dorsal/arxiv
View SchemaOn the weak solutions of the McKendrick equation: Existence of demography cycles
| Authors | Rui Dilao, Abdelkader Lakmeche |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0604035 |
| URL | https://arxiv.org/abs/q-bio/0604035 |
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.
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"abstract": "We develop the qualitative theory of the solutions of the McKendrick partial\ndifferential equation of population dynamics. We calculate explicitly the weak\nsolutions of the McKendrick equation and of the Lotka renewal integral equation\nwith time and age dependent birth rate. Mortality modulus is considered age\ndependent. We show the existence of demography cycles. For a population with\nonly one reproductive age class, independently of the stability of the weak\nsolutions and after a transient time, the temporal evolution of the number of\nindividuals of a population is always modulated by a time periodic function.\nThe periodicity of the cycles is equal to the age of the reproductive age\nclass, and a population retains the memory from the initial data through the\namplitude of oscillations. For a population with a continuous distribution of\nreproductive age classes, the amplitude of oscillation is damped. The\nperiodicity of the damped cycles is associated with the age of the first\nreproductive age class. Damping increases as the dispersion of the fertility\nfunction around the age class with maximal fertility increases. In general, the\nperiod of the demography cycles is associated with the time that a species\ntakes to reach the reproductive maturity.",
"arxiv_id": "q-bio/0604035",
"authors": [
"Rui Dilao",
"Abdelkader Lakmeche"
],
"categories": [
"q-bio.PE"
],
"title": "On the weak solutions of the McKendrick equation: Existence of demography cycles",
"url": "https://arxiv.org/abs/q-bio/0604035"
},
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