August  2003, 3(3): 439-456. doi: 10.3934/dcdsb.2003.3.439

Asymptotic behavior of a singular transport equation modelling cell division

1. 

Laboratoire de Mathématiques Appliquées, FRE 2570, Université de Pau et des Pays de l'Adour, Avenue de l'université, 64000 Pau, France

2. 

Department of Physiology, McGill University, McIntyre Medical Sciences Building, 3655 Promenade Sir William Osler, Montreal, QC, Canada H3G 1Y6, Canada

Received  May 2002 Revised  January 2003 Published  May 2003

This paper analyses the behavior of the solutions of a model of cells that are capable of simultaneous proliferation and maturation. This model is described by a first-order singular partial differential system with a retardation of the maturation variable and a time delay. Both delays are due to cell replication. We prove that uniqueness and asymptotic behavior of solutions depend only on cells with small maturity (stem cells).
Citation: Mostafa Adimy, Laurent Pujo-Menjouet. Asymptotic behavior of a singular transport equation modelling cell division. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 439-456. doi: 10.3934/dcdsb.2003.3.439
[1]

Mostafa Adimy, Abdennasser Chekroun, Tarik-Mohamed Touaoula. Age-structured and delay differential-difference model of hematopoietic stem cell dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2765-2791. doi: 10.3934/dcdsb.2015.20.2765

[2]

Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra, Maria dM Vivanco. A model for stem cell population dynamics with regulated maturation delay. Conference Publications, 2011, 2011 (Special) : 32-43. doi: 10.3934/proc.2011.2011.32

[3]

Frédérique Billy, Jean Clairambault, Franck Delaunay, Céline Feillet, Natalia Robert. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Mathematical Biosciences & Engineering, 2013, 10 (1) : 1-17. doi: 10.3934/mbe.2013.10.1

[4]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065

[5]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020042

[6]

Chengjun Guo, Chengxian Guo, Sameed Ahmed, Xinfeng Liu. Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2473-2489. doi: 10.3934/dcdsb.2016056

[7]

Ricardo Borges, Àngel Calsina, Sílvia Cuadrado. Equilibria of a cyclin structured cell population model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 613-627. doi: 10.3934/dcdsb.2009.11.613

[8]

A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701

[9]

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693

[10]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[11]

Dirk Hartmann, Isabella von Sivers. Structured first order conservation models for pedestrian dynamics. Networks & Heterogeneous Media, 2013, 8 (4) : 985-1007. doi: 10.3934/nhm.2013.8.985

[12]

Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117

[13]

Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19

[14]

Janet Dyson, Rosanna Villella-Bressan, G.F. Webb. The steady state of a maturity structured tumor cord cell population. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 115-134. doi: 10.3934/dcdsb.2004.4.115

[15]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657

[16]

E.V. Presnov, Z. Agur. The Role Of Time Delays, Slow Processes And Chaos In Modulating The Cell-Cycle Clock. Mathematical Biosciences & Engineering, 2005, 2 (3) : 625-642. doi: 10.3934/mbe.2005.2.625

[17]

Kim Knudsen, Mikko Salo. Determining nonsmooth first order terms from partial boundary measurements. Inverse Problems & Imaging, 2007, 1 (2) : 349-369. doi: 10.3934/ipi.2007.1.349

[18]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633

[19]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931

[20]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]