Skip to main content
SpringerLink
Log in

Nonlinear Renewal Equations

  • Chapter
  • First Online:
Selected Topics in Cancer Modeling

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  8. Calsina, À., Cuadrado, S., Asymptotic stability of equilibria of selectionmutation equations. J. Math. Biol., 54, no. 4, p489–511, (2007).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  Google Scholar 

  11. Chipot, M., On the equations of age dependent population dynamics, Arch. Rational Mech. Anal., 82, p13–25 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Diekmann, O., Gyllenberg, M., Metz, J.A.J., Steady state analysis of structured population models, Theoretical Population Biology, 63, p309–338 (2003).

    Article  MATH  Google Scholar 

  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. Diekmann, O., Heesterbeck, J.A.P.,Mathematical epidemiology of infectious diseases, Wiley, New York (2000).

    Google Scholar 

  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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MATH  Google Scholar 

  21. Feller, W.,An introduction to probability theory and applications. Wiley, New York (1966).

    MATH  Google Scholar 

  22. Gopalsamy, K., Age-specific coexistence in two-species competition, Mathematical Biosciences, 61, p101–122 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  23. Gurtin, M.E., MacCamy, R.C., Nonlinear age-dependent population dynamics, Arch. Rational Mech. Anal., 54, p281–300 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  25. Gyllenberg, M., Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Bios., 62, p45–74 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  26. Hoppensteadt, F.,Mathematical theories of populations: Demographics genetics and epidemics, SIAM Reg. Conf. Series in Appl. Math. (1975).

    Google Scholar 

  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. 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).

    Article  Google Scholar 

  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).

    Google Scholar 

  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).

    Article  Google Scholar 

  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. McKendrick, A.G., Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44, p98–130 (1926).

    Article  Google Scholar 

  33. Marcati, P., On the global stability of the logistic age-dependent population growth, J. Math. Biology, 15, p215–226 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Michel, P. Existence of a solution to the cell division eigenproblem. M3AS, Vol. 16, suppl. issue 1, p1125–1153 (2006).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  40. Murray, J.D.,Mathematical biology, Vol. 1 and 2, Second edition. Springer, New York (2002).

    Google Scholar 

  41. Öelz, D., Schmeiser, C. Modelling of the actin-cytoskeleton in symmetric lamellipodial fragments. Preprint 2007.

    Google Scholar 

  42. Pakdaman, K., Personal communication.

    Google Scholar 

  43. Perthame, B.,Transport Equations in Biology (LN Series Frontiers in Mathematics), Birkhauser (2007).

    Google Scholar 

  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. Rotenberg, M., Transport theory for growing cell populations, J. Theor. Biology, 103, p181–199 (1983).

    Article  MathSciNet  Google Scholar 

  46. Thieme, H.R.Mathematics in population biology. Woodstock Princeton University Press. Princeton, NJ (2003).

    MATH  Google Scholar 

  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. 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 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, 75230 Paris, France

    Benoît Perthame & Suman Kumar Tumuluri

  2. INRIA-Rocquencourt B.P. 105 Projet BANG F78153, Le Chesnay Cedex

    Benoît Perthame & Suman Kumar Tumuluri

  3. Institut Universitaire de France, 75005 Paris, France

    Benoît Perthame

Authors
  1. Benoît Perthame
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suman Kumar Tumuluri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benoît Perthame .

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Birkhäuser Boston

About this chapter

Cite this chapter

Perthame, B., Tumuluri, S.K. (2008). Nonlinear Renewal Equations. In: Selected Topics in Cancer Modeling. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4713-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation