Foundations of Ecology by Unknown

Author:Unknown
Language: eng
Format: epub
Publisher: University of Chicago Press
Published: 1991-08-14T16:00:00+00:00

PAPER 22

ON THE USE OF MATRICES IN CERTAIN POPULATION MATHEMATICS

BY P. H. LESLIE, Bureau of Animal Population, Oxford University

CONTENTS

1. Introduction

2. Derivation of the matrix elements

3. Numerical example

4. Properties of the basic matrix

5. Transformation of the co-ordinate system

6. Relation between the canonical form B and the Lxmx column

7. The stable age distribution

8. Properties of the stable vectors

9. The spectral set of operators

10. Reduction of B to classical canonical form

11. The relation between ϕ and ψ vectors

12. Case of repeated latent roots

13. The approach to the stable age distribution

14. Special case of the matrix with only a single non-zero Fx element

15. Numerical comparison with the usual methods of computation

16. Further practical applications

Appendix: (1) The tables of mortality and fertility

(2) Calculation of the rate of increase

(3) Numerical values of the matrix elements

References

1. INTRODUCTION

If we are given the age distribution of a population on a certain date, we may require to know the age distribution of the survivors and descendants of the original population at successive intervals of time, supposing that these individuals are subject to some given age-specific rates of fertility and mortality. In order to simplify the problem as much as possible, it will be assumed that the age-specific rates remain constant over a period of time, and the female population alone will be considered. The initial age distribution may be entirely arbitrary; thus, for instance, it might consist of a group of females confined to only one of the age classes.

The method of computing the female population in one unit’s time, given any arbitrary age distribution at time t, may be expressed in the form of m + 1 linear equations, where m to m + 1 is the last age group considered in the complete life table distribution, and when the same unit of age is adopted as that of time. If

nxt = the number of females alive in the age group x to x + 1 at time t,

Px = the probability that a female aged x to x + 1 at time t will be alive in the age group x + 1 to x + 2 at time t + 1,

Fx = the number of daughters born in the interval t to t + 1 per female alive aged x to x + 1 at time t, who will be alive in the age group 0-1 at time t + 1,

then, working from an origin of time, the age distribution at the end of one unit’s interval will be given by

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.