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Abstrakty
The present paper is focused on the analysis of three very simple models of carcinogenesis mutations that are based on reaction-diffusion systems and Lotka-Volterra food chains. We consider the case with two stages of mutations and study the systems of three reaction-diffusion equations with zero-flux boundary conditions. We focus on the Turing instability and show that this type of instability is not possible for these models. We also propose some modifications of the considered equations. Results are illustrated by computer simulations.
Wydawca
Czasopismo
Rocznik
Tom
Strony
283--302
Opis fizyczny
Bibliogr. 18 poz.
Twórcy
autor
- Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland, urszula@hydra.mimuw.edu.pl
Bibliografia
- [1] Ahangar R., Lin X. B., Multistage evolutionary model for carcinogenesis mutations, Electron. J. Differ. Equ. Conf. 10 (2003), 33-53.
- [2] Bhat N., Pande L., Three-step food chains in Gompertz and Lotka-Vollerra models, J. Theoret. Biol. 91(3) (1981), 429-435.
- [3] Britton, N. F., Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986.
- [4] Drasdo D., Höhme S., Individual based approaches to birth and death in avascular tumours, Math. Comput. Modelling 37(11) (2003), 1163-1175.
- [5] Foryś U., Marciniak-Czochra A., Logistic equation in tumour growth modelling, J. Appl. Math. Comput. 13 (3) (2003) 317-325.
- [6] Foryś, U., Comparison of the models for carcinogenesis mutations — one stage case, in Proceedings of the X National Conference on Mathematics Applied to Biology and Medicine (Święty Krzyż, September 2004), ed. by M. Ziółko, 2004.
- [7] Gantmacher, F. R., Applications of the Theory of Matrices, Interscience Publishers, Inc., New York, 1959.
- [8] Gard T., Hallam T., Persistence in food webs. I. Lotka-Volterra food chains, Bull. Math. Biol., 41(6) (1979), 877-891.
- [9] Henry D., Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
- [10] Hofbauer J., Sigmund K., The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1998.
- [11] Horn, M. A., Webb, G. (eds.), Special Issue on Mathematical Models in Cancer, Discrete Contin. Dyn. Syst. Ser. B 4(1) (2004).
- [12] Kruś S., Pathological Anathomy (in Polish), PZWL, Warsaw, 2001.
- [13] Michelson, S., Leith, J., Positive feedback and angiogenesis in tumor growth control, Bull. Math. Biol. 59(2) (1997), 233-254.
- [14] Murray, J. D., Mathematical Biology, Springer-Verlag, Berlin, 1993.
- [15] Preziosi, L. (ed.), Cancer Modelling and Simulation, Chapman & Hall/CRC, London, 2003.
- [16] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
- [17] So, J. W. H., A note on the global stability and bifurcation phenomenon of a Lotka-Volterra food chain, J. Theoret. Biol. 80(2) (1979), 185-187.
- [18] Turing, A. M., The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B 237 (1952), 37-72.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-LOD4-0001-0029