# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] O. Arino, E. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] H. Brézis, Functional Analysis. Theory and Applications, Collection of Applied Mathematics for the Master's Degree, Masson, Paris, 1983.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] O. Diekmann, H. 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.  Google Scholar [9] J. Dyson, R. 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.   Google Scholar [10] J. Dyson, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar [18] B. Lods, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] 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 [29] G. Schlüchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] T. M. Apostol, Mathematical Analysis, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974.  Google Scholar [2] O. Arino, E. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] H. Brézis, Functional Analysis. Theory and Applications, Collection of Applied Mathematics for the Master's Degree, Masson, Paris, 1983.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] O. Diekmann, H. 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.  Google Scholar [9] J. Dyson, R. 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.   Google Scholar [10] J. Dyson, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar [18] B. Lods, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] 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 [29] G. Schlüchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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