Loading [Contrib]/a11y/accessibility-menu.js
Search Advanced

Mathematical Biosciences and Engineering

2013, Volume 10, Issue 3: 861-872. doi: 10.3934/mbe.2013.10.861
Previous Article Next Article

A simple model of carcinogenic mutations with time delay and diffusion

  • 1.
    Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw
  • 2.
    College of Inter-faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Zwirki i Wigury 93, 02-089 Warsaw
  • Received: 01 June 2012 Accepted: 29 June 2018 Published: 01 April 2013

  • MSC : Primary: 34K20, 34K28, 37G35, 37N25; Secondary: 92B05, 92B25, 92C50.

  • In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy.Included delay stands for the interactions between benign and other cells.We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists.The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.

    Citation: Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion[J]. Mathematical Biosciences and Engineering, 2013, 10(3): 861-872. doi: 10.3934/mbe.2013.10.861

    Related Papers:

  • In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy.Included delay stands for the interactions between benign and other cells.We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists.The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.


    [1] Birkhäuser, Boston, 1997.
    [2] Electron. J. Diff. Eqns., 10 (2003), 33-53.
    [3] SIAM J. Appl. Math., 60 (1999), 371-391.
    [4] Funkcj. Ekvacioj, 29 (1986), 77-90.
    [5] J. Math. Anal. Appl., 254 (2001), 433-463.
    [6] in "Proceedings of the Tenth National Conference Application of Mathematics in Biology and Medicine," Świçety Krzy.z, (2004), 13-18.
    [7] in "Proceedings of the Eleventh National Conference Application of Mathematics in Biology and Medicine", Zawoja, (2005), 13-18.
    [8] J. Appl. Anal., 11 (2005), 200-281.
    [9] Math. Meth. Appl. Sci., 32 (2009), 2287-2308.
    [10] Springer, 1977.
    [11] Springer, 2002.
    [12] Springer, 2003.
    [13] Rev. Mod. Phys., 69 (1997), 1219-1267.

  • This article has been cited by:

    1. Antonella Belfatto, Marco Riboldi, Delia Ciardo, Federica Cattani, Agnese Cecconi, Roberta Lazzari, Barbara Alicja Jereczek-Fossa, Roberto Orecchia, Guido Baroni, Pietro Cerveri, Modeling the Interplay Between Tumor Volume Regression and Oxygenation in Uterine Cervical Cancer During Radiotherapy Treatment, 2016, 20, 2168-2194, 596, 10.1109/JBHI.2015.2398512
    2. Jean-Jacques Kengwoung-Keumo, Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation, 2016, 13, 1551-0018, 787, 10.3934/mbe.2016018
    3. Urszula Foryś, Beata Zduniak, Two-stage model of carcinogenic mutations with the influence of delays, 2014, 19, 1553-524X, 2501, 10.3934/dcdsb.2014.19.2501
    4. Marek Bodnar, Monika Joanna Piotrowska, Stability analysis of the family of tumour angiogenesis models with distributed time delays, 2016, 31, 10075704, 124, 10.1016/j.cnsns.2015.08.002
    5. Antonella Belfatto, Marco Riboldi, Delia Ciardo, Federica Cattani, Agnese Cecconi, Roberta Lazzari, Barbara Alicja Jereczek-Fossa, Roberto Orecchia, Guido Baroni, Pietro Cerveri, Kinetic Models for Predicting Cervical Cancer Response to Radiation Therapy on Individual Basis Using Tumor Regression MeasuredIn VivoWith Volumetric Imaging, 2016, 15, 1533-0346, 146, 10.1177/1533034615573796
    6. A. Belfatto, M. Riboldi, D. Ciardo, A. Cecconi, R. Lazzari, B. A. Jereczek-Fossa, R. Orecchia, G. Baroni, P. Cerveri, Adaptive Mathematical Model of Tumor Response to Radiotherapy Based on CBCT Data, 2016, 20, 2168-2194, 802, 10.1109/JBHI.2015.2453437
    7. Urszula Foryś, Monika J. Piotrowska, Analysis of the Hopf bifurcation for the family of angiogenesis models II: The case of two nonzero unequal delays, 2013, 220, 00963003, 277, 10.1016/j.amc.2013.05.077
    8. Ishtiaq Ali, On the Numerical Solutions of One and Two-Stage Model of Carcinogenesis Mutations with Time Delay and Diffusion, 2013, 04, 2152-7385, 118, 10.4236/am.2013.410A2012
    9. Vsevolod G. Sorokin, Andrei V. Vyazmin, Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration, 2022, 10, 2227-7390, 1886, 10.3390/math10111886
    10. Andrei D. Polyanin, Vsevolod G. Sorokin, Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays, 2023, 11, 2227-7390, 516, 10.3390/math11030516
    11. Larysa Dzyubak, Oleksandr Dzyubak, Jan Awrejcewicz, Nonlinear multiscale diffusion cancer invasion model with memory of states, 2023, 168, 09600779, 113091, 10.1016/j.chaos.2022.113091
    12. Andrei D. Polyanin, Vsevolod G. Sorokin, Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay, 2023, 11, 2227-7390, 3111, 10.3390/math11143111
    13. А. Д. Полянин, В. Г. Сорокин, РЕШЕНИЯ ЛИНЕЙНЫХ НАЧАЛЬНО-КРАЕВЫХ ЗАДАЧ РЕАКЦИОННО-ДИФФУЗИОННОГО ТИПА С ЗАПАЗДЫВАНИЕМ, 2023, 12, 2304-487X, 153, 10.26583/vestnik.2023.286
    14. Ali Sadiq Alabdrabalnabi, Ishtiaq Ali, Stability analysis and simulations of tumor growth model based on system of reaction-diffusion equation in two-dimensions, 2024, 9, 2473-6988, 11560, 10.3934/math.2024567
  • Reader Comments
  • © 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(3062) PDF downloads(528) Cited by(14)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog