×

On the weak solutions of the McKendrick equation: existence of demography cycles. (English) Zbl 1201.92054

Summary: We develop the qualitative theory of the solutions of the A.G. McKendrick [Proc. Edinburgh Math. Soc. 44, 98–130 (1926)] partial differential equation of population dynamics. We calculate explicitly the weak solutions of the McKendrick equation and of the Lotka renewal integral equation with time and age dependent birth rate. The mortality modulus is considered age dependent. We show the existence of demographic cycles. For a population with only one reproductive age class, independent of the stability of the weak solutions and after a transient time, the temporal evolution of the number of individuals of a population is always modulated by a time periodic function. The periodicity of the cycles is equal to the age of the reproductive age class, and a population retains the memory from the initial data through the amplitude of oscillations. For a population with a continuous distribution of reproductive age classes, the amplitude of oscillation is damped. The periodicity of the damped cycles is associated with the age of the first reproductive age class. Damping increases as the dispersion of the fertility function around the age class with maximal fertility increases. In general, the period of the demographic cycles is associated with the time that a species takes to reach the reproductive maturity.

MSC:

92D25 Population dynamics (general)
35B99 Qualitative properties of solutions to partial differential equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML