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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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Sys. Appl., 4 (2009), 19-33. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

W. Gurney and R. Nisbet, The regulation of inhomogeneneous population, J. Theor. Biol., 52 (1975), 441-457. Google Scholar

[16]

W. Gurney and R. Nisbet, A note on nonlinear population transport, J. Theor. Biol., 56 (1976), 249-251. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

J. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Sys. Appl., 4 (2009), 19-33. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

W. Gurney and R. Nisbet, The regulation of inhomogeneneous population, J. Theor. Biol., 52 (1975), 441-457. Google Scholar

[16]

W. Gurney and R. Nisbet, A note on nonlinear population transport, J. Theor. Biol., 56 (1976), 249-251. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

J. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196. Google Scholar

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