American Institute of Mathematical Sciences

Advanced
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

Lianzhang Bao , and  Wenjie Gao

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.
Keywords: Finite traveling wave, degenerate cross-diffusion, bacterial colony.
Mathematics Subject Classification: Primary:34B16, 35K65;Secondary:35K40.
Citation: Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & 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.  Google Scholar

[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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

E. O. Burdene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.   Google Scholar

[8]

E. O. Burdene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.   Google Scholar

[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.  Google Scholar

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 353-369.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  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., 22 (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]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

E. O. Burdene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.   Google Scholar

[8]

E. O. Burdene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.   Google Scholar

[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.  Google Scholar

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 353-369.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145
[2]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239
[3]

Yanxia Wu, Yaping Wu. Existence of traveling waves with transition layers for some degenerate cross-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 911-934. doi: 10.3934/cpaa.2012.11.911
[4]

Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145
[5]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[6]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[7]

Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : ⅰ-ⅰ. doi: 10.3934/dcdss.20202i
[8]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[9]

Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147
[10]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[11]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[12]

Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745
[13]

Daniel Ryan, Robert Stephen Cantrell. Avoidance behavior in intraguild predation communities: A cross-diffusion model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1641-1663. doi: 10.3934/dcds.2015.35.1641
[14]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185
[15]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019198
[16]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[17]

Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133
[18]

Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163
[19]

Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063
[20]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (18)
  • HTML views (22)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences