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Mathematical Biosciences and Engineering (MBE)
 

A simple model of carcinogenic mutations with time delay and diffusion

Pages: 861 - 872, Volume 10, Issue 3, June 2013      doi:10.3934/mbe.2013.10.861

 
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Monika Joanna Piotrowska - Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email)
Urszula Foryƛ - Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email)
Marek Bodnar - Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email)
Jan Poleszczuk - College of Inter-faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Zwirki i Wigury 93, 02-089 Warsaw, Poland (email)

Abstract: 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.

Keywords:  Delay equations, stability, diffusion, carcinogenic mutations.
Mathematics Subject Classification:  Primary: 34K20, 34K28, 37G35, 37N25; Secondary: 92B05, 92B25, 92C50.

Received: June 2012;      Accepted: August 2012;      Available Online: April 2013.

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