Abstract
The aim of this work is to give a direct and constructive proof of existence and uniqueness of a global solution to the equations of age-dependent population dynamics introduced and considered by M. E. Gurtin & R. C. MacCamy in [3]. The linear theory was developed by F. R. Sharpe & A. J. Lotka [10] and A. G. McKendrick [8] (see also [1], [9]) and extended to the nonlinear case by M. E. Gurtin & R. C. MacCamy in [3] (see also [4] [5] [6]). In [3], the key of the proof of existence and uniqueness was to reduce the problem to a pair of integral equations. In fact, as we shall see, the problem can also be solved by a simple fixed point argument. To outline more clearly the ideas of the proof, we will first discuss the setting and the resolution of the linear case, and then we will generalize the results of [3].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Feller: On the Integral Equation of Renewal Theory. Ann. Math. Statist. 12 (1941), 243–267.
M. E. Gurtin: The Mathematical Theory of Age-structured Populations. Book to appear.
M. E. Gurtin & R. C. MacCamy: Nonlinear Age-dependent Population Dynamics. Arch. Rational Mech. Anal. 54 (1974), 281–300.
M. E. Gurtin & R. C. MacCamy: On the Diffusion of Biological Populations. Math. Biosci. 33 (1977), 35–49.
M. E. Gurtin & R. C. MacCamy: Some Simple Models for Nonlinear Age-dependent Population Dynamics. Math. Biosci. 43 (1979), 199–211.
M. E. Gurtin & R. C. MacCamy: Diffusion Models for Age-structured Populations. Math. Biosci. 54 (1981), 49–59.
J. K. Hale: Ordinary Differential Equations. Wiley, 1969, Revised edition, Krieger, 1980.
A. G. McKendrick: Applications of Mathematics to Medical Problems: Proc. Edinburgh Math. Soc. 44, 1926, 98–130.
A. J. Lotka: Elements of Physical Biology, Baltimore, Williams & Wilkins (Republished as Element of Mathematical Biology, New York, Dover 1956).
F. R. Sharpe & A. J. Lotka: A Problem in Age Distribution. Phil. Mag. 21, 1911, 435–438.
H. VonFoerster: Some Remarks on Changing Populations. In the Kinetics of Cellular Proliferation. New York: Grune and Stratton (1959), 382–407.
Addendum After writing this paper, I have read the following very interesting works of G. F. Webb
G. F. Webb: Nonlinear Age-dependent Population Dynamics in L1 (to appear).
G. F. Webb: Nonlinear Age-dependent Population Dynamics with Continuous Age Distributions. Conference on Differential Equations and Applications, Schloss Retzhof, Austria, June 1–6, 1981. For his model, which extends the Gurtin-MacCamy theory, local existence and uniqueness can be extended with our technique in L ∞([0, T], L 1(R)). Then a regularity theory can be gotten as in [13]. Moreover, the complete bibliography of [12] introduced me to the following other interesting contributions and models
G. DiBlasio: Nonlinear Age-dependent Population Growth with History Dependent Birth Rate. Math. Biosci. 46 (1979), 279–291.
A. Haimovici: On the Age Dependent Growth of two Interacting Populations. Boll. Unione Mat. Ital. 15 (1979), 405–429.
M. Marcati & R. Serafini: Asymptotic Behaviour in Age Dependent Population Dynamics with Special Spread: Boll. Unione Math. Ital., 16B (1979), 734–753.
J. Prüss: Equilibrium Solutions of Age-Specific Population Dynamics of Several Species: J. of Math. Biol. 11 (1981), 65–84.
Author information
Authors and Affiliations
Additional information
Communicated by M. E. Gurtin
Rights and permissions
About this article
Cite this article
Chipot, M. On the equations of age-dependent population dynamics. Arch. Rational Mech. Anal. 82, 13–25 (1983). https://doi.org/10.1007/BF00251723
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00251723