Self-similarity in a general aggregation–fragmentation problem. Application to fitness analysis

Journal de Mathématiques Pures et Appliquées - Tập 98 - Trang 1-27 - 2012
Vincent Calvez1, Marie Doumic2, Pierre Gabriel3
1Ecole Normale Supérieure de Lyon, UMR CNRS 5669 ‘UMPA’, and INRIA Alpes, projet NUMED, 46 allée dʼItalie, F-69364 Lyon cedex 07, France
2INRIA Rocquencourt, projet BANG, Domaine de Voluceau, BP 105, F-78153 Rocquencourt, France
3Université Pierre et Marie Curie-Paris 6, UMR CNRS 7598 ‘LJLL’, BC187, 4 place de Jussieu, F-75252 Paris cedex 05, France

Tài liệu tham khảo

Doumic Jauffret, 2010, Eigenelements of a general aggregation–fragmentation model, Math. Models Methods Appl. Sci., 20, 757, 10.1142/S021820251000443X Escobedo, 2006, Dust and self-similarity for the Smoluchowski coagulation equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 331, 10.1016/j.anihpc.2005.05.001 Calvez, 2010, Prion dynamic with size dependency – strain phenomena, J. Biol. Dyn., 4, 28, 10.1080/17513750902935208 Calvez, 2009, Size distribution dependence of prion aggregates infectivity, Math. Biosci., 1, 88, 10.1016/j.mbs.2008.10.007 Bekkal Brikci, 2008, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Math. Comput. Modelling, 47, 699, 10.1016/j.mcm.2007.06.008 Bekkal Brikci, 2008, An age-and-cyclin-structured cell population model for healthy and tumoral tissues, J. Math. Biol., 57, 91, 10.1007/s00285-007-0147-x Perthame, 2007, Transport Equations in Biology, 10.1007/978-3-7643-7842-4 Banks, 2010, Label structured cell proliferation models, Appl. Math. Lett., 23, 1412, 10.1016/j.aml.2010.07.009 Banks, 2010, Estimation of cell proliferation dynamics using CFSE data, Bull. Math. Biol., 73, 116, 10.1007/s11538-010-9524-5 Doumic, 2007, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2, 121, 10.1051/mmnp:2007006 Magal, 2009, Analysis of a model for transfer phenomena in biological populations, SIAM J. Appl. Math., 70, 40, 10.1137/080732420 Bansaye, 2008, Proliferating parasites in dividing cells: Kimmelʼs branching model revisited, Ann. Appl. Probab., 18, 967, 10.1214/07-AAP465 Banasiak, 2009, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dynam. Systems Ser. B, 11, 563, 10.3934/dcdsb.2009.11.563 Michel, 2004, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338, 697, 10.1016/j.crma.2004.03.006 Michel, 2005, General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl. (9), 84, 1235, 10.1016/j.matpur.2005.04.001 Clairambault, 2009, Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models, Math. Model. Nat. Phenom., 4, 183, 10.1051/mmnp/20094308 N. Lenuzza, Modélisation de la réplication des prions: implication de la dépendance en taille des agrégats de PrP et de lʼhétérogénéité des populations cellulaires, Ph.D. thesis, Paris, 2009. Greer, 2006, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol., 242, 598, 10.1016/j.jtbi.2006.04.010 Gabriel, 2011, The shape of the polymerization rate in the prion equation, Math. Comput. Modelling, 53, 1451, 10.1016/j.mcm.2010.03.032 Prüss, 2006, Analysis of a model for the dynamics of prion, Discrete Contin. Dynam. Systems Ser. B, 6, 225, 10.3934/dcdsb.2006.6.225 Escobedo, 2005, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22, 99, 10.1016/j.anihpc.2004.06.001 Michel, 2006, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16, 1125, 10.1142/S0218202506001480 Goudon Michel, 2006, Optimal proliferation rate in a cell division model, Math. Model. Nat. Phenom., 1, 23, 10.1051/mmnp:2008002 Cáceres, 2011, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth–fragmentation equations, J. Math. Pures Appl., 96, 334, 10.1016/j.matpur.2011.01.003 Laurençot, 2009, Exponential decay for the growth–fragmentation/cell-division equation, Commun. Math. Sci., 7, 503, 10.4310/CMS.2009.v7.n2.a12 Laurençot, 2007, Well-posedness for a model of prion proliferation dynamics, J. Evol. Equ., 7, 241, 10.1007/s00028-006-0279-2 Collinge, 2007, A general model of prion strains and their pathogenicity, Science, 318, 930, 10.1126/science.1138718 P. Gabriel, Long-time asymptotics for nonlinear growth–fragmentation equations, Comm. Math. Sci. (2012), in press. Perthame, 2005, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210, 155, 10.1016/j.jde.2004.10.018 Silveira, 2005, The most infectious prion protein particles, Nature, 437, 257, 10.1038/nature03989 Engler, 2006, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl., 324, 98, 10.1016/j.jmaa.2005.11.021 Laurent, 1998, Bistability and the species barrier in prion diseases: stepping across the threshold or not, Biophys. Chem., 72, 211, 10.1016/S0301-4622(98)00135-5 Clairambault, 2007, An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations, C. R. Math. Acad. Sci. Paris, 345, 549, 10.1016/j.crma.2007.10.001 V. Calvez, P. Gabriel, Optimal control for a discrete aggregation–fragmentation model, in preparation. Doumic, 2010, On the calibration of a size-structured population model from experimental data, Acta Biotheor., 58, 405, 10.1007/s10441-010-9114-9 Perthame, 2007, On the inverse problem for a size-structured population model, Inverse Problems, 23, 1037, 10.1088/0266-5611/23/3/012 Doumic, 2009, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Problems, 25, 045008, 10.1088/0266-5611/25/4/045008 Gabriel, 2010, High-order WENO scheme for polymerization-type equations, ESAIM Proc., 30, 54, 10.1051/proc/2010006