August  2020, 25(8): 2969-3004. doi: 10.3934/dcdsb.2020048

Spectral theory and time asymptotics of size-structured two-phase population models

1. 

UMR 6623 Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, Besançon, 25000, France

2. 

UMR 5251 Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, 33400, France

Received  February 2019 Revised  October 2019 Published  August 2020 Early access  February 2020

This work provides a general spectral analysis of size-structured two-phase population models. Systematic functional analytic results are given. We deal first with the case of finite maximal size. We characterize the irreducibility of the corresponding $ L^{1} $ semigroup in terms of properties of the different parameters of the system. We characterize also the spectral gap property of the semigroup. It turns out that the irreducibility of the semigroup implies the existence of the spectral gap. In particular, we provide a general criterion for asynchronous exponential growth. We show also how to deal with time asymptotics in case of lack of irreducibility. Finally, we extend the theory to the case of infinite maximal size.

Citation: Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048
References:
[1]

T. M. Apostol, Mathematical Analysis, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974.

[2]

O. ArinoE. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl., 215 (1997), 499-513.  doi: 10.1006/jmaa.1997.5654.

[3]

M. Bai and S. Cui, Well-posedness and asynchronous exponential growth of solutions of a two-phase cell division model, Electron. J. Differential Equations, 2010 (2010), 1-12. 

[4]

E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, J. Evol. Equ., (2019). doi: 10.1007/s00028-019-00526-4.

[5]

H. Brézis, Functional Analysis. Theory and Applications, Collection of Applied Mathematics for the Master's Degree, Masson, Paris, 1983.

[6]

P. Clément, H. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-Parameter Semigroups, CWI Monographs, 5, North-Holland Publishing Co., Amsterdam, 1987.

[7]

J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 71, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.

[8]

O. DiekmannH. J. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248.  doi: 10.1007/BF00277748.

[9]

J. DysonR. Villella-Bressan and G. F. Webb, A maturity structured model of a population of proliferating and quiescent cells. Control and estimation in biological and medicine sciences, Arch. Control Sci., 9 (1999), 201-225. 

[10]

J. DysonR. Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Math. Biosci., 177/178 (2002), 73-83.  doi: 10.1016/S0025-5564(01)00097-9.

[11]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[12]

J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514.  doi: 10.1007/s11117-009-0033-4.

[13]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in Mathematics Applied to Science, Academic Press, Boston, MA, 1988, 79-105. doi: 10.1016/B978-0-12-289510-4.50012-4.

[14]

M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci., 86 (1987), 67-95.  doi: 10.1016/0025-5564(87)90064-2.

[15]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694.  doi: 10.1007/BF00160231.

[16]

M. Gyllenberg and G. F. Webb, Quiescence in structured population dynamics: Applications to tumor growth, in Mathematical Population Dynamics, Lecture Notes in Pure and Appl. Math., 131, Dekker, New York, 1991, 45–62.

[17]

H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.

[18]

B. LodsM. Mokhtar-Kharroubi and M. Sbihi, Spectral properties of general advection operators and weighted translation semigroups, Commun. Pure Appl. Anal., 8 (2009), 1469-1492.  doi: 10.3934/cpaa.2009.8.1469.

[19]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, 1936, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.

[20]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[21]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on Advances in Mathematics for Applied Sciences, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/3288.

[22]

M. Mokhtar-Kharroubi, On the convex compactness property for the strong operator topology and related topics, Math. Methods Appl. Sci., 27, (2004), 687–701. doi: 10.1002/mma.497.

[23]

M. Mokhtar-Kharroubi and Q. Richard, Time asymptotics of structured populations with diffusion and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4087-4116.  doi: 10.3934/dcdsb.2018127.

[24]

R. Nagel, W. Arendt, A. Grabosch, G. Greiner and U. Groh, et al., One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[25]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

B. Rossa, Quiescence as an explanation for asynchronous exponential growth in a size structured cell population of exponentially growing cells. {I}, in Advances in Mathematical Population Dynamics–-Molecules, Cells and Man, Ser. Math. Biol. Med., 6, World Sci. Publ., River Edge, NJ, 1997, 223–239.

[28]

M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.

[29]

G. Schlüchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.

[30]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.  doi: 10.2307/1934533.

[31]

J. Voigt, On resolvent positive operators and positive $C_0$-semigroups on $AL$-spaces, Semigroup Forum, 38 (1989), 263-266.  doi: 10.1007/BF02573236.

[32]

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.

[33]

G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., 1936, Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1.

[34]

L. W. Weis, A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory, J. Math. Anal. Appl., 129 (1988), 6-23.  doi: 10.1016/0022-247X(88)90230-2.

show all references

References:
[1]

T. M. Apostol, Mathematical Analysis, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974.

[2]

O. ArinoE. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl., 215 (1997), 499-513.  doi: 10.1006/jmaa.1997.5654.

[3]

M. Bai and S. Cui, Well-posedness and asynchronous exponential growth of solutions of a two-phase cell division model, Electron. J. Differential Equations, 2010 (2010), 1-12. 

[4]

E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, J. Evol. Equ., (2019). doi: 10.1007/s00028-019-00526-4.

[5]

H. Brézis, Functional Analysis. Theory and Applications, Collection of Applied Mathematics for the Master's Degree, Masson, Paris, 1983.

[6]

P. Clément, H. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-Parameter Semigroups, CWI Monographs, 5, North-Holland Publishing Co., Amsterdam, 1987.

[7]

J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 71, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.

[8]

O. DiekmannH. J. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248.  doi: 10.1007/BF00277748.

[9]

J. DysonR. Villella-Bressan and G. F. Webb, A maturity structured model of a population of proliferating and quiescent cells. Control and estimation in biological and medicine sciences, Arch. Control Sci., 9 (1999), 201-225. 

[10]

J. DysonR. Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Math. Biosci., 177/178 (2002), 73-83.  doi: 10.1016/S0025-5564(01)00097-9.

[11]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[12]

J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514.  doi: 10.1007/s11117-009-0033-4.

[13]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in Mathematics Applied to Science, Academic Press, Boston, MA, 1988, 79-105. doi: 10.1016/B978-0-12-289510-4.50012-4.

[14]

M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci., 86 (1987), 67-95.  doi: 10.1016/0025-5564(87)90064-2.

[15]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694.  doi: 10.1007/BF00160231.

[16]

M. Gyllenberg and G. F. Webb, Quiescence in structured population dynamics: Applications to tumor growth, in Mathematical Population Dynamics, Lecture Notes in Pure and Appl. Math., 131, Dekker, New York, 1991, 45–62.

[17]

H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.

[18]

B. LodsM. Mokhtar-Kharroubi and M. Sbihi, Spectral properties of general advection operators and weighted translation semigroups, Commun. Pure Appl. Anal., 8 (2009), 1469-1492.  doi: 10.3934/cpaa.2009.8.1469.

[19]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, 1936, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.

[20]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[21]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on Advances in Mathematics for Applied Sciences, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/3288.

[22]

M. Mokhtar-Kharroubi, On the convex compactness property for the strong operator topology and related topics, Math. Methods Appl. Sci., 27, (2004), 687–701. doi: 10.1002/mma.497.

[23]

M. Mokhtar-Kharroubi and Q. Richard, Time asymptotics of structured populations with diffusion and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4087-4116.  doi: 10.3934/dcdsb.2018127.

[24]

R. Nagel, W. Arendt, A. Grabosch, G. Greiner and U. Groh, et al., One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[25]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

B. Rossa, Quiescence as an explanation for asynchronous exponential growth in a size structured cell population of exponentially growing cells. {I}, in Advances in Mathematical Population Dynamics–-Molecules, Cells and Man, Ser. Math. Biol. Med., 6, World Sci. Publ., River Edge, NJ, 1997, 223–239.

[28]

M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.

[29]

G. Schlüchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.

[30]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.  doi: 10.2307/1934533.

[31]

J. Voigt, On resolvent positive operators and positive $C_0$-semigroups on $AL$-spaces, Semigroup Forum, 38 (1989), 263-266.  doi: 10.1007/BF02573236.

[32]

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.

[33]

G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., 1936, Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1.

[34]

L. W. Weis, A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory, J. Math. Anal. Appl., 129 (1988), 6-23.  doi: 10.1016/0022-247X(88)90230-2.

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