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Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility
Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling
College of Mathematics, Jilin University, Changchun, Jilin 130012, China |
$\begin{eqnarray*} \frac{\partial b}{\partial t} & = & D_b\nabla·\{n^pb(1-b)\nabla b\}+ n^qb^l, \\ \frac{\partial n}{\partial t} & = & D_n\nabla^2n-n^qb^l, \end{eqnarray*}$ |
$p≥q 0, q>1, l>1$ |
$n=b=0$ |
$b=1$ |
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., 22 (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] |
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. |
[5] |
J. W. Barrett and K. Deckelnick,
Existence, uniquesness and approximation of a doubly-degenerate nonlinear parabolic system modeling bacterial evolution, Math. Models Methods Appl. Sci., 17 (2007), 1095-1127.
doi: 10.1142/S0218202507002212. |
[6] |
H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik,
The non-local Fisher-KPP equation: Traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.
doi: 10.1088/0951-7715/22/12/002. |
[7] |
E. O. Burdene and H. C. Berg,
Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
|
[8] |
E. O. Burdene and H. C. Berg,
Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.
|
[9] |
P. Feng and Z. Zhou,
Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony, Commun. Pure Appl. Anal., 6 (2007), 1145-1165.
doi: 10.3934/cpaa.2007.6.1145. |
[10] |
R. A. Fisher,
The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 353-369.
|
[11] |
K. P. Hadeler, Travelling fronts and free boundary value problems, in Numerical Treatment of Free Boundary Value Problems (eds. J. Albretch, L. Collatz, K. H. Hoffman), Basel: Birkhauser, 1981. |
[12] |
K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Schigesada,
Modeling spatio-temporal patterns generated by bacillus subtilis, J. Theor. Biol., 188 (1997), 177-185.
|
[13] |
A. Kolmogorov, I. Petrovsky and I. N. 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. F. OLiveira-Pinto, B. W. Conolly), New York: Wiley, 1982. |
[14] |
L. Malaguti and C. Marcelli,
Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195 (2003), 471-496.
doi: 10.1016/j.jde.2003.06.005. |
[15] |
R. A. Satnoianu, P. K. Maini, F. S. Garduno and J. P. Armitage,
Traveling waves in a nonlinear degenerate diffusion model for bacterial pattern formation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 339-362.
doi: 10.3934/dcdsb.2001.1.339. |
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., 22 (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] |
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. |
[5] |
J. W. Barrett and K. Deckelnick,
Existence, uniquesness and approximation of a doubly-degenerate nonlinear parabolic system modeling bacterial evolution, Math. Models Methods Appl. Sci., 17 (2007), 1095-1127.
doi: 10.1142/S0218202507002212. |
[6] |
H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik,
The non-local Fisher-KPP equation: Traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.
doi: 10.1088/0951-7715/22/12/002. |
[7] |
E. O. Burdene and H. C. Berg,
Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
|
[8] |
E. O. Burdene and H. C. Berg,
Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.
|
[9] |
P. Feng and Z. Zhou,
Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony, Commun. Pure Appl. Anal., 6 (2007), 1145-1165.
doi: 10.3934/cpaa.2007.6.1145. |
[10] |
R. A. Fisher,
The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 353-369.
|
[11] |
K. P. Hadeler, Travelling fronts and free boundary value problems, in Numerical Treatment of Free Boundary Value Problems (eds. J. Albretch, L. Collatz, K. H. Hoffman), Basel: Birkhauser, 1981. |
[12] |
K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Schigesada,
Modeling spatio-temporal patterns generated by bacillus subtilis, J. Theor. Biol., 188 (1997), 177-185.
|
[13] |
A. Kolmogorov, I. Petrovsky and I. N. 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. F. OLiveira-Pinto, B. W. Conolly), New York: Wiley, 1982. |
[14] |
L. Malaguti and C. Marcelli,
Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195 (2003), 471-496.
doi: 10.1016/j.jde.2003.06.005. |
[15] |
R. A. Satnoianu, P. K. Maini, F. S. Garduno and J. P. Armitage,
Traveling waves in a nonlinear degenerate diffusion model for bacterial pattern formation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 339-362.
doi: 10.3934/dcdsb.2001.1.339. |
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