Long-time asymptotics for nonlinear growth-fragmentation equations

Communications in Mathematical Sciences

Volumn 10, Issue 3, 2012, Pages 787-820

  Gabriel, Pierre ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Eigenproblem; Growth fragmentation processes; Hopf bifurcation; Long time behavior; Periodic solution; Poincar Bendixon theorem; Relative entropy; Selfsimilarity; Size structured populations; Stability

Indexed keywords

EID: 84861719248     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2012.v10.n3.a4     Document Type: Article

References (61)

1. M. Adimy, F. Crauste, M.L. Hbid, and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70(5), 1611-1633, 2010.
2. H. Banks, F. Charles, M. Doumic, K. Sutton, and W. Thompson, Label structured cell proliferation models, Appl. Math. Letters, 23(12), 1412-1415, 2010.
3. V. Bansaye and V.C. Tran, Branching feller diffusion for cell division with parasite infection, ALEA, Lat. Am. J. Probab. Math. Stat., 8, 95-127, 2011.
4. 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(7-8), 699-713, 2008.
5. 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(1), 91-110, 2008.
6. S. Bertoni, Periodic solutions for non-linear equations of structured populations, J. Math. Anal. Appl., 220(1), 250-267, 1998.
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, 87-120, 2005.
8. M.J. Cáceres, J.A. Ca{ogonek}nizo, and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl., 96(4), 334-362, 2011.
9. V. Calvez, M. Doumic Jauffret, and P. Gabriel, Self-similarity in a general aggregationfragmentation problem; application to fitness analysis, J. Math. Pures Appl., to appear.
10. V. Calvez and P. Gabriel, Optimal control for a discrete aggregation-fragmentation model, in preparation.
11. V. Calvez, N. Lenuzza, M. Doumic, J.P. Deslys, F. Mouthon, and B. Perthame, Prion dynamic with size dependency - strain phenomena, J. Biol. Dyn., 4(1), 28-42, 2010.
12. 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, 88-99, 2009.
13. J. Clairambault, S. Gaubert, and T. Lepoutre, Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models, Math. Model. Nat. Phenom., 4(3), 183-209, 2009.
14. J. Clairambault, S. Gaubert, and T. Lepoutre, Circadian rhythm and cell population growth, Math. Comput. Modelling, 53(7-8), 1558-1567, 2011.
15. 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(10), 549-554, 2007.
16. J.M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics, Comput. Math. Appl., 9(3), 459-478, 1983.
17. 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. of the Cell, 14, 407-416, 2003.
18. O. Diekmann, M. Gyllenberg, and J.A.J. Metz, Steady-state analysis of structured population models, Theo. Pop. Bio., 63, 309-338, 2003.
19. M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2(3), 121-152, 2007.
20. M. Doumic, T. Goudon, and T. Lepoutre, Scaling limit of a discrete prion dynamics model, Commun. Math. Sci., 7(4), 839-865, 2009.
21. M. Doumic, P. Maia, and J. Zubelli, On the calibration of a size-structured population model from experimental data, Acta Biotheor., 58(4), 405-413, 2010.
22. M. Doumic, B. Perthame, and J. Zubelli, Numerical solution of an inverse problem in sizestructured population dynamics, Inv. Prob., 25(electronic version):045008, 2009.
23. M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Mode. Meth. Appl. Sci., 20(5), 757-783, 2010.
24. H. Engler, J. Prüss, and G. Webb, Analysis of a model for the dynamics of prions ii, J. Math. Anal. Appl., 324(1), 98-117, 2006.
25. M. Escobedo, S. Mischler, and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22(1), 99-125, 2005.
26. J.Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonlin. Anal. RealWorld Appl., 6, 962-969, 2005.
27. J.Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328(1), 119-136, 2007.
28. J.Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Commun. Pure Appl. Anal., 8, 1825-1839, 2009.
29. J.P. Françoise, Oscillations en Biologie, Math. Appl. (Berlin), Springer-Verlag, Berlin, 46, 2005.
30. P. Gabriel, Équations de Transport-Fragmentation et Applications aux Maladies 'a Prions [Transport-Fragmentation Equations and Applications to Prion Diseases], PhD thesis, Paris, 2011.
31. P. Gabriel, The shape of the polymerization rate in the prion equation, Math. Comput. Modelling, 53(7-8), 1451-1456, 2011.
32. P. Gabriel and L.M. Tine, High-order WENO scheme for polymerization-type equations, ESAIM Proc., 30, 54-70, 2010.
33. M.L. Greer, L. Pujo-Menjouet, and G.F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol., 242(3), 598-606, 2006.
34. M.L. Greer, P. van den Driessche, L. Wang, and G.F. Webb, Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation, SIAM J. Appl. Math., 68(1), 154-170, 2007.
35. M. Gyllenberg and G.F. Webb, A Nonlinear Structured Population Model of Tumor Growth With Quiescence, J. Math. Biol., 28, 671-694, 1990.
36. J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, in London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 7, 1988.
37. F. Hoppensteadt, Mathematical Ttheories of Populations: Demographics, Genetics and Epidemics, Society for Industrial and Applied Mathematics, Philadelphia, PA, Regional Conference Series in Applied Mathematics, 1975.
38. M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. math. Monographs C.N.R., Giardini editori e stampatori, Pisa, 7, 1995.
39. T. Kostova and J. Li, Oscillations and stability due to juvenile competitive effects on adult fertility, Comput. Math. Appl., BIOMATH-95, 32(11), 57-70, 1996.
40. P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7(2), 503-510, 2009.
41. N. Lenuzza, Modélisation de la Réplication des Prions: Implication de la dDépendance en Taille des Agrégats de PrP et de l'Hétérogénéité des Populations Cellulaires, PhD thesis, Paris, 2009.
42. Z. Liu, P. Magal, and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift für Angewandte Mathematik und Physik, 62, 191-222, 2011
43. P. Magal, P. Hinow, F. Le Foll, and G.F. Webb, Analysis of a model for transfer phenomena in biological populations, SIAM J. Appl. Math., 70, 40-62, 2009.
44. P. Magal and S. Ruan, Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models, Mem. Amer. Math. Soc., 202(951), 71, 2009.
45. P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466(2116), 965-992, 2010.
46. J.A.J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 68, 1986. Papers from the colloquium held in Amsterdam, 1983.
47. P. Michel, Existence of a solution to the cell division eigenproblem, Math. Mode. Meth. Appl. Sci., 16(7), 1125-1153, 2006.
48. P. Michel, General Relative Entropy in a nonlinear McKendrick model, G.Q. Chen, E. Hsu, and M. Pinsky (eds.), Stochastic Anal. Part. Diff. Eqs., AMS: Contemporary Mathematics, 429, 205-232, 2007.
49. P. Michel, S. Mischler, and B. Perthame, General entropy equations for structured population models and scattering, C.R. Math. Acad. Sci. Paris, 338(9), 697-702, 2004.
50. P. Michel, S. Mischler, and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84(9), 1235-1260, 2005.
51. S. Mischler, B. Perthame, and L. Ryzhik, Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci., 12(12), 1751-1772, 2002.
52. B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.
53. B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Diff. Eqs., 210(1), 155-177, 2005.
54. B. Perthame and S. Tumuluri, Nonlinear renewal equations, in Selected Topics in Cancer Modeling, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 65-96, 2008.
55. J. Prüss, On the qualitative behaviour of populations with age-specific interactions, Comput. Math. Appl., 9(3), 327-339, 1983.
56. J. Prüss, L. Pujo-Menjouet, G. Webb, and R. Zacher, Analysis of a model for the dynamics of prion, Dis. Cont. Dyn. Sys. Ser. B, 6(1), 225-235, 2006.
57. G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl., 324(1), 580-603, 2006.
58. J.H. Swart, Hopf bifurcation and the stability of nonlinear age-dependent population models, Comput. Math. Appl., 15(6-8), 555-564, 1988.
59. S.K. Tumuluri, Steady state analysis of a nonlinear renewal equation, Math. Comput. Modelling, 53(7-8), 1420-1435, 2011.
60. C. Walker, Prion proliferation with unbounded polymerization rates, in Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations, Electron. J. Diff. Equ. Conf., San Marcos, 15, 387-397, 2007.
61. G.F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 89, 1985