Eigenelements of a general aggregation-fragmentation model

Mathematical Models and Methods in Applied Sciences

Volumn 20, Issue 5, 2010, Pages 757-783

  Doumic Jauffret, Marie ^a   Gabriel, Pierre ^b  

^a INRIA ROCQUENCOURT   (France)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Aggregation fragmentation equations; Cell division; Eigenproblem; Long time asymptotic; Polymerization process; Size repartition

Indexed keywords

CELL DIVISIONS; EIGENPROBLEM; FRAGMENTATION EQUATION; POLYMERIZATION PROCESS; TIME ASYMPTOTIC;

CELL PROLIFERATION; EIGENVALUES AND EIGENFUNCTIONS; INTEGRODIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS; POLYMERIZATION;

ASYMPTOTIC ANALYSIS;

EID: 77953191728     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021820251000443X     Document Type: Article

References (37)

1. M. Adimy and L. Pujo-Menjouet, Asymptotic behaviour of a singular transport equation modelling cell division, Disc. Cont. Dyn. Syst. Ser. B 3 (2003) 439-456.
2. F. Baccelli, D. R. McDonald and J. Reynier, A mean-field model for multiple tcp connections through a buffer implementing red, Performance Evaluation 49 (2002) 77-97.
3. J. Banasiak and W. Lamb, On a coagulation and fragmentation equation with mass loss, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 1157-1173.
4. B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. Wake and D. J. N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol. 47 (2003) 295-312.
5. F. Bekkal Brikci, J. Clairambault and B. Perthame, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Math. Comput. Model. 47 (2008) 699-713.
6. F. Bekkal Brikci, J. Clairambault, B. Ribba and B. Perthame, An age-and-cyclin-structured cell population model for healthy and tumoral tissues, J. Math. Biol. 57 (2008) 91-110.
7. T. Biben, J.-C. Geminard and F. Melo, Dynamics of bio-polymeric brushes growing from a cellular membrane: Tentative modelling of the actin turnover within an adhesion unit; the Podosome, J. Biol. Phys. 31 (2005) 87-120.
8. H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise (Collection of Applied Mathematics for the Master's Degree) (Masson, 1983).
9. V. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame, Prion dynamic with size dependency - strain phenomena, J. Biol. Dyn., in press.
10. V. Calvez, N. Lenuzza, D. Oelz, J.-P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Size distribution dependence of prion aggregates infectivity, Math. Biosci. 1 (2009) 88-99.
11. J. Clairambault, S. Gaubert and T. Lepoutre, Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models, Math. Model. Nat. Phenom., in press, 2009.
12. J. Clairambault, S. Gaubert and B. Perthame, An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations, C. R. Math. Acad. Sci. Paris 345 (2007) 549-554.
13. O. Destaing, F. Saltel, J.-C. Geminard, P. Jurdic and F. Bard, Podosomes display actin turnover and dynamic self-organization in osteoclasts expressing actin-green fluorescent protein, Mol. Biol. Cell 14 (2003) 407-416.
14. M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom. 2 (2007) 121 - 152.
15. M. Doumic, B. Perthame and J. P. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Probl. 25 (electronic version) (2009) 045008.
16. H. Engler, J. Pruss and G. F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324 (2006) 98-117.
17. M. Escobedo, Ph. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Diff. Eqns. 195 (2003) 143-174.
18. M. Escobedo, S. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 (2002) 157-188.
19. J. Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonlin. Anal. Real World Appl. 6 (2005) 962-969.
20. M. L. Greer, L. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theor. Biol. 242 (2006) 598-606.
21. M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol. 28 (1990) 671-694.
22. P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, to appear in Comm. Math. Sci.
23. P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics, J. Evol. Eqns. 7 (2007) 241-264.
24. N. Lenuzza, Modelisation de la réplication des prions: Dépendance en taille des agrégats de PrP et de I'hétérogénéité des populations cellulaires, Ph.D. thesis, Paris, 2009.
25. Eds. J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Vol. 68 (Springer-Verlag, 1986). Papers from the colloquium held in Amsterdam, 1983.
26. P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Meth. Appl. Sci. 16 (2006) 1125-1153.
27. P. Michel, Optimal proliferation rate in a cell division model, Math. Model. Nat. Phenom. 1 (2006) 23-44.
28. P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris 338 (2004) 697-702.
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.
30. K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population, submitted, 2009.
31. B. Perthame, Transport Equations in Biology (Birkhäuser Verlag, 2007).
32. B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Diff. Eqns. 210 (2005) 155-177.
33. B. Perthame and T. M. Touaoula, Analysis of a cell system with finite divisions, Bol. SEMA (44) (2008) 55-79.
34. B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, in Selected Topics in Cancer Modeling, Model. Simul. Sci. Eng. Technol. (Birkhäuser, 2008).
35. B. Perthame and J. P. Zubelli, On the inverse problem for a size-structured population model, Inv. Probl. 23 (2007) 1037-1052.
36. J. Pruss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamics of prion, Disc. Cont. Dyn. Sys. Ser. B 6 (2006) 225-235.
37. J. R. Silveira, G. J. Raymond, A. G. Hughson, R. E. Race, V. L. Sim, S. F. Hayes and B. Caughey, The most infectious prion protein particles, Nature 437 (2005) 257-261.