Foryś, Urszula; Zduniak, Beata Two-stage model of carcinogenic mutations with the influence of delays. (English) Zbl 1304.35734 Discrete Contin. Dyn. Syst., Ser. B 19, No. 8, 2501-2519 (2014). Summary: In this paper, we make an attempt to study the influence of time delays combined with diffusion on the dynamics of the two-stage carcinogenic mutations model. The included delays represent the time needed for transformation from one type of cells to the other one. In the presented analysis we focus on possible stability switches due to increasing delays and diffusion driven instability. It occurs that diffusion has no significant impact on the asymptotic behaviour of the model solutions, while one of the present delays has a destabilising effect in most of the cases we study. The analytical results are illustrated by numerical examples of the model dynamics. Cited in 1 Document MSC: 35R10 Partial functional-differential equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 34K20 Stability theory of functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 37G35 Dynamical aspects of attractors and their bifurcations 37N25 Dynamical systems in biology 92B05 General biology and biomathematics 92B25 Biological rhythms and synchronization 92C50 Medical applications (general) Keywords:delay equations; stability switches; diffusion; carcinogenic mutations PDFBibTeX XMLCite \textit{U. Foryś} and \textit{B. Zduniak}, Discrete Contin. Dyn. Syst., Ser. B 19, No. 8, 2501--2519 (2014; Zbl 1304.35734) Full Text: DOI References: [1] R. Ahangar, Multistage evolutionary model for carcinogenesis mutations,, Electron. J. Diff. Eqns., 10, 33 (2003) · Zbl 1014.92018 [2] P. K. Brazhnik, On travelling wave solutions of Fisher’s equation in two spatial dimensions,, SIAM J. Appl. Math., 60, 371 (1999) · Zbl 0957.35065 · doi:10.1137/S0036139997325497 [3] C. M. Beauséjour, Reversal of human cellular senescence: roles of the p53 and p16 pathways,, EMBO J, 22, 4212 (2003) [4] K. Camphausen, Radiation abscopal antitumor effect is mediated through p53,, Cancer Res., 63, 1990 (2003) [5] Z. Chen, Crucial role of p53-dependent cellular senescence in suppression of Pten-deficient tumorigenesis,, Nature, 436, 725 (2005) · doi:10.1038/nature03918 [6] K. L. Cooke, On Zeroes of Some Transcendental Equations,, Funkcj. Ekvacioj, 29, 77 (1986) · Zbl 0603.34069 [7] J. Coppé, Senescence-associated secretory phenotypes reveal cell-nonautonomous functions of oncogenic RAS and the p53 tumor suppressor,, PLoS Biol., 6 (2008) · doi:10.1371/journal.pbio.0060301 [8] E. R. Fearon, A genetic model for colorectal tumorigenesis,, Cell, 61, 759 (1990) · doi:10.1016/0092-8674(90)90186-I [9] U. Foryś, Biological delay systems and the Mikhailov criterion of stability,, J. Biol. Sys., 12, 45 (2004) · Zbl 1101.92001 [10] U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations,, J. Appl. Anal., 11, 200 (2005) · Zbl 1101.34033 · doi:10.1515/JAA.2005.283 [11] U. Foryś, Multi-dimensional Lotka-{Volterra} system for carcinogenesis mutations,, Math. Meth. Appl. Sci., 32, 2287 (2009) · Zbl 1181.35129 · doi:10.1002/mma.1137 [12] U. Foryś, Influence of time delays on a two-stage mutations model,, in Proceedings of the XIX National Conference Applications of Mathematics in Biology and Medicine (2013) [13] J. S. Fridman, Control of apoptosis by p53,, Oncogene, 22, 9030 (2003) · doi:10.1038/sj.onc.1207116 [14] M. S. Greenblatt, Mutations in the p53 tumor suppressor gene: clues to cancer etiology and molecular pathogenesis,, Cancer Res., 54, 4855 (1994) [15] J. K. Hale, <em>Introduction to Functional Differential Equations</em>,, Springer (1993) · Zbl 0787.34002 · doi:10.1007/978-1-4612-4342-7 [16] B. Hat, Exploring mechanisms of oscillations in p53 and nuclear factor-\( \kappa B\) systems,, Systems Biology, 3, 342 (2009) · doi:10.1049/iet-syb.2008.0156 [17] E. Michalak, Death squads enlisted by the tumour suppressor p53,, Biochem. Bioph. Res. Co., 331, 786 (2005) · doi:10.1016/j.bbrc.2005.03.183 [18] M. J. Piotrowska, A simple model of carcinogenic mutations with time delay and diffusion,, Math. Biosci. Eng., 10, 861 (2013) · Zbl 1268.92068 · doi:10.3934/mbe.2013.10.861 [19] K. Puszyński, Oscillations and bistability in the stochastic model of p53 regulation,, Journal of Theoretical Biology, 254, 452 (2008) · Zbl 1400.92200 [20] L. D. Wood, The genomic landscapes of human breast and colorectal cancers,, Science, 318, 1108 (2007) · doi:10.1126/science.1145720 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.