# American Institute of Mathematical Sciences

January  2011, 29(1): 67-79. doi: 10.3934/dcds.2011.29.67

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

 1 School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000 2 Wydział Matematyki Informatyki i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa

Received  January 2010 Revised  May 2010 Published  September 2010

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.
Citation: Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67
##### References:
 [1] J. Banasiak, Chaotic linear systems in mathematical biology,, South African Journal of Science, 104 (2008), 173.   Google Scholar [2] J. Banasiak and M. Lachowicz, Chaos for a class of linear kinetic models,, Compt. Rend. Acad. Sci. Paris, 329 (2001), 439.   Google Scholar [3] J. Banasiak and M. Lachowicz, Chaotic linear dynamical systems with applications,, in: C. Kubrusly, (2001), 32.   Google Scholar [4] J. Banasiak and M. Lachowicz, Topological chaos for birth-and-death-type models with proliferation,, Math. Models Methods Appl. Sci., 12 (2002), 755.  doi: doi:10.1142/S021820250200188X.  Google Scholar [5] 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.  doi: doi:10.1016/j.mbs.2005.08.004.  Google Scholar [6] J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos,, Discrete and Continuous Dynamical Systems - A, 12 (2005), 959.   Google Scholar [7] J. Banasiak and M. Moszyński, Hypercyclicity and chaoticity spaces of $C_0$-semigroups,, Discr. Cont. Dyn. Sys.-A, 20 (2008), 577.   Google Scholar [8] 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.  doi: doi:10.1142/S0218339099000048.  Google Scholar [9] W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators,, Ergodic Theory Dynam. Systems, 17 (1997), 793.  doi: doi:10.1017/S0143385797084976.  Google Scholar [10] S. El Mourchid, The imaginary point spectrum and hypercyclicity,, Semigroup Forum, 73 (2006), 313.  doi: doi:10.1007/s00233-005-0533-x.  Google Scholar [11] S. N. Elaydi, "An Introduction to Difference Equations,", Springer Verlag, (1999).   Google Scholar [12] K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000).   Google Scholar [13] K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups,", Springer Science+Business Media, (2006).   Google Scholar [14] W. Feller, "An Introduction to Probability and its Applications,", vol. 1, 1 (1968).   Google Scholar [15] M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification,, Bull. Math. Biol., 50 (1994), 337.   Google Scholar [16] 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.   Google Scholar [17] E. H. Spanier, "Algebraic Topology,", McGraw-Hill, (1966).   Google Scholar [18] 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, (1996), 411.   Google Scholar

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##### References:
 [1] J. Banasiak, Chaotic linear systems in mathematical biology,, South African Journal of Science, 104 (2008), 173.   Google Scholar [2] J. Banasiak and M. Lachowicz, Chaos for a class of linear kinetic models,, Compt. Rend. Acad. Sci. Paris, 329 (2001), 439.   Google Scholar [3] J. Banasiak and M. Lachowicz, Chaotic linear dynamical systems with applications,, in: C. Kubrusly, (2001), 32.   Google Scholar [4] J. Banasiak and M. Lachowicz, Topological chaos for birth-and-death-type models with proliferation,, Math. Models Methods Appl. Sci., 12 (2002), 755.  doi: doi:10.1142/S021820250200188X.  Google Scholar [5] 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.  doi: doi:10.1016/j.mbs.2005.08.004.  Google Scholar [6] J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos,, Discrete and Continuous Dynamical Systems - A, 12 (2005), 959.   Google Scholar [7] J. Banasiak and M. Moszyński, Hypercyclicity and chaoticity spaces of $C_0$-semigroups,, Discr. Cont. Dyn. Sys.-A, 20 (2008), 577.   Google Scholar [8] 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.  doi: doi:10.1142/S0218339099000048.  Google Scholar [9] W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators,, Ergodic Theory Dynam. Systems, 17 (1997), 793.  doi: doi:10.1017/S0143385797084976.  Google Scholar [10] S. El Mourchid, The imaginary point spectrum and hypercyclicity,, Semigroup Forum, 73 (2006), 313.  doi: doi:10.1007/s00233-005-0533-x.  Google Scholar [11] S. N. Elaydi, "An Introduction to Difference Equations,", Springer Verlag, (1999).   Google Scholar [12] K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000).   Google Scholar [13] K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups,", Springer Science+Business Media, (2006).   Google Scholar [14] W. Feller, "An Introduction to Probability and its Applications,", vol. 1, 1 (1968).   Google Scholar [15] M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification,, Bull. Math. Biol., 50 (1994), 337.   Google Scholar [16] 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.   Google Scholar [17] E. H. Spanier, "Algebraic Topology,", McGraw-Hill, (1966).   Google Scholar [18] 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, (1996), 411.   Google Scholar
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