doi: 10.3934/eect.2022021
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. 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.

[2] J. BanasiakW. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅰ, CRC Press, Boca Raton, FL, 2020. 
[3] J. BanasiakW. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅱ, CRC Press, Boca Raton, FL, 2020. 
[4]

J. BanasiakK. 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. BanksF. 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. BarbuM. 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.

show all references

References:
[1]

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.

[2] J. BanasiakW. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅰ, CRC Press, Boca Raton, FL, 2020. 
[3] J. BanasiakW. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models. Ⅱ, CRC Press, Boca Raton, FL, 2020. 
[4]

J. BanasiakK. 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. BanksF. 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. BarbuM. 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.

Figure 1.  Schematic representation of the cell transit between (P) and (Q)
[1]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024

[2]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

[3]

Nguyen Dinh Cong. Semigroup property of fractional differential operators and its applications. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022064

[4]

Alessia Andò, Dimitri Breda, Francesca Scarabel. Numerical continuation and delay equations: A novel approach for complex models of structured populations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2619-2640. doi: 10.3934/dcdss.2020165

[5]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[6]

Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control and Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013

[7]

Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643

[8]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[9]

Karl Peter Hadeler. Structured populations with diffusion in state space. Mathematical Biosciences & Engineering, 2010, 7 (1) : 37-49. doi: 10.3934/mbe.2010.7.37

[10]

Agnieszka Bartłomiejczyk, Henryk Leszczyński. Structured populations with diffusion and Feller conditions. Mathematical Biosciences & Engineering, 2016, 13 (2) : 261-279. doi: 10.3934/mbe.2015002

[11]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations and Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028

[12]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481

[13]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[14]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[15]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[16]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

[17]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[18]

Runzhang Xu. Preface: Special issue on advances in partial differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : i-i. doi: 10.3934/dcdss.2021137

[19]

Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems and Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020

[20]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (119)
  • HTML views (61)
  • Cited by (0)

Other articles
by authors

[Back to Top]