Stochastic Abundance Models: With Emphasis on Biological Communities and Species Diversity

131 Pages·1978·8.2 MB·English

by  S. Engen (auth.)

MONOGRAPHS ON
APPLIED PROBABILITY AND STATISTICS
General Editors
M.S. BARTLETT, F.R.S., and D.R. COX, F.R.S.
STOCHASTIC ABUNDANCE
MODELS
Stochastic Abundance
Models
WITH EMPHASIS ON BIOLOGICAL COMMUNITIES
AND SPECIES DIVERSITY
S. ENGEN
Department of Mathematics
University of Trondheim
LONDON
CHAPMAN AND HALL
A Halsted Press Book
John Wiley & Sons, New York
First published 1978
by Chapman and Hall Ltd.
11 New Fetter Lane, London EC4P 4EE
© 1978 S. Engen
Sojtcover reprint ojthe hardcover lst edition 1978
Photosetting by Thomson Press (India) Limited, New Delhi
ISBN-13:978-94-009-5786-2 e-ISBN-13:978-94-009-5784-8
DOl: /{}.I0071978-94-009-5784-8
All rights reserved. No part of this book
may be reprinted, or reproduced or utilized in any
form or by any electronic, mechanical or other means,
now known or hereafter invented, including photocopying and
recording, or in any information storage or retrieval
system, without permission in writing from
the publisher
Distributed in the U.S.A. by Halsted Press,
a Division of John Wiley & Sons, Inc., New York
Library of Congress Cataloging in Publication Data
Engen, S
Stochastic abundance models, with emphasis on
biological communities and species diversity.
(Monographs on applied probability and statistics)
Based on the author's thesis, Oxford.
"A Halsted Press book."
Bibliography: p.
Includes indexes.
1. Animal populations-Statistical methods.
2. Stochastic processes. I. Title.
QH352.E53 1978 574.5'24'0182 77-28215
Contents
Preface Page vii
Acknowledgments IX
PART I: THEORETICAL TREATEMENT
1. Introduction 3
1.1 Abundance models 3
1.2 Some distributions used in the
statistical analysis of abundance and diversity 4
1.3 Description of fixed populations 7
1.4 Description of random populations 12
2. Sampling from a Population of Classes 16
2.1 Introduction 16
2.2 The limitation of inference drawn from one sample 18
2.3 Sampling from a finite population 23
2.4 Sampling from an infinite population 25
2.5 Quadrat sampling: presence absence data 28
2.6 Species-area curves 29
2.7 Good's (empirical) Bayesian approach 30
3. Abundance Models 34
3.1 General introduction 34
3.2 Maximum Iiklihood theory applied to
symmetric models 37
3.3 Fisher's logarithmic series model 38
3.4 The negative binomial model 45
3.5 The geometric series model 55
3.6 The Poisson lognormal model 57
3.7 Zipfs model 60
3.8 Some other models 63
3.9 Some concluding remarks 64
4. Sample Coverage 67
4.1 Introduction 67
VI CONTENTS
4.2 Finite populations 68
4.3 Infinite populations 70
4.4 Some results for the negative binomial model 71
5. Indices of Diversity and Equitability 74
5.1 Introduction 74
5.2 Finite populations 74
5.3 Infinite populations 75
5.4 Parameter transformation 77
PART II: ECOLOGICAL APPLICATIONS
6. Abundance Models in Ecology 85
6.1 Introduction 85
6.2 Some historical remarks 85
6.3 Interpretations of fixed and random models in ecology 87
6.4 The stability of population parameters 92
7. Abundance Models in Ecology-Examples 94
7.1 Discussion of assumptions 94
7.2 Judging goodness of fit 97
7.3 Quadrat sampling \02
7.4 Higher order classification \07
7.5 Species diversity of chironomid communities-
an example 108
References III
Appendix A: The Structural Distribution for the Model
dealt with in Example 2.4 116
Appendix B: Tables 118
Author Index 123
Subject Index 125
Preface
This monograph deals with the analysis of populations of elements.
Each element is a member of one and only one class, and we shall
mainly be concerned with populations with a large number of
classes. No doubt the present theory has its outspring in ecology,
where the elements symbolize the individual animals or plants,
while the classes are the various species of the ecological community
under consideration. Some basic ideas point back to a classical
contribution by R.A. Fisher (1943, in collaboration with A.S. Corbet
and c.B. Williams) representing a breakthrough for the theoretical
analysis of diverse populations. Though most of the work in this
field has been carried out by ecologists, statisticians and biometri
cians have, over the past 15 years, shown an ever increasing interest
in the topic.
Besides being directed towards biometricians and statisticians,
this monograph may hopefully be of interest for any research worker
dealing with the classification of units into a large number of
classes, in particular ecologists, sociologists and linguists. However,
some background in statistics and probability theory is required.
It would be unless to read the present book without some knowledge
of the continuous and discrete probability distributions summarized
in section 1.1, and the use of generating functions. In particular,
a clear intuitive and formal understanding of the concept of condi
tional probability and conditional distributions is required in
order to interpret the various models correctly.
The analysis of diverse populations is naturally divided into
three parts: (I) Description of populations of classes, abundance
models, and parameter estimation; (ii) generating processes;
(iii) environmental influence on the diversity (of classes) and the
role of interaction between classes. This monograph deals with (i)
only. Generating processes will not be dealt with, except for some
comments on the subject at the end of Chapter 3. We may consider
(iii) as a further development of (i). In fact, one main purpose of
viii PREFACE
this text is to present a theory that can form a basis for a study of
factors controlling the diversity. However, it would be too large
an area to discuss here, nor would the present author be the right
person to do so.
The simultaneous description of more than one population is
not dealt with. Such models, including the measurement of simi
larity between populations, are as yet r3ther poorly developed.
It is, however, an important field, and its problems represent a
challenge to research workers in biometry, ecology and the other
branches of science mentioned above.
As regards the material included in the monograph, Part I,
the general theory, deals with the basic description of populations
of classes, abundance models, and the estimation of population
parameters. We also examine possible deductions relating to the
'species-area' curve and the concept of sample coverage; equations
that establish relations between model parameters and generally
applicable indices of diversity and equitability will be derived in
Chapter 5. Part I is written without reference to particular appli
cations, and it is hoped the presentation may prove useful to any
scientist faced with populations consisting of a large number of
classes.
Part II is devoted to ecology. We here discuss the realism of the
various assumptions upon which different types of models are
based. Through examples of the analysis of real biological data
we show how to assess the applicability of some of the models
presented in Part I. The typical ecological (botanical) sampling
scheme known as 'quadrat sampling' is mentioned briefly in Part I
and dealt with in greater detail in Part II.
Mathematics Department, Steinar Engen
University of Trondheim
September 1977
Acknowledgments
I gratefully acknowledge the help and advice of a number of friends
and scholars in the preparation of this book. T. Str~mgren in
particular deserves my thanks for arousing my interest in diversity
problems and for the encouragement and stimulation he has provid
ed over the years. I am also glad to take this opportunity of expres
sing my gratitude to Professor A. H~yland, who willingly backed
up all my applications for financial support.
The present monograph is based on my D. Phil. thesis Statistical
Analysis of Species Diversity. Professor M.S. Bartlett and Dr M.G.
Bulmer supervised my work for the thesis, at the Department of
Biomathematics, University of Oxford, during the period 1972-74.
I am greatly indebted to both for their sound advice and penetrating
criticism at all stages. The research carried out in connection with
my thesis was supported by grants from the Norwegian Research
Council for Science and the Humanities in collaboration with
The Royal Society, as part of the European Science Exchange
Programme.
Further, I wish to acknowledge my debt to Eva Seim, who read
parts of the draft and suggested a number of improvements in the
presentation; to H. Waadeland for valuable suggestions concerning
the proof in Appendix A; to S. Svensson, who kindly placed some of
his Christmas data at my disposal; to Professor S. Haftorn, who
established this contact; and to K. Aagaard, who helped me with
the example on Chironomid communities. The tables in Appendix B
are reproduced by kind permission from Biometrics.
I. Siegismund typed the manuscript of the present book. I am
indebted to her for her patient and meticulous work, much of it
undertaken at short notice and under great pressure. Finally, I would
like to thank L.E. Breivik for reading the draft in its entirety and
for checking my English.
S.E.
We must attend to the quantitative aspect of a situation or problem
and make a basic quantitative analysis. Every quality manifests itself
in a certain quantity, and without quantity there can be no quality.
To this day many of our comrades still do not understand that thc:y
must attend to the quantitative aspect of things-the basic statistics,
the main percentages and the quantitative limits that determine the
qualities of things. They have no figures in their heads and as a result
cannot help making mistakes.
Mao Tse-Tung: 'Methods of Work of
Party Committees' (13 March, 1949)
PART I
Theoretical treatment