Skip to main content

Iterative Methods for Generalized von Foerster Equations with Functional Dependence

Abstract

We investigate when a natural iterative method converges to the exact solution of a differential-functional von Foerster-type equation which describes a single population depending on its past time and state densities, and on its total size. On the right-hand side, we assume either Perron comparison conditions or some monotonicity.

[12345678910111213141516]

References

  1. 1.

    Brauer F, Castillo-Chávez C: Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics. Volume 40. Springer, New York, NY, USA; 2001:xxiv+416.

    Google Scholar 

  2. 2.

    Keyfitz N: Introduction to the Mathematics of Population. Addison-Wesley, Reading, Mass, USA; 1968.

    Google Scholar 

  3. 3.

    Lotka AJ: Elements of Physical Biology. Dover, New York, NY, USA; 1956. Wiliams and Wilkins, Baltimore 1925, republished as Elements of Mathematical Biology Wiliams and Wilkins, Baltimore 1925, republished as Elements of Mathematical Biology

    Google Scholar 

  4. 4.

    Nakhushev AM: Equations of Mathematical Biology. Vysshaya Shkola, Moscow, Russia; 1995.

    Google Scholar 

  5. 5.

    Verhulst PF: Recherches mathématiques sur la loi d'accroissement de la population. Mémoires de l'Académie Royale des Sciences et des Belles-Lettres de Bruxelles 1845,18(1):1–45.

    Google Scholar 

  6. 6.

    Dawidowicz AL: Existence and uniqueness of solutions of generalized von Foerster integro-differential equation with multidimensional space of characteristics of maturity. Bulletin of the Polish Academy of Sciences. Mathematics 1990,38(1–12):65–70.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Dawidowicz AL, Łoskot K: Existence and uniqueness of solution of some integro-differential equation. Annales Polonici Mathematici 1986,47(1):79–87.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    von Foerster H: Some remarks on changing populations. In The Kinetics of Cellular Proliferation. Grune and Stratton, New York, NY, USA; 1959.

    Google Scholar 

  9. 9.

    Gurtin ME: A system of equations for age-dependent population diffusion. Journal of Theoretical Biology 1973,40(2):389–392. 10.1016/0022-5193(73)90139-2

    Article  Google Scholar 

  10. 10.

    Gurtin ME, MacCamy RC: Non-linear age-dependent population dynamics. Archive for Rational Mechanics and Analysis 1974, 54: 281–300.

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Kamont Z: Hyperbolic Functional Differential Inequalities and Applications, Mathematics and Its Applications. Volume 486. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xiv+304.

    Google Scholar 

  12. 12.

    Kamont Z, Leszczyński H: Uniqueness result for the generalized entropy solutions to the Cauchy problem for first-order partial differential-functional equations. Zeitschrift für Analysis und ihre Anwendungen 1994,13(3):477–491.

    MATH  Google Scholar 

  13. 13.

    Leszczyński H: On CC-solutions to the initial-boundary-value problem for first-order partial differential-functional equations. Rendiconti di Matematica e delle sue Applicazioni. Serie VII 1995,15(2):173–209.

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Leszczyński H: Fast convergent iterative methods for some problems of mathematical biology. In Differential & Difference Equations and Applications. Hindawi, New York, NY, USA; 2006:661–666.

    Google Scholar 

  15. 15.

    Hale JK: Functional Differential Equations. Volume 99. Springer, New York, NY, USA; 1993.

    Google Scholar 

  16. 16.

    Leszczyński H, Zwierkowski P: Existence of solutions to generalized von Foerster equations with functional dependence. Annales Polonici Mathematici 2004,83(3):201–210. 10.4064/ap83-3-2

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Henryk Leszczyński.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Leszczyński, H., Zwierkowski, P. Iterative Methods for Generalized von Foerster Equations with Functional Dependence. J Inequal Appl 2007, 012324 (2007). https://doi.org/10.1155/2007/12324

Download citation

Keywords

  • Exact Solution
  • Iterative Method
  • Functional Dependence
  • State Density
  • Total Size
\