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Preface
Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term
1. | College of Mathematics, Jilin University, Changchun, Jilin 130012, China |
2. | Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA |
$\begin{equation*} ρ_t = [ D(ρ)ρ_x]_x + g(ρ)\ \ \ t≥0,\ \ \ x∈ \mathbb{R},\end{equation*}$ |
$ D(ρ) $ |
$ ρ $ |
$ α ∈ [0,1] $ |
$ ρ\equiv0 $ |
$ ρ\equiv 1 $ |
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. |
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