**Export file:**

**Format**

- RIS(for EndNote,Reference Manager,ProCite)
- BibTex
- Text

**Content**

- Citation Only
- Citation and Abstract

A singular limit for an age structured mutation problem

^{setArticleTag('','1,2','','');},^{setArticleTag('','2','','');}

1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

2. Institute of Mathematics, Technical University of Łódź, Łódź, Poland

Received: , Accepted: , Published:

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genome, mostly occurring during the cell division at the end of its life cycle. In this context, the process is described by a system of McKendrick type equations which resembles a network transport problem. In this paper we show that, under an appropriate scaling of the latter, these two descriptions are asymptotically equivalent.

# References

[1] H. Amann and J. Escher,
*Analysis II* Birkhäuser, Basel 2008.

[2] W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987): 321-349.

[3] J. Banasiak,L. Arlotti, null, Positive Perturbations of Semigroups with Applications, Springer Verlag, London, 2006.

[4] J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability *Appl. Math. Lett. *,**45** (2015), 25-30

[5] J. Banasiak, A. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks *Semigroup Forum*. [DOI 10.1007/s00233-015-9730-4]

[6] J. Banasiak,A. Falkiewicz,P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016): 215-247.

[7] J. Banasiak and M. Lachowicz,
*Methods of Small Parameter in Mathematical Biology* Birkhäuser/Springer, Cham, 2014.

[8] J. Banasiak,M. Moszyński, Dynamics of birth-and-death processes with proliferation -stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011): 67-79.

[9] A. Bobrowski, null, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005.

[10] A. Bobrowski, null, Convergence of One-parameter Operator Semigroups. In Models of Mathematical Biology and Elsewhere, Cambridge University Press, Cambridge, 2016.

[11] A. Bobrowski, On Hille-type approximation of degenerate semigroups of operators, Linear Algebra and its Applications, 511 (2016): 31-53.

[12] A. Bobrowski,M. Kimmel, Asymptotic behaviour of an operator exponential related to branching random walk models of DNA repeats, J. Biol. Systems, 7 (1999): 33-43.

[13] B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008): 341-356.

[14] K. J. Engel,R. Nagel, null, One-Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 1999.

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

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

[17] M. Kramar,E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005): 139-162.

[18] J. L. Lebowitz,S. I. Rubinov, A theory for the age and generation time distribution of a microbial population, J. Theor. Biol., 1 (1974): 17-36.

[19] C. D. Meyer,
*Matrix Analysis and Applied Linear Algebra* SIAM, Philadelphia, 2000.

[20] P. Namayanja,
*Transport on Network Structures* Ph. D thesis, UKZN, 2012.

[21] M. Rotenberg, Transport theory for growing cell population, J. Theor. Biol., 103 (1983): 181-199.

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

[23] H. T. K. Tse, W. McConnell Weaver and D. Di Carlo, Increased asymmetric and multi-daughter cell division in mechanically confined microenvironments *PLoS ONE*, **7** (2012), e38986.

Copyright Info: © 2017, Jacek Banasiak, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)