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

Jacek Banasiak; Marcin Moszyński

doi:10.3934/dcds.2011.29.67

Article Contents

2011, Volume 29, Issue 1: 67-79. Doi: 10.3934/dcds.2011.29.67

This issue Previous Article Classification of local asymptotics for solutions to heat equations with inverse-square potentials Next Article The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms

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

Jacek Banasiak^1, and
Marcin Moszyński^2,

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: December 31, 2009

Revised: April 30, 2010

Published: December 2010

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Abstract

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.

Keywords:

Mathematics Subject Classification: 34G10, 37B99, 47A16, 47D03.

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References

[1]	J. Banasiak, Chaotic linear systems in mathematical biology, South African Journal of Science, 104 (2008), 173-179.
[2]	J. Banasiak and M. Lachowicz, Chaos for a class of linear kinetic models, Compt. Rend. Acad. Sci. Paris, 329, Série IIb, (2001), 439-444.
[3]	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.
[4]	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.
[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), 200-215.doi: doi:10.1016/j.mbs.2005.08.004.
[6]	J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete and Continuous Dynamical Systems - A, 12 (2005), 959-972.
[7]	J. Banasiak and M. Moszyński, Hypercyclicity and chaoticity spaces of $C_0$-semigroups, Discr. Cont. Dyn. Sys.-A, 20 (2008), 577-587.
[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), 33-43.doi: doi:10.1142/S0218339099000048.
[9]	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.
[10]	S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum, 73 (2006), 313-316.doi: doi:10.1007/s00233-005-0533-x.
[11]	S. N. Elaydi, "An Introduction to Difference Equations," Springer Verlag, New York, 1999.
[12]	K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.
[13]	K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Springer Science+Business Media, New York, 2006.
[14]	W. Feller, "An Introduction to Probability and its Applications," vol. 1, 3rd edition, J. Wiley, New York, 1968.
[15]	M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification, Bull. Math. Biol., 50 (1994), 337-357.
[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), 1-16.
[17]	E. H. Spanier, "Algebraic Topology," McGraw-Hill, New York-Toronto, Ont.-London, 1966.
[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, Volume B, 411-416.