Dynamics and density evolution in piecewise deterministic growth processes

Annales Polonici Mathematici

Volumn 94, Issue 2, 2008, Pages 111-129

  Mackey, Michael C ^a   Tyran Kamińska, Marta ^b,c  

^a MCGILL UNIVERSITY   (Canada)

^b INSTITUTE OF MATHEMATICS   (Poland)

^c UNIVERSITY OF SILESIA   (Poland)

Author keywords

Asymptotic stability; First order partial differential equation; Invariant density; Stochastic semigroup

Indexed keywords

EID: 67349175815     PISSN: 00662216     EISSN: 17306272     Source Type: Journal    
DOI: 10.4064/ap94-2-2     Document Type: Article

References (25)

1. J. Banasiak, On an extension of the Kato-Voigt perturbation theorem for substochastic semigroups and its application, Taiwanese J. Math. 5 (2001), 169-191.
2. J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monogr. Math., Springer, London, 2006.
3. K. Baron and A. Lasota, Asymptotic properties of Markov operators defined by Volterra type integrals, Ann. Polon. Math. 58 (1993), 161-175.
4. A. Bobrowski, T. Lipniacki, K. Pichór, and 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.
5. O. Diekmann, H. J. A. M. Heijmans, and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 (1984), 227-248.
6. N. Friedman, L. Cai, and X. Xie, Linking stochastic dynamics to population distribution: An analytical framework of gene expression, Phys. Rev. Lett. 97 (2006), 168302-1/4.
7. G. Frosali, C. V. M. van der Mee, and F. Mugelli, A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators, Math. Methods Appl. Sci. 27 (2004), 669-685.
8. H. Gacki and A. Lasota, Markov operators defined by Volterra type integrals with advanced argument, Ann. Polon. Math. 51 (1990), 155-166.
9. D. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81 (1977), 2340-2361.
10. T. Kato, On the semi-groups generated by Kolmogoroff's differential equations, J. Math. Soc. Japan 6 (1954), 1-15.
11. A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 43-62.
12. A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Appl. Math. Sci. 97, Springer, New York, 1994.
13. A. Lasota, M. C. Mackey, and J. Tyrcha, The statistical dynamics of recurrent biological events, J. Math. Biol. 30 (1992), 775-800.
14. A. Lasota and J. Traple, Invariant measures related with Poisson driven stochastic differential equation, Stochastic Process. Appl. 106 (2003), 81-93.
15. T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier, and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theoret. Biol. 238 (2006), 348-367.
16. M. C. Mackey, M. Santavy, and P. Selepova, A mitotic oscillator with a strange attractor and distributions of cell cycle times, in: Nonlinear Oscillations in Biology and Chemistry (Salt Lake City, UT, 1985), Lecture Notes in Biomath. 66, Springer, Berlin, 1986, 34-45.
17. K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), 668-685.
18. R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262.
19. R. Rudnicki and K. Pichór, Markov semigroups and stability of the cell maturity distribution, J. Biol. Systems 8 (2000), 69-94.
20. R. Rudnicki, K. Pichór, and M. Tyran-Kaminska, Markov semigroups and their applications, in: Dynamics of Dissipation, P. Garbaczewski and R. Olkiewicz (eds.), Lecture Notes in Phys. 597, Springer, Berlin, 2002, 215-238.
21. J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Polish Acad. Sci. Math. 44 (1996), 161-182.
22. M. Tyran-Kaminska, Substochastic semigroups and densities of piecewise deterministic Markov processes, http://arxiv.org/abs/0804.4889, preprint, 2008.
23. J. Tyrcha, Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle, J. Math. Biol. 26 (1988), 465-475.
24. J. J. Tyson and K. B. Hannsgen, Cell growth and division: a deterministic/probabilistic model of the cell cycle, ibid. 23 (1986), 231-246.
25. J. Voigt, On substochastic C0-semigroups and their generators, Transport Theory Statist. Phys. 16 (1987), 453-466.