Advanced Search
Article Contents
Article Contents

Dynamics of birth-and-death processes with proliferation - stability and chaos

Abstract Related Papers Cited by
  • We provide a detailed description of long time dynamics in $l^1$ of the semigroup associated with constant coefficient infinite birth-and-death systems with proliferation. In particular, we discuss and slightly extend earlier stability results of [8] and also identify a range of parameters for which the semigroup is both stable in the sense of op. cit. and topologically chaotic. Moreover, for a range of parameters, we provide an explicit description of subspaces of $l^1$ which cannot generate chaotic orbits.
    Mathematics Subject Classification: 34G10, 37B99, 47A16, 47D03.


    \begin{equation} \\ \end{equation}
  • [1]

    J. Banasiak, Chaotic linear systems in mathematical biology, South African Journal of Science, 104 (2008), 173-179.


    J. Banasiak and M. Lachowicz, Chaos for a class of linear kinetic models, Compt. Rend. Acad. Sci. Paris, 329, Série IIb, (2001), 439-444.


    J. Banasiak and M. Lachowicz, Chaotic linear dynamical systems with applications, in: C. Kubrusly, N. Levan, M. da Silveira (Eds), "Semigroups of Operators: Theory and Applications," 2nd International Conference, Rio de Janeiro, 2001, Optimization Software Inc., Publications, New York - Los Angeles, 2002, 32-44.


    J. Banasiak and M. Lachowicz, Topological chaos for birth-and-death-type models with proliferation, Math. Models Methods Appl. Sci., 12 (2002), 755-775.doi: doi:10.1142/S021820250200188X.


    J. Banasiak, M. Lachowicz and M. Moszyński, Chaotic behavior of semigroups related to the process of gene amplification-deamplification with cells' proliferation, Math. Biosciences, 206 (2007), 200-215.doi: doi:10.1016/j.mbs.2005.08.004.


    J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete and Continuous Dynamical Systems - A, 12 (2005), 959-972.


    J. Banasiak and M. Moszyński, Hypercyclicity and chaoticity spaces of $C_0$-semigroups, Discr. Cont. Dyn. Sys.-A, 20 (2008), 577-587.


    A. Bobrowski and M. Kimmel, Asymptotic behaviour of an operator exponential related to branching random walk models of DNA repeats, Journal of Biological Systems, 7 (1999), 33-43.doi: doi:10.1142/S0218339099000048.


    W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819.doi: doi:10.1017/S0143385797084976.


    S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum, 73 (2006), 313-316.doi: doi:10.1007/s00233-005-0533-x.


    S. N. Elaydi, "An Introduction to Difference Equations," Springer Verlag, New York, 1999.


    K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.


    K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Springer Science+Business Media, New York, 2006.


    W. Feller, "An Introduction to Probability and its Applications," vol. 1, 3rd edition, J. Wiley, New York, 1968.


    M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification, Bull. Math. Biol., 50 (1994), 337-357.


    M. Kimmel, A. Świerniak and A. Polański, Infinite-dimensional model of evolution of drug resistance of cancer cells, J. Math. Systems Estimation Control, 8 (1998), 1-16.


    E. H. Spanier, "Algebraic Topology," McGraw-Hill, New York-Toronto, Ont.-London, 1966.


    A. Świerniak, A. Polański and M. Kimmel, Control problems arising in chemotherapy under evolving drug resisitance, Preprints of the 13th World Congress of IFAC 1996, Volume B, 411-416.

  • 加载中

Article Metrics

HTML views() PDF downloads(166) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint