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Boundary perturbations and steady states of structured populations
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Continuity for the rotation-two-component Camassa-Holm system
GRE methods for nonlinear model of evolution equation and limited ressource environment
1. | Ecole Centrale de Lyon, University Claude Bernard Lyon 1, CNRS UMR 5208, Ecully 69130, France |
2. | T.I.F.R. Centre for Applicable Mathematics, Bangalore 560065, India |
In this paper, we consider nonlocal nonlinear renewal equation (Markov chain, Ordinary differential equation and Partial Differential Equation). We show that the General Relative Entropy [
References:
[1] |
B. Abdellaoui and T. M. Touaoula,
Decay solution for the renewal equation with diffusion, Nonlinear Differ. Equ. Appl. (Nodea), 17 (2010), 271-288.
doi: 10.1007/s00030-009-0053-6. |
[2] |
H. Behncke and S. Al-Nassir,
On the Harvesting of Age Structured of Fish Populations, Communications in Mathematics and Applications, 8 (2017), 139-156.
|
[3] |
P. Billingsley, Probability and Measure (3rd ed.), Wiley, New York, 1995. |
[4] |
J. W. Brewer,
The age-dependent eigenfunctions of certain Kolmogorov equations of engineering, economics, and biology, Applied Mathematical Modeling, 13 (1989), 47-57.
doi: 10.1016/0307-904X(89)90197-2. |
[5] |
V. Calvez, N. Lenuzza, D. Oelz, J. P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Bimodality, prion aggregates infectivity and prediction of strain phenomenon, arXiv: preprint, 2008. |
[6] |
J. Clairambault, P. Michel and B. Perthame,
A mathematical model of the cell cycle and its circadian control, Mathematical Modeling of Biological Systems, 1 (2006), 239-251.
doi: 10.1007/978-0-8176-4558-8_21. |
[7] |
J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998.
doi: 10.1137/1.9781611970005. |
[8] |
R. Dautray and J. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences Et les Techniques, Masson, Paris, 1987. |
[9] |
A. Devys, T. Goudon and P. Lafitte, A model describing the growth and the size distribution of multiple metastatic tumors, AIMS, 12 (2009), 731–767, Available from: http://hal.inria.fr/inria-00351489/fr/.
doi: 10.3934/dcdsb.2009.12.731. |
[10] |
M. Doumic, B. Perthame and J. P. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Problems, 25 (2009), 045008, 25 pp.
doi: 10.1088/0266-5611/25/4/045008. |
[11] |
N. Echenim, Modelisation et Controle Multi-echelles du Processus de Selection des Follicules Ovulatoires, Phd Thesis, Universit Paris Sud-Ⅺ, 2006. |
[12] |
N. Echenim, D. Monniaux, M. Sorine and F. Clement,
Multi-scale modeling of the follicle selection process in the ovary, Math. Biosci., 198 (2005), 57-79.
doi: 10.1016/j.mbs.2005.05.003. |
[13] |
N. Echenim, F. Clément and M. Sorine,
Multiscale modeling of follicular ovulation as a reachability problem, Multiscale Modeling and Simulation, 6 (2007), 895-912.
doi: 10.1137/060664495. |
[14] |
K. -J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, 2006. |
[15] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[16] |
P. Gwiazda and B. Perthame,
Invariants and exponential rate of convergence to steady state in the renewal equation, Markov Processes and Related Fields (MPRF), 12 (2006), 413-424.
|
[17] |
M. Iannelli, Age-structured population. In encyclopedia of mathematics, Supplement Ⅱ. Hazewinkel M. (a cura di), Kluwer Academics, (2000), 21–23. |
[18] |
M. Iannelli, Mathematical theory of age-structured population dynamics, Applied Mathematics Monograph C.N.R., 7 (1995), In Pisa: Giardini editori e stampatori. |
[19] |
M. Iannelli and J. Ripoll,
Two-sex age structured dynamics in a fixed sex-ratio population, Nonlinear Analysis: Real World Applications, 13 (2012), 2562-2577.
doi: 10.1016/j.nonrwa.2012.03.002. |
[20] |
M. Iosifsecu, Finite Markov Processes and their Applications, John Wiley, New York, 1980. |
[21] |
B. K. Kakumani and S. K. Tumuluri,
On a nonlinear renewal equation with diffusion, Math. Meth. Appl. Sci., 39 (2016), 697-708.
doi: 10.1002/mma.3511. |
[22] |
B. K. Kakumani and S. K. Tumuluri,
Extinction and blow-up phenomena in a non-linear gender structured population model, Nonlinear Analysis: Real World Applications, 28 (2016), 290-299.
doi: 10.1016/j.nonrwa.2015.10.005. |
[23] |
M. G. Kreǐn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., (1950), 128 pp. |
[24] |
P. Laurencot and B. Perthame,
Exponential decay for the growth-fragmentation/cell-division equation, Comm. Math. Sci., 7 (2009), 503-510.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[25] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-662-13159-6. |
[26] |
P. Michel, General relative entropy in a nonlinear McKendrick model, Stochastic Analysis and Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 429 (2007), 205–232.
doi: 10.1090/conm/429/08238. |
[27] |
P. Michel,
Optimal proliferation rate in a cell division model, Mathematical Modelling of Natural Phenomen, 1 (2006), 23-44.
doi: 10.1051/mmnp:2008002. |
[28] |
P. Michel,
Fitness optimization in a cell division model, Comptes Rendus Mathematique, 341 (2005), 731-736.
doi: 10.1016/j.crma.2005.10.012. |
[29] |
P. Michel, S. Mischler and B. Perthame,
General relative entropy inequality: An illustration on growth models., J. Math. Pures Appl., 84 (2005), 1235-1260.
doi: 10.1016/j.matpur.2005.04.001. |
[30] |
P. Michel and T. M. Touaoula,
Asymptotic behavior for a class of the renewal nonlinear equation with diffusion, Mathematical Methods in the Applied Sciences, 36 (2012), 323-335.
doi: 10.1002/mma.2591. |
[31] |
S. Mischler, B. Perthame and L. Ryzhik,
Stability in a nonlinear population maturation model, Mathematical Models and Methods in Applid Sciences, 12 (2002), 1751-1772.
doi: 10.1142/S021820250200232X. |
[32] |
J. D. Murray, Mathematical Biology, I, An introduction, Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002. |
[33] |
R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math., Springer-Verlag, 1986. |
[34] |
B. Perthame, Transport Equations in Biology. Frontiers in Mathematics, Birkhauser Verlag, Basel, 2007. |
[35] |
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205–244 (electronic).
doi: 10.1090/S0273-0979-04-01004-3. |
[36] |
B. Perthame,
The general relative entropy principle applications in Perron-Frobenius and Floquet theories and a parabolic system for biomotors, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 307-325.
|
[37] |
B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, in: N. Bellomo, M. Chaplain, E. De Angelis (Eds.), Selected Topics on Cancer Modeling Genesis-Evolution-Immune Competition-Therapy, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, 2008, 65–96. |
[38] |
J. A. Silva and T. G. Hallam,
Compensation and stability in nonlinear matrix models, Math Biosci., 110 (1992), 67-101.
doi: 10.1016/0025-5564(92)90015-O. |
[39] |
H. R. Thieme, Mathematics in Population Biology, University Press, Princeton, NJ, 2003.
![]() ![]() |
[40] |
T. M. Touaoula and B. Abdellaoui,
Decay solution for the renewal equation with diffusion, Nonlinear Differential Equations and Applications NoDEA, 17 (2010), 271-288.
doi: 10.1007/s00030-009-0053-6. |
[41] |
S. K. Tumuluri,
Steady state analysis of a non-linear renewal equation, Mathematical and Computer Modeling, 53 (2011), 1420-1435.
doi: 10.1016/j.mcm.2010.02.050. |
[42] |
N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Math., 888, North-Holland Publishing Co., Amsterdam-New York, 1981. |
[43] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Pure and Applied Mathematics, 89, Marcel Dekker, New York, 1985. |
[44] |
A. Wikan and O. Kristensen, Nonstationary and chaotic dynamics in age-structured population models, Discrete Dynamics in Nature and Society, 8 (2017), Art. ID 1964286, 11 pp.
doi: 10.1155/2017/1964286. |
[45] |
K. Yosida, Functional Analysis (Classics in Mathematics), Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-3-642-61859-8. |
show all references
References:
[1] |
B. Abdellaoui and T. M. Touaoula,
Decay solution for the renewal equation with diffusion, Nonlinear Differ. Equ. Appl. (Nodea), 17 (2010), 271-288.
doi: 10.1007/s00030-009-0053-6. |
[2] |
H. Behncke and S. Al-Nassir,
On the Harvesting of Age Structured of Fish Populations, Communications in Mathematics and Applications, 8 (2017), 139-156.
|
[3] |
P. Billingsley, Probability and Measure (3rd ed.), Wiley, New York, 1995. |
[4] |
J. W. Brewer,
The age-dependent eigenfunctions of certain Kolmogorov equations of engineering, economics, and biology, Applied Mathematical Modeling, 13 (1989), 47-57.
doi: 10.1016/0307-904X(89)90197-2. |
[5] |
V. Calvez, N. Lenuzza, D. Oelz, J. P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Bimodality, prion aggregates infectivity and prediction of strain phenomenon, arXiv: preprint, 2008. |
[6] |
J. Clairambault, P. Michel and B. Perthame,
A mathematical model of the cell cycle and its circadian control, Mathematical Modeling of Biological Systems, 1 (2006), 239-251.
doi: 10.1007/978-0-8176-4558-8_21. |
[7] |
J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998.
doi: 10.1137/1.9781611970005. |
[8] |
R. Dautray and J. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences Et les Techniques, Masson, Paris, 1987. |
[9] |
A. Devys, T. Goudon and P. Lafitte, A model describing the growth and the size distribution of multiple metastatic tumors, AIMS, 12 (2009), 731–767, Available from: http://hal.inria.fr/inria-00351489/fr/.
doi: 10.3934/dcdsb.2009.12.731. |
[10] |
M. Doumic, B. Perthame and J. P. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Problems, 25 (2009), 045008, 25 pp.
doi: 10.1088/0266-5611/25/4/045008. |
[11] |
N. Echenim, Modelisation et Controle Multi-echelles du Processus de Selection des Follicules Ovulatoires, Phd Thesis, Universit Paris Sud-Ⅺ, 2006. |
[12] |
N. Echenim, D. Monniaux, M. Sorine and F. Clement,
Multi-scale modeling of the follicle selection process in the ovary, Math. Biosci., 198 (2005), 57-79.
doi: 10.1016/j.mbs.2005.05.003. |
[13] |
N. Echenim, F. Clément and M. Sorine,
Multiscale modeling of follicular ovulation as a reachability problem, Multiscale Modeling and Simulation, 6 (2007), 895-912.
doi: 10.1137/060664495. |
[14] |
K. -J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, 2006. |
[15] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[16] |
P. Gwiazda and B. Perthame,
Invariants and exponential rate of convergence to steady state in the renewal equation, Markov Processes and Related Fields (MPRF), 12 (2006), 413-424.
|
[17] |
M. Iannelli, Age-structured population. In encyclopedia of mathematics, Supplement Ⅱ. Hazewinkel M. (a cura di), Kluwer Academics, (2000), 21–23. |
[18] |
M. Iannelli, Mathematical theory of age-structured population dynamics, Applied Mathematics Monograph C.N.R., 7 (1995), In Pisa: Giardini editori e stampatori. |
[19] |
M. Iannelli and J. Ripoll,
Two-sex age structured dynamics in a fixed sex-ratio population, Nonlinear Analysis: Real World Applications, 13 (2012), 2562-2577.
doi: 10.1016/j.nonrwa.2012.03.002. |
[20] |
M. Iosifsecu, Finite Markov Processes and their Applications, John Wiley, New York, 1980. |
[21] |
B. K. Kakumani and S. K. Tumuluri,
On a nonlinear renewal equation with diffusion, Math. Meth. Appl. Sci., 39 (2016), 697-708.
doi: 10.1002/mma.3511. |
[22] |
B. K. Kakumani and S. K. Tumuluri,
Extinction and blow-up phenomena in a non-linear gender structured population model, Nonlinear Analysis: Real World Applications, 28 (2016), 290-299.
doi: 10.1016/j.nonrwa.2015.10.005. |
[23] |
M. G. Kreǐn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., (1950), 128 pp. |
[24] |
P. Laurencot and B. Perthame,
Exponential decay for the growth-fragmentation/cell-division equation, Comm. Math. Sci., 7 (2009), 503-510.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[25] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-662-13159-6. |
[26] |
P. Michel, General relative entropy in a nonlinear McKendrick model, Stochastic Analysis and Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 429 (2007), 205–232.
doi: 10.1090/conm/429/08238. |
[27] |
P. Michel,
Optimal proliferation rate in a cell division model, Mathematical Modelling of Natural Phenomen, 1 (2006), 23-44.
doi: 10.1051/mmnp:2008002. |
[28] |
P. Michel,
Fitness optimization in a cell division model, Comptes Rendus Mathematique, 341 (2005), 731-736.
doi: 10.1016/j.crma.2005.10.012. |
[29] |
P. Michel, S. Mischler and B. Perthame,
General relative entropy inequality: An illustration on growth models., J. Math. Pures Appl., 84 (2005), 1235-1260.
doi: 10.1016/j.matpur.2005.04.001. |
[30] |
P. Michel and T. M. Touaoula,
Asymptotic behavior for a class of the renewal nonlinear equation with diffusion, Mathematical Methods in the Applied Sciences, 36 (2012), 323-335.
doi: 10.1002/mma.2591. |
[31] |
S. Mischler, B. Perthame and L. Ryzhik,
Stability in a nonlinear population maturation model, Mathematical Models and Methods in Applid Sciences, 12 (2002), 1751-1772.
doi: 10.1142/S021820250200232X. |
[32] |
J. D. Murray, Mathematical Biology, I, An introduction, Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002. |
[33] |
R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math., Springer-Verlag, 1986. |
[34] |
B. Perthame, Transport Equations in Biology. Frontiers in Mathematics, Birkhauser Verlag, Basel, 2007. |
[35] |
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205–244 (electronic).
doi: 10.1090/S0273-0979-04-01004-3. |
[36] |
B. Perthame,
The general relative entropy principle applications in Perron-Frobenius and Floquet theories and a parabolic system for biomotors, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 307-325.
|
[37] |
B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, in: N. Bellomo, M. Chaplain, E. De Angelis (Eds.), Selected Topics on Cancer Modeling Genesis-Evolution-Immune Competition-Therapy, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, 2008, 65–96. |
[38] |
J. A. Silva and T. G. Hallam,
Compensation and stability in nonlinear matrix models, Math Biosci., 110 (1992), 67-101.
doi: 10.1016/0025-5564(92)90015-O. |
[39] |
H. R. Thieme, Mathematics in Population Biology, University Press, Princeton, NJ, 2003.
![]() ![]() |
[40] |
T. M. Touaoula and B. Abdellaoui,
Decay solution for the renewal equation with diffusion, Nonlinear Differential Equations and Applications NoDEA, 17 (2010), 271-288.
doi: 10.1007/s00030-009-0053-6. |
[41] |
S. K. Tumuluri,
Steady state analysis of a non-linear renewal equation, Mathematical and Computer Modeling, 53 (2011), 1420-1435.
doi: 10.1016/j.mcm.2010.02.050. |
[42] |
N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Math., 888, North-Holland Publishing Co., Amsterdam-New York, 1981. |
[43] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Pure and Applied Mathematics, 89, Marcel Dekker, New York, 1985. |
[44] |
A. Wikan and O. Kristensen, Nonstationary and chaotic dynamics in age-structured population models, Discrete Dynamics in Nature and Society, 8 (2017), Art. ID 1964286, 11 pp.
doi: 10.1155/2017/1964286. |
[45] |
K. Yosida, Functional Analysis (Classics in Mathematics), Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-3-642-61859-8. |
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