-
Previous Article
A mathematical model studying mosquito-stage transmission-blocking vaccines
- MBE Home
- This Issue
-
Next Article
Effect of residual stress on peak cap stress in arteries
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 |
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. |
[1] |
Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243 |
[2] |
Sophia A. Maggelakis. Modeling the role of angiogenesis in epidermal wound healing. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 267-273. doi: 10.3934/dcdsb.2004.4.267 |
[3] |
Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 |
[4] |
Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242 |
[5] |
Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure and Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019 |
[6] |
Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067 |
[7] |
Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663 |
[8] |
Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591 |
[9] |
Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441 |
[10] |
Shuang-Ming Wang, Zhaosheng Feng, Zhi-Cheng Wang, Liang Zhang. Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2005-2034. doi: 10.3934/cpaa.2021145 |
[11] |
Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199 |
[12] |
Avner Friedman, Bei Hu, Chuan Xue. A three dimensional model of wound healing: Analysis and computation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2691-2712. doi: 10.3934/dcdsb.2012.17.2691 |
[13] |
Yang Liu, Zhiying Liu, Kaifei Xu. Imitative innovation or independent innovation strategic choice of emerging economies in non-cooperative innovation competition. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022023 |
[14] |
Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 |
[15] |
Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921 |
[16] |
John A. Adam. Inside mathematical modeling: building models in the context of wound healing in bone. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 1-24. doi: 10.3934/dcdsb.2004.4.1 |
[17] |
Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21 |
[18] |
Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258 |
[19] |
Guo Lin, Yahui Wang. Spreading speed in a non-monotonic Ricker competitive integrodifference system. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022108 |
[20] |
Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]