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January  2014, 13(1): 347-369. doi: 10.3934/cpaa.2014.13.347

Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis

Fengqi Yi 1, , Hua Zhang 2, , Alhaji Cherif 3, and  Wenying Zhang 2,

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

Received  March 2013 Revised  April 2013 Published  July 2013

In this paper, a homogeneous reaction-diffusion model describing the control growth of mammalian hair is investigated. We provide some global analyses of the model depending upon some parametric thresholds/constraints. We find that when one of the dimensionless parameter is less than one, then the unique positive equilibrium is globally asymptotically stable. On the contrary, when this threshold is greater than one, the existence of both steady-state and Hopf bifurcations can be observed under further parametric constraints. In addition, we find that both spatially homogeneous and heterogeneous oscillatory solutions can be seen for some spatially independent parameters provided that some conditions are met. Under these conditions, the direction and stability of these oscillatory behaviors, global stability of the unique constant steady state and the local orbital asymptotic stability of the spatially homogeneous periodic orbits are also investigated.
Keywords: Lyapunov function., spatiotemporal patterns, steady state and Hopf bifurcations, Diffusive hair growth model, globally asymptotically stable.
Mathematics Subject Classification: Primary: 3532, 35J55, 35K57, 92C15; Secondary: 92C4.
Citation: Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[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.  Google Scholar

[6]

B. D. Hassard, N. D. Kazarinoff and Y. Wan, "Theory and Application of Hopf Bifurcation," Cambridge Univ. Press, Cambridge, 1981. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

B. N. Nagorcka, Evidence for a reaction-diffusion system as a mechanism controlling mammalian hair growth, BioSystems, (1984), 323-332. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," Springer-Verlag, Berlin, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Natural Resource Modelling, 11 (1998), 131-142.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[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.  Google Scholar

[6]

B. D. Hassard, N. D. Kazarinoff and Y. Wan, "Theory and Application of Hopf Bifurcation," Cambridge Univ. Press, Cambridge, 1981. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

B. N. Nagorcka, Evidence for a reaction-diffusion system as a mechanism controlling mammalian hair growth, BioSystems, (1984), 323-332. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," Springer-Verlag, Berlin, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Natural Resource Modelling, 11 (1998), 131-142.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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