American Institute of Mathematical Sciences

doi: 10.3934/eect.2022021
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Mathematical analysis of an abstract model and its applications to structured populations (I)

 LMCM-RSA, 22 Rue Des Canadiens, Poitiers 86000, France

To the memory of the professor Ovide Arino. He was a philanthropist and a great mathematician.

Received  September 2021 Revised  March 2022 Early access April 2022

The first part of this works deals with an integro–differential operator with boundary condition related to the interior solution. We prove that the model is governed by a strongly continuous semigroup and we precise its growth inequality. In the second part of this works, we model the proliferation-quiescence phases through a system of first order equations. We also prove that the proliferation-quiescence model is governed by a strongly continuous semigroup and we precise its growth inequality. In the last part, we give some applications in Demography and Biology.

Citation: Mohamed Boulanouar. Mathematical analysis of an abstract model and its applications to structured populations (I). Evolution Equations and Control Theory, doi: 10.3934/eect.2022021
References:
 [1] 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. [2] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅰ, CRC Press, Boca Raton, FL, 2020. [3] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅱ, CRC Press, Boca Raton, FL, 2020. [4] J. Banasiak, K. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta. Appl. Math., 119 (2012), 149-166.  doi: 10.1007/s10440-011-9666-y. [5] H. T. Banks, F. Kappel and C Wang, Weak solutions and differentiability for size structure population models, Estimation and Control of Distributed Parameter Systems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 100 (1991), 35-50.  doi: 10.1007/978-3-0348-6418-3_2. [6] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Boca Raton, FL, 2009. [7] V. Barbu, M. Iannelli and M. Martcheva, On the controllability of the Lotka-McKendrick model of population dynamics, Journ. Math. Ana. Appl., 253 (2001), 142-165.  doi: 10.1006/jmaa.2000.7075. [8] M. Boulanouar, On a mathematical model of age-cycle length structured cell population with non-compact boundary conditions, Math. Meth. Appl. Sci., 38 (2015), 2081-2104.  doi: 10.1002/mma.3206. [9] M. Boulanouar, A mathematical study in the theory of dynamic population, Journ. Math. Anal. Appl., 255 (2001), 230-259.  doi: 10.1006/jmaa.2000.7237. [10] M. Boulanouar, A transport equation in cell population dynamics, Diff. Int. Equa., 13 (2000), 125-144. [11] M. Boulanouar, Mathematical analysis of an abstract model and its applications to structured populations. Ⅱ, In preparation. [12] M. Boulanouar, Mathematical analysis of an abstract model and its applications to structured populations. Ⅲ, In preparation. [13] M. Boulanouar, Mathematical analysis of an abstract model and its applications to structured populations. Ⅳ, In preparation. [14] M. Boulanouar and L. Leboucher, A transport equation in cell population dynamics, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 305-308. [15] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [16] Ph. Clément and al, One-Parameter Semigroups, North-Hollandn, Amerterdam, New York, 1987. [17] R. Dilão and A. Lakmeche, On the weak solutions of the McKendrick equation: Existence of demography cycles, Math. Model. Nat. Phenom., 1 (2006), 1-32.  doi: 10.1051/mmnp:2006001. [18] K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. [19] 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. [20] M. Iannelli, Mathematical Theory of age-structured population dynamics, Giardini Editory e Stampatori, Pisa, 1995. [21] M. Iannelli and F. Milner, On the approximation of Lotka McKendrick equation with finite life span, Jour. Comput. Appl. Math., 136 (2001), 245-254.  doi: 10.1016/S0377-0427(00)00616-6. [22] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. [23] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130.  doi: 10.1017/S0013091500034428. [24] J. A. J. Metz and O. Diekmann, The dynamics of physiologically structured populations, Lecture Notes in Biomathematics, 68 (1986). [25] M. Mokhtar-Kharroubi and Q. Richard, Spectral theory and time asymptotics of size-structured two-phase population models, Discrete Contin. Dyn. Syst. Serie B, 25 (2020), 2969-3004.  doi: 10.3934/dcdsb.2020048. [26] G. Pelovska and M. Iannelli, Numerical methods for the Lotka McKendrick's equation, Jour. Comp. Appl. Math., 197 (2006), 534-557.  doi: 10.1016/j.cam.2005.11.033. [27] O. Scherbaum and G. Rasch, Cell size distribution and single cell growth in Tetrahymena pyriformis, GL. Arch. Pathol. Microbiol. Scand., 41 (1957), 161-182.  doi: 10.1111/j.1699-0463.1957.tb01014.x. [28] F. R. Sharpe and A. J. Lotka, A problem in age distribution, Phil. Mag., 21 (1911), 435-438.  doi: 10.1007/978-3-642-81046-6_13. [29] J. W. Sinko and W. Streifer, A new model for age-size structure for a population, Ecology, 48 (1967), 910-918.  doi: 10.2307/1934533. [30] H. Von Foerster, Some remarks on changing populations, The Kinetics of Cellular Proliferation (Grune and Stratton, NY), (1959), 382–407. [31] G. F. Webb, Dynamics of structured populations with inherited properties, Comput. Math. Appl., 13 (1987), 749-757.  doi: 10.1016/0898-1221(87)90160-X.

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References:
 [1] 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. [2] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅰ, CRC Press, Boca Raton, FL, 2020. [3] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅱ, CRC Press, Boca Raton, FL, 2020. [4] J. Banasiak, K. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta. Appl. Math., 119 (2012), 149-166.  doi: 10.1007/s10440-011-9666-y. [5] H. T. Banks, F. Kappel and C Wang, Weak solutions and differentiability for size structure population models, Estimation and Control of Distributed Parameter Systems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 100 (1991), 35-50.  doi: 10.1007/978-3-0348-6418-3_2. [6] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Boca Raton, FL, 2009. [7] V. Barbu, M. Iannelli and M. Martcheva, On the controllability of the Lotka-McKendrick model of population dynamics, Journ. Math. Ana. Appl., 253 (2001), 142-165.  doi: 10.1006/jmaa.2000.7075. [8] M. Boulanouar, On a mathematical model of age-cycle length structured cell population with non-compact boundary conditions, Math. Meth. Appl. Sci., 38 (2015), 2081-2104.  doi: 10.1002/mma.3206. [9] M. Boulanouar, A mathematical study in the theory of dynamic population, Journ. Math. Anal. Appl., 255 (2001), 230-259.  doi: 10.1006/jmaa.2000.7237. [10] M. Boulanouar, A transport equation in cell population dynamics, Diff. Int. Equa., 13 (2000), 125-144. [11] M. Boulanouar, Mathematical analysis of an abstract model and its applications to structured populations. Ⅱ, In preparation. [12] M. Boulanouar, Mathematical analysis of an abstract model and its applications to structured populations. Ⅲ, In preparation. [13] M. Boulanouar, Mathematical analysis of an abstract model and its applications to structured populations. Ⅳ, In preparation. [14] M. Boulanouar and L. Leboucher, A transport equation in cell population dynamics, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 305-308. [15] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [16] Ph. Clément and al, One-Parameter Semigroups, North-Hollandn, Amerterdam, New York, 1987. [17] R. Dilão and A. Lakmeche, On the weak solutions of the McKendrick equation: Existence of demography cycles, Math. Model. Nat. Phenom., 1 (2006), 1-32.  doi: 10.1051/mmnp:2006001. [18] K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. [19] 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. [20] M. Iannelli, Mathematical Theory of age-structured population dynamics, Giardini Editory e Stampatori, Pisa, 1995. [21] M. Iannelli and F. Milner, On the approximation of Lotka McKendrick equation with finite life span, Jour. Comput. Appl. Math., 136 (2001), 245-254.  doi: 10.1016/S0377-0427(00)00616-6. [22] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. [23] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130.  doi: 10.1017/S0013091500034428. [24] J. A. J. Metz and O. Diekmann, The dynamics of physiologically structured populations, Lecture Notes in Biomathematics, 68 (1986). [25] M. Mokhtar-Kharroubi and Q. Richard, Spectral theory and time asymptotics of size-structured two-phase population models, Discrete Contin. Dyn. Syst. Serie B, 25 (2020), 2969-3004.  doi: 10.3934/dcdsb.2020048. [26] G. Pelovska and M. Iannelli, Numerical methods for the Lotka McKendrick's equation, Jour. Comp. Appl. Math., 197 (2006), 534-557.  doi: 10.1016/j.cam.2005.11.033. [27] O. Scherbaum and G. Rasch, Cell size distribution and single cell growth in Tetrahymena pyriformis, GL. Arch. Pathol. Microbiol. Scand., 41 (1957), 161-182.  doi: 10.1111/j.1699-0463.1957.tb01014.x. [28] F. R. Sharpe and A. J. Lotka, A problem in age distribution, Phil. Mag., 21 (1911), 435-438.  doi: 10.1007/978-3-642-81046-6_13. [29] J. W. Sinko and W. Streifer, A new model for age-size structure for a population, Ecology, 48 (1967), 910-918.  doi: 10.2307/1934533. [30] H. Von Foerster, Some remarks on changing populations, The Kinetics of Cellular Proliferation (Grune and Stratton, NY), (1959), 382–407. [31] G. F. Webb, Dynamics of structured populations with inherited properties, Comput. Math. Appl., 13 (1987), 749-757.  doi: 10.1016/0898-1221(87)90160-X.
Schematic representation of the cell transit between (P) and (Q)
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