On the Weak Solutions of the McKendrick Equation: Existence of Demography Cycles

Dilão, R., and Lakmeche, A.. "On the Weak Solutions of the McKendrick Equation: Existence of Demography Cycles." Mathematical Modelling of Natural Phenomena 1.1 (2010): 1-30. <http://eudml.org/doc/222222>.

@article{Dilão2010,
abstract = { We develop the qualitative theory of the solutions of the McKendrick 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. Mortality modulus is considered age dependent. We show the existence of demography cycles. For a population with only one reproductive age class, independently 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 demography cycles is associated with the time that a species takes to reach the reproductive maturity. },
author = {Dilão, R., Lakmeche, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {McKendrick equation; renewal equation; demography cycles; periodic solutions; age-structure; demography cycles},
language = {eng},
month = {3},
number = {1},
pages = {1-30},
publisher = {EDP Sciences},
title = {On the Weak Solutions of the McKendrick Equation: Existence of Demography Cycles},
url = {http://eudml.org/doc/222222},
volume = {1},
year = {2010},
}

TY - JOUR
AU - Dilão, R.
AU - Lakmeche, A.
TI - On the Weak Solutions of the McKendrick Equation: Existence of Demography Cycles
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/3//
PB - EDP Sciences
VL - 1
IS - 1
SP - 1
EP - 30
AB - We develop the qualitative theory of the solutions of the McKendrick 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. Mortality modulus is considered age dependent. We show the existence of demography cycles. For a population with only one reproductive age class, independently 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 demography cycles is associated with the time that a species takes to reach the reproductive maturity.
LA - eng
KW - McKendrick equation; renewal equation; demography cycles; periodic solutions; age-structure; demography cycles
UR - http://eudml.org/doc/222222
ER -