Nonlinear renewal equations

Modeling and Simulation in Science, Engineering and Technology

Volumn , Issue 9780817647124, 2008, Pages

  Perthame, Benoît ^a,b,c   Tumuluri, Suman Kumar ^a,b  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b INRIA ROCQUENCOURT   (France)

^c INSTITUT UNIVERSITAIRE DE FRANCE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84881064212     PISSN: 21643679     EISSN: 21643725     Source Type: Book Series    
DOI: 10.1007/978-0-8176-4713-1_4     Document Type: Chapter

References (48)

1. Adimy, M., Crauste, F., Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis, 54, p1469–1491 (2003).
2. Adimy, M., Pujo-Menjouet, L., Asymptotic behavior of a singular transport equation modelling cell division, Dis. Cont. Dyn. Sys. Ser. B 3, 3, p439–456 (2003).
3. Arino, O., Sanchez, E., Webb, G.F., Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl. 215 p499–513 (1997).
4. Bees, M.A., Angulo, O., López-Marcos, J.C., Schley, D., Dynamics of a structured slug population model in the absence of seasonal variation, Mathematical Models and Methods in Applied Sciences, Vol. 16, No. 12, p1961–1985, (2006).
5. Bekkal Brikci, F., Boushaba, K., Arino, O., Nonlinear age structured model with cannibalism, Discrete and Continuous Dynamical Systems, 2, 7, p251–273, March (2007).
6. Bekkal Brikci, F., Clairambault, J., Perthame, B., Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Mathematical and Computer Modelling, (2007), doi:10.1016/j.mcm.2007.06.008.
7. Busenberg, S., Iannelli, M., Class of nonlinear diffusion problems in age dependent population dynamics, Nonlinear Analy. Theory Meth. & Applic., Vol. 7, No. 5, p501–529 (1983).
8. Calsina, À., Cuadrado, S., Asymptotic stability of equilibria of selection-mutation equations. J. Math. Biol., 54, no. 4, p489–511, (2007).
9. Carrillo, J.A., Cuadrado, S., Perthame, B., Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model, Mathematical Biosciences, vol. 205(1), p137–161 (2007).
10. Chiorino, G., Metz, J.A.J., Tomasoni D., Ubezio, P., Desynchronization rate in cell populations: mathematical modeling and experimental data, J. Theor. Biol. 208, p185–199 (2001).
11. Chipot, M., On the equations of age dependent population dynamics, Arch. Rational Mech. Anal., 82, p13–25 (1983).
12. Clairambault, J., Gaubert, S., Perthame, B., An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations. Preprint (2007).
13. Diekmann, O., A beginner’s guide to adaptive dynamics. In Rudnicki, R. (Ed.), Mathematical modeling of population dynamics, Banach Centre Publications, Warsaw, Poland, Vol. 63, p47–86 (2004).
14. Diekmann, O., Gyllenberg, M., Metz, J.A.J., Steady state analysis of structured population models, Theoretical Population Biology, 63, p309–338 (2003).
15. Diekmann, O., Gyllenberg, M., Metz, J.A.J., Physiologically structured population models: towards a general mathematical theory. In Mathematics for ecology and environmental sciences Springer, New York, p5–20 (2007).
16. Diekmann, O., Heesterbeck, J.A.P., Mathematical epidemiology of infectious diseases, Wiley, New York (2000).
17. Ducrot, A., Travelling wave solutions for a scalar age structured equation, Discrete and Continuous Dynamical Systems, Volume 7, p251–273, March (2007).
18. Dyson, J., Villella-Bressan, R. and Webb, G., A nonlinear age and maturity structured model of population dynamics. II. Chaos, J. Math. Anal. Appl., 242, No. 2, p255–270 (2000).
19. Echenim, N., Monniaux, D., Sorine, M., Clément F., Multi-scale modeling of the follicle selection process in the ovary, Math. Biosci., 198, No. 1, 57–79 (2005).
20. Fasano, A., Yashima, H.F., Equazioni integrali per un modello di simbiosi di età: caso di pino cembro e nocciolaia, Rend. Sem. Mat. Univ. Padova, Vol. 111, p205–238 (2004).
21. Feller, W., An introduction to probability theory and applications. Wiley, New York (1966).
22. Gopalsamy, K., Age-specific coexistence in two-species competition, Mathematical Biosciences, 61, p101–122 (1982).
23. Gurtin, M.E., MacCamy, R.C., Nonlinear age-dependent population dynamics, Arch. Rational Mech. Anal., 54, p281–300 (1974).
24. Gwiazda, P., Perthame, B., Invariants and exponential rate of convergence to steady state in renewal equation, Markov Processes and Related Fields (MPRF) 2, p413–424 (2006).
25. Gyllenberg, M., Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Bios., 62, p45–74 (1982).
26. Hoppensteadt, F., Mathematical theories of populations: Demographics genetics and epidemics, SIAM Reg. Conf. Series in Appl. Math. (1975).
27. Iannelli, M., Mathematical theory of age-structured population dynamics, Applied Mathematics Monograph C.N.R. Vol. 7, In Pisa: Giardini editori e stam-patori (1995).
28. Iwata, K., Kawasaki, K., Shigesada, N., A dynamical model for the growth and size distribution of multiple metastatic tumors, J. Theor. Biol., 203, p177–186 (2000).
29. Jost, J., Mathematical Methods in Biology and Neurobiology, Lecture Notes given at ENS, http://www.mis.mpg.de/jjost/publications/mathematical_ methods.pdf (2006).
30. Kermack, W.O., McKendrick, A.G., A contribution to the mathematical theory of epidemics, Proc. Roy. Soc., A 115, p700–721 (1927).
31. Mackey, M. C., Rey, A., Multistability and boundary layer development in a transport equation with retarded arguments, Can. Appl. Math. Quart. 1, p1–21 (1993).
32. McKendrick, A.G., Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44, p98–130 (1926).
33. Marcati, P., On the global stability of the logistic age-dependent population growth, J. Math. Biology, 15, p215–226 (1982).
34. Metz, J.A.J., Diekmann, O., The dynamics of physiologically structured population, LN in Biomathematics 68, Springer-Verlag, New York (1986).
35. Michel, P., General Relative Entropy in a Nonlinear McKendrick Model, Stochastic Analysis and Partial Differential Equations, Editors: Gui-Qiang Chen, Elton Hsu, and Mark Pinsky. Contemp. Math (2007).
36. Michel, P. Existence of a solution to the cell division eigenproblem. M3AS, Vol. 16, suppl. issue 1, p1125–1153 (2006).
37. Michel, P., Mischler, S., Perthame, B., General entropy equations for structured population models and scattering, C.R. Acad. Sc Paris, Sér.1, 338, p697–702 (2004).
38. Michel, P., Mischler, S., Perthame, B., General relative entropy inequality: an illustration on growth models, J. Math. Pures et Appl., 9, p1235–1260 (2005).
39. Mischler, S., Perthame, B., Ryzhik, L., Stability in a nonlinear population maturation model, Math. Models Meth. Appli. Sci., 12, No. 12, p1751–1772 (2002).
40. Murray, J.D., Mathematical biology, Vol. 1 and 2, Second edition. Springer, New York (2002).
41. Öelz, D., Schmeiser, C. Modelling of the actin-cytoskeleton in symmetric lamel-lipodial fragments. Preprint 2007.
42. Pakdaman, K., Personal communication.
43. Perthame, B., Transport Equations in Biology (LN Series Frontiers in Mathe-matics), Birkhauser (2007).
44. Perthame, B., Ryzhik, L., Exponential decay for the fragmentation or cell-division equation, Journal of Differential Equations, 210, 1, p155–177 (March 2005).
45. Rotenberg, M., Transport theory for growing cell populations, J. Theor. Biology, 103, p181–199 (1983).
46. Thieme, H.R. Mathematics in population biology. Woodstock Princeton University Press. Princeton, NJ (2003).
47. VonFoerster, H., Some remarks on changing populations. The kinetics of cellular proliferation (ed. F. Stohlman), Grune and Strutton, New York (1959).
48. Webb, G. F., Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker Inc., New York and Basel (1985).