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A simple model of carcinogenic mutations with time delay and diffusion

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


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  • [1]

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


    R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations, Electron. J. Diff. Eqns., 10 (2003), 33-53.


    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), 371-391.doi: 10.1137/S0036139997325497.


    K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90.


    T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.doi: 10.1006/jmaa.2000.7182.


    U. Foryś, Comparison of the models for carcinogenesis mutations - one-stage case, in "Proceedings of the Tenth National Conference Application of Mathematics in Biology and Medicine," Świçety Krzy.z, (2004), 13-18.


    U. Foryś, Time delays in one-stage models for carcinogenesis mutations, in "Proceedings of the Eleventh National Conference Application of Mathematics in Biology and Medicine", Zawoja, (2005), 13-18.


    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), 200-281.doi: 10.1515/JAA.2005.283.


    U. Foryś, Multi-dimensional Lotka-Volterra system for carcinogenesis mutations, Math. Meth. Appl. Sci., 32 (2009), 2287-2308.doi: 10.1002/mma.1137.


    J. K. Hale, "Theory of Functional Differential Equations," Springer, 1977.


    J. D. Murray, "Mathematical Biology I: An Introduction," Springer, 2002.


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


    A. S. Perelson and G. Weisbuch, Immunology for physicists, Rev. Mod. Phys., 69 (1997), 1219-1267.doi: 10.1103/RevModPhys.69.1219.

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