December  2019, 24(12): 6653-6673. doi: 10.3934/dcdsb.2019161

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

* Corresponding author: Philippe Michel

Received  June 2018 Revised  November 2018 Published  December 2019 Early access  July 2019

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 [29] can be extend to nonlinear problems and under some assumptions on the nonlinearity we prove the convergence of the solution to its steady state as time tends to infinity.

Citation: Philippe Michel, Bhargav Kumar Kakumani. GRE methods for nonlinear model of evolution equation and limited ressource environment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6653-6673. doi: 10.3934/dcdsb.2019161
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. ClairambaultP. 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. EchenimD. MonniauxM. 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. EchenimF. 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. MichelS. 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. MischlerB. 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. ClairambaultP. 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. EchenimD. MonniauxM. 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. EchenimF. 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. MichelS. 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. MischlerB. 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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