2014, 11(5): 1215-1227. doi: 10.3934/mbe.2014.11.1215

Spatial dynamics for a model of epidermal wound healing

1. 

Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100

2. 

School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, China

Received  April 2013 Revised  February 2014 Published  June 2014

In this paper, we consider the spatial dynamics for a non-cooperative diffusion system arising from epidermal wound healing. We shall establish the spreading speed and existence of traveling waves and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. We also construct some new types of entire solutions which are different from the traveling wave solutions and spatial variable independent solutions. The traveling wave solutions provide the healing speed and describe how wound healing process spreads from one side of the wound. The entire solution exhibits the interaction of several waves originated from different locations of the wound. To the best of knowledge of the authors, it is the first time that it is shown that there is an entire solution in the model for epidermal wound healing.
Citation: Haiyan Wang, Shiliang Wu. Spatial dynamics for a model of epidermal wound healing. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1215-1227. doi: 10.3934/mbe.2014.11.1215
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in Partial Differential Equations and Related Topics (ed. J. A. Goldstein), (1975), 5.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics,, Adv. Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

P. D. Dale, P. K. Maini and J. A. Sherratt, Mathematical modelling of corneal epithelial wound healing,, Math. Biosci., 124 (1994), 127.  doi: 10.1016/0025-5564(94)90040-X.  Google Scholar

[4]

S. I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar

[5]

S. I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems,, Phys. D, 165 (2002), 176.  doi: 10.1016/S0167-2789(02)00379-2.  Google Scholar

[6]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[7]

J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations,, Discrete Contin. Dyn. Syst., 12 (2005), 193.   Google Scholar

[8]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation,, Comm. Pure Appl. Math., 52 (1999), 1255.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[9]

R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (1985).  doi: 10.1017/CBO9780511810817.  Google Scholar

[10]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[11]

A. Kolmogorov, I. Petrovsky and N. Piscounoff, Etude de l'equation de la diffusion avec croissance de la quantite de matière et son application a un problème biologique., Bull. Moscow Univ. Math. Mech., 1 (1937), 1.   Google Scholar

[12]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Commun. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[13]

W. T. Li, N. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders,, J. Math. Pures Appl., 90 (2008), 492.  doi: 10.1016/j.matpur.2008.07.002.  Google Scholar

[14]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[15]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[16]

B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[17]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[18]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, Springer-Verlag, (2003).   Google Scholar

[19]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation,, J. Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[20]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217.  doi: 10.1137/080723715.  Google Scholar

[21]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[22]

J. Sherratt and J. D. Murray, Models of epidermal wound healing,, Proc. R. Soc. London B, 241 (1990), 29.  doi: 10.1098/rspb.1990.0061.  Google Scholar

[23]

J. Sherratt and J. Murray, Mathematical analysis of a basic model for epidermal wound healing,, J. Math. Biol., 29 (1991), 389.  doi: 10.1007/BF00160468.  Google Scholar

[24]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[25]

H. R. Thieme, Density-Dependent Regulation of Spatially Distributed Populations and their Asymptotic speed of Spread,, J. Math. Biol., 8 (1979), 173.  doi: 10.1007/BF00279720.  Google Scholar

[26]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887.  doi: 10.1016/j.jde.2009.04.002.  Google Scholar

[27]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems,, Discrete and Continuous Dynamical Systems B, 17 (2012), 2243.  doi: 10.3934/dcdsb.2012.17.2243.  Google Scholar

[28]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, J. Nonlinear Sci., 21 (2011), 747.  doi: 10.1007/s00332-011-9099-9.  Google Scholar

[29]

M. X. Wang and G. Y. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay,, Nonlinearity, 23 (2010), 1609.  doi: 10.1088/0951-7715/23/7/005.  Google Scholar

[30]

Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[31]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183.  doi: 10.1007/s002850200145.  Google Scholar

[32]

H. F. Weinberger, Asymptotic behavior of a model in population genetics,, in Nonlinear Partial Differential Equations and Applications (ed. J. M. Chadam), (1978), 47.   Google Scholar

[33]

H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds for a partially cooperative 2-species reaction-diffusion model,, Discrete Contin. Dyn. Syst., 23 (2009), 1087.  doi: 10.3934/dcds.2009.23.1087.  Google Scholar

[34]

S. L. Wu and C.-H. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays,, Nonlinearity, 25 (2012), 2785.  doi: 10.1088/0951-7715/25/9/2785.  Google Scholar

[35]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems,, J. Dynam. Diff. Eqns., 25 (2013), 505.  doi: 10.1007/s10884-013-9293-6.  Google Scholar

[36]

S. L. Wu, Y. J. Sun and S. Y. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity,, Discrete Contin. Dyn. Syst., 33 (2013), 921.  doi: 10.3934/dcds.2013.33.921.  Google Scholar

[37]

P. X. Weng, Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 883.  doi: 10.3934/dcdsb.2009.12.883.  Google Scholar

[38]

H. Q. Zhao, S. L. Wu and S. Y. Liu, Entire solutions of a monostable age-structured population model in a 2D lattice strip,, J. Math. Anal. Appl., 401 (2013), 85.  doi: 10.1016/j.jmaa.2012.11.032.  Google Scholar

[39]

H. Q. Zhao, S. L. Wu and S. Y. Liu, Pulsating traveling fronts and entire solutions in a discrete periodic system with a quiescent stage,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 2164.  doi: 10.1016/j.cnsns.2012.12.033.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in Partial Differential Equations and Related Topics (ed. J. A. Goldstein), (1975), 5.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics,, Adv. Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

P. D. Dale, P. K. Maini and J. A. Sherratt, Mathematical modelling of corneal epithelial wound healing,, Math. Biosci., 124 (1994), 127.  doi: 10.1016/0025-5564(94)90040-X.  Google Scholar

[4]

S. I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar

[5]

S. I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems,, Phys. D, 165 (2002), 176.  doi: 10.1016/S0167-2789(02)00379-2.  Google Scholar

[6]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[7]

J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations,, Discrete Contin. Dyn. Syst., 12 (2005), 193.   Google Scholar

[8]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation,, Comm. Pure Appl. Math., 52 (1999), 1255.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[9]

R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (1985).  doi: 10.1017/CBO9780511810817.  Google Scholar

[10]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[11]

A. Kolmogorov, I. Petrovsky and N. Piscounoff, Etude de l'equation de la diffusion avec croissance de la quantite de matière et son application a un problème biologique., Bull. Moscow Univ. Math. Mech., 1 (1937), 1.   Google Scholar

[12]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Commun. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[13]

W. T. Li, N. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders,, J. Math. Pures Appl., 90 (2008), 492.  doi: 10.1016/j.matpur.2008.07.002.  Google Scholar

[14]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[15]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[16]

B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[17]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[18]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, Springer-Verlag, (2003).   Google Scholar

[19]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation,, J. Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[20]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217.  doi: 10.1137/080723715.  Google Scholar

[21]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[22]

J. Sherratt and J. D. Murray, Models of epidermal wound healing,, Proc. R. Soc. London B, 241 (1990), 29.  doi: 10.1098/rspb.1990.0061.  Google Scholar

[23]

J. Sherratt and J. Murray, Mathematical analysis of a basic model for epidermal wound healing,, J. Math. Biol., 29 (1991), 389.  doi: 10.1007/BF00160468.  Google Scholar

[24]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[25]

H. R. Thieme, Density-Dependent Regulation of Spatially Distributed Populations and their Asymptotic speed of Spread,, J. Math. Biol., 8 (1979), 173.  doi: 10.1007/BF00279720.  Google Scholar

[26]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887.  doi: 10.1016/j.jde.2009.04.002.  Google Scholar

[27]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems,, Discrete and Continuous Dynamical Systems B, 17 (2012), 2243.  doi: 10.3934/dcdsb.2012.17.2243.  Google Scholar

[28]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, J. Nonlinear Sci., 21 (2011), 747.  doi: 10.1007/s00332-011-9099-9.  Google Scholar

[29]

M. X. Wang and G. Y. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay,, Nonlinearity, 23 (2010), 1609.  doi: 10.1088/0951-7715/23/7/005.  Google Scholar

[30]

Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[31]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183.  doi: 10.1007/s002850200145.  Google Scholar

[32]

H. F. Weinberger, Asymptotic behavior of a model in population genetics,, in Nonlinear Partial Differential Equations and Applications (ed. J. M. Chadam), (1978), 47.   Google Scholar

[33]

H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds for a partially cooperative 2-species reaction-diffusion model,, Discrete Contin. Dyn. Syst., 23 (2009), 1087.  doi: 10.3934/dcds.2009.23.1087.  Google Scholar

[34]

S. L. Wu and C.-H. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays,, Nonlinearity, 25 (2012), 2785.  doi: 10.1088/0951-7715/25/9/2785.  Google Scholar

[35]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems,, J. Dynam. Diff. Eqns., 25 (2013), 505.  doi: 10.1007/s10884-013-9293-6.  Google Scholar

[36]

S. L. Wu, Y. J. Sun and S. Y. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity,, Discrete Contin. Dyn. Syst., 33 (2013), 921.  doi: 10.3934/dcds.2013.33.921.  Google Scholar

[37]

P. X. Weng, Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 883.  doi: 10.3934/dcdsb.2009.12.883.  Google Scholar

[38]

H. Q. Zhao, S. L. Wu and S. Y. Liu, Entire solutions of a monostable age-structured population model in a 2D lattice strip,, J. Math. Anal. Appl., 401 (2013), 85.  doi: 10.1016/j.jmaa.2012.11.032.  Google Scholar

[39]

H. Q. Zhao, S. L. Wu and S. Y. Liu, Pulsating traveling fronts and entire solutions in a discrete periodic system with a quiescent stage,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 2164.  doi: 10.1016/j.cnsns.2012.12.033.  Google Scholar

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