September  2017, 22(7): 2813-2829. doi: 10.3934/dcdsb.2017152

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

* Corresponding author: Lianzhang Bao

Received  November 2014 Revised  March 2017 Published  May 2017

This work deals with the properties of the traveling wave solutions of a double degenerate cross-diffusion model
$\begin{eqnarray*} \frac{\partial b}{\partial t} & = & D_b\nabla·\{n^pb(1-b)\nabla b\}+ n^qb^l, \\ \frac{\partial n}{\partial t} & = & D_n\nabla^2n-n^qb^l, \end{eqnarray*}$
where
$p≥q 0, q>1, l>1$
. This system accounts for degenerate diffusion at the population density
$n=b=0$
and
$b=1$
modeling the growth of certain bacteria colony with volume filling. The existence of the finite traveling wave solutions is proven which provides partial answers to the spatial patterns of the colony. In order to overcome the difficulty of traditional phase plane analysis on higher dimension, we use Schauder fixed point theorem and shooting arguments in our paper.
Citation: Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152
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. BerestyckiG. NadinB. 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. KawasakiA. MochizukiM. MatsushitaT. 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. SatnoianuP. K. MainiF. 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. BerestyckiG. NadinB. 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. KawasakiA. MochizukiM. MatsushitaT. 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. SatnoianuP. K. MainiF. 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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