American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Analysis of a predator-prey model with predators impulsively diffusing between two patches
  • DCDS-B Home
  • This Issue
  • Next Article
    A spectral collocation method for solving initial value problems of first order ordinary differential equations
October  2010, 14(3): 1055-1080. doi: 10.3934/dcdsb.2010.14.1055

Mathematical analysis of a kinetic model for cell movement in network tissues

Thomas Hillen 1, , Peter Hinow 2, and  Zhi-An Wang 3,

1. 

Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, T6G 2G1
2. 

Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413, United States
3. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States

Received  March 2009 Revised  March 2010 Published  July 2010

Mesenchymal motion describes the movement of cells in biological tissues formed by fibre networks. An important example is the migration of tumour cells through collagen networks during the process of metastasis formation. We investigate the mesenchymal motion model proposed by T. Hillen in [14] in higher dimensions. We formulate the problem as an evolution equation in a Banach space of measure-valued functions and use methods from semigroup theory to show the global existence of mild and classical solutions. We investigate steady states of the model and show that patterns of network type exist as steady states. For the case of constant fibre distribution, we find an explicit solution and we prove the convergence to the parabolic limit.
Keywords: pointwise steady states., parabolic limits, Mesenchymal motion, kinetic theory.
Mathematics Subject Classification: Primary: 35L03; Secondary: 92C1.
Citation: Thomas Hillen, Peter Hinow, Zhi-An Wang. Mathematical analysis of a kinetic model for cell movement in network tissues. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1055-1080. doi: 10.3934/dcdsb.2010.14.1055
[1]

Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic & Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013
[2]

Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[3]

Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21
[4]

Catherine Ha Ta, Qing Nie, Tian Hong. Controlling stochasticity in epithelial-mesenchymal transition through multiple intermediate cellular states. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2275-2291. doi: 10.3934/dcdsb.2016047
[5]

József Z. Farkas, Peter Hinow. Steady states in hierarchical structured populations with distributed states at birth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2671-2689. doi: 10.3934/dcdsb.2012.17.2671
[6]

Àngel Calsina, József Z. Farkas. Boundary perturbations and steady states of structured populations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6675-6691. doi: 10.3934/dcdsb.2019162
[7]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
[8]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621
[9]

H.J. Hwang, K. Kang, A. Stevens. Drift-diffusion limits of kinetic models for chemotaxis: A generalization. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 319-334. doi: 10.3934/dcdsb.2005.5.319
[10]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029
[11]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
[12]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[13]

Anton Arnold, Laurent Desvillettes, Céline Prévost. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Communications on Pure & Applied Analysis, 2012, 11 (1) : 83-96. doi: 10.3934/cpaa.2012.11.83
[14]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026
[15]

Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231
[16]

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173
[17]

Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic & Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040
[18]

Tian Xiang. A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2457-2485. doi: 10.3934/dcdsb.2013.18.2457
[19]

Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467
[20]

Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy