# American Institute of Mathematical Sciences

June  2019, 24(6): 2443-2472. doi: 10.3934/dcdsb.2018260

## Asymptotics of the Lebowitz-Rubinow-Rotenberg model of population development

 1 Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland 2 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-656 Warsaw, Poland

Received  August 2017 Published  June 2019 Early access  October 2018

Fund Project: This research was partly supported by Polish National Science Centre grant 2014/15/N/ST1/03110.

We study a mathematical model of cell populations dynamics proposed by J. Lebowitz and S. Rubinow, and analysed by M. Rotenberg. Here, a cell is characterized by her maturity and speed of maturation. The growth of cell populations is described by a partial differential equation with a boundary condition. In the first part of the paper we exploit semigroup theory approach and apply Lord Kelvin's method of images in order to give a new proof that the model is well posed. A semi-explicit formula for the semigroup related to the model obtained by the method of images allows two types of new results. First of all, we give growth order estimates for the semigroup, applicable also in the case of decaying populations. Secondly, we study asymptotic behavior of the semigroup in the case of approximately constant population size. More specifically, we formulate conditions for the asymptotic stability of the semigroup in the case in which the average number of viable daughters per mitosis equals one. To this end we use methods developed by K. Pichór and R. Rudnicki.

Citation: Adam Gregosiewicz. Asymptotics of the Lebowitz-Rubinow-Rotenberg model of population development. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2443-2472. doi: 10.3934/dcdsb.2018260
##### References:
 [1] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, volume 233, Springer, New York, 2006. [2] J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability, Appl. Math. Lett., 45 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.006. [3] J. Banasiak, A. Falkiewicz and 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.  doi: 10.1142/S0218202516400017. [4] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583. [5] A. Bobrowski, Generation of cosine families via Lord Kelvin's method of images, J. Evol. Equ., 10 (2010), 663-675.  doi: 10.1007/s00028-010-0065-z. [6] A. Bobrowski, Lord Kelvin's method of images in semigroup theory, Semigroup Forum, 81 (2010), 435-445.  doi: 10.1007/s00233-010-9230-5. [7] A. Bobrowski and A. Gregosiewicz, A general theorem on generation of moments-preserving cosine families by Laplace operators in $C[0, 1]$, Semigroup Forum, 88 (2014), 689-701.  doi: 10.1007/s00233-013-9561-0. [8] A. Bobrowski, A. Gregosiewicz and M. Murat, Functionals-preserving cosine families generated by Laplace operators in $C[0, 1]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1877-1895.  doi: 10.3934/dcdsb.2015.20.1877. [9] M. Boulanouar, A mathematical study for a Rotenberg model, J. Math. Anal. Appl., 265 (2002), 371-394.  doi: 10.1006/jmaa.2001.7721. [10] Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-Parameter Semigroups, volume 5, North-Holland Publishing Co., Amsterdam, 1987. [11] N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, Inc., New York, 1988. [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, volume 194. Springer-Verlag, New York, 2000. [13] J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985. [14] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1994. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators. I, volume 256, Springer-Verlag, Berlin, second edition, 2003. doi: 10.1007/978-3-642-61497-2. [16] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, volume 97, Springer-Verlag, New York, second edition, 1994. doi: 10.1007/978-1-4612-4286-4. [17] K. Latrach and M. Mokhtar-Kharroubi, On an unbounded linear operator arising in the theory of growing cell population, J. Math. Anal. Appl., 211 (1997), 273-294.  doi: 10.1006/jmaa.1997.5460. [18] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, J. Math. Biol., 1 (1974), 17-36.  doi: 10.1007/BF02339486. [19] B. Lods and M. Mokhtar-Kharroubi, On the theory of a growing cell population with zero minimum cycle length, J. Math. Anal. Appl., 266 (2002), 70-99.  doi: 10.1006/jmaa.2001.7712. [20] M. Mokhtar-Kharroubi and R. Rudnicki, On asymptotic stability and sweeping of collisionless kinetic equations, Acta Appl. Math., 147 (2017), 19-38.  doi: 10.1007/s10440-016-0066-1. [21] K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl., 436 (2016), 305-321.  doi: 10.1016/j.jmaa.2015.12.009. [22] K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn., 18 (2018), 1850001, 18 pp. doi: 10.1142/S0219493718500016. [23] M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3. [24] S. I. Rubinow, A maturity-time representation for cell populations, Biophysical Journal, 8 (1968), 1055-1073.  doi: 10.1016/S0006-3495(68)86539-7. [25] R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology. Springer, Cham, 2017. doi: 10.1007/978-3-319-61295-9. [26] H. H. Schaefer, Banach Lattices and Positive Operators, volume 215, Springer-Verlag, New York-Heidelberg, 1974. [27] G. F. Webb, A model of proliferating cell populations with inherited cycle length, J. Math. Biol., 23 (1986), 269-282.  doi: 10.1007/BF00276962. [28] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.  doi: 10.1090/S0002-9947-1987-0902796-7.

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##### References:
 [1] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, volume 233, Springer, New York, 2006. [2] J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability, Appl. Math. Lett., 45 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.006. [3] J. Banasiak, A. Falkiewicz and 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.  doi: 10.1142/S0218202516400017. [4] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583. [5] A. Bobrowski, Generation of cosine families via Lord Kelvin's method of images, J. Evol. Equ., 10 (2010), 663-675.  doi: 10.1007/s00028-010-0065-z. [6] A. Bobrowski, Lord Kelvin's method of images in semigroup theory, Semigroup Forum, 81 (2010), 435-445.  doi: 10.1007/s00233-010-9230-5. [7] A. Bobrowski and A. Gregosiewicz, A general theorem on generation of moments-preserving cosine families by Laplace operators in $C[0, 1]$, Semigroup Forum, 88 (2014), 689-701.  doi: 10.1007/s00233-013-9561-0. [8] A. Bobrowski, A. Gregosiewicz and M. Murat, Functionals-preserving cosine families generated by Laplace operators in $C[0, 1]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1877-1895.  doi: 10.3934/dcdsb.2015.20.1877. [9] M. Boulanouar, A mathematical study for a Rotenberg model, J. Math. Anal. Appl., 265 (2002), 371-394.  doi: 10.1006/jmaa.2001.7721. [10] Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-Parameter Semigroups, volume 5, North-Holland Publishing Co., Amsterdam, 1987. [11] N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, Inc., New York, 1988. [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, volume 194. Springer-Verlag, New York, 2000. [13] J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985. [14] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1994. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators. I, volume 256, Springer-Verlag, Berlin, second edition, 2003. doi: 10.1007/978-3-642-61497-2. [16] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, volume 97, Springer-Verlag, New York, second edition, 1994. doi: 10.1007/978-1-4612-4286-4. [17] K. Latrach and M. Mokhtar-Kharroubi, On an unbounded linear operator arising in the theory of growing cell population, J. Math. Anal. Appl., 211 (1997), 273-294.  doi: 10.1006/jmaa.1997.5460. [18] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, J. Math. Biol., 1 (1974), 17-36.  doi: 10.1007/BF02339486. [19] B. Lods and M. Mokhtar-Kharroubi, On the theory of a growing cell population with zero minimum cycle length, J. Math. Anal. Appl., 266 (2002), 70-99.  doi: 10.1006/jmaa.2001.7712. [20] M. Mokhtar-Kharroubi and R. Rudnicki, On asymptotic stability and sweeping of collisionless kinetic equations, Acta Appl. Math., 147 (2017), 19-38.  doi: 10.1007/s10440-016-0066-1. [21] K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl., 436 (2016), 305-321.  doi: 10.1016/j.jmaa.2015.12.009. [22] K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn., 18 (2018), 1850001, 18 pp. doi: 10.1142/S0219493718500016. [23] M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3. [24] S. I. Rubinow, A maturity-time representation for cell populations, Biophysical Journal, 8 (1968), 1055-1073.  doi: 10.1016/S0006-3495(68)86539-7. [25] R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology. Springer, Cham, 2017. doi: 10.1007/978-3-319-61295-9. [26] H. H. Schaefer, Banach Lattices and Positive Operators, volume 215, Springer-Verlag, New York-Heidelberg, 1974. [27] G. F. Webb, A model of proliferating cell populations with inherited cycle length, J. Math. Biol., 23 (1986), 269-282.  doi: 10.1007/BF00276962. [28] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.  doi: 10.1090/S0002-9947-1987-0902796-7.
The set Ω
${\tilde{\Omega }}$ is the union of $\Omega_i$'s
The set $\bigcup_{i = 1}^4 (1+\Omega_i)$ is shaded
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