American Institute of Mathematical Sciences

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

Spatial dynamics for a model of epidermal wound healing

Haiyan Wang 1, and  Shiliang Wu 2,

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.
Keywords: non-cooperative diffusion systems, epidermal wound healing., entire solution, Traveling waves, spreading speed.
Mathematics Subject Classification: 35K57, 92C5.
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), Lecture Notes in Mathematics Ser., 446, Springer-Verlag, Berlin, 1975, 5-49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[3]

P. D. Dale, P. K. Maini and J. A. Sherratt, Mathematical modelling of corneal epithelial wound healing, Math. Biosci., 124 (1994), 127-147. doi: 10.1016/0025-5564(94)90040-X.

[4]

S. I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137. doi: 10.1023/A:1012980128575.

[5]

S. I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Phys. D, 165 (2002), 176-198. doi: 10.1016/S0167-2789(02)00379-2.

[6]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[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), 193-212.

[8]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276. doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.

[9]

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

[10]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.

[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), 1-26.

[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), 1-40. doi: 10.1002/cpa.20154.

[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), 492-504. doi: 10.1016/j.matpur.2008.07.002.

[14]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.

[15]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.

[16]

B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338. doi: 10.1007/s00285-008-0175-1.

[17]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.

[18]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003.

[19]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014.

[20]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240. doi: 10.1137/080723715.

[21]

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

[22]

J. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 29-36. doi: 10.1098/rspb.1990.0061.

[23]

J. Sherratt and J. Murray, Mathematical analysis of a basic model for epidermal wound healing, J. Math. Biol., 29 (1991), 389-404. doi: 10.1007/BF00160468.

[24]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[25]

H. R. Thieme, Density-Dependent Regulation of Spatially Distributed Populations and their Asymptotic speed of Spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.

[26]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, J. Differential Equations, 247 (2009), 887-905. doi: 10.1016/j.jde.2009.04.002.

[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), 2243-2266. doi: 10.3934/dcdsb.2012.17.2243.

[28]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.

[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), 1609-1630. doi: 10.1088/0951-7715/23/7/005.

[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), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1.

[31]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[32]

H. F. Weinberger, Asymptotic behavior of a model in population genetics, in Nonlinear Partial Differential Equations and Applications (ed. J. M. Chadam), Lecture Notes in Mathematics, 648, Springer-Verlag, Berlin, 1978, 47-96.

[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), 1087-1098. doi: 10.3934/dcds.2009.23.1087.

[34]

S. L. Wu and C.-H. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (2012), 2785-2801. doi: 10.1088/0951-7715/25/9/2785.

[35]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Diff. Eqns., 25 (2013), 505-533. doi: 10.1007/s10884-013-9293-6.

[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), 921-946. doi: 10.3934/dcds.2013.33.921.

[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), 883-904. doi: 10.3934/dcdsb.2009.12.883.

[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), 85-97. doi: 10.1016/j.jmaa.2012.11.032.

[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), 2164-2176. doi: 10.1016/j.cnsns.2012.12.033.

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), Lecture Notes in Mathematics Ser., 446, Springer-Verlag, Berlin, 1975, 5-49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[3]

P. D. Dale, P. K. Maini and J. A. Sherratt, Mathematical modelling of corneal epithelial wound healing, Math. Biosci., 124 (1994), 127-147. doi: 10.1016/0025-5564(94)90040-X.

[4]

S. I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137. doi: 10.1023/A:1012980128575.

[5]

S. I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Phys. D, 165 (2002), 176-198. doi: 10.1016/S0167-2789(02)00379-2.

[6]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[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), 193-212.

[8]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276. doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.

[9]

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

[10]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.

[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), 1-26.

[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), 1-40. doi: 10.1002/cpa.20154.

[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), 492-504. doi: 10.1016/j.matpur.2008.07.002.

[14]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.

[15]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.

[16]

B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338. doi: 10.1007/s00285-008-0175-1.

[17]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.

[18]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003.

[19]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014.

[20]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240. doi: 10.1137/080723715.

[21]

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

[22]

J. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 29-36. doi: 10.1098/rspb.1990.0061.

[23]

J. Sherratt and J. Murray, Mathematical analysis of a basic model for epidermal wound healing, J. Math. Biol., 29 (1991), 389-404. doi: 10.1007/BF00160468.

[24]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[25]

H. R. Thieme, Density-Dependent Regulation of Spatially Distributed Populations and their Asymptotic speed of Spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.

[26]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, J. Differential Equations, 247 (2009), 887-905. doi: 10.1016/j.jde.2009.04.002.

[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), 2243-2266. doi: 10.3934/dcdsb.2012.17.2243.

[28]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.

[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), 1609-1630. doi: 10.1088/0951-7715/23/7/005.

[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), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1.

[31]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[32]

H. F. Weinberger, Asymptotic behavior of a model in population genetics, in Nonlinear Partial Differential Equations and Applications (ed. J. M. Chadam), Lecture Notes in Mathematics, 648, Springer-Verlag, Berlin, 1978, 47-96.

[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), 1087-1098. doi: 10.3934/dcds.2009.23.1087.

[34]

S. L. Wu and C.-H. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (2012), 2785-2801. doi: 10.1088/0951-7715/25/9/2785.

[35]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Diff. Eqns., 25 (2013), 505-533. doi: 10.1007/s10884-013-9293-6.

[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), 921-946. doi: 10.3934/dcds.2013.33.921.

[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), 883-904. doi: 10.3934/dcdsb.2009.12.883.

[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), 85-97. doi: 10.1016/j.jmaa.2012.11.032.

[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), 2164-2176. doi: 10.1016/j.cnsns.2012.12.033.

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