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

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 1, Issue: 1, page 1-30
- ISSN: 0973-5348

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topDilão, 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 -

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