Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term

Previous Article
State transformations of time-varying delay systems and their applications to state observer design
DCDS-S Home
This Issue
Next Article
Preface

June 2017, 10(3): 395-412. doi: 10.3934/dcdss.2017019

Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term

Lianzhang Bao ^1,, and Zhengfang Zhou ^2,

1.	College of Mathematics, Jilin University, Changchun, Jilin 130012, China
2.	Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

^* Corresponding author

Received December 2015 Revised November 2016 Published February 2017

Fund Project: The work of L. Bao is sponsored by SRF for ROCS, SEM., the Twelfth Five-Year Plan project of Jilin Province's Educational Science.

This work is concerned with the properties of the traveling wave solutions of a one dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion with net birth term

$\begin{equation*} ρ_t = [ D(ρ)ρ_x]_x + g(ρ)\ \ \ t≥0,\ \ \ x∈ \mathbb{R},\end{equation*}$

where

$ D(ρ) $

may take positive or negative values with different population density

$ ρ $

and adhesion coefficient

$ α ∈ [0,1] $

, and the negative one will lead to the ill-posedness of the equation. In all these cases we prove the existence of infinitely many traveling wave solutions, where these solutions are parameterized by their wave speed and monotonically connect the stationary states

$ ρ\equiv0 $

and

$ ρ\equiv 1 $

.

Keywords: Traveling wave, diffusion, aggregation, cell-to-cell adhesion.

Mathematics Subject Classification: Primary: 35K57, 92D25; Secondary: 34B16.

Citation: Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019

References:

[1]	K. Anguige, Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion, European J. Appl. Math., 21 (2010), 109-136. doi: 10.1017/S0956792509990167.
[2]	K. Anguige, A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling, European J. Appl. Math., 4 (2011), 291-316. doi: 10.1017/S0956792511000040.
[3]	K. Anguige and C. Schmeiser, A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Math. Biol., 58 (2009), 395-427. doi: 10.1007/s00285-008-0197-8.
[4]	N. Armstrong, K. Painter and J. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theor. Biol., 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030.
[5]	L. Bao and Z. Zhou, Traveling wave in backward and forward parabolic equations from population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1507-1522. doi: 10.3934/dcdsb.2014.19.1507.
[6]	D. Dicarlo, R. Juanes, T. LaForce and T. Witelski, Nonmonotonic traveling wave solutions of infiltration into porous media, Water Resources Research, 44 (2008). doi: 10.1029/2007WR005975.
[7]	H. Engler, Relations between travelling wave solutions of quasilinear parabolic equations, Proc. Amer. Math. Soc., 93 (1985), 297-302. doi: 10.1090/S0002-9939-1985-0770540-6.
[8]	P. Feng and Z. Zhou, Finite traveling wave solutions in a denegerate cross-diffusion model for bacterial colony, Commun. Pure Appl. Anal., 6 (2007), 1145-1165. doi: 10.3934/cpaa.2007.6.1145.
[9]	Z. Feng and G. Chen, Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth, Discrete Contin. Dyn. Syst., 24 (2009), 763-780. doi: 10.3934/dcds.2009.24.763.
[10]	L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Sys. Appl., 4 (2009), 19-33.
[11]	R. Fisher, The wave of advance of advantageous genes, Ann. Eugen, 7 (1937), 353-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[12]	F. S. Garduño and P. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178.
[13]	F. S. Garduño and P. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Diff. Eqns., 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.
[14]	F. S. Garduño, P. Maini and J. Velázquez, A non-linear degenerate equation for direct aggregation and taravelling wave dynamics, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 455-487. doi: 10.3934/dcdsb.2010.13.455.
[15]	W. Gurney and R. Nisbet, The regulation of inhomogeneneous population, J. Theor. Biol., 52 (1975), 441-457.
[16]	W. Gurney and R. Nisbet, A note on nonlinear population transport, J. Theor. Biol., 56 (1976), 249-251.
[17]	K. Hadeler, Travelling fronts and free boundary value problems, in Numerical Treatment of Free Boundary Value Problems (eds. Albretch, J. , Collatz, L. , Hoffman, K. H. ), Basel: Birkhauser, 58 (1982), 90-107. doi: 10.1007/978-3-0348-6563-0_7.
[18]	D. Horstmann, K. Painter and H. Othmer, Aggregation under local reinforcement: From lattice to continuum, European J. Appl. Math., 15 (2004), 546-576. doi: 10.1017/S0956792504005571.
[19]	A. Kolmogorov, I. Petrovsky and I. Piskounov, Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem, Applicable mathematics of non-physical phenomena, (eds. OLiveira-Pinto, F. , Conolly, B. W. ) New York: Wiley, 1982.
[20]	M. Kuzmin and S. Ruggerini, Front Propagation in Diffusion-Aggregation Models with Bi-Stable Reaction, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 819-833. doi: 10.3934/dcdsb.2011.16.819.
[21]	P. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1175-1189. doi: 10.3934/dcdsb.2006.6.1175.
[22]	L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Diff. Eqns., 195 (2003), 471-496. doi: 10.1016/j.jde.2003.06.005.
[23]	J. Sherratt, On the form of smooth-front travelling waves in a diffusion equation with degenerate nonlinear diffusion, Mathematical Modelling of Natural Phenomena, 5 (2010), 64-79. doi: 10.1051/mmnp/20105505.
[24]	V. Pandrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Diff. Eqns., 23 (1998), 457-486. doi: 10.1080/03605309808821353.
[25]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Math. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.
[26]	J. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

show all references

References:

[1]	K. Anguige, Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion, European J. Appl. Math., 21 (2010), 109-136. doi: 10.1017/S0956792509990167.
[2]	K. Anguige, A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling, European J. Appl. Math., 4 (2011), 291-316. doi: 10.1017/S0956792511000040.
[3]	K. Anguige and C. Schmeiser, A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Math. Biol., 58 (2009), 395-427. doi: 10.1007/s00285-008-0197-8.
[4]	N. Armstrong, K. Painter and J. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theor. Biol., 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030.
[5]	L. Bao and Z. Zhou, Traveling wave in backward and forward parabolic equations from population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1507-1522. doi: 10.3934/dcdsb.2014.19.1507.
[6]	D. Dicarlo, R. Juanes, T. LaForce and T. Witelski, Nonmonotonic traveling wave solutions of infiltration into porous media, Water Resources Research, 44 (2008). doi: 10.1029/2007WR005975.
[7]	H. Engler, Relations between travelling wave solutions of quasilinear parabolic equations, Proc. Amer. Math. Soc., 93 (1985), 297-302. doi: 10.1090/S0002-9939-1985-0770540-6.
[8]	P. Feng and Z. Zhou, Finite traveling wave solutions in a denegerate cross-diffusion model for bacterial colony, Commun. Pure Appl. Anal., 6 (2007), 1145-1165. doi: 10.3934/cpaa.2007.6.1145.
[9]	Z. Feng and G. Chen, Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth, Discrete Contin. Dyn. Syst., 24 (2009), 763-780. doi: 10.3934/dcds.2009.24.763.
[10]	L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Sys. Appl., 4 (2009), 19-33.
[11]	R. Fisher, The wave of advance of advantageous genes, Ann. Eugen, 7 (1937), 353-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[12]	F. S. Garduño and P. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178.
[13]	F. S. Garduño and P. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Diff. Eqns., 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.
[14]	F. S. Garduño, P. Maini and J. Velázquez, A non-linear degenerate equation for direct aggregation and taravelling wave dynamics, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 455-487. doi: 10.3934/dcdsb.2010.13.455.
[15]	W. Gurney and R. Nisbet, The regulation of inhomogeneneous population, J. Theor. Biol., 52 (1975), 441-457.
[16]	W. Gurney and R. Nisbet, A note on nonlinear population transport, J. Theor. Biol., 56 (1976), 249-251.
[17]	K. Hadeler, Travelling fronts and free boundary value problems, in Numerical Treatment of Free Boundary Value Problems (eds. Albretch, J. , Collatz, L. , Hoffman, K. H. ), Basel: Birkhauser, 58 (1982), 90-107. doi: 10.1007/978-3-0348-6563-0_7.
[18]	D. Horstmann, K. Painter and H. Othmer, Aggregation under local reinforcement: From lattice to continuum, European J. Appl. Math., 15 (2004), 546-576. doi: 10.1017/S0956792504005571.
[19]	A. Kolmogorov, I. Petrovsky and I. Piskounov, Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem, Applicable mathematics of non-physical phenomena, (eds. OLiveira-Pinto, F. , Conolly, B. W. ) New York: Wiley, 1982.
[20]	M. Kuzmin and S. Ruggerini, Front Propagation in Diffusion-Aggregation Models with Bi-Stable Reaction, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 819-833. doi: 10.3934/dcdsb.2011.16.819.
[21]	P. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1175-1189. doi: 10.3934/dcdsb.2006.6.1175.
[22]	L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Diff. Eqns., 195 (2003), 471-496. doi: 10.1016/j.jde.2003.06.005.
[23]	J. Sherratt, On the form of smooth-front travelling waves in a diffusion equation with degenerate nonlinear diffusion, Mathematical Modelling of Natural Phenomena, 5 (2010), 64-79. doi: 10.1051/mmnp/20105505.
[24]	V. Pandrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Diff. Eqns., 23 (1998), 457-486. doi: 10.1080/03605309808821353.
[25]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Math. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.
[26]	J. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[1]	Caleb Mayer, Eric Stachura. Traveling wave solutions for a cancer stem cell invasion model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5067-5093. doi: 10.3934/dcdsb.2020333
[2]	Alessandro Cucchi, Antoine Mellet, Nicolas Meunier. Self polarization and traveling wave in a model for cell crawling migration. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2381-2407. doi: 10.3934/dcds.2021194
[3]	Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261
[4]	Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186
[5]	Hossein Pourbashash, Sergei S. Pilyugin, Patrick De Leenheer, Connell McCluskey. Global analysis of within host virus models with cell-to-cell viral transmission. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3341-3357. doi: 10.3934/dcdsb.2014.19.3341
[6]	Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006
[7]	Matthew S. Mizuhara, Peng Zhang. Uniqueness and traveling waves in a cell motility model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2811-2835. doi: 10.3934/dcdsb.2018315
[8]	Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683
[9]	Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861
[10]	Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789
[11]	Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic and Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035
[12]	Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
[13]	Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309
[14]	Ali Ashher Zaidi, Bruce Van Brunt, Graeme Charles Wake. A model for asymmetrical cell division. Mathematical Biosciences & Engineering, 2015, 12 (3) : 491-501. doi: 10.3934/mbe.2015.12.491
[15]	Deborah C. Markham, Ruth E. Baker, Philip K. Maini. Modelling collective cell behaviour. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123
[16]	Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure and Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243
[17]	H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3785-3801. doi: 10.3934/dcdss.2020433
[18]	Paolo Ubezio. Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 323-335. doi: 10.3934/dcdsb.2004.4.323
[19]	Russell Betteridge, Markus R. Owen, H.M. Byrne, Tomás Alarcón, Philip K. Maini. The impact of cell crowding and active cell movement on vascular tumour growth. Networks and Heterogeneous Media, 2006, 1 (4) : 515-535. doi: 10.3934/nhm.2006.1.515
[20]	Lisette dePillis, Trevor Caldwell, Elizabeth Sarapata, Heather Williams. Mathematical modeling of regulatory T cell effects on renal cell carcinoma treatment. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 915-943. doi: 10.3934/dcdsb.2013.18.915

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Lianzhang Bao Zhengfang Zhou
on Google Scholar
Lianzhang Bao Zhengfang Zhou

2020 Impact Factor: 2.425

Tools

Article outline

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Lianzhang Bao Zhengfang Zhou
on Google Scholar
Lianzhang Bao Zhengfang Zhou

[Back to Top]