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Stochastic semigroups and their applications to biological models

Wybrane pełne teksty z tego czasopisma
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Some recent results concerning generation and asymptotic properties of stochastic semigroups are presented. The general results are applied to biological models described by piecewise deterministic Markov processes: birth-death processes, the evolution of the genome, genes expression and physiologically structured models.
Wydawca
Rocznik
Strony
463--494
Opis fizyczny
Bibliogr. 46 poz.
Twórcy
autor
  • Institute of Mathematics Silesian University Bankowa 14 40 007 Katowice, Poland
autor
  • Institute of Mathematics Polish Academy of Science Bankowa 14 40 007 Katowice, Poland
  • Institute of Mathematics Silesian University Bankowa 14 40 007 Katowice, Poland
  • Institute of Mathematics Silesian University Bankowa 14 40 007 Katowice, Poland
Bibliografia
  • [1] A. S. Ackleh, B. G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, J. Math. Biol. 35 (1997), 480–502.
  • [2] A. S. Ackleh, K. Deng, On the first order hyperbolic coagulation model, Math. Methods Appl. Sci. 26 (2003), 703–715.
  • [3] O. Arino, M. Kimmel, Comparison of approaches to modeling of cell population dynamics, SIAM J. Appl. Math. 53 (1993), 1480–1504.
  • [4] O. Arino, R. Rudnicki, Phytoplankton dynamics, C. R. Biologies 327 (2004), 961–969.
  • [5] J. Banasiak, On an extension of the Kato–Voigt perturbation theorem for substochastic semigroups and its application, Taiwanese J. Math. 5 (2001), 169–191.
  • [6] J. Banasiak, On conservativity and shattering for an equation of phytoplankton dynamics, C. R. Biologies 327 (2004), 1025–1036.
  • [7] J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 2006.
  • [8] J. Banasiak, W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Syst. - B 11 (2009), 563–585.
  • [9] J. Banasiak, K. Pichór, R. Rudnicki, Asynchronous exponential growth of a structured population model, Acta Appl. Math., DOI:10.1007/s10440-011-9666-y.
  • [10] G. I. Bell, E. C. Anderson, Cell growth and division: I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophysical Journal 7 (1967), 329–351.
  • [11] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. An Introduction, Cambridge University Press, Cambridge, 2005.
  • [12] A. Bobrowski, T. Lipniacki, K. Pichór, R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl. 333 (2007), 753–769.
  • [13] M. H. A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), 353–388.
  • [14] M. H. A. Davis, Markov models and optimization, Monographs on Statistics and Applied Probability, vol. 49, Chapman & Hall, London, 1993.
  • [15] O. Diekmann, H. J. A. M. Heijmans, H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 (1984), 227–248.
  • [16] N. Friedman, L. Cai, X. Xie, Linking stochastic dynamics to population distribution: An analytical framework of gene expression, Phys. Rev. Lett. 97 (2006), 168302–1/4.
  • [17] M. Gyllenberg, H. J. A. M. Heijmans, An abstract delay-differential equation modeling size dependent cell growth and division, SIAM J. Math. Anal. 18 (1987), 74–88.
  • [18] M. Gyllenberg, G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci. 86 (1987), 67–95.
  • [19] M. Gyllenberg, G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol. 28 (1990), 671–694.
  • [20] H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Math. Biosci. 72 (1984), 19–50.
  • [21] E. Hille, R. S. Phillips, Functional Analysis and Semi-groups, AmericanMathematical Society Colloquium Publications 31, American Mathematical Society, Providence, R. I., 1957.
  • [22] M. A. Huynen, E. van Nimwegen, The frequency distribution of gene family size in complete genomes, Molecular Biology Evolution 15 (1998), 583–589.
  • [23] T. Kato, On the semi-groups generated by Kolmogoroff’s differential equations, J. Math. Soc. Japan 6 (1954), 1–15.
  • [24] M. Kimmel, Z. Darzynkiewicz, O. Arino, F. Traganos, Analysis of a cell cycle model based on unequal division of metabolic constituents to daughter cells during cytokinesis, J. Theor. Biol. 110 (1984), 637–664.
  • [25] M. Komorowski, J. Miękisz, A. M. Kierzek, Translational repression contributes greater noise to gene expression than transcriptional repression, Biophysical Journal 96 (2009), 372–384.
  • [26] A. L. Koch, J. V. Holtje, A physical basis for the precise location of the division site of rod-shaped bacteria: the central stress model, Microbiology 13 (1995), 3171–3180.
  • [27] A. Lasota, M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 43–62.
  • [28] A. Lasota, M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Springer Applied Mathematical Sciences 97, New York, 1994.
  • [29] T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier, M. Kimmel, ranscriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006), 348–367.
  • [30] M. C. Mackey, R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 (1994), 89–109.
  • [31] M. C. Mackey, M. Tyran-Kamińska, Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math. 94 (2008), 111–129.
  • [32] M. C. Mackey, M. Tyran-Kamińska, R. Yvinec, Molecular distributions in gene regulatory dynamics, J. Theor. Biol. 274 (2011), 84–96.
  • [33] J. A. J. Metz, O. Diekmann (eds.), The Dynamics of Physiologically Structured Populations, Springer Lecture Notes in Biomathematics 68, New York, 1986.
  • [34] K. Pichór, R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), 668–685.
  • [35] K. Pichór, Asymptotic stability and sweeping of substochastic semigroups, Ann. Polon. Math. 103 (2012), 123–134.
  • [36] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245–262.
  • [37] R. Rudnicki, K. Pichór, Markov semigroups and stability of the cell maturation distribution, J. Biol. Systems 8 (2000), 69–94.
  • [38] R. Rudnicki, J. Tiuryn, D. Wójtowicz, A model for the evolution of paralog families in genomes, J. Math. Biology 53 (2006), 759–770.
  • [39] R. Rudnicki, R. Wieczorek, Fragmentation – coagulation models of phytoplankton, Bull. Pol. Acad. Sci. Math. 54 (2006), 175–191.
  • [40] R. Rudnicki, R. Wieczorek, Phytoplankton dynamics: from the behaviour of cells to a transport equation, Math. Mod. Nat. Phenomena 1 (2006), 83–100.
  • [41] R. Rudnicki. R. Wieczorek, Mathematical models of phytoplankton dynamics, in: Russo R. (Ed.) Aquaculture I. Dynamic Biochemistry, Process Biotechnology and Molecular Biology 2 (Special Issue 1), (2008), 55–63.
  • [42] P. P. Slonimski, M. O. Mosse, P. Golik, A. Henaût, Y. Diaz, J. L. Risler, J. P. Comet, J. C. Aude, A. Wozniak, E. Glemet, J. J. Codani, The first laws of genomics, Microbial Comp. Genomics 3 (1998), 46.
  • [43] M. Tyran-Kamińska, Substochastic semigroups and densities of piecewise deterministic Markov processes, J. Math. Anal. Appl. 357 (2009), 385–402.
  • [44] M. Tyran-Kamińska, Ergodic theorems and perturbations of contraction semigroups, Studia Math. 195 (2009), 147–155.
  • [45] J. Voigt, On substochastic C0-semigroups and their generators, Transport Theory Statist. Phys. 16 (1987), 453–466.
  • [46] G. W. Webb, Structured population dynamics, in: R. Rudnicki (eds.), Mathematical Modelling of Population Dynamics, Banach Center Publ. 63, 123–163, Warszawa (2004).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA4-0034-0039
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