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2013, 10(3): 861-872. doi: 10.3934/mbe.2013.10.861

A simple model of carcinogenic mutations with time delay and diffusion

Monika Joanna Piotrowska 1, , Urszula Foryś 1, , Marek Bodnar 1, and  Jan Poleszczuk 2,

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, Poland

Received  June 2012 Revised  August 2012 Published  April 2013

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, carcinogenic mutations., diffusion.
Mathematics Subject Classification: Primary: 34K20, 34K28, 37G35, 37N25; Secondary: 92B05, 92B25, 92C5.
Citation: Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861
References:
[1]

J. A. Adam and N. Bellomo, "A Survey of Models for Tumor-imune System Synamics,", Birkhäuser, (1997).   Google Scholar

[2]

R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations,, Electron. J. Diff. Eqns., 10 (2003), 33.   Google Scholar

[3]

P. K. Brazhnik and J. J. Tyson, On travelling wave solutions of Fisher's equation in two spatial dimensions,, SIAM J. Appl. Math., 60 (1999), 371.  doi: 10.1137/S0036139997325497.  Google Scholar

[4]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations,, Funkcj. Ekvacioj, 29 (1986), 77.   Google Scholar

[5]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion,, J. Math. Anal. Appl., 254 (2001), 433.  doi: 10.1006/jmaa.2000.7182.  Google Scholar

[6]

U. Foryś, Comparison of the models for carcinogenesis mutations - one-stage case,, in, (2004), 13.   Google Scholar

[7]

U. Foryś, Time delays in one-stage models for carcinogenesis mutations,, in, (2005), 13.   Google Scholar

[8]

U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations,, J. Appl. Anal., 11 (2005), 200.  doi: 10.1515/JAA.2005.283.  Google Scholar

[9]

U. Foryś, Multi-dimensional Lotka-Volterra system for carcinogenesis mutations,, Math. Meth. Appl. Sci., 32 (2009), 2287.  doi: 10.1002/mma.1137.  Google Scholar

[10]

J. K. Hale, "Theory of Functional Differential Equations,", Springer, (1977).   Google Scholar

[11]

J. D. Murray, "Mathematical Biology I: An Introduction,", Springer, (2002).   Google Scholar

[12]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", Springer, (2003).   Google Scholar

[13]

A. S. Perelson and G. Weisbuch, Immunology for physicists,, Rev. Mod. Phys., 69 (1997), 1219.  doi: 10.1103/RevModPhys.69.1219.  Google Scholar

show all references

References:
[1]

J. A. Adam and N. Bellomo, "A Survey of Models for Tumor-imune System Synamics,", Birkhäuser, (1997).   Google Scholar

[2]

R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations,, Electron. J. Diff. Eqns., 10 (2003), 33.   Google Scholar

[3]

P. K. Brazhnik and J. J. Tyson, On travelling wave solutions of Fisher's equation in two spatial dimensions,, SIAM J. Appl. Math., 60 (1999), 371.  doi: 10.1137/S0036139997325497.  Google Scholar

[4]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations,, Funkcj. Ekvacioj, 29 (1986), 77.   Google Scholar

[5]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion,, J. Math. Anal. Appl., 254 (2001), 433.  doi: 10.1006/jmaa.2000.7182.  Google Scholar

[6]

U. Foryś, Comparison of the models for carcinogenesis mutations - one-stage case,, in, (2004), 13.   Google Scholar

[7]

U. Foryś, Time delays in one-stage models for carcinogenesis mutations,, in, (2005), 13.   Google Scholar

[8]

U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations,, J. Appl. Anal., 11 (2005), 200.  doi: 10.1515/JAA.2005.283.  Google Scholar

[9]

U. Foryś, Multi-dimensional Lotka-Volterra system for carcinogenesis mutations,, Math. Meth. Appl. Sci., 32 (2009), 2287.  doi: 10.1002/mma.1137.  Google Scholar

[10]

J. K. Hale, "Theory of Functional Differential Equations,", Springer, (1977).   Google Scholar

[11]

J. D. Murray, "Mathematical Biology I: An Introduction,", Springer, (2002).   Google Scholar

[12]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", Springer, (2003).   Google Scholar

[13]

A. S. Perelson and G. Weisbuch, Immunology for physicists,, Rev. Mod. Phys., 69 (1997), 1219.  doi: 10.1103/RevModPhys.69.1219.  Google Scholar

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