Heilongjiang, China, China |
3. | Centre for Mathematical Biology, Mathematical Institute, University of Oxford, St Giles 24-29, OX1 3LB, United Kingdom |
References:
[1] |
R. A. Barrio, R. E. Baker, B. Vaughan, K. Tribuzy, M. R. de Carvalho, R. Bassanezi and P. K. Maini, Modelling the skin pattern of fishes, Phys. Rev. E., 79 (2009), 031908.
doi: 10.1103/PhysRevE.79.031908. |
[2] |
R. E. Baker, S. Schnell and P. K. Maini, A mathematical investigation of a new model for somitogenesis, J. Math. Biol., 52 (2006), 458-482.
doi: 10.1007/s00285-005-0362-2. |
[3] |
R. E. Baker, S. Schnell, S. and P. K. Maini, A clock and wavefront mechanism for somite formation, Dev. Biol., 293 (2006), 116-126.
doi: 10.1016/j.ydbio.2006.01.018. |
[4] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[5] |
E. A. Gaffney, K. Pugh, P. K. Maini and F. Arnold, Investigating a simple model of cutaneous wound healing angiogenesis, J. Math. Biol., 45 (2002), 337-374.
doi: 10.1007/s002850200161. |
[6] |
B. D. Hassard, N. D. Kazarinoff and Y. Wan, "Theory and Application of Hopf Bifurcation," Cambridge Univ. Press, Cambridge, 1981. |
[7] |
S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (2002), 765-768.
doi: 10.1038/376765a0. |
[8] |
P. De Kepper, V. Castets, E. Dulos and J. Boissonade, Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D., 49 (1991), 161-169.
doi: 10.1016/0167-2789(91)90204-M. |
[9] |
S. A. Kauffman, R. M. Shymko and K. Trabert, Control of sequential compartment formation in drosophila, Science, 199 (1978), 259-270.
doi: 10.1126/science.413193. |
[10] |
I. Lengyel and I. R. Epstein, Modeling of Turing structure in the Chloride-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650652.
doi: 10.1126/science.251.4994.650. |
[11] |
I. Lengyel and I. R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci., USA, 89 (1992), 3977-3979.
doi: 10.1073/pnas.89.9.3977. |
[12] |
C.-M. Lin, T.-X. Jiang, R. E. Baker, P. K. Maini, R. B. Widelitz and C.-M. Chuong, Spots versus stripes: FGF/ERK signalling and mesenchymal condensation during feather pattern formation, Dev. Biol., 334 (2009), 369-382.
doi: 10.1016/j.ydbio.2009.07.036. |
[13] |
P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.
doi: 10.1016/j.jfa.2007.06.015. |
[14] |
P. Liu, J. Shi and Y. Wang, Bifurcation from a degenerate simple eigenvalue, J. Funct. Anal., 264 (2013), 2269-2299.
doi: http://dx.doi.org/10.1016/j.jfa.2013.02.009. |
[15] |
S. S. Liaw, C. C. Yang, R. T. Liu and J. T. Hong, Turing model for the patterns of lady beetles, Phys. Rev., 64 (2001), 041909. |
[16] |
J. R. Mooney, Steady states of a reaction-diffusion system on the off-centre annulus, SIAM J. Appl. Math., 44 (1984), 745-761.
doi: 10.1137/0144053. |
[17] |
J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[18] |
P. K. Maini, R. E. Baker and C. M. Chuong, The Turing model comes of molecular age, Science, 314 (2006), 397-1398.
doi: 10.1126/science.1136396. |
[19] |
D.G. Miguez, M. Dolnik, A. P. Munuzuri and L. Kramer, Effect of axial growth on Turing pattern formation, Phys. Rev. L., 96 (2006), 048304.
doi: 10.1103/PhysRevLett.96.048304. |
[20] |
S. McDougall, J. Dallon, J. A. Sherratt and P. K. Maini, Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications, Phil. Trans. Roy. Soc., A 364 (2006), 1385-1405. |
[21] |
P. K. Maini, D. S. L. McElwain and S. Leavesley, Travelling waves in a wound healing assay, Appl. Math. Lett., 17 (2004), 575-580.
doi: 10.1016/S0893-9659(04)90128-0. |
[22] |
H. Meinhardt, P. Prusinkiewicz and D. R. Fowler, "The Algorithmic Beauty of Sea Shells," Springer Verlag, Berlin, 2003.
doi: 10.1007/978-3-662-05291-4. |
[23] |
P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648-663.
doi: 10.1137/0137048. |
[24] |
B. N. Nagorcka, Evidence for a reaction-diffusion system as a mechanism controlling mammalian hair growth, BioSystems, (1984), 323-332. |
[25] |
B. N. Nagorcka and J. R. Mooney, The role of a reaction-diffusion system in the formation of hair fibres, J. Theor. Biol., 98 (1982), 5757-607.
doi: 10.1016/0022-5193(82)90139-4. |
[26] |
W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Tran. Amer. Math. Soc., 357 (2005), 3953-3969.
doi: 10.1090/S0002-9947-05-04010-9. |
[27] |
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," Springer-Verlag, Berlin, 1980. |
[28] |
M. V. Plikus, D. De La Cruz, J. Mayer, R. E. Baker, R. Maxon, P. K. Maini and C.-M. Chuong, Cyclic dermal BMP signalling regulates stem cell activation during hair regeneration, Nature, 451 (2008), 340-344
doi: 10.1038/nature06457. |
[29] |
R. Peng, F. Yi and X. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, Jour. Diff. Equa., 254 (2013), 2465-2498.
doi: 10.1016/j.jde.2012.12.009. |
[30] |
S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Natural Resource Modelling, 11 (1998), 131-142. |
[31] |
S. Ruan, Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling, IMA J. Appl. Math., 60 (1998), 15-32.
doi: 10.1093/imamat/61.1.15. |
[32] |
M. B. Short et al, A statistical model of criminal behavior, Math. Models Method. in Appl. Sci., 18 (2008), 1249-1267.
doi: 10.1142/S0218202508003029. |
[33] |
M. B. Short et al, Dissipation and displacement of hotspots in reaction-diffusion models of crime, Proc. Nat. Acad. Sci., 107 (2010), 3961-3965.
doi: 10.1073/pnas.0910921107. |
[34] |
S. Sick, S. Reinker, J. 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. |
[35] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic system on bounded domains, Jour. Diff. Equa., 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[36] |
M. J. Tindall, S. L. Porter, P. K. Maini, G. Gaglia and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell, Bull. Math. Biol., 70 (2008), 1525-1569.
doi: 10.1007/s11538-008-9322-5. |
[37] |
M. J. Tindall, S. L. Porter, P. K. Maini, G. Gaglia and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: the single cell, Bull. Math. Biol., 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
[38] |
A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. |
[39] |
J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Jour. Diff. Equa., 25 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[40] |
J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 3 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
[41] |
F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogenous diffusive predator-prey system, Jour. Diff. Equa., 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024. |
[42] |
F. Yi, J. Wei and J. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett., 22 (2009), 52-55.
doi: 10.1016/j.aml.2008.02.003. |
show all references
References:
[1] |
R. A. Barrio, R. E. Baker, B. Vaughan, K. Tribuzy, M. R. de Carvalho, R. Bassanezi and P. K. Maini, Modelling the skin pattern of fishes, Phys. Rev. E., 79 (2009), 031908.
doi: 10.1103/PhysRevE.79.031908. |
[2] |
R. E. Baker, S. Schnell and P. K. Maini, A mathematical investigation of a new model for somitogenesis, J. Math. Biol., 52 (2006), 458-482.
doi: 10.1007/s00285-005-0362-2. |
[3] |
R. E. Baker, S. Schnell, S. and P. K. Maini, A clock and wavefront mechanism for somite formation, Dev. Biol., 293 (2006), 116-126.
doi: 10.1016/j.ydbio.2006.01.018. |
[4] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[5] |
E. A. Gaffney, K. Pugh, P. K. Maini and F. Arnold, Investigating a simple model of cutaneous wound healing angiogenesis, J. Math. Biol., 45 (2002), 337-374.
doi: 10.1007/s002850200161. |
[6] |
B. D. Hassard, N. D. Kazarinoff and Y. Wan, "Theory and Application of Hopf Bifurcation," Cambridge Univ. Press, Cambridge, 1981. |
[7] |
S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (2002), 765-768.
doi: 10.1038/376765a0. |
[8] |
P. De Kepper, V. Castets, E. Dulos and J. Boissonade, Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D., 49 (1991), 161-169.
doi: 10.1016/0167-2789(91)90204-M. |
[9] |
S. A. Kauffman, R. M. Shymko and K. Trabert, Control of sequential compartment formation in drosophila, Science, 199 (1978), 259-270.
doi: 10.1126/science.413193. |
[10] |
I. Lengyel and I. R. Epstein, Modeling of Turing structure in the Chloride-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650652.
doi: 10.1126/science.251.4994.650. |
[11] |
I. Lengyel and I. R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci., USA, 89 (1992), 3977-3979.
doi: 10.1073/pnas.89.9.3977. |
[12] |
C.-M. Lin, T.-X. Jiang, R. E. Baker, P. K. Maini, R. B. Widelitz and C.-M. Chuong, Spots versus stripes: FGF/ERK signalling and mesenchymal condensation during feather pattern formation, Dev. Biol., 334 (2009), 369-382.
doi: 10.1016/j.ydbio.2009.07.036. |
[13] |
P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.
doi: 10.1016/j.jfa.2007.06.015. |
[14] |
P. Liu, J. Shi and Y. Wang, Bifurcation from a degenerate simple eigenvalue, J. Funct. Anal., 264 (2013), 2269-2299.
doi: http://dx.doi.org/10.1016/j.jfa.2013.02.009. |
[15] |
S. S. Liaw, C. C. Yang, R. T. Liu and J. T. Hong, Turing model for the patterns of lady beetles, Phys. Rev., 64 (2001), 041909. |
[16] |
J. R. Mooney, Steady states of a reaction-diffusion system on the off-centre annulus, SIAM J. Appl. Math., 44 (1984), 745-761.
doi: 10.1137/0144053. |
[17] |
J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[18] |
P. K. Maini, R. E. Baker and C. M. Chuong, The Turing model comes of molecular age, Science, 314 (2006), 397-1398.
doi: 10.1126/science.1136396. |
[19] |
D.G. Miguez, M. Dolnik, A. P. Munuzuri and L. Kramer, Effect of axial growth on Turing pattern formation, Phys. Rev. L., 96 (2006), 048304.
doi: 10.1103/PhysRevLett.96.048304. |
[20] |
S. McDougall, J. Dallon, J. A. Sherratt and P. K. Maini, Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications, Phil. Trans. Roy. Soc., A 364 (2006), 1385-1405. |
[21] |
P. K. Maini, D. S. L. McElwain and S. Leavesley, Travelling waves in a wound healing assay, Appl. Math. Lett., 17 (2004), 575-580.
doi: 10.1016/S0893-9659(04)90128-0. |
[22] |
H. Meinhardt, P. Prusinkiewicz and D. R. Fowler, "The Algorithmic Beauty of Sea Shells," Springer Verlag, Berlin, 2003.
doi: 10.1007/978-3-662-05291-4. |
[23] |
P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648-663.
doi: 10.1137/0137048. |
[24] |
B. N. Nagorcka, Evidence for a reaction-diffusion system as a mechanism controlling mammalian hair growth, BioSystems, (1984), 323-332. |
[25] |
B. N. Nagorcka and J. R. Mooney, The role of a reaction-diffusion system in the formation of hair fibres, J. Theor. Biol., 98 (1982), 5757-607.
doi: 10.1016/0022-5193(82)90139-4. |
[26] |
W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Tran. Amer. Math. Soc., 357 (2005), 3953-3969.
doi: 10.1090/S0002-9947-05-04010-9. |
[27] |
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," Springer-Verlag, Berlin, 1980. |
[28] |
M. V. Plikus, D. De La Cruz, J. Mayer, R. E. Baker, R. Maxon, P. K. Maini and C.-M. Chuong, Cyclic dermal BMP signalling regulates stem cell activation during hair regeneration, Nature, 451 (2008), 340-344
doi: 10.1038/nature06457. |
[29] |
R. Peng, F. Yi and X. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, Jour. Diff. Equa., 254 (2013), 2465-2498.
doi: 10.1016/j.jde.2012.12.009. |
[30] |
S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Natural Resource Modelling, 11 (1998), 131-142. |
[31] |
S. Ruan, Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling, IMA J. Appl. Math., 60 (1998), 15-32.
doi: 10.1093/imamat/61.1.15. |
[32] |
M. B. Short et al, A statistical model of criminal behavior, Math. Models Method. in Appl. Sci., 18 (2008), 1249-1267.
doi: 10.1142/S0218202508003029. |
[33] |
M. B. Short et al, Dissipation and displacement of hotspots in reaction-diffusion models of crime, Proc. Nat. Acad. Sci., 107 (2010), 3961-3965.
doi: 10.1073/pnas.0910921107. |
[34] |
S. Sick, S. Reinker, J. 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. |
[35] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic system on bounded domains, Jour. Diff. Equa., 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[36] |
M. J. Tindall, S. L. Porter, P. K. Maini, G. Gaglia and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell, Bull. Math. Biol., 70 (2008), 1525-1569.
doi: 10.1007/s11538-008-9322-5. |
[37] |
M. J. Tindall, S. L. Porter, P. K. Maini, G. Gaglia and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: the single cell, Bull. Math. Biol., 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
[38] |
A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. |
[39] |
J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Jour. Diff. Equa., 25 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[40] |
J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 3 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
[41] |
F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogenous diffusive predator-prey system, Jour. Diff. Equa., 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024. |
[42] |
F. Yi, J. Wei and J. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett., 22 (2009), 52-55.
doi: 10.1016/j.aml.2008.02.003. |
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