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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] M. Boulanouar, A mathematical study for a Rotenberg model, J. Math. Anal. Appl., 265 (2002), 371-394.  doi: 10.1006/jmaa.2001.7721.  Google Scholar [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.  Google Scholar [11] N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, Inc., New York, 1988.  Google Scholar [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, volume 194. Springer-Verlag, New York, 2000.  Google Scholar [13] J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.  Google Scholar [14] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn., 18 (2018), 1850001, 18 pp. doi: 10.1142/S0219493718500016.  Google Scholar [23] M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] H. H. Schaefer, Banach Lattices and Positive Operators, volume 215, Springer-Verlag, New York-Heidelberg, 1974.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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References:
 [1] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, volume 233, Springer, New York, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] M. Boulanouar, A mathematical study for a Rotenberg model, J. Math. Anal. Appl., 265 (2002), 371-394.  doi: 10.1006/jmaa.2001.7721.  Google Scholar [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.  Google Scholar [11] N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, Inc., New York, 1988.  Google Scholar [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, volume 194. Springer-Verlag, New York, 2000.  Google Scholar [13] J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.  Google Scholar [14] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn., 18 (2018), 1850001, 18 pp. doi: 10.1142/S0219493718500016.  Google Scholar [23] M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] H. H. Schaefer, Banach Lattices and Positive Operators, volume 215, Springer-Verlag, New York-Heidelberg, 1974.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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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