Nonlinear Renewal Equations

Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Nonlinear Stability Exponential Convergence Adjoint Problem Time Asymptotics Birth Term 


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  1. 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).MATHCrossRefMathSciNetGoogle Scholar
  2. 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).MATHMathSciNetGoogle Scholar
  3. 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).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bees, M.A., Angulo, O., L’opez-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).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bekkal Brikci, F., Boushaba, K., Arino, O., Nonlinear age structured model with cannibalism, Discrete and Continuous Dynamical Systems, 2, Volume 7, p251–273, March (2007).Google Scholar
  6. 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.Google Scholar
  7. 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).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Calsina, À., Cuadrado, S., Asymptotic stability of equilibria of selectionmutation equations. J. Math. Biol., 54, no. 4, p489–511, (2007).MATHCrossRefMathSciNetGoogle Scholar
  9. 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).MATHCrossRefMathSciNetGoogle Scholar
  10. 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).CrossRefGoogle Scholar
  11. 11.
    Chipot, M., On the equations of age dependent population dynamics, Arch. Rational Mech. Anal., 82, p13–25 (1983).MATHCrossRefMathSciNetGoogle Scholar
  12. 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).Google Scholar
  13. 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).Google Scholar
  14. 14.
    Diekmann, O., Gyllenberg, M., Metz, J.A.J., Steady state analysis of structured population models, Theoretical Population Biology, 63, p309–338 (2003).MATHCrossRefGoogle Scholar
  15. 15.
    Diekmann, O., Gyllenberg, M., Metz, J.A.J., Physiologically structured population models: towards a general mathematical theory. InMathematics for ecology and environmental sciences Springer, New York, p5–20 (2007).Google Scholar
  16. 16.
    Diekmann, O., Heesterbeck, J.A.P.,Mathematical epidemiology of infectious diseases, Wiley, New York (2000).Google Scholar
  17. 17.
    Ducrot, A., Travelling wave solutions for a scalar age structured equation, Discrete and Continuous Dynamical Systems, 2, Volume 7, p251–273, March (2007).Google Scholar
  18. 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).MATHCrossRefMathSciNetGoogle Scholar
  19. 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).MATHCrossRefMathSciNetGoogle Scholar
  20. 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).MATHGoogle Scholar
  21. 21.
    Feller, W.,An introduction to probability theory and applications. Wiley, New York (1966).MATHGoogle Scholar
  22. 22.
    Gopalsamy, K., Age-specific coexistence in two-species competition, Mathematical Biosciences, 61, p101–122 (1982).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Gurtin, M.E., MacCamy, R.C., Nonlinear age-dependent population dynamics, Arch. Rational Mech. Anal., 54, p281–300 (1974).MATHCrossRefMathSciNetGoogle Scholar
  24. 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).MathSciNetGoogle Scholar
  25. 25.
    Gyllenberg, M., Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Bios., 62, p45–74 (1982).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hoppensteadt, F.,Mathematical theories of populations: Demographics genetics and epidemics, SIAM Reg. Conf. Series in Appl. Math. (1975).Google Scholar
  27. 27.
    Iannelli, M.,Mathematical theory of age-structured population dynamics, Applied Mathematics Monograph C.N.R. Vol. 7, In Pisa: Giardini editori e stampatori (1995).Google Scholar
  28. 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).CrossRefGoogle Scholar
  29. 29.
    Jost, J., Mathematical Methods in Biology and Neurobiology, Lecture Notes given at ENS, methods.pdf (2006).Google Scholar
  30. 30.
    Kermack, W.O., McKendrick, A.G., A contribution to the mathematical theory of epidemics, Proc. Roy. Soc., A 115, p700–721 (1927). Part II, Proc. Roy. Soc., A 138, p55–83 (1932).CrossRefGoogle Scholar
  31. 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).Google Scholar
  32. 32.
    McKendrick, A.G., Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44, p98–130 (1926).CrossRefGoogle Scholar
  33. 33.
    Marcati, P., On the global stability of the logistic age-dependent population growth, J. Math. Biology, 15, p215–226 (1982).MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Metz, J.A.J., Diekmann, O.,The dynamics of physiologically structured population, LN in Biomathematics 68, Springer-Verlag, New York (1986).Google Scholar
  35. 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).Google Scholar
  36. 36.
    Michel, P. Existence of a solution to the cell division eigenproblem. M3AS, Vol. 16, suppl. issue 1, p1125–1153 (2006).Google Scholar
  37. 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).MATHMathSciNetGoogle Scholar
  38. 38.
    Michel, P., Mischler, S., Perthame, B., General relative entropy inequality: an illustration on growth models, J. Math. Pures et Appl., 84, Issue 9, p1235–1260 (2005).MathSciNetGoogle Scholar
  39. 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).MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Murray, J.D.,Mathematical biology, Vol. 1 and 2, Second edition. Springer, New York (2002).Google Scholar
  41. 41.
    Öelz, D., Schmeiser, C. Modelling of the actin-cytoskeleton in symmetric lamellipodial fragments. Preprint 2007.Google Scholar
  42. 42.
    Pakdaman, K., Personal communication.Google Scholar
  43. 43.
    Perthame, B.,Transport Equations in Biology (LN Series Frontiers in Mathematics), Birkhauser (2007).Google Scholar
  44. 44.
    Perthame, B., Ryzhik, L., Exponential decay for the fragmentation or celldivision equation, Journal of Differential Equations, 210, Issue 1, p155–177 (March 2005).Google Scholar
  45. 45.
    Rotenberg, M., Transport theory for growing cell populations, J. Theor. Biology, 103, p181–199 (1983).CrossRefMathSciNetGoogle Scholar
  46. 46.
    Thieme, H.R.Mathematics in population biology. Woodstock Princeton University Press. Princeton, NJ (2003).MATHGoogle Scholar
  47. 47.
    VonFoerster, H., Some remarks on changing populations.The kinetics of cellular proliferation (ed. F. Stohlman), Grune and Strutton, New York (1959).Google Scholar
  48. 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).Google Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Département de Mathématiques et ApplicationsÉcole Normale SupérieureFrance
  2. 2.INRIA-Rocquencourt B.P. 105 Projet BANG F78153Le Chesnay Cedex
  3. 3.Institut Universitaire de FranceFrance

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