A simple model of carcinogenic mutations with time delay and diffusion

  • 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.


    加载中
    [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.
  • 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

Metrics

Article views(1823) PDF downloads(522) Cited by(13)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog