[Un principe général d'entropie pour les modèles de population structurés et les équations de scattering]
Nous considérons divers modèles de populations structurées (en âge, en taille ou en maturité) et aussi l'équation de scattering. Ces modèles ne sont pas conservatifs, néammoins nous montrons qu'ils vérifient tous une structure d'entropie relative commune qui utilise les premiers éléments propres du problème et qui, dans le cas du scattering, généralise le « principe d'équilibre en détail » habituel. Trois types de conséquences découlent de cette structure entropique : des estimations a priori, la convergence en temps grand vers un état stationnaire et parfois des taux exponentiels de convergence.
We consider several structured population models (age structured, size structured, maturity structured) and the general scattering equation. These models are not conservation laws, nevertheless, we show that they admit a common relative entropy structure which uses the first eigenelements of the problem. In case of scattering, it is more general than the usual ‘detailed balance principle’. Three types of consequences are deduced from this entropy structure: a priori bounds, large time convergence to the steady state and in some cases, exponential rates of convergence.
Accepté le :
Publié le :
@article{CRMATH_2004__338_9_697_0,
author = {Michel, Philippe and Mischler, St\'ephane and Perthame, Beno{\i}̂t},
title = {General entropy equations for structured population models and scattering},
journal = {Comptes Rendus. Math\'ematique},
pages = {697--702},
publisher = {Elsevier},
volume = {338},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.03.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.03.006/}
}
TY - JOUR AU - Michel, Philippe AU - Mischler, Stéphane AU - Perthame, Benoı̂t TI - General entropy equations for structured population models and scattering JO - Comptes Rendus. Mathématique PY - 2004 SP - 697 EP - 702 VL - 338 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.03.006/ DO - 10.1016/j.crma.2004.03.006 LA - en ID - CRMATH_2004__338_9_697_0 ER -
%0 Journal Article %A Michel, Philippe %A Mischler, Stéphane %A Perthame, Benoı̂t %T General entropy equations for structured population models and scattering %J Comptes Rendus. Mathématique %D 2004 %P 697-702 %V 338 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.03.006/ %R 10.1016/j.crma.2004.03.006 %G en %F CRMATH_2004__338_9_697_0
Michel, Philippe; Mischler, Stéphane; Perthame, Benoı̂t. General entropy equations for structured population models and scattering. Comptes Rendus. Mathématique, Tome 338 (2004) no. 9, pp. 697-702. doi : 10.1016/j.crma.2004.03.006. http://www.numdam.org/articles/10.1016/j.crma.2004.03.006/
[1] Asymptotic behavior of a singular transport equation modelling cell division, DCDS(B), Volume 3 (2003) no. 3, pp. 439-456
[2] A survey of structured cell population dynamics, Acta Biotheor., Volume 43 (1995), pp. 3-25
[3] A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol. (2003)
[4] F. Chalub, P. Markowich, B. Perthame, C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monat. Math., in press
[5] Desynchronization rate in cell populations: mathematical modeling and experimental data, J. Theor. Biol., Volume 208 (2001), pp. 185-199
[6] Equilibrium rate for the linear inhomogeneous relaxation time Boltzmann equation for charged particles, Comm. Partial Differential Equations, Volume 28 (2003) no. 1–2, pp. 969-989
[7] Mathematical Analysis and Numerical Methods for Sciences and Technology, Springer, 1990
[8] Diffusion approximation for non homogeneous and non microreversible processes, Indiana Univ. Math. J., Volume 49 (2000), pp. 1175-1198
[9] Entropic methods for the study of the longtime behavior of kinetic equations, The Sixteenth International Conference on Transport Theory, Part I (Atlanta, GA, 1999) (Transport Theory Statist. Phys.), Volume 30 (2001) no. 2–3, pp. 155-168
[10] N. Fournier, S. Mischler, Trend to equilibrium for discrete coagulation equations with strong fragmentation and without balance condition, Preprint, 2003
[11] On periodic cohort solutions of a size-structured population model, J. Math. Biol., Volume 35 (1997) no. 8, pp. 908-934
[12] A theory for the age and generation time distribution of a microbial population, J. Math. Biol., Volume 1 (1977), pp. 17-36
[13] B. Lods, G. Toscani, The dissipative linear Boltzmann equation for hard spheres, Appl. Math. Lett., in press
[14] Diffusion limit of a non linear kinetic model without the detailed balance principle, Monat. Math., Volume 134 (2002), pp. 305-329
[15] The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath., vol. 68, Springer-Verlag, 1986
[16] P. Michel, S. Mischler, B. Perthame, The entropy structure of models of structured population dynamics, in preparation
[17] Stability in a nonlinear population maturation model, Math. Models Meth. Appl. Sci., Volume 12 (2002) no. 12, pp. 1751-1772
[18] B. Perthame, L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, Preprint DMA, 2003
[19] Transport theory for growing cell populations, J. Theor. Biol., Volume 103 (1983), pp. 181-199
[20] Topics in Optimal Transportation, Graduate Stud. Math., vol. 58, American Mathematical Society, Providence, RI, 2003
[21] C. Villani, A review of mathematical topics in collisional kinetic theory, in: S. Friedlander, D. Serre (Eds.), Handbook of Mathematical Fluid Dynamics Tome I, Chapter 2
Cité par Sources :
