Ecological Methods

Ecological Methods T.R.E. Southwood Kt, DSc, PhD, FRS Lineacre Professor of Zoology, University of Oxford, South Parks Road, Oxford

P.A. Henderson PhD Pisces Conservation Limited, IRC House, The Square, Pennington, Lymington, Hants

Third Edition

This book has an active website that provides additional illustrations, details of equipment and computer programs as well as references to work published since the manuscript was completed. The site is located at http://www.blackwell-science.com/ southwood

Ecological Methods

Ecological Methods T.R.E. Southwood Kt, DSc, PhD, FRS Lineacre Professor of Zoology, University of Oxford, South Parks Road, Oxford

P.A. Henderson PhD Pisces Conservation Limited, IRC House, The Square, Pennington, Lymington, Hants

Third Edition

This book has an active website that provides additional illustrations, details of equipment and computer programs as well as references to work published since the manuscript was completed. The site is located at http://www.blackwell-science.com/ southwood

© 2000 by T.R.E. Southwood and P.A. Henderson Published by Blackwell Science Ltd Editorial Offices: Osney Mead, Oxford OX2 0EL 25 John Street, London WC1N 2BL 23 Ainslie Place, Edinburgh EH3 6AJ 350 Main Street, Malden MA 02148-5018, USA 54 University Street, Carlton Victoria 3053, Australia 10, rue Casimir Delavigne 75006 Paris, France Other Editorial Offices: Blackwell Wissenschafts-Verlag GmbH Kurfürstendamm 57 10707 Berlin, Germany Blackwell Science KK MG Kodenmacho Building 7–10 Kodenmacho Nihombashi Chuo-ku, Tokyo 104, Japan First published 1966 by Methuen & Co Ltd Second edition published 1978 by Chapman & Hall Third edition published 2000 by Blackwell Science Ltd Set by Excel Typesetters Co., Hong Kong Printed and bound in Great Britan at the University Press, Cambridge. The Blackwell Science logo is a trade mark of Blackwell Science Ltd, registered at the United Kingdom Trade Marks Registry

The right of the authors to be identified as the Authors of this Work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the copyright owner. A catalogue record for this title is available from the British Library ISBN 0-632-05477-8 Library of Congress Cataloging-in-publication Data Southwood, Richard, Sir. Ecological methods / T.R.E. Southwood. — 3rd ed. / revised by P.A. Henderson and T.R.E. Southwood. p. cm. Includes bibliographical references and index. ISBN 0-632-05477-8 1. Ecology — Technique. 2. Animal populations. I. Henderson, P.A. II. Title. QH541.28.S68 2000 577¢.028 — dc21 99-39859 CIP

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Contents

Prefaces, xi 1 Introduction to the Study of Animal Populations, 1 1.1 Population estimates, 2 1.1.1 Absolute and related estimates, 2 1.1.2 Relative estimates, 3 1.1.3 Population indices, 4 1.2 Errors and confidence, 4 References, 5

2 The Sampling Programme and the Measurement and Description of Dispersion, 7 2.1 Preliminary sampling, 7 2.1.1 Planning and fieldwork, 7 2.1.2 Statistical aspects, 10 2.2 The sampling programme, 15 2.2.1 The number of samples per habitat unit (e.g. plant, host, or puddle), 15 2.2.2 The sampling unit, its selection, size, and shape, 18 2.2.3 The number of samples, 20 2.2.4 The pattern of sampling, 23 2.2.5 The timing of sampling, 25 2.3 Data processing, 26 2.4 Jackknife and bootstrap techniques, 27 2.5 Dispersion, 29 2.5.1 Mathematical distributions that serve as models, 29 2.5.2 Biological interpretation of dispersion parameters, 42 2.5.3 Nearest-neighbour and related techniques: measures of departure from randomness of the distribution, 50 2.6 Sequential sampling, 53 2.6.1 Sampling numbers, 53 2.6.2 Presence or absence sampling, 57 2.6.3 Sampling a fauna, 59 References, 62

3 Absolute Population Estimates Using Capture–Recapture Experiments, 73 3.1 Capture–recapture methods, 74 3.1.1 Assumptions common to most methods, 75 3.1.2 Estimating closed populations, 82 3.1.3 Estimations for open populations, 88

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CONTENTS 3.2 Methods of marking animals, 102 3.2.1 Handling techniques, 104 3.2.2 Release, 106 3.2.3 Surface marks using paints and solutions of dyes, 107 3.2.4 Dyes and fluorescent substances in powder form, 112 3.2.5 Marking formed by ingestion or absorption of dyes, 114 3.2.6 Marking by injection, Panjet, or tattooing, 115 3.2.7 External tags, 116 3.2.8 Branding, 117 3.2.9 Mutilation, 118 3.2.10 Natural marks, parasites, and genes, 118 3.2.11 Rare elements, 119 3.2.12 Radioactive isotopes, 120 3.2.13 Radio and sonic tags, 127 References, 127

4 Absolute Population Estimates by Sampling a Unit of Habitat: Air, Plants, Plant Products, and Vertebrate Hosts, 141 4.1 Sampling from the air, 141 4.1.1 Sampling apparatus, 142 4.1.2 Rotary and other traps, 145 4.1.3 Comparison and efficiencies of the different types of suction traps, 146 4.1.4 Conversion of catch to aerial density, 147 4.1.5 Conversion of density to total aerial population, 147 4.2 Sampling from plants, 148 4.2.1 Assessing the plant, 149 4.2.2 Determining the numbers of invertebrates, 150 4.2.3 Special sampling problems with animals in plant material, 167 4.3 Sampling from vertebrate hosts, 167 4.3.1 Sampling from living hosts, 168 4.3.2 Sampling from dead hosts, 170 4.3.3 Sampling from vertebrate ‘homes’, 171 References, 172

5 Absolute Population Estimates by Sampling a Unit of Aquatic Habitat, 180 5.1 Open water, 180 5.1.1 Nets, 180 5.1.2 Pumps, 183 5.1.3 Water-sampling bottles, 184 5.1.4 The Patalas–Schindler volume sampler, 185 5.1.5 Particular methods for insects, 185 5.2 Vegetation, 187 5.2.1 Emergent vegetation, 187 5.2.2 Submerged vegetation, 189 5.2.3 Sampling floating vegetation, 191 5.3 Bottom fauna, 191 5.3.1 Hand-net sampling of forest litter, 192 5.3.2 Lifting stones, 193 5.3.3 The planting of removable portions of the substrate, 193 5.3.4 Cylinders and boxes for delimiting an area, 196 5.3.5 Trawls, bottom sledges, and dredges, 197 5.3.6 Grabs, 199 5.3.7 Dendy (1944) inverting sampler, 202 5.3.8 Box samplers and corers, 203 5.3.9 Airlift and suction devices, 206

CONTENTS 5.4 Poisons and anaesthetics used for sampling fish in rock pools and small ponds, 207 References, 208

6 Absolute Population Estimates by Sampling a Unit of Soil or Litter Habitat: Extraction Techniques, 213 6.1 Sampling, 213 6.2 Bulk staining, 215 6.3 Mechanical methods of extraction, 215 6.3.1 Dry sieving, 216 6.3.2 Wet sieving, 216 6.3.3 Soil washing and flotation, 218 6.3.4 Flotation, 221 6.3.5 The separation of plants and insects by differential wetting, 222 6.3.6 Centrifugation, 224 6.3.7 Sedimentation, 225 6.3.8 Elutriation, 226 6.3.9 Sectioning, 228 6.4 Behavioural or dynamic methods, 228 6.4.1 Dry extractors, 229 6.4.2 Wet extractors, 234 6.5 Summary of the applicability of the methods, 239 6.5.1 Substrate type, 239 6.5.2 Animal type, 240 6.5.3 Cost, 241 References, 241

7 Relative Methods of Population Measurement and the Derivation of Absolute Estimates, 247 7.1 Factors affecting the size of relative estimates, 247 7.1.1 The ‘phase’ of the animal, 248 7.1.2 The activity of the animal, 249 7.1.3 Differences in the response between species, sexes, and individuals, 250 7.1.4 The efficiency of the trap or searching method, 252 7.2 The uses of relative methods, 253 7.2.1 Measures of the availability, 253 7.2.2 Indices of absolute population, 254 7.2.3 Estimates of absolute population, 254 7.2.4 Collecting, 261 7.3 Relative methods: catch per unit effort, 262 7.3.1 Observation by radar, 262 7.3.2 Hydroacoustic methods, 263 7.3.3 Fish counters, 263 7.3.4 Electric fishing, 264 7.3.5 Aural detection, 264 7.3.6 Exposure by plough, 265 7.3.7 Collecting with a net or similar device, 265 7.3.8 Visual searching and pooting, 268 7.4 Relative methods: trapping, 269 7.4.1 Interception traps, 270 7.4.2 Flight traps combining interception and attraction, 279 7.4.3 Light and other visual traps, 283 7.5 Traps that attract the animals by some natural stimulus or a substitute, 293 7.5.1 Shelter traps, 293 7.5.2 Host-plant traps, 294 7.5.3 Baited traps, 294 7.5.4 The use of vertebrate hosts or substitutes as bait for insects, 297

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CONTENTS 7.6 Using sound, 302 References, 302

8 Indices: the Use of Signs, Products, and Effects, 319 8.1 Arthropods, 319 8.1.1 Exuviae, 319 8.1.2 Frass, 320 8.1.3 Other products, 323 8.2 Vertebrate products, 323 8.3 Effects, 324 8.3.1 Effects due to an individual insect, 324 8.3.2 General effects — damage, 325 References, 329

9 Wildlife Population Estimates by Census and Distance-measuring Techniques, 334 9.1 Census methods, 335 9.2 Point- and line-survey methods, 336 9.2.1 Indices of abundance using transects, 336 9.2.2 Methods based on flushing, 336 9.2.3 Line-transect methods: the Fourier-series estimator, 339 9.2.4 Point transects, 343 9.3 Spatial distribution and plotless density estimators, 344 9.3.1 Closest-individual or distance method, 345 9.3.2 Nearest-neighbour methods, 345 References, 346

10 Observational and Experimental Methods for the Estimation of Natality, Mortality, and Dispersal, 348 10.1 Natality, 348 10.1.1 Fertility, 348 10.1.2 Numbers entering a stage, 350 10.1.3 The birth-rate from mark-and-recapture data, 357 10.2 Mortality, 357 10.2.1 Total, 357 10.2.2 The death-rate from mark-and-recapture data, 358 10.2.3 Climatic factors, 358 10.2.4 Biotic factors, 358 10.3 Dispersal, 377 10.3.1 Detecting and quantifying jump dispersal, 377 10.3.2 Quantifying neighbourhood dispersal, 377 References, 392

11 The Construction, Description, and Analysis of Age-specific Life-tables, 404 11.1 Types of life-table and the budget, 404 11.2 The construction of a budget, 405 11.2.1 Southwood’s graphical method, 406 11.2.2 Richard and Waloff ’s first method, 407 11.2.3 Manly’s method, 409 11.2.4 Ruesink’s method, 409 11.2.5 Dempster’s method, 410 11.2.6 Richard and Waloff ’s second method, 411

CONTENTS 11.2.7 Kiritani, Nakasuji and Manly’s method, 411 11.2.8 Kempton’s method, 412 11.2.9 The Bellows and Birley method, 413 11.3 The description of budgets and life-tables, 414 11.3.1 Survivorship curves, 414 11.3.2 Stock-recruitment (Moran–Ricker) curves, 415 11.3.3 The life-table and life expectancy, 417 11.3.4 Life and fertility tables and the net reproductive rate, 418 11.3.5 Population growth rates, 420 11.3.6 The calculation of r, 422 11.4 The analysis of life-table data, 423 11.4.1 The comparison of mortality factors within a generation, 424 11.4.2 The simple statistical relationship of population size to a factor, 425 11.4.3 Survival and life-budget analysis, 425 References, 433

12 Age-grouping,Time-specific Life-tables, and Predictive Population Models, 437 12.1 Age-grouping, 437 12.1.1 Ageing young by developmental stage, 438 12.1.2 Ageing by using structures, 439 12.2 Time-specific life-tables and survival rates, 447 12.2.1 Physiological time, 450 12.2.2 Life-table parameters, 450 12.2.3 Recruitment in the field, 451 12.2.4 Empirical models, 451 12.2.5 Intrinsic-rate models and variable life-tables, 452 12.2.6 Lewis–Leslie matrices, 452 References, 454

13 Species Richness, Diversity, and Packing, 462 13.1 Diversity, 462 13.1.1 Description of a and g diversity, 463 13.1.2 Species richness, 464 13.1.3 Models for the S: N relationship, 471 13.1.4 Non-parametric indices of diversity, 474 13.1.5 Which model or index?, 477 13.1.6 Comparing communities — diversity ordering, 478 13.1.7 Procedure to determine a diversity, 480 13.1.8 Determining b diversity, 481 13.2 Similarity and the comparison and classification of samples, 484 13.2.1 Measures of complementarity, 485 13.2.2 Similarity indices, 485 13.2.3 Multivariate analysis, 488 13.3 Species packing, 490 13.3.1 Measurement of interspecific association, 491 13.3.2 Measurement of resource utilization, 494 13.3.3 Niche size and competition coefficients, 501 References, 502

14 The Estimation of Productivity and the Construction of Energy Budgets, 507 14.1 Estimation of standing crop, 509 14.1.1 Measurement of biomass, 509 14.1.2 Determination of energy density, 510

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CONTENTS 14.2 Estimation of energy flow, 511 14.2.1 The measurement of production, 513 14.2.2 The measurement of feeding and assimilation, 516 14.2.3 The measurement of the energy loss due to respiration and metabolic process, 521 14.3 The energy budget, efficiencies, and transfer coefficients, 528 14.3.1 The energy budget of a population (or trophic level), 528 14.3.2 Energy transfer across trophic links, 530 14.4 Identification of ecological pathways using stable isotopes, 532 14.5 Assessment of energy and time costs of strategies, 532 References, 533

15 Large Scale Spatial and Temporal Studies and the Classification of Habitats, 540 15.1 Remote sensing, 541 15.2 Long-term studies, 545 15.2.1 Planning spatial and temporal sampling, 546 15.2.2 The classification of time series, 547 15.2.3 Time-series analysis, 547 15.2.4 Detecting synchrony, 554 15.2.5 Measuring temporal variability, 554 15.2.6 Detection of density dependence, 556 15.3 Geographical information systems, 558 15.4 Habitat classification, 559 15.4.1 Qualitative, 559 15.4.2 Quantitative, 561 References, 562 Index, 565

Prefaces

Preface to the Third Edition We have been encouraged to prepare this third edition by the continuing use of the earlier editions. In doing so, we have been struck by the contrast between the advances in some areas, especially data handling, and the enduring value of various other techniques. Ecology has continued its advance into the popular and political domain, though far from everything that has gained the ‘eco-’ prefix falls within our purview. The underlying paradigm of ecology has, however, shifted. In particular, the concept of the metapopulation is now recognized as central to the understanding of the distribution and abundance of animals and its exploration is aided by the accessibility of numerous data sets, often with large temporal or spatial scales. The availability of molecular techniques has encouraged the consideration of genetic and phylogenetic aspects and has permitted the growth of quantitative comparative analyses, described by P.H. Harvey and M.D. Pagel (1991) The Comparative Method in Evolutionary Biology (Oxford). The extent to which we have felt that revision was necessary has varied greatly in different parts of the book. Where there has been little change in the method, we have retained the early references. We have done this on the bases that journal editors formerly permitted more detailed description of methods, that these papers will not be located by computer searches, and that these pioneers continue to deserve credit. Other portions have required considerable modification; we have deleted one chapter (12) and added a new one (15), as well as reorganizing the structure in some places. Although the primary focus remains on insects, which are in terms of species the dominant animals, we have taken the opportunity to explicitly expand the coverage to all major macroscopic groups. The widespread availability of high-capacity personal computers (PCs), with software packages and access to the internet, has totally changed the speed and ease of handling (and sometimes accessing) data. We have therefore given references to some relevant software packages and websites, whilst eliminating many descriptions of time-consuming graphical methods. However, we believe that the advice given in the preface to the second edition is even more applicable today. The researcher who relies entirely on the output of a computer is in danger of drawing false conclusions xi

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and overlooking possible insights. It is essential to understand the features of the data (are there any outliers?), the assumptions of the methods, and the biological basis of the analysis and to acquire a feel for the capabilities and responses of the species under study. The book has an active website that provides additional illustrations, details of equipment and computer programs and references to work published since the manuscript was completed. We aim to keep the revision up to date through this site; it would greatly assist us if ecologists could send us explanatory paragraphs and references to new work. These should be posted on http://www.blackwellscience.com/southwood. We would also be most grateful to have any errors drawn to our attention through the same route. The interpretation of ‘ecological methods’ remains as described in the Preface to the First Edition, namely those methods peculiar to ecologists, either in their origin or in modification. Just as the measurement of physical factors, using the methods of the physical sciences, has always been outside the book’s scope, so are the methods of molecular biology. These are described in a number of works, such as Snow A.A. & Parker P.G. (1998) Molecular markers for population ecology. Ecology 79, 359–425. We are most grateful to many ecologists who sent comments on the second edition. In particular, generous help has been given by Drs C. Henderson, D.J. Rogers, A.E. Magurran, W.D. Hamilton and G.R.W. Wint and Mr C. Hambler. T.R.E. Southwood P.A. Henderson Oxford, October 1999

Preface to the Second Edition In the twelve years since the First Edition was prepared there have been remarkable developments in ecology. The subject has changed its lay image, from a rather recondite branch of biology, to something that is widely considered ‘good’, but only vaguely understood. The public’s focus on environmental problems and the insights into these that ecology can provide are a great challenge to ecologists to develop their subject: they need to be able to provide reliable quantitative inputs for the management of the biosphere. The enormous volume of work that it has been necessary to review for this edition is evidence of the extent to which ecologists are seeking to meet this challenge. I believe that the theme of the first edition, the need for precise measurement and critical analysis, is equally valid today; although many recent studies show levels of sophistication that were beyond my wildest hopes when I embarked on the preparation of the first edition. In his review of the first edition Dr R.R. Sokal was kind enough to say it was an ‘unusual book’ for it covered both traps and mathematical

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formulae, topics that were usually of interest to different people. This, I am glad to say, is now no longer generally true. The computer and the electronic calculator have revolutionized the handling of ecological data, but neither can make a ‘silk purse’ of sound insight, out of a ‘sow’s ear’ of unreliable raw data or confused analytical procedure. More than ever the ecologist needs to keep his biological assumptions in mind and remember the value of preliminary simple graphical analysis as a means of recognizing new patterns and gaining fresh insights. It has been a gratifying, though exhausting, experience preparing the new edition! Progress has been so rapid in several areas that some chapters have been completely or largely rewritten (e.g. 11, 12, 13), whilst most have large new sections. It has been necessary to be highly selective in the additions to the bibliographies, even so there are nearly a thousand new entries and only a few older references could be deleted. I hope that, with the advent of Ecological Abstracts, the selective nature of the bibliographies will not handicap workers. As the mathematical side of ecology has grown, so have the problems of notation and it is now quite impossible (without extending far beyond the roman and greek alphabets!) to retain a unique notation throughout. Apart from widespread and generally accepted symbols, I have merely aimed to be consistent within a section. I am most grateful to many ecologists who have helped me in this revision by sending me reprints of their papers or notes on difficulties and errors in the first edition. Detailed criticism, advice and help, including access to unpublished work has been generously given by M.H. Birley, P.F.L. Boreham, M.J.W. Cock, G.R. Conway, M.P. Hassell, R.M. May, A. Milne, S. McNeill, G. Murdie, G.A. Norton, S. Parry, P.M. Reader, G. Seber, N.E.A. Scopes, L.R. Taylor, R.A.J. Taylor, T.W. Tinsley, J.M. Webster, R.G. Wiegert and I.J. Wyatt. I am deeply indebted to them, and to others, especially Margaret Clements and my wife, who have assisted with patience and forbearance in the essential tasks associated with the preparation of the manuscript of the new edition. T. R. E. Southwood Imperial College Silwood Park, Ascot. October 1977

Preface to the First Edition This volume aims to provide a handbook of ecological methods pertinent for the study of animals. Emphasis is placed on those most relevant to work on insects and other non-microscopic invertebrates of terrestrial and aquatic environments, but it is believed that the principles and general techniques will be found of value in studies on vertebrates and marine animals. The term ecology is now widely used in the field of social, as well as biological, science; whilst the subject of ecology, covering as it does the relationship of

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the organism to its environment, has many facets. It is, in fact, true to say that the ecologist may have need of recourse to almost all of the methods of the biologist and many of those of the physical scientist: the measurement of the physical factors of the environment may be a particularly important part of an ecologist’s work and he will refer to books such as R.B. Platt & J.E. Griffiths’ (1964) Environmental Measurement and Interpretation. There are, however, certain methods that are peculiar to the ecologist, those concerning the central themes of his subject, the measurement, description and analysis of both the population and the community. These are ecological methods (as opposed to ‘methods for ecologists’ which would need to cover everything from laboratory workshop practice to information theory); they are the topic of this book. During the 10 years that I have been giving advanced and elementary courses on ecological methods at Imperial College, London, and at various Field Centres, the number and range of techniques available to the ecologist have increased enormously. It has been the comments of past students on the utility of these courses in helping to overcome the difficulties of coping with the scattered and growing literature that have encouraged me to attempt the present compilation. I am grateful to many former students for their criticisms and comments, as I am to the members of classes I was privileged to teach at the University of California, Berkeley, and at the Escuela Nacional de Agricultura, Mexico, whilst writing this book. Although the general principles of most methods are of wide application, the study of a particular animal in a particular habitat may require certain special modifications. It is clearly impossible to cover all variants and therefore the reader is urged to consult the original papers that appear relevant to his problem. I am grateful to my publishers for agreeing to the publication of the extensive bibliographies, it is hoped that these will provide many leads on specific problems; they are, however, by no means exhaustive. The present book is designed to be of use to those who teach the practical aspects of animal ecology in schools, training colleges and universities; insects, being numerically the dominant component of the macrofauna of terrestrial and many aquatic habitats, almost invariably come to the forefront of ecological field work. This volume is intended as an aid to all who need to measure and compare populations and communities of animals, not only for the research ecologist, but also for the conservationist and the economic entomologist. Population measurement is as necessary in the assessment of the effects of a pesticide and in the determination of the need for control measures, as it is in intensive ecological studies. It is frequently pointed out that ecological theories have outstripped facts about animal populations and I trust that it is not too presumptuous of me to hope that this collection of methods may encourage more precise studies and more critical analysis of the assembled data so that, in the words of O.W. Richards, we may have ‘more light and less heat’, in our discussions. The topics have been arranged on a functional basis, that is, according to the type of information given by a particular method. As a result some techniques are

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discussed in several places, e.g. radiotracers will be found under marking methods for absolute population estimates (chapter 3), the measurement of predation and dispersal (chapter 9) and the construction of energy budgets (chapter 14). By its very nature ecology cannot be divided into rigid compartments, but frequent crossreferences in the text, together with the detailed contents list and index, should enable the reader to find the information he needs. The sequence of chapters parallels, to a large extent, the succession of operations in a piece of intensive research. It is a pleasure to express my great indebtedness to colleagues who have read and criticized various chapters in draft: Dr N.H. Anderson (ch. 6), Dr R.E. Blackith (ch. 2 & 13), Dr J.P. Dempster (ch. 1, 2 & 3), Mr G.R. Gradwell (ch. 10), Dr C.S. Holling (ch. 12), Mr S. Hubbell (ch. 14), Dr C.B. Huffaker (ch. 2), Dr G.M. Jolly, (section II of ch. 3), Dr C.T. Lewis, (section I of ch. 3), Dr R.F. Morris (ch. 10), Dr O.H. Paris (ch. 9 & 14), Mr L.R. Taylor (ch. 2 & 4), Professor G.C. Varley (ch. 10) and Dr N. Waloff (ch. 1, 2 & 10); frequently these colleagues have also made available unpublished material; they are of course in no way responsible for the views I have expressed or any errors. For access to ‘in press’ manuscripts, for unpublished data and for advice on specific points I am grateful to: Drs J.R. Anderson, R. Craig and D.J. Cross, Mr R.J. Dalleske, Drs W. Danthanarayana, H.V. Daly, E.A.G. Duffey, P.J.M. Greenslade, M.P. Hassell, P.H. Leslie, J. MacLeod, C.O. Mohr, W.W. Murdoch and F. Sonleitner, Mr W.O. Steel, Drs A.J. Thorsteinson, R.L. Usinger, H.F. van Emden, E.G. White, D.L. Wood and E.C. Young. Ecologists in all parts of the world have greatly helped by sending me reprints of their papers. I have been extremely fortunate too in the assistance I have received in translating; Mrs M. Van Emden has generously made extensive translations of works in German, and with other languages I have been helped by Dr F. Baranyovits (Hungarian), Dr T. Bilewicz-Pawinska (Polish), Mr Guro Kuno (Japanese), Dr P. Stys (Czechoslovakian) and Dr N. Waloff (Russian). Much of the manuscript was prepared whilst I held a visiting professorship in the Department of Entomology and Parasitology of the University of California, Berkeley; I am indebted to the Chairman of that Department, Dr Ray F. Smith, for his interest and the many kindnesses and facilities extended to me and to the Head of my own Department, Professor O.W. Richards, F.R.S., for his support and advice. I wish to thank Mrs M.P. Candey and Mrs C.A. Lunn for assisting me greatly in the tedious tasks of preparing the bibliographies and checking the manuscript. My wife has encouraged me throughout and helped in many ways, including typing the manuscript. T.R.E. Southwood London, October 1965

1

Introduction to the Study of Animal Populations

Information about animal populations is sought for a variety of purposes; but the object of a study will largely determine the methods used and thus this must be clearly defined at the outset. Very broadly, studies may be divided into extensive and intensive (Morris, 1960). Extensive studies are carried out over larger areas than intensive studies. They are frequently used to provide information on distribution and abundance for conservation or management programmes (e.g. monitoring the status of mammal populations (Macdonald et al., 1998)). Recent developments in remote sensing capability and geographical information system software have given great impetus to extensive studies in recent years (Chapter 15). For pests or parasites, they provide assessments of incidence or damage and may guide the application of control measures. Aphids, for example, are regularly monitored by the Rothamsted suction trap survey (Knight et al., 1992). In extensive surveys, an area will often be sampled once or at the most a few times per study period. The timing of sampling in relation to the life cycle of the animal is obviously of critical importance and, for many species, only limited stages in the life cycle can be sampled. Extensive studies produce information about the spatial pattern of populations, and it is often possible to relate the level of the population to edaphic, oceanographic, or climatic factors. Intensive studies involve the repeated observation of the population of an animal. Usually information is acquired on the sizes of the populations of successive developmental stages so that a life-table or budget may be constructed. Then, using this table, an attempt is made to determine the factors that influence population size and those that govern or regulate it (Varley & Gradwell, 1963; Sibley & Smith, 1998). It is important to consider at the start the type of analysis (see Chapter 11) that will be applied and so ensure that the necessary data are collected in the best manner. Intensive studies may have even more limited objectives, such as the determination of the level of parasitism, the amount of dispersal, or the overall rate of population change. The census of populations and the stages at which mortality factors operate are necessary first stages in the estimation of the productivity (Chapter 14) of ecosystems. In survey and conservation work, the species make-up of the community and changes in its diversity associated with human activities are most frequently the 1

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features it is desired to measure. Special methods of analysis need to be used (Chapter 13), but difficulties usually arise because of the impossibility of recording the abundance of all the species living within a habitat with equal efficiency.

1.1 Population estimates Population estimates can be classified into a number of different types; the most convenient classification is that adopted by Morris (1955), although he used the terms somewhat differently in a later paper (1960). 1.1.1 Absolute and related estimates For large animals that are easily observed and have small, countable populations, such as rhinos, elephants, whales, and some birds, it may be possible to express the global or metapopulation size as a total number of individuals. However, for most animals, numbers will be expressed as a density per unit area or volume or density per unit of the habitat. Such estimates are given by distance sampling and related techniques (Chapter 9), marking and recapture (Chapter 3), sampling a known fraction of the habitat (Chapters 4–6), and removal sampling and random-walk techniques (Chapter 7). 1.1.1.1 Absolute population This is defined as the number of animals per unit area. For planktonic animals, the number per unit volume can be more appropriate. It is almost impossible to construct a budget or to study mortality factors without the conversion of population estimates to absolute figures. This is because other measures of habitat are variable. For example, the amount of plant available to an insect is always changing; further, insects often move from the plant to the soil at different developmental stages. The importance of obtaining absolute estimates cannot be overemphasized. 1.1.1.2 Population intensity This is the number of animals per unit of habitat, e.g. per leaf, per shoot, per plant, per host. Such a measure is often, from the nature of the sampling, the type first obtained (see also p. 148). When the level of the animal population is being related to habitat availability or plant or host damage, it is more meaningful than an estimate in absolute terms. It is also valuable in comparing the densities of natural enemies and their prey. However, the number of habitat units/area should be assessed, for differences in plant density can easily lead to the most intense population being the least dense in absolute terms (Pimentel, 1961). When dealing with different varieties of plants, differences in leaf area may account for apparently denser populations, in absolute terms, on certain varieties (Bradley, 1952). Thus the choice of the leaf or of the plant as the unit for expressing population intensity can affect the relative population estimate obtained (Broadbent, 1948; Fig. 1.1). Similarly, with litter fauna, owing to the effects of seasonal leaf fall, the intensity measure (in animals/weight of

INTRODUCTION

3

2400

Variety Majestic

2200

Variety Arran Pilot

2000 Number of M. persicae

1800 1600 1400 1200 1000 800 600 400 200 0 29 9 June (a)

19 29 July

8

18 28 August

29 9 June (b)

19 29 July

8

18 28 August

29 9 June

19 29 July

8

18 28 August

(c)

Fig. 1.1 The influence of habitat unit on relative population levels when these are measured in terms of population intensity: the populations of Myzus persicae on different varieties of potato. (a) Per 100 leaves; (b) per plant; (c) per plant, corrected for proportions of upper, middle, and lower leaves (after Broadbent, 1948).

litter) will give a different seasonal picture from an absolute estimate per square metre (Gabbutt, 1958). These examples also underline the importance of absolute estimates when interest lies primarily in the animal population. 1.1.1.3 Basic population In some habitats, especially forests and orchards, it is often convenient to have an intermediate unit between that used for measuring intensity and absolute measures of ground area, e.g. 1 m2 of branch surface (Morris, 1955) or branches of apple trees (Lord, 1968). 1.1.2 Relative estimates These estimates, in which the number caught cannot be expressed as a density or intensity per area or habitat unit, only allow comparisons in space or time. They are especially useful in extensive work on species distributions, monitoring changes in species richness, environmental assessments, recording patterns of animal activity, and the investigation of the constitution of a polymorphic population. The methods employed are either the catch per unit effort type or various forms of trapping, in which the number of individuals caught depends on a number of factors besides population density (Chapter 7). There is no hard and fast line between relative and absolute methods, for absolute methods of sampling are seldom 100% efficient and relative methods can sometimes be corrected in various ways to give density estimates.

4

CHAPTER 1

Relative methods are important in applied areas, such as fisheries or game management, where most of the information available may be derived from fishing or hunting returns. In fisheries research, catch per unit effort is often difficult to calculate from landing statistics because of changes in catch efficiency as fishing technology and the economy changes. 1.1.3 Population indices The animals themselves are not counted, but their products (e.g. frass, webs, exuviae, tubes, nests) or effects (especially plant damage) are recorded. Both population indices and relative estimates of population can sometimes be related to absolute population (if this is measured at the same time) by regression analysis. If such a study has been based on sufficient data, subsequent estimates from relative methods or indices can be converted to absolute terms, using various correction factors; such an approach is common in fisheries research (e.g. Beverton and Holt, 1957).

1.2 Errors and confidence The statistical errors of various estimates can usually be calculated and are referred to as the fiducial limits (the estimate (x) being expressed as x ± y, where y = fiducial limits). These are sometimes incorrectly referred to as ‘confidence’ limits, but the distinction between the two terms is in practice unimportant. The fiducial limits are calculated for a given probability level, normally 0.05, which means that there are only five chances in 100 that the range given by the fiducial limits does not include the true value (hence the expressions 5% probability level and 95% fiducial limits). If more samples are taken, the limits will be narrower, but the estimate may not move closer to the actual value. The biologist is often worried that some of the assumptions about sampling efficiency (e.g. does the ‘knock-down’ method really collect all the weevils on a tree?) may be incorrect. Intuitively, most believe, quite correctly, that this estimate should be compared with another method that has different assumptions. If the estimates are of the same order of magnitude, then the investigator can have much greater confidence that the result of his/her study is not misleading. It is therefore sound practice for the ecologist to contemporaneously estimate the population or other variables by more than one method. When two estimates have been obtained from different sampling procedures, provided they are internally consistent (e.g. a t-test shows that the means are not significantly different), they may be combined to give a weighted mean, weighting each estimate inversely to its variance (Cochran, 1954). Under some circumstances, Bayes’ theorem could be used (see p. 84) to give the combined estimate. Laughlin (1976) has suggested that the ecologist may be satisfied with a higher probability level (say 0.2) and thus narrower fiducial limits for estimates based on more than one method, because such estimates have a qualitative, biological assurance,

INTRODUCTION

5

in additional to that from the consistency of the data, that the true mean lies close to the estimate. Most population studies are based on sampling and the values obtained are considered to have a generality that scales with the area from which the samples were drawn. All estimates have fiducial limits and the level of accuracy that should be aimed at is difficult to determine. Morris (1960) has aptly said that ‘we are not likely to learn what precision is required by pessimistic contemplation of individual fiducial limits’. Excessive concerns about accuracy can always be used by those who favour the warmth of the hearth to being in the field. As the amount of time and labour that can be put into a problem is limited, it should always be borne in mind that the law of diminishing returns applies to the reduction of the statistical errors of sampling. In the long run, more knowledge of the ecology of the animal may be gained by studying other areas, by making other estimates, or by taking further samples than by straining for a very high level of accuracy in each operation. Against this must be set the fact that, when animals are being extracted from samples, the errors all lie below the true value, as animals will occasionally be missed. A number of very carefully conducted control samples may allow a correction factor to be applied, but the percentage of animals missed may vary with density; sometimes more are overlooked at the lowest densities (Morris, 1955). An alternative to sampling is the continuous, or regularly repeated, study of a restricted cohort, e.g. the population of an aphid on a particular leaf or leaf-miners on a bough. Such studies have a very high level of accuracy, but they sacrifice generality. A combination of some cohort studies with larger-scale sampling often provides valuable insights. References Beverton, R.J.H. & Holt, S.J. (1957) On the Dynamics of Exploited Fish Populations. Ministry of Agriculture, Fisheries and Food, HMSO, London. Bradley, R.H.E. (1952) Methods of recording aphid (Homoptera: Aphidae) populations on potatoes and the distribution of species on the plant. Can. Entomol. 84, 93–102. Broadbent, L. (1948) Methods of recording aphid populations for use in research on potato virus diseases. Ann. Appl. Biol. 35, 551–66. Cochran, W.G. (1954) The combination of estimates from different experiments. Biometrics 10, 101–29. Gabbutt, P.D. (1958). The seasonal abundance of some arthropods collected from oak leaf litter in S.E. Devon Proc. X Int. Congr. Entomol. 2, 717. Knight, J.D., Tatchell, G.M., Norton, G.A., &

Harrington, R. (1992) FLYPAST: an information management system for the Rothamsted Aphid Database to aid pest control research and advice. Crop Prot. 11(5), 419–26. Laughlin, R. (1976) Counting the flowers in the forest: combining two population estimates. Aust. J. Ecol. 1, 97–101. Lord, F.T. (1968) An appraisal of methods of sampling apple trees and results of some tests using a sampling unit common to insect predators and their prey. Can. Entomol. 100, 23–33. Macdonald, D.W., Mace, G., & Rushton, S. (1998) Proposals for Future Monitoring of British Mammals. Department of the Environment, London, 374 pp. Morris, R.F. (1955) The development of sampling techniques for forest insect defoliators, with particular reference to the spruce budworm. Can. J. Zool. 33, 225–94.

6

CHAPTER 1 Morris, R.F. (1960) Sampling insect populations. Ann. Rev. Entomol. 5, 243–64. Pimentel, D. (1961) The influence of plant spatial patterns on insect populations. Ann. Entomol. Soc. Am. 54, 61–9. Sibley, R.M. & Smith, R.H. (1998) Identifying

key factors using lambda contribution analysis. J. Anim. Ecol. 67, 17–24. Varley, G.C. & Gradwell, G.R. (1963) The interpretation of insect population changes. Proc. Ceylon Assoc. Adv. Sci. 18(D), 142–56.

2

The Sampling Programme and the Measurement and Description of Dispersion

2.1 Preliminary sampling 2.1.1 Planning and fieldwork At the outset, the worker must be quite clear as to the problem he/she is proposing to investigate (Morris, 1960). As it is normally impossible to count and identify all the animals in a habitat, it is necessary to estimate the population by sampling. Naturally, these estimates should have the highest accuracy commensurate with the effort expended. This requires a plan that includes a sampling programme laying down the number of samples, their distribution, and their size. While the statistical principles for sampling are given in texts such as Hansen et al. (1953), Yates (1953), Stuart (1962), Cochran (1977), Elliott (1977), Green (1979), Seber (1982), Scheiner and Gurevitch (1993), and Underwood (1997), there is no universal sampling method. The importance of careful formulation of the hypothesis for test cannot be overstressed. In community studies, preliminary work should consider species richness and potential problems with species identification. The appropriate degree of taxonomic discrimination must be decided, as it is important to maintain a consistent taxonomy. Sample sorting and species identification are often the most labour-intensive parts of a study and it may be useful to process a trial sample to assess the effort required. Saila et al. (1976) found that a single marine grab sample cost $25 to collect and $300 to sort. The planning of the timing of sampling requires knowledge of life cycles. Preliminary work will be necessary to gain some knowledge of the distribution of the animal(s). From these observations, an estimated cost per unit sample in terms of time, resources and money can be made. The first decision concerns the scale of the universe to be sampled. Whether this is to be a single habitat (e.g. field, woodland, or pond) or representatives of the habitat type from a wide geographical area will depend on whether an intensive or an extensive study is planned. The correct definition of the target population or community is essential: if too small, it may not produce results representative of the structure as a whole; if too large, it will waste resources. A good example of this decision process for plankton is given in Omori and Ikeda (1984). The second decision must be to define the accuracy or precision of the popula7

8

CHAPTER 2

tion estimates required. Accuracy measures how close an estimate is to the real value and precision measures reproducibility of the estimate. These decisions must be taken by considering both the objectives of the study and the variability of the system under study. For example, many species of insect pest exhibit a 10-fold or even 100-fold population change within a single season and recruitment in fish stocks can vary 10-fold between years (Cushing, 1992). For such species, an estimate of population density with a standard error of about 25% of the mean, enabling the detection of a doubling or halving of the population, is sufficiently accurate for damage control or stock assessment studies (Church & Strickland, 1954). For lifetable studies, more especially on natural populations, a higher level of accuracy, frequently set at 10%, will be necessary. Conservationists often seek to build species inventories. Here completeness will replace accuracy as a measure of quality. While for large mammals or birds the aim might be to record all resident species, for highdiversity groups such as beetles the objective may be set at only 5–10%. Community studies often aim to generate summarizing statistics, such as measures of diversity or species richness (Chapter 13), which can be used to compare localities or changes through time. The accuracy and precision of these estimates must be carefully considered if changes are to be detected at the desired resolution. In extensive work the amount of sampling in a particular locality will be limited and therefore a further decision concerning both the age-group to be studied and the timing of sampling will need to be made. For insects, the best stage for sampling may be that most closely correlated with the amount of damage (Burrage & Gyrisco, 1954), or, if the purpose of the survey is to assess the necessity for control, the timing should be such that it will give advanced information of an outbreak (Gonzalez, 1970). For aquatic organisms, timing is often determined by the reproductive cycle. For example, in temperate waters benthic surveys carried out in autumn will show a population dominated by recent recruits. The same survey carried out in the spring will show the resident community which has survived both competition for space and the rigours of the winter (Bamber & Spencer, 1984; Bamber, 1993). While the life-history stage to be sampled will depend on the objectives, the stage must not be one whose numbers change greatly with time and must be present for a period sufficient to allow the survey to be completed. Further, the easier the stage is to sample and count, the better. A problem with extensive surveys can be that different areas hold populations that differ in their position in the reproductive cycle. The reliability of samples in an extensive survey may be particularly sensitive to current weather conditions (Harris et al., 1972). The influence of latitudinal gradients on the timing of sampling can be important. In extensive surveys of marine plankton, the timing of blooms or larval production can vary by 1 month over 1° of latitude. Although the preliminary sampling and the analysis of the assembled data will provide a measure of many of the variables, the actual decisions must still, in many cases, be a matter of judgement. Furthermore, as the density changes, so will many of the statistical parameters, and a method that is suitable at a higher density may be found inadequate if the population level drops. Shaw (1955) found that Thomas and

THE SAMPLING PROGRAMME

9

Jacob’s (1949) recommendation for sampling potato aphids — one upper, middle, and lower leaf from each of 50 plants — was unsatisfactory in Scotland, in certain years, because of the lower densities. Detailed and now classic accounts of the development of sampling programmes for various insects are given by Morris (1955, 1960), LeRoux and Reimer (1959), Harcourt (1961a, 1964), Lyons (1964), and Coulson et al. (1975, 1976), amongst others. Terrestrial sampling programmes for vertebrates, small mammals and birds are described by Caughley (1977), Smith et al. (1975), and Blondel and Frochet (1987), respectively. For aquatic systems, estuarine and coastal sampling programmes are described for many groups in Baker and Wolff (1987), sampling marine benthos by McIntyre et al. (1984) and zooplankton by Omori and Ikeda (1984). The next step is to subdivide for sampling purposes. These subdivisions may be orientated with respect to an environmental gradient. If none is apparent, it is often convenient to divide the area into regions of regular shape, but this may be neither possible nor desirable. Ideally, the within-subdivision variability should be much less than the between-subdivision variability in habitat. The habitat must be considered from the biological angle and a decision made as to whether it might need further division. If it is woodland, for example, the various levels of the tree — upper, middle, and lower canopy and probably the tips and bases of the branches — would, on a priori grounds, be considered as potentially different divisions. The aspect of the tree might also be important. In herbage or grassland, if leaves or another small sampling unit is being taken, the upper and lower parts of the plants should be treated separately. Similarly, aquatic samples may be taken from different depths, from different positions with respect to flow, and from different plants or substrates. It is often of value to take at least two different-sized sampling units (Waters & Hensen, 1959). In benthic surveys, a 0.1 m2 grab sample suitable for sampling macroinvertebrates can itself be core-sampled for the meiofauna prior to processing (McIntyre & Warwick, 1984). In an insect study, one size of sample should be towards the smallest possible limit, e.g. a leaf blade or half a leaf. As a general principle, a higher level of reproducibility is obtained (for the same cost) by taking more smaller units than by taking fewer large ones.1 Small sampling units may also enable precision to be increased by distinguishing between favourable and unfavourable microhabitats; Condrashoff (1964) found with a leaf miner that the upper and lower leaf surfaces should be considered as separate units. Two examples of any one sizesampling unit should be taken within each sampling plot or subsection. For example, for a field crop a preliminary plan could be as in Fig. 2.1, which gives a total of 160 samples. In community studies, where it is essential to gain some idea of species diversity 1

The only disadvantage of sampling by small units is the number of zeros that may result at low densities; this truncation may make analysis difficult and has led to the suggestion that larger-sized samples should be taken (Pradhan & Menon, 1945; Spiller, 1952). The decision must be related to the density of the animal, although moderate truncation can be overcome by suitable transformation; in other cases, it may be necessary to increase the size of the unit.

10

CHAPTER 2

1

/ 2 upper leaf/plant

2 plants/plot 1

/ 2 lower leaf/plant

Field–20 plots 1 upper shoot/plant 2 plants/plot 1 lower shoot/plant Fig. 2.1 Example sampling plan.

and the species–area relationship, a number of sampling methods may be needed. In benthic sampling, for example, a pilot study might use dredges to obtain a general idea of animal presence and distribution and a limited number of grab samples to estimate densities. During a pilot study, a record should be kept of the cost of each part of the sampling routine, normally expressed in man-hours. 2.1.2 Statistical aspects Before the data gathered in the preliminary samples can be analysed, some aspects of statistics need to be considered. 2.1.2.1 The normal distribution and transformations The most important and commonly used of the theoretical distributions is the normal or Gaussian distribution. This has a probability curve that is a symmetrical bell shape. The probability density of a normal variable P(x) is: P ( x)

È 1 Ê x - m ˆ 2˘ 1 exp Í˙ 2ps Î 2Ë s ¯ ˚

(2.1)

where m is the mean and s the standard deviation. This distribution is symmetrical about the mean and the shape is determined by the standard deviation. It is important to ecologists because most of the common statistical tests assume the data to be normally distributed. While many distributions obtained from observation: e.g. the heights of humans — are approximately normal, the spatial distribution of individuals is seldom, if ever, normal. The distribution of a field population could approach normality only if the individuals were randomly scattered and the population was so dense or the size of sampling unit so large that considerable numbers were present in each sample. Therefore, in contrast to other distributions described below, the normal distribution is not of interest to ecologists as a means of describing dispersion. Its importance arises solely from the fact that, for most statistical methods to be applied to a set of data, the frequency distribution

THE SAMPLING PROGRAMME

11

must be normal. The data should possess the associated properties that the variance is independent of the mean (or, more strictly, the variance is homogeneous, all errors coming from the same frequency curve) and its components are additive. Although analysis of variance is quite robust to deviations from normality — in some respects, more so than the c2 test (Reimer, 1959; Abrahamsen & Strand, 1970) — data whose frequency distribution is considerably skewed and with the variance closely related to the mean cannot be analysed without the risk of errors (Beall, 1942). Ecological data are generally skewed. Such distributions are usually transformed by taking logarithms or square roots. For example, if the square-root transformation were applied to 9, 16, and 64, they would become 3, 4, and 8, and it will be observed that this tends to reduce the spread of the larger values. The interval between the second and third observations (16 and 64) is on the first scale nearly seven times that between the first and second observations; when transformed, the interval between the second and third observations is only four times that between the first and second. It is thus easy to visualize that a transformation of this type would tend to ‘push’ the long tail of a skew distribution in, so that the curve becomes more symmetrically bell-shaped. If all the observations (x) are positive and no a priori reason exists to choose between the many types of transformation possible, the Box–Cox empirical method can be used (Box & Cox, 1964). The general transformation function from x to x¢ is: x¢ =

( x g - 1) g

x¢ = ln x,

, g π0

g=0

(2.2)

The best transformation to normality is the value of exponent, g, which maximizes the log-likelihood (L) equation: L=

n g ln s x2¢ + (g - 1) Â ln x 2 n

(2.3)

where sx¢2 is the variance of the transformed distribution and n the degrees of freedom (number of observations - 1). Examination of the general transformation function (2.2) will show that certain values of g correspond to commonly used transformations; these are listed in Table 2.1. If the calculated value for g falls close to that corresponding to a common transformation, then this transformation should be used. Thus, if g = 0.1, a logarithmic transformation might be considered appropriate. It must be stressed, however, that transformation does lead to difficulties, particularly in the consideration of the mean and other estimates (see below). It should not be undertaken routinely, but only when the conditions for statistical tests are grossly violated (LeRoux & Reimer, 1959; Finney, 1973). The distribution of individuals in natural populations is such that the variance is

12

CHAPTER 2

Table 2.1 Values of g or 1 - 1/2b obtained by fitting either the Box–Cox transformation function or Taylor’s power law and their corresponding transformations.

g or 11 0.5 0 -0.5 -1

1 2b

Transformation linear y = ax + b – square root y = √x logarithmic reciprocal square root reciprocal y = 1/x

not independent of the mean. If the mean and variance of a series of samples are plotted, they tend to increase. This relationship has been shown by Taylor (1961, 1965) and Taylor et al. (1976, 1978) to fit a power law:

_

s 2 = ax b

(2.4)

where a and b are constants; a is largely a sampling factor, while b appears to be an index of aggregation characteristic of the species. This relationship holds for a variety of distributions, from regular through _ random to highly aggregated. The same relationship between the mean x and the variance s2 was discovered independently by Fracker and Brischle (1944) for quadrat counts of the currant, Ribes, and by Hayman and Lowe (1961) for counts of the cabbage aphid, Brevicoryne brassicae. The applicability of Taylor’s power law is discussed by Rutledge and Swartz (1991). They held that it had no unique advantage over a quadratic of the form:

_

_

s 2 = a1x + a2x 2

(2.5)

and noted that Taylor’s power law was particularly poor in predicting the variance for means close to zero. If the goal is to minimize the squared errors in predicting the mean, then better methods are sometimes available. For example, if replicate counts are taken from a Poisson-distributed population, but sampling error allows the mean to vary from count to count, then it can be shown that the variance–mean relationship is:

s 2 = m + c2m2

(2.6)

where c is the coefficient of variation of the Poisson mean. Taylor et al. (1978) had suggested a split domain model: s2

=

am b,

a

1 1- b

<m

and s2

= m, m £

1 1 a b

(2.7)

THE SAMPLING PROGRAMME

13

which would address the poor fit at low mean densities. However, the application of this model can be time-consuming and needs care (Perry & Woiwod, 1992). These authors compared a number of models, including the generalized form: s 2 = m + em f, e ≥ 0

(2.8)

They concluded that no approach was outstandingly superior, but noted that, with the development of generalized linear models, in which all additive effects and the relationship between mean and variance were specified separately, the need for variance-stabilizing transformations has been made unnecessary. It seems that Taylor’s power law, used with care, offers an approach to finding a general transformation which makes the variance independent of the mean. To do this, the series of means and variances necessary to calculate a and b may be obtained from several sets of samples from different areas, from sets of samples of different sizes, or by combining samples to form different-sized sampling units. The mean and the variance are calculated from the raw data by the usual methods. The constants a and b in equation (2.4) can then be found by non-linear regression, using computer statistics packages, such as Genstat or SAS. An approximate value of b can be found using linear regression to solve: log s 2 = log a + b log x

(2.9)

Having determined b each variable, x, is transformed to z using:

z=x

1-

1 2b

(2.10)

An evaluation of this transformation and the determination of its parameters is given by Taylor et al. (1978). As in the case of the Box–Cox method, an appropriate transformation for different values of 1 - 1/2b is given in Table 2.1. Healey and Taylor (1962) give tables for 1 - 1/2b = 0.2, 0.4, 0.6, 0.8 and for the negative powers. Downing (1979), following an examination of benthic data, noted that b was frequently close to 1.5 and proposed a fourth-root transformation. This proposal was criticized by Taylor (1980). These are precise transformations, but in practice, where sampling and other errors are fairly large, it will usually be found adequate to transform the data from a regular population by using squares, for a slightly contagious (aggregated) one by using square roots and for a distinctly contagious one by a logarithmic transformation. In order to overcome difficulties with zero counts in log transformations, a constant (normally 1) is generally added to the original count (x); this is expressed as ‘log(x + 1)’. Anscombe (1948) has suggested that a better transformation would be obtained by taking log(x + k/2), where k is the dispersion parameter of the negative binomial (see below). As k is frequently in the region of 2, in many cases this refine-

14

CHAPTER 2

ment would have little effect. Andersen (1965) has shown that, if the mean and k are very small (less than 3 and approaching zero, respectively), then the variance will not be stabilized by k1/2 or any of the common transformations. However, if independent samples are pooled or the size of the sampling unit increased, the data may be satisfactorily transformed. It is customary to transform percentages to angles (arc sin), but Reimer (1959) has suggested this is only of value when the probability of finding the individual bearing the attribute (e.g. the leaf having a gall) is uniform within each area (for which a % infestation has been calculated) but varies considerably between the different classes (whose various % infestations one wishes to analyse). Even in such cases, it is necessary only when several of the % points lie outside the 20–80 range. When the various percentages are based on grossly unequal numbers of individuals, they will need to be weighted before the analysis of variance can be applied (Reimer, 1959). The use of transformations can lead to problems and, as stressed above, transformation should not be routinely undertaken. If the fiducial limits are calculated from the transformed mean, this may be erroneous (Abrahamsen & Strand, 1970). The biological interpretation of estimates based on transformed data is often difficult and may lead to insurmountable problems in the construction of life budgets. There is indeed much to commend the use of the arithmetic mean (i.e. that based on the untransformed data) in population studies (van Emden et al., 1961; Lyons, 1964) and, if the distribution of the animal is random (see below), the fiducial limits are available in tables (Pearson & Hartley, 1958). If data have been transformed, the means of the untransformed and transformed values should be provided; backtransforms of, say, geometric means from logarithmic transformations are more difficult to interpret and contain biases unless the variances are small (Finney, 1973). Beauchamp and Olson (1973) suggest that the bias will be corrected by using the log-normal distribution for detransforming. Checking the adequacy of the transformation. An adequate transformation should eliminate or considerably reduce two attributes of the data that are easily tested for. These are as follows. 1 Excess skewness of the frequency curve causing large deviations from normality, which can be detected either from a probit plot or by a statistical test (Snedecor & Cochran, 1989). 2 The dependence of the variance on the mean (instability or non-homogeneity of the variance), which may be shown graphically (Fig. 2.2). Generally, as Hayman and Lowe (1961) have pointed out, ‘as non-normality must be extreme to invalidate the analysis of variance it is better to concentrate on stabilising the variance of the samples’. A correct transformation for this property will also ensure the third property necessary for the analysis of variance: the additivity of the variance (Bliss & Owen, 1958). Indeed, all three properties are related and, for practical purposes, the distinction between transformation for normality and

THE SAMPLING PROGRAMME

15

Same habitat, but different sized means obtained by bulking samples 2.0 +

+

+

+

100

1.5 +

Other habitats or other years

Variance (s 2)

Variance (s 2)

+

+

1000

1.0 +

10

+

0.5 (a) 0 (a)

1.0

10

0

100

Mean ( –x)

(b)

1.0

3.0 Mean ( –x)

5.0

7.0

Fig. 2.2 (a) The plot of variance against the mean on a log/log scale to obtain the constant a of Taylor’s power law, data from samples of olive scales, Parlatoria oleae, per twig. (b) The same data transformed to x0.4, showing the relative independence of the variance from the mean.

that for stabilizing the variance need not be emphasized. The adequacy of a transformation in stabilizing the variance may be tested for graphically (Fig. 2.2b) or by calculating the correlation coefficient for the two terms (Harcourt, 1961b, 1965). Alternatively, more precise methods may be used, as described by Forsythe and Gyrisco (1961), who used a non-parametric rank-correlation coefficient to test for non-normality. When normality has been established, the heterogeneity and nonadditivity of the variance can be determined, using the tests of Hartley (1950) and Tukey (1949), respectively.

2.2 The sampling programme 2.2.1 The number of samples per habitat unit (e.g. plant, host, or puddle) There are two aspects to consider: firstly, whether different regions of the unit need to be sampled separately and, secondly, the number of samples within each unit or subunit (if these are necessary) that should be taken for maximum efficiency. Although the habitat unit could, for example, be the fleece of a sheep, a bag of grain, a rock in a stream, or an area of seabed, for convenience, the word plant will, in general, be used in its place in the discussion below. 2.2.1.1 Subdivision of the habitat If the distribution of the population throughout the habitat is biased towards certain subdivisions but the samples are taken randomly, what LeRoux and Reimer (1959) aptly term ‘systematic errors’ will arise. This can be overcome either by sampling so that the differential number of samples from each subdivision reproduces in the

16

CHAPTER 2

samples the gradient in the habitat or by regarding each part separately and subsequently correcting. The question of the estimation of the area or volume of the plant is discussed in Chapter 4. The amount of subdivision of the plant that various workers have found necessary varies greatly. On apple, the eggs, larvae, and pupae of the tortricid moth, Archips argyrospilus, were found for most of the year to be randomly distributed over the tree, so that only one level (the lower for ease) needed to be sampled (Paradis & LeRoux, 1962). In contrast, on the same trees and in the same years, the immature stages of two other moths showed marked differences between levels at all seasons (LeRoux & Reimer, 1959). With the spruce budworm, Choristoneura fumiferana, Morris (1955) found that there were ‘substantial and significant differences from one crown level to another’ and that there was a tendency for eggs and larvae to be more abundant at the top levels, but there was no significant difference associated with different sides of the same tree. A similar variation with height was found with the eggs of the larch sawfly, Pristiphora erichsonii, although here it was concluded that, in view of the cost and mechanical difficulties of stratified sampling at different heights, a reasonable index of the population would be obtained by sampling the mid-crown only (Ives, 1955). Studying all the organisms on aspen, Populus tremuloides, Henson (1954) found it necessary to sample at three different levels of the crown. Even with field crops, height often needs to be considered: Broadbent (1948) recommended that potato aphids be estimated by picking three leaves — lower, middle, and upper — from each plant and, when estimating the population of the European corn-borer, Ostrinia nubilalis, Hudson and LeRoux (1961) showed that the lower and upper halves of the maize stem needed to be considered separately, the former containing the majority of the larvae. The age of the plant may also affect dispersion (Hirata, 1962; Peng & Brewer, 1994). In contrast, Teulon and Cameron (1995) found few significant differences in the distribution of thrips eggs and adults at different heights in sugar-maple trees. Aspect is sometimes important; in Nova Scotia in the early part of the season, the codling moth lays mostly on the south-east of the apple trees, but later this bias disappears (MacLellan, 1962). Aspect has also been found to influence the distribution on citrus of the long-tailed mealy bug, Pseudococcus adonidum (Browning, 1959) and the eggs of the oak leaf-roller moth, Archips (Ellenberger & Cameron, 1977) and of three species of mite, each of which was most prevalent on a different side (Dean, 1959), but not that of the pine beetle, Dendroctonus (Dudley, 1971). Variations in the spatial distribution of similar species in the same habitat, which complicates a sampling programme designed to record both, has also been recorded for two potato aphids, by Helson (1958). Some insects are distributed without bias on either side of the mid-vein of leaves, so that they may be conveniently subsampled. Similarly, Nelson et al. (1957) record that, when estimating populations of sheep keds, the fleece of only one side need be sampled. Aspect can also be important in aquatic systems. For example, in the northern hemisphere north-facing rocks in the intertidal zone offer the coolest, most shady habitat available, and thus tend to favour red algae and its associated fauna (Baker & Crothers, 1987). Conversely, in Caithness, at

THE SAMPLING PROGRAMME

17

the northern limit of its range, the barnacle Chthamalus is more abundant on southfacing slopes. In aquatic habitats, orientation with respect to flow and the degree of exposure are important (Baker & Crothers, 1987). Intertidal communities were classified in terms of their exposure to wave action by Ballentine (1961). Plankton sampling may be undertaken at set depths. However, when a general measure of abundance throughout the water column is required a common practice is to collect what are termed double-oblique plankton-net hauls by lowering and raising the net while fishing (Omori & Ikeda, 1984). For example, Henderson et al. (1984) used samples collected in this way to estimate the growth and mortality of larval herring, Clupea harengus, which were known to be unevenly distributed with respect to depth (Henderson, 1987). Occasionally, sampling may be restricted to one part of a plant upon which a large and constant proportion of the population of the animal lives. Wilson (1959) showed that, in Minnesota, 84% of the eggs of the spruce budworm, C. fumiferana, are laid on the tips of the branches and, if sampling is confined to these rather than entire branches, sampling time may be reduced by up to 40%. A hierarchical design is one in which, for example, a number of plants are sampled from each of a number of plots from within a number of fields. If a certain number of samples are collected randomly within each strata this is often termed nested random sampling and is analysed by a nested analysis of variance. There may be two or three hierarchical levels, rarely more. Examples of such sampling programmes are in Bancroft and Brindley (1958), Harcourt (1961a), Buntin and Pedigo (1981), and Steffey et al. (1982). 2.2.1.2 The number of samples per subdivision To determine the optimum number of samples per plant (or part of it) (n), the variance of within-plant samples (s2s ) must be compared with the variance of the between-plant samples (sp2) and set against the cost of sampling within the same plant (cs) or of moving to another plant and sampling within it (cp): ns =

s 2s c p ¥ s 2p c s

(2.11)

If the interplant variance, s2p, is the major source of variance and unless the cost of moving from plant to plant is very high, n will be of the order of 1 or less (which means 1 in practice). Interplant variance has been found to be much greater than within-plant variance in many insect populations, e.g. the spruce sawfly, Diprion hercyniae (Prebble, 1943), the lodgepole needle miner, Recurvaria starki (Stark, 1952), the cabbage aphid, B. brassicae (Church & Strickland, 1954), the spruce budworm, C. fumiferana (Morris, 1955), the diamondback moth, Plutella maculipennis (Harcourt, 1961a), the cabbage butterfly, Pieris rapae (Harcourt, 1962), the western pine beetle, Dendroctonus brevicomis (Dudley, 1971), the pine chermid, Pineus pinifoliae (Ford & Dimond, 1973), and the spider mite, Panonychus ulmi (Herbert & Butler, 1973). In

18

CHAPTER 2

most of these examples, the within-plant variance was small, so that only one sample was taken per plant or per stratum of that plant, although, of course, when this is done, within- and between-tree variances cannot be separated. However, with Eoreuma loftini on sugar cane, within-plant variance, from stalk to stalk, was found to be high, as was field-to-field variance, the lowest variance being found between samples from within a field (Meagher et al., 1996). On apples, the within-tree variance of insect numbers may also be high, particularly at certain seasons, when as many as seven samples may be taken from a single tree (LeRoux & Reimer, 1959; LeRoux, 1961; Paradis & LeRoux, 1962). Often, a considerable saving in cost without loss of accuracy in the estimation of the population, but with loss of information on the sampling error, may be obtained by taking randomly a number of subsamples which are bulked before sorting and counting. This is especially true where the extraction process is complex, as with soil samples; Jepson and Southwood (1958) bulked four random, 76.25 mm (3 inch) row samples of young oat plants and soil to make a single 305 mm (1 ft) row sample, which was then washed and the eggs of the frit fly, Oscinella frit, extracted. Such a process gave a mean as accurate as that obtained by washing all the 76.25 mm samples separately (at greater cost). Paradis and LeRoux (1962) sampled the eggs of a tortricid moth, A. argyrospilus, on apple by bulking 25 cluster samples. 2.2.2 The sampling unit, its selection, size, and shape The criteria for the sample unit are broadly those of Morris (1955): 1 It must be such that all units of the universe have an equal chance of selection. 2 It must have stability (or, if not, its changes should be easily and continuously measured — as with the number of shoots in a cereal crop). 3 The proportion of the population using the sample unit as a habitat must remain constant. 4 The sampling unit must lend itself to conversion to unit areas. 5 The sampling unit must be easily delineated in the field. 6 The sampling unit should be of such a size as to provide a reasonable balance between the variance and the cost. 7 The sampling unit must not be too small in relation to the animal’s size, as this would increase edge-effect errors. 8 The sampling unit for mobile animals should approximate to the average ambit of an individual. This ‘condition’ is particularly significant in studies on dispersion involving contiguous sampling units, Lloyd (1967) has suggested that a test of the appropriate size would be provided by several series of counts of animals in contiguous quadrats conforming to a Poisson series. A sampling unit defined in relation to the animal’s ambit or territory (e.g. gallery of a bark-beetle) will give different information from one defined in terms of the habitat (e.g. Cole, 1970). This re-emphasizes the need to be very clear as to objectives and hypothesis before commencing a sampling programme. To compare various sampling units in respect of variance and cost, it is generally

THE SAMPLING PROGRAMME

19

convenient to keep one or other constant. The same method of sampling must, of course, be used throughout. From preliminary sampling, the variances of each of the different units (su2) can be calculated; these should then be computed to a common basis, which is often conveniently the size of the smallest unit. For example, if the smallest unit is 1 m of row, then the variance of a 2 m row unit will be divided by 2 and that of 4 m by 4. The costs will similarly be reduced to a common basis (Cu). Then the relative net cost for the same precision for each unit will be proportional to: Cusu2

(2.12)

where Cu = cost per unit on a common basis and su2 variance per unit on a common basis. Alternatively, the relative net precision of each will be proportional to the reciprocal of equation 2.12. The higher this reciprocal, the greater the precision for the same cost. A full treatment of the methods of selecting the optimum-size sampling unit is given in Cochran (1977) and other textbooks, but as population density, and hence variance, is always fluctuating, too much stress should not be placed on a precise determination of the optimum size of the sampling unit. An example of the determination of optimal sample size is Zehnder’s (1990) study of larval and adult Colorado potato beetle Leptinotarsa decemlineata (Say). He found that, while a five-stem sampling unit was most efficient, a three-stem unit would reduce costs with only a small reduction in sampling precision. Pieters (1977) compared the estimates of mean density and precision of arthropod samples collected from cotton using a D-Vac suction sample and found that mean estimates increased with reduced sample size while precision remained unchanged, leading to the conclusion that small sample sizes are better. Similarly, when sampling for benthic macrofauna, no advantage has been generally shown for 0.2 m2 over 0.1 m2 grab samplers (McIntyre et al., 1984). With insects on plants, the nature of the plant usually restricts the possible sizes to, for example, half leaf, single leaf, or shoot (see p. 149). Because the sample size range initially chosen for examination is often arbitrary, it may not include the optimal sample size. The shape of the sampling unit when this is of the quadrat type, rather than a biological unit, is theoretically of importance because of the bias introduced by edge effects. These are minimal with circles, maximal with squares and rectangles and intermediate with hexagons (Seber, 1982), because they are proportional to the ratio of sample unit boundary length to sampling unit area. If the total habitat is to be divided into numbered sampling units (for random number selection), then circular units are impractical, because of the gaps, and it is doubtful if the reduction of error from the use of hexagons normally justifies the difficulties of layout. Clearly, the larger the sampling unit, the proportionally smaller the boundary edge effect. The size of the organism will also influence this effect: the larger it is in relation to the sample size, the greater the chance of an individual lying across a boundary. This problem has been investigated for subcortical insects, where the damage to the edge individuals by the punch and the curved nature of the sampled substrate pose special

20

CHAPTER 2

problems (Safranyik & Graham, 1971). In general, edge effects can be minimized by a convention (e.g. of the animals crossing the boundaries, only those on the top and left-hand boundaries are counted). 2.2.3 The number of samples Precision is measured by the scatter about the mean of the results obtained and is sometimes expressed as the coefficient of variation of the samples. This is different from the accuracy of a method, which measures how close the estimate is to the actual population. The total number of samples depends on the degree of precision required. This may be expressed either in terms of achieving a standard error of the mean within a predetermined magnitude or as a probability that the estimated mean is within a selected value of the mean (Karandinos, 1976; Ruesink, 1980). For many purposes, a standard error of 5% of the mean is satisfactory. Within a homogeneous habitat, the number of samples (n) required is given by: n=

2 Ê s ˆ Ë Ex ¯

(2.13)

_

where s = standard deviation, x = mean, and E = the predetermined standard error as a decimal of the mean (i.e. for a standard error ±5%, then E = 0.05). This expression compares the standard deviation (s) of the observations with the standard error (E) acceptable for the contrasts we need to make; it will be noted from this equation, that, in any given situation, the value of the standard error will change with the square root of the number of samples: thus a large increase in n is necessary to bring about a small improvement in s. Where sampling is necessary at two levels, e.g. a number of clusters per tree, the number of units (n) that need to be sampled at the higher level, e.g. trees (LeRoux & Reimer, 1959; Harcourt, 1961a), is given by:

nt =

Ê s 2s ˆ 2 Á ˜ + sp Ë ns ¯

( x + E )2

(2.14)

where nt = the number of samples within the habitat unit (calculated as above), s2 = variance within the habitat unit, sp2 = variance between the habitat unit (= interplant _ variance), x = mean per sample (calculated from the transformed data and given in this form), and E is as above. If the dispersion of the population is well described by either the Poisson or negative binomial distribution, the desired number of samples is given, respectively, by: N= and:

1 (E 2 x )

THE SAMPLING PROGRAMME

1 1 + n = x 2k E

21

(2.15)

where k is the dispersion parameter of the negative binomial (see p. 31). The second approach uses confidence limits, so that the estimate is within a selected distance from the mean with a given probability. The general formula then becomes: n=

2 Ê ts ˆ Ë Dx ¯

(2.16)

where t = ‘Student’s t’ of standard statistical tables: its value depends on the number of samples and approximates to 2 for more than 10 samples at the 5% level — and D = the predetermined half-width of the confidence limits for the estimation of the mean expressed as a decimal (e.g. ±10% = 0.1). It will be seen that normally this gives a similar estimate to equation 2.13, provided the values of E and D are adjusted to accord with their meanings. The procedure is, perforce, somewhat approximate, depending on the preliminary estimates of the mean and standard deviation, and the inclusion of t does perhaps give it a slightly bogus air of precision! Additionally, there is normally a biological aspect to consider: as population characteristics change with time, so will the optimal number of samples (e.g. Bryant, 1976; Kapatos et al., 1977). If the distribution of the animal can be described by a Poisson or negative binomial distribution or if Taylor’s power law applies, then s2 can be substituted by other expressions (Buntin, 1994). For example, the formula when Taylor’s power law applies is:

_

n = t 2ax b-2D-2

(2.17)

where parameters a and b are those in equation (2.4). Equation 2.17 should be used with caution, because, as discussed in section 2.1.2, Taylor’s power law may give a poor description of the variance–mean relationship at low densities; Riddle (1989) showed that equation (2.17) substantially underestimated the required sample size for low-density benthic animals. Shelton and Trumble (1991) concluded that equation 2.17 should only be used when b < 2.0. For convenience, Table 2.2 gives the number of samples for different levels of precision, mean density, and clumping, as calculated by Elliott and Drake (1981) using equation 2.17 with a = 1. Because the estimated number of samples required is the product of the tabulated value and the power law parameter, a, the value of which can itself change with sampling conditions and sample size (Taylor, 1984), the estimated sample number is sensitive to sample size, which must be chosen appropriately. Another type of sampling programme concerns the measurement of the frequency of occurrence of a particular organism or event — for example, the frequency of occurrence of galls on a leaf or of a certain genotype in the population (Oakland, 1950; Cornfield, 1951; Henson, 1954; Cochran, 1977). Before an estimate can be

22

CHAPTER 2 Table 2.2 Number (n) of sampling units per sample required for different levels of precision (95% confidence limits (CL) expressed as a percentage of the sample mean) for different values of the mean number caught per sample (x) and for different values of b (equation 2.4).The value of a in equation 2.17 is assumed to be 1.To find n, multiply the value of n found in the table by the actual value of a (after Elliott & Drake, 1981). Sample mean (x) Precision D%

0.5

1

5

10

20

50

100

1000

Index of aggregation (b)

10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100

800 200 50 24 14 10 566 140 35 17 11 8 400 100 26 13 8 6 283 70 19 10 6 4

400 100 26 13 8 6 400 100 26 13 8 6 400 100 26 13 8 6 400 100 26 13 8 6

80 22 7 3 1 1 178 45 13 7 5 2 400 100 26 13 8 6 894 224 56 26 14 9

40 12 4 1 1 1 126 32 10 5 3 2 400 100 26 13 8 6 1265 316 79 35 20 13

22 7 1 1 1 1 90 24 8 4 2 1 400 100 26 13 8 6 1789 447 112 50 28 18

10 3 1 1 1 1 56 16 6 2 1 1 400 100 26 13 8 6 2828 707 177 79 44 28

6 1 1 1 1 1 40 12 5 1 1 1 400 100 26 13 8 6 4000 1000 250 111 63 40

1 1 1 1 1 1 15 5 1 1 1 1 400 100 26 13 8 6 12 649 3162 791 351 198 126

1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2 2.5 2.5 2.5 2.5 2.5 2.5

For definition of D, see equation 2.16.

made of the total number of samples required, an approximate value of the probability of occurrence must be obtained. For example, if it is found in a preliminary survey that 25% of the leaves of oak trees bear galls, the probability is 0.25. The number of samples (N ) is given by: N=

t 2 p(1 - p) D2

(2.18)

where p = the probability of occurrence (i.e. 0.25 in the above example) and D is as defined for equation (2.16). If it is found that the leaves (or other units) are distributed differently in the different parts of the habitat, they should be sampled in proportion to the variances. For example, Henson (1954) found from an analysis of variance of the distribution of the leaf bunches of aspen that the level of the crown from which the leaves had been drawn caused a significant variation and, when this variance was portioned into

THE SAMPLING PROGRAMME

23

levels, the values were: lower 112 993, middle 68 012, upper 39 436. Therefore leaf bunches were sampled in the ratio of 3 : 2 : 1 from these three levels of the crown. Note that, if the Taylor’s power law constants a and b are substituted in equation _ (2.17), using the equalities p = a and 1 - p = x -b-2, we obtain equation (2.8). Worked examples of the calculation of sample number for a variety of sampling programmes are given by Greenwood (1996). 2.2.4 The pattern of sampling Again, it is important to consider the object of the programme carefully. If the aim is to obtain estimates of the mean density for use in, for example, life-tables, then it is desirable to minimize variance. But, if the dispersion (= distribution = pattern) of the animal is of prime interest, then there is no virtue in a small variance. In order to obtain an unbiased estimate of the population, the sampling data should be collected at random, that is, so that every sampling unit in the universe has an equal chance of selection. In the simplest form — the unrestricted random sample — the samples are selected by the use of random numbers from the whole area (universe) being studied (random number tables are in many statistical works or may be generated using a computer or, failing these sources, the last two figures in the columns of numbers in most telephone books provide a substitute). The position of the sample site is selected on the basis of two random numbers giving the distances along two coordinates; the point of intersection is taken as the centre or a specified corner of the sample. If the size of the sample is large compared with the total area, then the area should be divided into plots, which will be numbered and selected using a single random number (e.g. Lloyd, 1967). Such a method eliminates any personal choice by the worker, whose bias in selecting sampling sites may lead to large errors (Handford, 1956). However, a random-choice method is not an efficient way to minimize the variance, since the majority of the samples may turn out to come from one area of the field. The method of stratified random sampling is therefore to be preferred for most ecological work (Yates & Finney, 1942; Abrahamsen, 1969); here the area is divided up into a number of equal-sized subdivisions or strata and one sample is randomly selected from each stratum. Alternatively, if the strata are unequal in size, the number of units taken in each part is proportional to the size of the part; this is referred to as self-weighting (Wadley, 1952). Such an approach maximizes the accuracy of the estimate of the population, but an exact estimate of sampling error can only be obtained if additional samples are taken from one (or two) strata (Yates & Finney, 1942). The taking of one sample randomly and the other a fixed distance from it has been recommended by Hughes (1962) as a method of mapping aggregations. The fixed distance must be less than the diameter of the aggregations, which are assumed to be circular; the standard error cannot be calculated. However, the method has been found useful for soil and benthic faunas (Gardefors & Orrage, 1968). When it is apparent that there are systematic variations in the density of the

24

CHAPTER 2

study animal across the study area, stratified sampling should be used. In general, the strata are chosen to minimize within-stratum variance. Individual strata need not form continuous patches within the study area. When the habitat is stratified, biological knowledge can often be used to eliminate strata in which few animals would be found. Such a restricted universe will give a greater level of precision for the calculation of a mean than an unrestricted and completely random sample with a wide variance. Prebble (1943) found that, with a pine sawfly, satisfactory estimates of the pupae were only obtained if sampling was limited to the areas around the bases of the trees. The variance of completely random sampling throughout the whole forest was too great, as many areas were included that were unsuitable pupation sites (see also Stark & Dahlsten, 1961). The other approach is the systematic sample, taken at a fixed interval in space (or time). In general, such spatial data cannot be analysed statistically, but Milne (1959) has shown that, if the centric systematic area sample is analysed as if it were a random sample, the resulting statistics are ‘at least as good, if not rather better’ than those obtained from random sampling. The centric systematic sample is the one drawn from the exact centre of each area or stratum and its theoretical weakness is that it might coincide with some unsuspected systematic distribution pattern. As Milne points out, the biologist should, and probably would, always watch for any systematic pattern, either disclosing itself as the samples are recorded on the sampling plan or apparent from other knowledge. Such a sampling programme may be carried out more quickly than the random method and so has a distinct advantage from the aspect of cost (see also p. 19). Systematic sampling is often used in marine studies where the primary aim is to map distribution with respect to environmental gradients or suspected sources of pollution. In the absence of environmental gradients, marine benthic or plankton sampling is often undertaken at regularly distributed stations (McIntyre et al., 1984; Omori & Ikeda, 1984). This allows easier contour mapping of animal density, particularly if computers are used. An example of an unbiased systematic method is given by Anscombe (1948). All the units (e.g. leaves) are counted systematically (e.g. from top to bottom and each stem in turn); then, every time a certain number (say, 50) is reached, that unit is sampled and the numbering is commenced again from 1; only one allocation of a random number is needed and that is the number (say, somewhere between 1 and 20) allotted to the first unit. Biologists often use methods for random sampling that are less precise than the use of random numbers, such as throwing a stick or quadrant or the haphazard selection of sites. Such methods are not strictly random; their most serious objection is that they allow the intrusion of a personal bias and quite frequently marginal areas tend to be undersampled. It may be worthwhile doing an extensive trial comparing a simple haphazard method with a fully randomized or systematic one, especially if the cost of the latter is high when compared with the former. Spiller (1952) found that scale insects on citrus leaves could be satisfactorily sampled by walking round the tree, clockwise and

THE SAMPLING PROGRAMME

25

then anticlockwise, with the eyes shut and picking leaves haphazardly. For assessing the level of eggs of the red boll-worm, Diparopsis castanea, to determine the application of control measures, Tunstall and Matthews (1961) recommended two diagonal traverses across the field, counting the eggs at regular intervals. Bias may intrude due to causes other than personal selection by the worker. Grains of wheat that contain the older larvae or pupae of the grain weevil, Sitophilus granarius, are lighter than uninfested grains. The most widespread method of sampling is to spread the grains over the bottom of a glass dish and then scoop up samples of a certain volume; as the lighter infected grains tend to be at the top, this method can easily overestimate the population of these stages (Howe, 1963). In contrast, for the earlier larval instars before they have appreciably altered the weight of the grain, such a simple method gives reliable results (Howe, 1963); it was undoubtedly this difference that led to Krause and Pedersen (1960) stressing the need for samples to contain a relatively high proportion of the same stage if good replication was to be obtained. The distribution of parasitized animals may be such that they are more vulnerable to capture. For example, roach infested with Ligula intestinalis swim close to the surface. When choosing a sampling method, it is important to choose a method with a low and stable level of bias, for this cannot be reduced by increasing sample number (Dahlsten et al., 1990). 2.2.5 The timing of sampling In the absence of sampling constraints, the seasonal timing of sampling will be determined by the life cycle (e.g. Morris, 1955). Marine sampling is often constrained by climatic conditions, as fishing and bottom-sampling cannot be done in heavy seas. In extensive work, when only a single stage is being sampled, it is obviously most important that this operation should coincide with peak numbers (e.g. Edwards, 1962). This can sometimes be determined by phenological considerations (Unterstenhöfer, 1957), but the possibility of a control population in an outdoor cage (Harcourt, 1961a) to act as an indicator should be borne in mind. The faster the development rate, the more critical the timing. With intensive studies that are designed to provide a life-table, regular sampling will be needed throughout the season. It is not always realized that the time of day at which the samples are taken may also have a considerable effect. The diurnal rhythms of the insects may cause them to move from one part of the habitat to another, as Dempster (1957) found with the Moroccan locust, Dociostaurus maroccanus (Fig. 2.3). Many grassland insects move up and down the vegetation, not only in response to weather changes, but also at certain times of the day or night (p. 268), and during the day quite a proportion of active insects may be airborne (see the observations of Southwood et al. (1961) on the numbers of adults of the frit fly, O. frit, on oats). There is a marked periodicity of host-seeking behaviour in many bloodsucking invertebrates (e.g. Camin et al., 1971; Corbet & Smith, 1974). Diurnal changes in activity and distribution are common in aquatic organisms. Plankton may become

CHAPTER 2

50

(a)

(b)

40 Locusts occurence (%)

26

30

20

10

0

0%

50%

100%

0%

50%

100%

Vegetative cover Fig. 2.3 The variation in the distribution of adults of the Moroccan locust Dociostaurus maroccanus at different times of the day: (a) early morning before 9.30 hours; (b) morning, after 9.30 hours and after noon. The histograms show the relative numbers on bare ground and areas with moderate and dense vegetation (after Dempster, 1957).

concentrated in surface waters at night and fish that remain hidden by day become active and vulnerable to trapping at night. Aquatic insects may emerge at a particular time of day. The ecologist may find that some of his/her sampling problems can be overcome, or at least additional information gained, if the work is done at night, dusk or dawn, rather than during conventional working hours.

2.3 Data processing A sampling plan needs to consider the processes involved in data acquisition, organization, analysis, and presentation. Smooth and rapid progress along this chain is aided by the use of computers, but only if the data can be easily transferred between software. If different software products are used, then compatibility must be considered. The rapid improvement in the power of computers such as the IBM personal computer (PC) and the Apple Macintosh has allowed ecologists to undertake most of their computation on small, often portable, machines. These PCs can use powerful statistical and data manipulation packages, such as SAS, SPSS, Systat, or Genstat. For an ecologist, the statistics package of choice should handle non-linear models, time-series analysis, ordination techniques and a range of cluster analyses, as well as general statistical procedures. Both software and hardware capability needs to be considered for each stage of the study. During data acquisition it is important to assess the data storage and processing requirements. Data have only been acquired when they have been processed into a usable form. Data processing and input rates are often different. An example of a common problem is from a study of salmon movement through a fish pass (P.A. Henderson, unpublished). Over a 10-week period, about 1680 h of video film was

THE SAMPLING PROGRAMME

27

recorded of the water in the fish pass. It was originally planned to use an imageanalysis program to identify and count the fish on the video. However, at the end of the fieldwork, it was found that the image-analysis software worked at less than onetenth of real time, requiring >16 800 h of computer time to analyse the images. In practice, this meant that image analysis would only just be complete by the beginning of the next salmon migratory season. The only way to acquire the fish-count data within the required time was to employ three people for 6 weeks to view the video film. Many software products are available for data organization. Those most frequently used by ecologists are termed spreadsheets. Examples of these programs are EXCEL and LOTUS 1-2-3. When choosing the product to use, consider carefully its capability to hold, manipulate, and convert the data. Ecological data sets of animal abundance can often be organized within a three-dimensional array of species (or other taxa) by station (locality) by time. This is the data structure spreadsheets are designed to handle. However, spreadsheets often have a limit on the number of rows or columns that can be used. Ecological data sets from highdiversity communities may easily exceed the upper limit of the system. If a spreadsheet has the capability to hold the data sets, then it is often the preferred software for data organization and elementary analysis, for spreadsheets allow rapid production of summary statistics, tables, and graphs. Specialist database applications, such as ACCESS, DB4, or PARADOX, can also be used to hold and organize data. These programs are better than spreadsheets if the worker needs to relate together different types of data and undertake operations such as sorting the data into different groups. They are usually inferior to spreadsheets in their ability to undertake calculations and plot graphs. Before committing to a particular database or spreadsheet application, the ability to move data into other programs must be considered. The three most common requirements are to export data to statistical, graphical, or word-processing packages. Manufacturers generally claim this is easier than it is. Large statistical software packages often offer their own spreadsheet capability. There is an advantage in using a group of programs all written for the same operating system, such as Windows 95 or 98. A common problem is backward compatibility. Most new software is designed to use data exported from existing programs. Often, however, data cannot be exported from recent to older programs.

2.4 Jackknife and bootstrap techniques It is frequently difficult to estimate the accuracy of ecological indices because no equation exists for the variance of the statistic and the sampling distribution is unknown. Jackknife and bootstrap methods, which offer a non-parametric means of estimating parameters, are now widely used by ecologists to estimate the accuracy of population parameter estimates and indices for diversity or similarity. Both techniques are based on the repeated estimation of the parameters of interest on a subsample of the data, for which a computer is essential. Perhaps their most important

28

CHAPTER 2

application is for the estimation of error limits and confidence intervals for estimated parameters. The jackknife method uses less computational effort than a bootstrap analysis and is thus often quicker and easier to undertake. However, it cannot be used to estimate confidence intervals. The general scheme is as follows. 1 Consider a situation where a parameter is to be estimated from n samples, e.g. a species diversity index from 10 kicknet samples from a stream. 2 Use all n samples to calculate the parameter of interest, E, the diversity index. 3 Now remove the samples one at a time and recalculate the parameter of interest, Ei. 4 Calculate the pseudovalues for the parameter of interest: fi = nE - (n - 1)Ei, where f = pseudovalue for jackknife estimate i, n = the total number of replicates, E = the estimate for all n samples, Ei = the estimate with sample i removed. 5 Estimate the mean and standard error of the parameter of interest from the n pseudovalues. The jackknife estimate of bias is E minus the mean of the pseudovalues, and the estimate of the standard error of the sample is simply the standard error of the pseudovalues: SE

=

 (fi - f )2

SE

=

 (fi - f )2

n(n - 1) _ where f is the mean of the pseudovalues. The bootstrap method differs from the jackknife in the means of sampling the original data set. The subsets of observations used to repeatedly calculate the parameter of interest are each selected at random, with replacement from the original data set of n observations and with each observation only included once. This procedure is repeated, often hundreds of times, so that many combinations of the original observations are generated. For example, if a sample comprised 10 species and 100 individuals, then each subset of data would be a randomly selected 100 individuals taken from a population that represented all the species according to their original frequency of observation. Each of these samples is then used to generate estimates of the parameter of interest (e.g. a diversity index), as for the jackknife method. The bootstrap estimate of the bias is simply found by subtracting the observed parameter value from the mean parameter obtained from the bootstrap replicates. The estimate of the standard error of the sample is simply the standard error of the replicate parameter estimates:

b -1 _ where f is the mean of the replicates and b is the number of bootstrap samples taken. A number of methods can be used to estimate confidence intervals. The percentile bootstrap is the simplest and most commonly used. The 2.5 and 97.5 percentiles of the bootstrap distribution of the parameter are used as the 95% confidence intervals. For example, if 1000 bootstrap replicates have been taken, then these are arranged in

THE SAMPLING PROGRAMME

29

order of magnitude and the 2.5 percentile calculated as the average of the 25th and 26th smallest values and the 97.5 percentile as the average of the 975th and 976th smallest values. Bootstrap and jackknife estimates are used to estimate species richness (Chapter 13), population size (Chapter 3), and population growth rates (Chapter 12). Introductions to bootstrap and jackknife methods are given in Efron and Tibshirani (1986), Potvin and Roff (1993), and Scheiner and Gurevitch (1993).

2.5 Dispersion The dispersion of a population — the description of the pattern of the distribution or disposition of the animals in space — is of considerable ecological significance. Not only does it affect the sampling programme and the method of analysis of the data, but it may be used to give a measure of population size (nearest-neighbour and related techniques) and, in its own right, is a description of the condition of the population. Changes in the dispersion pattern should be considered alongside changes in size when interpreting population dynamics. For example, if a mortality factor reduces the clumping of a sessile organism, this is an indication that it acts most severely on the highest densities, or, if the dispersion of a population becomes more regular, then intensification of competition should be suspected (Iwao, 1970a). An understanding of dispersion is vital in the analysis of predator–prey and host– parasite relationships (Crofton, 1971; Murdie & Hassell, 1973; Anderson, 1974; Hassell & May, 1974). 2.5.1 Mathematical distributions that serve as models It is necessary to outline some of the mathematical models that have been proposed to describe the distribution of organisms in space; for a fuller treatment, reference should be made to early works, such as Anscombe (1950), Wadley (1950), Cassie (1962), and Katti (1996), to other papers cited in the text, to textbooks, e.g. Bliss and Calhoun (1954), and to Patil and Joshi’s (1968) dictionary of distributions. 2.5.1.1 Binomial family In the binomial distribution the variance is less than the mean, for the Poisson it is equal to the mean, and for the negative binomial it is greater than the mean. A full discussion of variance and its calculation will be found in most statistical textbooks (e.g. Bailey, 1959; Zar, 1984), but the estimated variance (S2) of a distribution may be calculated using:

s2 =

È (Â x)2 ˘ ˙ N ˙ Î ˚ N -1

 x 2 - ÍÍ

(2.19)

where S = the sum of, x = various values of the number of animal/sample, and N = number of samples.

30

CHAPTER 2

Random

Contagious

Regular

Fig. 2.4 Different types of spatial distribution.

The central place in this family is occupied by the Poisson series, which describes a random distribution (Fig. 2.4). It is important to realize that this does not mean an even or uniform distribution (Fig. 2.4), but that there is an equal probability of an organism occupying any point in space and that the presence of one individual does not influence the distribution of another. Because for the Poisson distribution the _ variance (S2) is equal to the mean ( x ), the probability of finding a certain number (x) of animals is described by one parameter, as follows: px = e - x

xx x!

(2.20)

where e = base of natural (Napierian) logarithms. The goodness of fit of a set of data to the Poisson distribution may be tested by a c2 test on the observed and expected values. Occasionally, it may be found that the variance is less than the mean; this implies a more regular (or uniform or even) distribution than that described by a Poisson series (Fig. 2.4). Most commonly in ecological studies the variance will be found to be larger than the mean, that is, the distribution is contagious2 (Fig. 2.4) — the population is

2

The term ‘contagious’ is a mathematical one coined in connection with work on epidemiology and has certain implications that, to some extent, make its use in ecology inappropriate (Waters & Henson, 1959). An alternative is the term ‘overdispersion’, first introduced into ecology by Romell (1930), with its opposite — for more uniform spacing — ‘underdispersion’; unfortunately, however, the use of these terms has been reversed by some ecologists. Therefore, the terms used here are contagious and regular, which are also those commonly used in plant ecology (Greig-Smith, 1978).

THE SAMPLING PROGRAMME

31

clumped or aggregated. Many contagious populations that have been studied can be adequately expressed by the negative binomial distribution. Examples of its use in insect and marine benthic populations are given in Bliss and Owen (1958), Lyons (1964) and Harcourt (1965). This distribution is described by two positive parameters, c and the exponent k, which is a measure of the amount of clumping and is often referred to as the dispersion parameter. The mean and variance of the negative binomial are given in terms of c and k by: m=

k c

s2 =

(2.21) k k + c c2

(2.22)

Defining for convenience p = c/(c + 1) and q = 1 - p, the probabilities of 0, 1, 2, 3, . . . events (animals per sample) are given by the terms of the expansion of: p k (1 - q) - k

(2.23)

for which the first few terms are

(k + 2) 3 ¸ k(k + 1) 2 Ï p k Ì1, kp, q , k(k + 1) q , . . .˝. 2! 3! ˛ Ó

(2.24)

Generally, for natural populations, values of k are in the region of 2; as k Æ •, the distribution tends to the Poisson, whilst, as k Æ 0, the distribution tends to the logarithmic series. The value of k is not a constant for a population, but often increases with the mean (Anscombe, 1949; Bliss & Owen, 1958; Waters & Hensen, 1959) (see p. 44). The truncated Poisson may be of value where the distribution is non-random, but the data are too limited to allow the fitting of the negative binomial (Finney & Varley, 1955). Calculating k of the negative binomial. The value of k may be computed by several methods (Anscombe, 1949, 1950; Bliss & Fisher, 1953; Debauche, 1962; Legay, 1963). Three are presented here, two approximate and one more accurate. Another method is described by Katti (1966). This diversity of methods is because the maximum-likelihood estimation does not lead to a simple formula. 1 Having calculated the mean and the variance of the population using equation (2.19), k can be estimated using equations (2.21) and (2.22). The efficiency of this estimate for different values of k and x can be seen in Fig. 2.5. It is most useful for low-density populations. 2 k can be found iteratively by solving on a computer: ÊNˆ Ê xˆ logÁ ˜ = k log 1 + Ë k¯ Ë n0 ¯

(2.25)

where N = total number of samples, n0 = number of samples containing no animals, other parameters as above. A reasonable value of k to start with is 2. It will be seen from Fig. 2.5 that this method is efficient for most populations with very small

CHAPTER 2

100 40

98%

20 10 Exponent k

32

90%

4 2 1

90%

0.4 98%

0.2 0.1 0.04

0.1

0.2

0.4

1

2

4 10 Mean –x

20

40

100 200

400

Fig. 2.5 Large sample efficiencies of the estimation of k of the negative binomial distribution by methods 1 ( — ) and 2 (---) (redrawn from Anscombe, 1950).

means, but for large ones only where there is extensive clumping. This can be expressed another way by saying that about one-third of the samples must be blank (empty) if the mean is below 10; with larger means, more blank samples must have been found for the method to be reasonably efficient. 3 The maximum-likelihood estimate of k is obtained by solving iteratively on a computer: Ê xˆ Ê Ax ˆ N log e 1+ = Ë k ¯ ÂË k + x¯

(2.26)

where log e = natural logarithms, Ax = the sum of all frequencies of sampling units containing more than x individuals (e.g. A6 = Sf7 + f8 + f9). Shenton and Wallington (1962) have shown that even this maximum-likelihood estimator of k will have a bias if the mean is small and k large. Testing the fit of the negative binomial. When the value of k has been found, the agreement between the negative binomial series as a model and the actual distribution can be tested in three ways. The expected frequencies of each value calculated using equation (2.24) may be compared by a c2 test with the actual values (Bliss & Fisher, 1953). An alternative expression for px using the gamma function G(k) is: px =

G(k + x) Ê x ˆ x Ê k ˆ k x! G(k) Ë x + k ¯ Ë k + x ¯

(2.27)

THE SAMPLING PROGRAMME

33 SE(

50

80

U)

E

40

20

20 10

10

5

SE 10 (T) 00

2

5

32

1 x–/k

0

0.5 80 40

0. 2 10

5

5

0.0

20

0. 1

1

2

0.5

1

0 .5

0.2 0.1

0

16

E 0.2

0.5

1

2

–x

5

10

20

50

100

Fig. 2.6 Testing the fit of the negative binomial distribution — the standard errors of T and U for N = 100. For — other values of N multiply the standard error by 10/÷N . The line E–E represents the contour along which the efficiencies of the methods are equal; the use of the second-moment test is preferable in the area to the left of the line and of the third-moment test in the area to the right (after Evans, 1953).

The c2 test has three fewer degrees of freedom than the number of comparisons that are made between expected and actual frequencies; those with small expectations are generally pooled, but Pahl (1969) considers that this is not necessary. Computer programs to estimate k by iteration and test the fit to a negative binomial model are widely available. Such a comparison between actual and expected frequencies may be distorted by irregularities due to chance; there are two alternative tests based on the difference between the actual and expected moments (mean, variance, or skewness) compared with their standard errors (Anscombe, 1950; Bliss & Fisher, 1953). Which of these _ tests is most efficient will depend on the sizes of x and k. Evans (1953) has shown that with small means, unless k is large, the most efficient test is that based on the _ second moment (variance); for other values of x and k, the test based on the third moment (skewness) should be used. The actual dividing line between the two tests (E–E) is given in Fig. 2.6. The test based on the second moment involves the calculation of the statistic U, which is the difference between the actual variance and the expected variance, given by:

34

CHAPTER 2

x2 ˆ Ê U = s2 - Á x + ˜ Ë k¯

(2.28)

the value of k being derived by the second method above. If U is significantly less than its standard error (SE(U )), most conveniently calculated from Fig. 2.6 (the exact formula is given by Evans (1953)), then the negative binomial may be taken as a satisfactory model. The third-moment test involves the calculation of T, the difference between the actual third moment (skewness) of the data and its value predicted from the first two moments (mean and variance) of the same sample: Ê Â fx 3 - 3x  fx 2 - 2x 2  fx ˆ 2 Ê 2s 2 ˆ T =Á ˜ - s ÁË x - 1˜¯ N Ë ¯

(2.29)

This should be compared with its standard error (the square root of its large variance) and, if the negative binomial is a satisfactory model, T will be significantly smaller than the SE. The standard error SE(T ) may be approximately read off from Fig. 2.6 or calculated using: 2x (k + 1) SE (T ) =

x2 Ê 1+ k2 Ë

x ˆ2È Ê x 2 3 + 5 ˆ + 3kÊ1 Í ¯ Ë Ë k Î k¯ N

xˆ˘ k ¯ ˙˚

(2.30)

where the symbols are as above and the value of k is found using equation (2.26). A large positive value of U or T means that the actual distribution is more skew than that described by the negative binomial, and a large negative value means that the actual distribution is less skew. Calculating a common k. Samples may be taken from various fields or other units and each will have a separate k. The comparison of these and the calculation of a common k (if there is one) will be of value in transforming the data for the analysis of variance and for sequential sampling (see p. 53). The simplest method is the moment or regression method (Bliss, 1958; Bliss & Owen, 1958). Two statistics are calculated for each unit: Ê s2 ˆ x¢ = x 2 - Á ˜ ËN¯

_

y¢ = s 2 - x

_

(2.31) (2.32)

_

where x = the mean, s 2 = variance, and N = number of individual counts on which x is based. When y¢ is plotted against x¢ (Fig. 2.7) (including occasional negative or zero values of y¢), the regression line of y¢ on x¢ passes through the origin and has the slope 1. An approximate estimate of the common k (kc) is given by:

THE SAMPLING PROGRAMME

35

700

60

600

50

500 40 400

Y'

Y'

30

300 200

20

100

10

0 0 (a)

1000 2000 3000 4000 5000 6000 7000 X'

0 (b)

20

40

60 X'

80

100

120

Fig. 2.7 Regression estimate of a common k for: (a) Colorado beetle, Leptinotarsa, in eight plots within each of 16 blocks; (b) wireworms (Coleoptera, Elateridae) in 175 sampling units in each of 24 irrigated fields (after Bliss & Owen, 1958).

1 = kc

 y¢  x¢

(2.33)

It may be apparent from the plotting of y¢ on x¢ that a few points lie completely outside the main trend, and therefore, although their exclusion will mean that the resultant k is not common to the whole series of samples, it is doubtful if the k derived by including them would really be meaningful. A further graphical test of the homogeneity of the samples is obtained by plot_ ting (1/k) ( y¢/x¢) against the mean, x , for each sub-area or group of samples. If there is neither trend nor clustering (Fig. 2.8), we may regard the fitting of a common k as justified. Fairly rough estimates of kc are usually adequate, but in critical cases a weighted estimate of kc should be obtained and whether or not this lies within the sampling error can be tested by the computation of c2. The method is given by Bliss (1958) and Bliss and Owen (1958). The latter authors describe another method of calculating a common k that is especially suitable for field experiments arranged in restricted designs (e.g. randomized blocks). 2.5.1.2 Logarithmic and other contagious models A number of other mathematical models have been developed to describe various non-random distributions, and several of these may have more than one mode. Anscombe (1950) and Evans (1953) have reviewed these and they include the Thomas, Neyman’s types A, B, and C, and the Polya–Aeppli distributions. The Thomas (1949) type is based on the assumption of randomly distributed colonies

36

CHAPTER 2

0.3

1.4 1.2 1.0

0.1

1/k

1/k

0.2

1/kc

0.8 0.6 0.4

0

0.2 0.1 0 (a)

10

20

30

40 50 Log –x

60

70

80

0 –0.6 –0.4 –0.2 0

90 (b)

0.2 0.4 0.6 0.8 1.0 1.2 Log –x

Fig. 2.8 The relation of 1/k to the mean for: (a) Colorada beetle; (b) wireworms, based on the same data as in Fig. 2.7 (after Bliss & Owen, 1958).

whose individual populations are values plus one, from a Poisson series. Neyman (1939) distributions are similar and are intended to describe conditions found soon after insect larvae hatched from egg batches; the modes are equally spaced. Skellam (1958) has shown that Neyman’s type A is particularly applicable where the organisms occur in compact clusters, but that it can be used as an approximation in certain conditions when the clustering is less compact. The Polya–Aeppli distribution describes the situation when an initial wave of simultaneous invaders have settled and produced clusters of offspring; it may have one or two modes. As mentioned above, the logarithmic model describes situations for which the negative binomial would give a very small value of k. The logarithmic series (Fisher et al., 1943), which is derived from the negative binomial with k tending to zero and with the zero readings neglected, and the discrete and truncated (and censored) lognormal distributions (Preston, 1948; Grundy, 1952) have been found of most value in the description of the relationship between numbers of species and numbers of individuals and are discussed later (p. 471). Also they have been found to give a reasonable description of the distribution of the individuals of some insects (e.g. the citrus scale insect, Aonidiella ornatum (Spiller, 1952) or the eggs of the larch sawfly, P. erichsonii (Ives, 1955). Working with plankton, with large mean values, Cassie (1962) suggested that the action of a series of environmental factors led to the population being distributed in a succession of Poisson series, the means of the series being themselves distributed according to the log-normal model. He called this the Poisson log normal; it differs from the negative binomial mainly in the left-hand flank, as ordinarily plotted, where it allows for fewer zero values. 2.5.1.3 Implications of the distribution models and of changes in the type of distribution Neyman’s distributions are based on precise models and, as Upholt and Craig (1940) found, if the biological assumptions underlying the distribution are not fulfilled, it

THE SAMPLING PROGRAMME

37

will not adequately describe the dispersion of the population. It is perhaps for this reason that Neyman’s distributions have been used relatively little by ecologists, most preferring the other distributions, which can be derived from a number of different hypotheses. The negative binomial can arise in at least five different ways (Anscombe, 1950; Waters & Hensen, 1959). 1 Inverse binomial sampling: if a proportion of individuals in a population possesses a certain character, the number of samples in excess of k that have to be taken to obtain k individuals with this character will have a negative binomial distribution with exponent k. 2 Heterogeneous Poisson sampling: if the mean of a Poisson distribution varies randomly between samples, under certain conditions a negative binomial results (Pielou, 1969). A biological example of this is the observation of Ito et al. (1962) that a series of counts of a gall-wasp on chestnut trees were distributed as a Poisson for each single tree, but when the counts from all trees were combined they were described by a negative binomial. The distribution of oribatid mites in the soil (Berthet & Gerard, 1965) and that of a tapeworm in fish (Anderson, 1974) also appear to arise according to this model. 3 Compounding of Poisson and logarithmic distributions: if a number of colonies are distributed as a Poisson but the number of individuals per colony follows a logarithmic distribution, the resulting distribution per unit area (i.e. independent of colonies) will be a negative binomial. Counts of bacteria (Quenouille, 1949) and the dispersion of eggs of the cabbage butterfly, P. rapae (Kobayashi, 1966) have been shown to satisfy this model. 4 Constant birth–death–immigration rates, the former two expressed per individual and the immigration rate per unit of time, will lead to a population whose size will form a negative binomial series. 5 True contagion: where the presence of one individual in a unit increases the chance that another will occur there also. The logarithmic series (Fisher et al., 1943) can also be derived from several modes of population growth (Kendall, 1948; Shinozaki & Urata, 1953). It is clear, therefore, that, from mathematical considerations alone, it is unsound to attempt to analyse the details of the biological processes involved in generating a distribution from the mathematical model it can be shown to fit or, more often, approximately fit (Waters & Hensen, 1959). There is value in expressing the various possible mechanisms in biological terms. This has been done for parasite–host interactions by Crofton (1971) and Anderson (1974). These then provide alternative hypotheses, which may be tested by other observations. The extent of clumping and the changes in it provide important evidence about the population. The uses and interpretation of the parameters will be discussed later (p. 42); here, changes in the actual type of distribution will be discussed. The main distinction lies between regular, random, and contagious distributions, with the respective implications that the animals compete (or at least tend to keep apart), have no effect on each other, or are aggregated or clumped. One must be careful to

38

CHAPTER 2

define the spatial scale over which the dispersion is described. The behavioural significance of the spacing pattern within the colony area (see below) is different from that between colonies. The first difficulty is that the sampling method chosen by the experimenter may affect the apparent distribution (Waters & Hensen, 1959), as is shown by the example from the work of Ito et al. (1962) quoted above and that of Shibuya & Ouchi (1955), who found that the distribution of the gall-midge, Asphondylia, on soy bean was contagious if the plant was taken as the sampling unit, but random if the numbers per pod were considered. The contagion appeared to be due to there being more eggs on those plants with more pods. The effect of the size of the sampling unit on the apparent distribution has also been demonstrated for chafer-beetle larvae (Burrage & Gyrisco, 1954) and grasshopper egg pods (Putnam & Shklov, 1956). Careful testing of the size and pattern of sampling, as suggested above, will help to discover whether the contagion is an artefact or a reflection of the mixture of dispersion within ‘colony areas’ and the pattern of ‘colony’ distribution. Kennedy and Crawley (1967) aptly describe the dispersion of the sycamore aphid, Drephanosiphon platanoides, as ‘spaced out gregarious’; they point out that, at close proximity — how close depending on the size of their ‘reactive envelopes’ — animals may repel each other (e.g. caddis larvae (Glass & Boubjerg, 1969)), but on a larger scale they will often show grouping or contagion. This may arise from patchiness of the habitat (Ferguson et al., 1992), including differential predation (Waters & Hensen, 1959), or from the behaviour of the animals themselves, or a combination of both. The behaviour leading to aggregation, in the absence of special attractive areas in the habitat, may be of two types: inter-individual attraction or the laying of eggs (or young) in groups (Cole, 1946). The dispersion of the initial insect invaders of a crop is often random; this randomness may be real or an artefact due to the low density relative to the sample size (see below). The distribution of aphids in a field during the initial phase of infestation is random, becoming contagious as each aphid reproduces (Sylvester & Cox, 1961; Shiyomi & Nakamura, 1964), although there may be differences between species (Kieckhefer, 1975; Badenhauser, 1994). The egg masses of many insects are randomly distributed (e.g. Chiang & Hodson, 1959) and the individual eggs and young larvae are clumped; however, the dispersion of the larvae may become random, or approach it, in later instars. Such changes in distribution with the age of the population have been observed in wireworms (Salt & Hollick, 1946), the rice stem borer, Chilo simplex (Kono, 1953), a chafer beetle, Amphimallon majalis (Burrage & Gyrisco, 1954), the cabbage butterfly, P. rapae (Ito et al., 1960), the diamondback moth. P. maculipennis (Harcourt, 1961a), white grubs (Phyllophaga) (Guppy & Harcourt, 1970), leafhoppers (Nestel & Klein, 1995), and others. It is tempting to infer from such changes that either mortality or emigration or both are density-dependent; however, as Morisita (1962) has shown, this conclusion may not be justified with non-sedentary animals. Such changes could result from the alteration of the size of the area occupied by the colony relative to that of the sample or

THE SAMPLING PROGRAMME

39

from the decrease in population density (see below). In contrast, the tendency to aggregate may be such that density-dependent mortality is masked, as in the pear lace-bug, Stephanitis nashi, where, even after the bugs have been artificially removed so as to produce a random distribution, it returns to non-randomness in a few days (Takeda & Hukusima, 1961), and, in the cabbage-root fly, Erioschia brassicae, k is similar for all immature stages (Mukerji & Harcourt, 1970). Changes in the density of an animal often lead to changes, or at least apparent changes, in the distribution. When the population is very sparse, the chances of individuals occurring in any sampling unit are so low that their distribution is effectively random. The random distribution of low populations and the contagious distributions of higher ones have often been observed — for example, with wireworms (Finney, 1941), grasshopper egg pods (Davis & Wadley, 1949), a ladybird beetle, Epilachna 28-maculata (Iwao, 1956), the cabbage butterfly, P. rapae (Harcourt, 1961b), the pea aphid, Acyrthosiphon pisum (Forsythe & Gyrisco, 1963), and cotton insects (Kuehl & Fye, 1972). However, with the Nantucket pine-tip moth, Rhyacionia frustrana, as the population density increases still further, the distribution tends towards the Poisson again, i.e. the k value becomes high (Waters & Hensen, 1959). Populations of several other forest insects (Waters & Hensen, 1959) and of the periodical cicadas (Dybas & Davis, 1962) show a tendency to become more random at higher densities (p. 29). Indeed, one can envisage that, at even higher densities, the distribution would pass beyond the Poisson and become regular. Dispersions approaching a regular distribution have been but rarely observed; examples include the bivalve, Tellina (Holme, 1950), ants (Waloff & Blackith, 1962), aphids (Kennedy & Crawley, 1967), and a coral (Stimson, 1974). In different habitats, the type of distribution may change, as Yoshihara (1953) found with populations of the winkle, Tectarius granularis (Mollusca), on rough and smooth rocks, although it is difficult to separate this habitat effect from associated changes in density. Within the same habitat, different species will usually show different dispersion patterns, as has been found with leafhoppers (Homoptera: Auchenorrhyncha) on rice plants (Kuno, 1963). These differences can arise from several biological causes: one species may aggregate more than another in the same habitat because it disperses less, because it reproduces more, or because only certain parts of the habitat are suitable for it. However, Abrahamsen and Strand (1970) suggest that, for many soil organisms, there may be a relationship between k and x that can be expressed by the regression: k = 0.27 + 0.334 x 2.5.1.4 Comparison of aggregation indices When a population is sampled, four basic bits of information are available. 1 The estimate (x) of the true mean, m. 2 The estimate (s 2) of the true variance, s 2.

(2.34)

40

CHAPTER 2

3 The locality of the sample. 4 The size (unit). The indices used for the description of animal populations are derived from various arrangements of this information. The simplest approach is the variance/mean ratio, _ s 2/x , which, as described above, is unity for a Poisson (random) distribution. The index of clumping of David and Moore (1954) is simply: I DM =

s2 -1 x

(2.35)

which gives a value of zero for a random (non-clumped) population. A more general approach to the variance–mean relationship is given by Taylor’s power law:

_

s 2 = ax b This empirical relationship holds for many species (Taylor et al., 1978; Perry, 1981; Taylor, 1984) and has been found to be of considerable practical value in designing sampling programmes. Such a relationship can be shown to arise as a result of a variety of mechanisms, both stochastic demographic events and density-related behavioural processes (Anderson et al., 1982; Perry, 1988), and from its underlying mathematical basis (Tokeshi, 1995). Pattern analysis in plant communities was developed by P. Greig-Smith (see Greig-Smith, 1978). The method is based on increasing quadrat size; where a peak arises in the plot of a measure of variance against quadrat size, this is considered as a ‘clump area’. This method can only be applied to contiguous quadrats. Morisita (1954, 1959, 1962, 1964) developed an index, Id, that provided similar information but could be used on scattered quadrats. Zahl (1974) has suggested that GreigSmith’s approach can be made more robust by the use of the S-method of Scheffé. With mobile animals, the degree of crowding experienced by an individual inter. ested Lloyd (1967), who devised an index of mean crowding (m): Ês2 ˆ ˙ = m +Á m - 1˜ Ë m ¯

(2.36) .

where m and s 2 are the true mean and variance, respectively. Thus m is the amount by which the variance mean ratio exceeds unity added to the mean. If the distribus2 . tion conforms to a Poisson (random), then = 1 and m = m. With estimates based x . on samples, the index of mean crowding, x, is given by: Ê s2 ˆ x˙ ª x + Á - 1˜ Ëx ¯

(2.37)

For contagious dispersions described by the negative binomial (equations 2.25 and 2.26): s2 = x +

x2 k

(2.38)

THE SAMPLING PROGRAMME

41

so equation (2.37) may be written: x˙ = x +

x k

(2.39)

Rearranging equation (2.39): x˙ 1 = 1+ x k

(2.40)

one sees that the reciprocal of k is the proportion by which mean crowding exceeds x˙ mean density. Lloyd (1967) termed the ‘patchiness’. x Clarification and unification of these various approaches were achieved by Iwao (1968, 1970a,b,c, 1972), who initially demonstrated that the relationship of mean _ . crowding, x, to mean density, x , for a species could be expressed over a range of densities by a linear regression:

_

.

(2.41)

x = a + bx

.

(Iwao used the expression m–m method, but in reality one is normally working with _ . estimates (x and x ), rather than the true mean and its derivatives. a and b are characteristic for the species and a particular habitat (Fig. 2.9) and their interpretation will be discussed later. Subsequently, he developed the r-index (Iwao, 1972): r1 =

x˙ i - x˙ i -1 xi - xi -1 .

(2.42)

_

where xi and x i are the mean crowding index and mean density for the ith-sized quadrat sample. Thus, with a series of different-sized quadrats, values of r may be

1.8 Mean crowding (x )

.

10 1.0

10

0 0.2

1.0

0

10

0

10

Mean density (x– ) (a)

(b)

(c)

Fig. 2.9 Iwao’s patchiness regression: (a) potato beetle, Epilachna, number of egg masses per plant () and of individual eggs ( ); (b) spruce budworm, Choristoneura, numbers of larvae/twig; (c) pea aphid, Macrosiphum, total aphids (excluding first instar) per shoot (after Iwao, 1968).

•

42

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plotted for the second and subsequent quadrat sizes (Fig. 2.10). Changes in the value of the r-index will indicate clump or territory size in a manner analogous, but theoretically preferable, to Greig-Smith’s method. . x expresses ‘crowding’. Lloyd (1967) derives it by estimating the mean number, less one, present in the area. Working with mites on leaves, Tanigoshi et al. (1975) . restored the one to give an index they called ‘mean colony size’ (C): . . C=x+1 (2.43) Lloyd (1967) termed this ‘mean demand’ — the mean number of individuals per quadrat per individual — and used it as an expression of trophic demand. The concept of mean colony size is useful, but is likely to be meaningful as formulated only when the sampling unit, as with spider mites on a leaf, contains an entire colony. Thus, all the above indices are related to each other and to the simple concept of _ the comparison of the estimate of the variance (s 2) with that for the mean x : 1+

s2 - x 1 x˙ C˙ - 1 = 1+ = = ª Id 2 k x x x

Bringing in the third piece of information, quadrat size, the r-index is approximately the same as plotting the changes in Id with quadrat size; it is the proportion by which mean crowding changes against mean density with increased sample area. Essentially, in Greig-Smith’s method, variance is plotted against sample area. Perry and Hewitt (1991) criticized the above methods because there is no direct relationship between the mean and variance and the movements of the individual animals. Futher, they do not use the available spatial information. A methodology called spatial analysis by distance indices (SADIE) was developed (Perry & Hewitt, 1991; Aliston, 1994; Perry, 1994, 1995). The method involves the use of a computer simulation. The approach is to calculate the total distance by which the individuals need to be moved to make their distribution as crowded, regular, or random as possible. Perry (1994) gives examples where the method has been applied to a variety of animals and plants over a range of spatial scales; the software can be obtained at nominal cost (Perry, 1995).

2.5.2 Biological interpretation of dispersion parameters 2.5.2.1 Index of dispersion — the departure of the distribution from randomness If the dispersion follows the Poisson distribution (p. 30), the mean will equal the _ variance and therefore departures of the coefficient of dispersion (or variation) s 2/x from unity will be a measure of the departure from the Poisson (from randomness). This is tested by calculating the index: ID =

s 2 (n - 1) x

(2.44)

Random (Poisson)

Contagious

Random

Contagious

Random components

–x

x

0

0

–1

1

1

1

0

0

0

ρ

0

0

–x

0

ρ

1

Quadrat size

Regular components

.

.

.

x

0 0

ρ-index

Regular

x

.

Patchiness regression (x – x– )

Randomly distributed loose colonies

Compact colonies

THE SAMPLING PROGRAMME

Individuals

0

–x

–1

ρ

Quadrat size

1

1

0

0

1

Quadrat size

0

Fig. 2.10 The expected patchiness regressions and r-index plots for various patterns of dispersion (after Iwao, 1972). 43

44

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where n = the number of samples and ID is approximately distributed as c2 with n - 1 degrees of freedom, so that, if the distribution is in fact Poisson, the value of ID will not lie outside the limits (taken as 0.95 and 0.05) of c2 for n - 1, as given in standard tables.3 The index is therefore used as a test criterion for the null hypothesis that the pattern is random. The coefficient of variation or mean/variance ratio is a descriptive sample statistic that will approach zero for regularly distributed organisms, whilst a large value implies aggregation. As Naylor (1959) has pointed out, the indices may be added if they are from the same-sized samples. Early examples of the use of this index are in Salt and Hollick’s (1946) studies on wireworms, Naylor’s (1959) on Tribolium, Overgaard Nielsen’s (1963) on Culicoides and Milne’s (1964) on the chafer beetle, Phyllopertha. In contagious populations, the value of this index is influenced by the density and size of the sampling unit (i.e. by the size of the mean). 2.5.2.2 ‘k’ of the negative binomial — an index of aggregation in the population If the negative binomial (p. 32) can be fitted to the data, the value of k gives a measure of dispersion; the smaller the value of k, the greater the extent of aggregation, whereas a large value (over about 8) indicates that the distribution is approaching a Poisson, i.e. is virtually random. This may be appreciated from the relationship of k to the coefficient of variation, cv: 1

s È1 1 ˘ 2 cv = = Í + ˙ x Îk x ˚

(2.45)

Clearly, the smaller the k, the larger the cv. The value of k is often influenced by the size of the sampling unit (Cole, 1946; Morris & Reeks, 1954; Waters & Hensen, 1959; Harcourt, 1961a; Ho, 1993) and therefore comparisons can be made only when using the same-sized unit. But, within this restriction, it does provide a useful measure of the degree of aggregation of the particular population, varying with the habitat and the developmental stage (Hairston, 1959; Waters & Henson, 1959; Harcourt, 1961a; Dybas & Davis, 1962). Examples of these variations are given in Table 2.3. As Waters and Henson (1959) have pointed out, the degree of aggregation of a population, which k expresses, could well affect the influence of predators and parasites. The aggregation recognized by the negative binomial may be due either to active aggregation by the insects or to some heterogeneity of the environment at large (microclimate, soil, plant, natural enemies; p. 374). Dr R.E. Blackith has suggested that, if the mean size of a clump is calculated using Arbous and Kerrich’s (1951) formula and this is found to be less than 2, then the ‘aggregation’ would seem to be due to some environmental effect and not to an active process. Sedentary animals, a large number of which are killed between settling and sampling, could be an excep3

For values of n larger than those given in tables, χ2 may be calculated (see p. 82).

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45

Table 2.3 The variation k of the negative binomial with some factors, a, b, and c, with sampling unit, d, with developmental stage and change in density. a. Eriophyes leaf galls on Populus tremuloides. b. Nantucket pine-tip moth, Rhyacionia frustrana (from Waters & Henson, 1959). c and d. Cabbage white butterfly, Pieris rapae (from Harcourt, 1961b).

a

b

c

d

Sampling unit

Mean number per unit

k

Single leaf 1 leaf bunch 2 leaf bunches 5 leaf bunches Branch Tip of shoot Branch whorl Tree

0.21 + 0.01 0.88 + 0.04 1.65 + 0.08 3.78 + 0.26 14.23 + 1.68 0.49 + 0.03 2.36 + 0.36 13.13 + 4.21

0.0611 + 0.0013 0.8830 + 0.0024 0.1150 + 0.0035 0.1740 + 0.0064 0.2000 + 0.0119 0.449 + 0.021 0.214 + 0.035 0.253 + 0.074

Sampling unit

k for lst-instar larvae

k for 4th-instar larvae

Quadrant of cabbage Half cabbage Whole cabbage

1.38 2.28 2.32

1.96 4.24 4.28

Stage

Mean density

k

Egg Ist instar 2nd instar 3rd instar 4th instar 5th instar Pupa

9.5 5.6 4.4 4.0 3.6 2.6 1.7

3.1 2.8 2.8 4.6 5.1 7.8 2.3

tion, in having a behavioural cause for an aggregation size of less than 2. Aggregations of two or more insects could be caused by either factor. The mean number of individuals in the aggregation is calculated by: l=

x n 2k

(2.46)

where n is a function with a c2 distribution with 2k degrees of freedom and l = the number of individuals in the aggregation for the probability level allocated to n. To find the mean size of the ‘aggregate’, the value at the 0.5 probability level is used. Usually, 2k degrees of freedom will be fractional, but an adequate c2 can be calculated by reference to graphs or by proportionality. Let us take two examples from Table 2.3d, the eggs and pupae of P. rapae: Eggs: l =

9.5 ¥ 5.55 = 8.5 3.1 ¥ 2

Pupae: l =

1.7 ¥ 3.54 = 1.3 2.3 ¥ 2

CHAPTER 2

8.0 7.0 6.0 5.0 Mean (x– )

46

4.0 3.0 λ=2 Contagion due to environment

2.0 1.0

0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

k Fig. 2.11 The cause of contagion in the data — the plot of the mean aggregation (l) of the two individuals for various values of the mean and k of the negative binomial. Populations whose value for the mean plotted against k lies below the line may be considered to exhibit contagion because of some environmental factor; in those populations whose value lies above the line, contagion could be due either to an active behavioural process or to the environment.

From this it can be concluded that the ‘clumping’ of the eggs could be due to a behavioural cause — the females tending to lay a number of eggs in close proximity or, because of environmental heterogeneity, only certain areas being suitable for oviposition. But ‘clumping’ of the pupae is due to environmental causes. The values of the k and the mean for l = 2 are plotted in Fig. 2.11, from which it can be ascertained for a particular population whose k and mean are known whether the mean ‘aggregation’ size is above or below 2. The negative binomial has been found useful in describing the pattern of predator attacks on patches of prey (Hassell, 1978). May (1978) has shown that the k of the attack pattern is the reciprocal of the coefficient of dispersion (or variation) (s/x) of the numbers of predators per patch. It has been argued that it is a useful model for the dynamics of animal aggregation and density dependence (Taylor et al., 1979). 2.5.2.3 ‘b’ of Taylor’s power law — a coefficient of spatial variability for the species The extent to which the parameter b of Taylor’s power law (p. 12) is a constant for a species has been a matter of controversy. It may be concluded that it provides a description of the spatial structure of a species in a particular environment. There are many examples of it being relatively constant for a species (Taylor, 1961; Banerjee, 1976; Taylor et al., 1978; Perry, 1988; Sawyer, 1989). However, as Riley et al. (1992) found when working on a weevil on peppers, whilst b did not vary signi-

THE SAMPLING PROGRAMME

47

ficantly over time or with changes in population density, it was sensitive to the variety of the host plant and the type of sample. Bliss (1971) has pointed out that special techniques should be applied for the calculation of this regression, as the independent variate is not error-free. The same criticisms apply even more forcefully to Iwao’s patchiness regression (see below), but in practice the more accurate approach is seldom required. As shown by Iwao (1970c), Taylor’s power law will not fit certain theoretical distributions, although the biological frequency (and significance) of these is uncertain. It remains a simple and useful description of species distribution (Taylor, 1970; Usher, 1971; Egwuatu & Taylor, 1975; Kieckhefer, 1975; Croft et al., 1976; Ng et al., 1983; Pena & Baranowski, 1990; Steiner, 1990; Pringle & Giliomee, 1992; Peng & Brewer, 1994). High values of b show strong contagion. But, if the regression line crosses the Poisson line (slope b = 1), this shows a change to random distribution at lower densities. Hence, unless a = 1, the value of b in itself cannot be taken as a test of randomness, as shown by George (1974) for zooplankton. The constant a is largely a scaling factor related to sample size, but, if sample size is larger than the colony size and yet refers to a biologically meaningful entity (e.g. a leaf for spider mites), then a is also related to colony size. It would seem that the amount of scatter of the points around the line (see Fig. 2.2), which can be approximately estimated as the variance around the appropriate regression line (p. 12), is a measure of the effect of habitat variation on the extent of aggregation. 2.5.2.4 Lloyd’s mean crowding and Iwao’s patchiness regression indices for the population and species . The index of mean crowding x (Lloyd, 1967) is given approximately in equation (2.37). If the underlying distribution is known to fit the negative binomial, then equation (2.39) is applicable. This is an efficient estimate provided the number of samples (N) is large. When using equation (2.39), the most accurate method for estimating k (method 3, p. 321) should be used. A general formula is: s2 ˆ Ê s2 ˆÊ x˙ = x + Á - 1˜ Á1 + ˜ Ëx ¯ Ë N x2 ¯

(2.47)

the second term in parentheses correcting for sampling bias and k being calculated by moments (method 1, p. 31). The appropriate large-sample variance of the estimate is: È1 var( x˙ ) ª Í ÎN

2 2 Ê x˙ ˆ Ê s ˆ Ê ˙ 2s ˆ ˘ x+ ˜ Ë x ¯ ÁË x ˜¯ ÁË x ¯ ˙˚

(2.48)

Lloyd (1967) discusses the problems of applying these to rare species and the statistical requirement — difficult for the biologist to meet — that one is a priori assured . of encountering the species. The index x is the mean number of other individuals per sample unit per individual and is thus an expression of the intensity of interaction between individuals. In an extensive comparison of all the indices described hitherto . applied to samples of Heliothis (corn-borer) eggs, Terry et al. (1989) found that x agreed least with the other indices, but it has been found useful with adult pear

48

CHAPTER 2

thrips, Psylla pyricola, which do not form colonies but have a contagious distribution (Burts & Brunner, 1981). . As will be seen from the first formula for the calculation of x, when the population is randomly distributed the mean crowding index equals the mean density. Many examples have been given in the previous section of increased contagion with _ . increased density. Iwao (1968) showed that x is related to the mean (x ) over a series of densities: .

_

x= a + bx

(2.49)

x˙ , and x may be termed Iwao’s patchiness regression. The constant a indicates the tendency to crowding (+ve) or repulsion (-ve) and Iwao (1970b) termed it the ‘index of basic contagion’. It is a property of the species and the mathematical basis suggests that it may warrant a biological interpretation (Tokeshi, 1995). The coefficient b is related to the pattern in which the organism utilizes its habitat and Iwao named it the ‘density contagiousness coefficient’. The theoretical effects of various underlying mathematical distributions on a and b and their application to many sets of field data are described by Iwao (1968, 1970a,b, 1972) and by Iwao and Kuno (1971) and they provide a measure of the numbers in the minimum clump in the sample unit, e.g. 38 for the individual eggs of Epilachna (Fig. 2.9) or 21.5 for Macrosiphon (excluding first instar) (Fig. 2.9). A negative value of α shows there is a tendency for the animals to repel each other, although, of course, there will not be actual points in this region (see Iwao & Kuno, 1971, Fig. 3). Coefficient b expresses the extent to which the colonies (as defined by b) are contagious at higher densities, e.g. in Choristoneura (Fig. 2.9b) or Macrosiphum (Fig. 2.9c). Although a single ‘patchiness regression’ can often be fitted for data from different areas, significantly different values of b could theoretically arise where the pattern of spatial heterogeneity differed between the two areas. Many comparative applications of Taylor’s power law and Iwao’s patchiness regression to insect sampling data have found the former to provide a more consistent fit (e.g. Pena & Duncan, 1992). Strong density-dependent mortality would influence the value of b. Density-independent mortality should alter the value of a but leave b unaffected (Iwao, 1970a,b). This is based on the changing relationship of Lloyd’s index of patchiness,

2.5.2.5 Iwao’s r index — a measure of colony area The area occupied by a colony may be determined by the use of Iwao’s (1972) index, given in equation (2.42). Clearly, this approach is only possible with animals living in habitats where sample size can be regularly increased. The index shows a sharp change in magnitude at the quadrat size corresponding to colony area (Fig. 2.10). 2.5.2.6 Breder’s equations — a measure of the cohesion of aggregations Working on fish schools, Breder (1954) found that the distance between individuals varied only slightly, and such uniform or regular distribution is indeed characteristic of the distribution of crowded individuals (p. 30). With such schools or social

THE SAMPLING PROGRAMME

49

swarms, when the individuals are their normal distance (d) apart, the attraction (a) between individuals can be said to equal the repulsion (r), so that c, a measure of the cohesion in the equation c = (a - r)/d2, is zero. Negative values of c will show ‘overcrowding’ and there will be a tendency for the individuals to move apart; positive values will show that normal aggregation has not been obtained — these will approach +1 asymptotically if the attractive force is taken as 1. If r1 and r2 are the repulsive actions of two animals on each other, then the equation becomes: c=

1 - r1r2 d2

(2.50)

or r1r2 = d2s

(2.51)

where ds = the mean normal distance between individuals in the aggregation, because, by definition, c = 0 under these conditions. The mean distance between individuals in a moving school may be measured from photographs and so the value for r found. It is then possible to plot c against d for a series of values (see Fig. 2.12). The angle at which the curves cross the 0 value is of interest: the steeper the line, the tighter the aggregation, angles of intersection of 40–50° being characteristic of fish that keep apart by a distance considerably less than their own length; smaller angles indicate looser aggregations, both spatially and, probably, Breder suggests, in their tendency to split up. The values of r given in — Fig. 2.12 represents ÷r1r2.

1 0 5 10 25

–5 c

50 –10 75

100

–15

0

5

10

15

20

25

d Fig. 2.12 Breder’s curves for the cohesion of aggregations. The normal distance between individuals as a decimal fraction of the animal total length gives the value of d for c = 0, the appropriate curve for r (= r1r2) may be read off (after Breder, 1954).

50

CHAPTER 2

Some insects, such as locusts and army worms, form swarms, many of the characters of which would seem to parallel fish schools, and therefore the equations developed by Breder might well be used for comparisons with these insects. From Breder’s equations, it would seem that the smaller the mean distance, in terms of their total lengths, between individuals, the more stable the aggregation. Extending this approach, various models have been developed for simulating the dynamics of fish schools and these may be used to analyse behaviour (Gotceitas & Colgan, 1991; Niwa, 1994). 2.5.2.7 Deevey’s coefficient of crowding This measure was used by Deevey (1947) to express the extent of crowding in barnacle populations, where the individuals actually impinge on one another as they grow, and is given by: Cc = 2pr 2N 2

(2.52)

where r = the radius of the fully grown animal and N = the density of the animal per unit area. The units of r and N must of course be comparable (e.g. cm and cm2) and then Cc will define the number of contacts per unit area. This coefficient could be used with any sedentary, approximately circular animal or with plant galls on leaves; when a realistic value can be given to r, for example the radius of a circular home range (p. 390), it could also be used with mobile forms. 2.5.3 Nearest-neighbour and related techniques: measures of departure from randomness of the distribution There are basically two separate approaches: one may either select an individual at random and measure the distance between it and its nearest neighbour (true nearestneighbour techniques) or one may select a point and measure the distance between this point and the nearest or nth-nearest individuals (sometimes referred to as closest-individual techniques). From each of these, either conclusions may be drawn about the departure of the distribution from random, if the population density is known from some other method, or alternatively, if one can assume the distribution to be random, its density can be estimated (see Chapter 9 for a discussion of the use of these methods for the estimation of density). These techniques are most easily used with stationary, discrete, easily mapped organisms, e.g. trees, and have been used extensively by botanists (Greig-Smith, 1978). With animals, their mobility and the risk that one will fail to find the nearest neighbour limit their application (Turner, 1960). However, such methods have been applied to studies on grasshoppers (Blackith, 1958) and frogs (Turner, 1960) and, with more success, to studies of fairly conspicuous and relatively stationary animals (e.g. snails, Keuls et al., 1963) or of well-marked colonies (e.g. ant mounds, Waloff & Blackith, 1962; Blackith et al., 1963). Nearest-neighbour and related techniques could also be used with other relatively sedentary animals, such as barnacles, limpets, scale insects, tube-building animals, and gall-formers (especially on leaves).

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51

The use of photography in mapping natural distributions of snails has been described by Heywood and Edwards (1961). This can be a valuable aid to the application of these techniques, not only to fairly sedentary organisms, but also, where the substratum is suitable, to far more mobile forms. The various dolichopodid and other flies on mud are an example of a community that might be studied in this way. There are, furthermore, a few other techniques, such as the ‘squashing’ or ‘imprinting’ method for estimating mite numbers on leaves (p. 155) or the deadheart method for estimating stem borers (p. 325), that do, or could easily, provide maps of the distribution of the individuals to which nearest-neighbour or closest-individual methods could be applied. Henson (1961) has used a combination of photography and nearest-neighbour methods to study the aggregations of a scolytid beetle in the laboratory (see also p. 151). 2.5.3.1 Nearest-neighbour methods In this method, a point is selected at random, and then one searches around in tight concentric rings until an animal is found. The searching is then continued until its nearest neighbour is found and the distance between these two animals is measured. Strictly speaking, this is not a random method (Pielou, 1969); a truly random method would involve numbering all the animals in advance. The extent to which the method is robust in respect of this violation is not clear and thus, in general, the closest-individual method is to be preferred (Seber, 1982). There are two expressions; the simplest is due to Clark and Evans (1954): D=

1 4r 2

(2.53)

_

where D = density per unit area and x = mean distance between nearest neighbours. As has been pointed out by Blackith (1958), this formula is based on the mean x _ squared (x )2, whereas Craig’s (1953) formula is based more correctly on the sum of x squared (Σx)2; the latter formula, however, is less easy to use and describes the density in a quarter segment. If the density has been measured by some other method then the departure of the distribution from randomness will be given by the value of the numerator in the expressions (Clark & Evans, 1954; Waloff & Blackith, 1962): uniform, hexagonal spacing competition: x2 =

1.154 D

(2.54)

random: x2 =

0.25 D

_

(2.55)

where x = mean distance between nearest neighbours and D = number of individuals per unit area. Smaller values of the numerator will indicate aggregation or clumping

52

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due to environmental or behavioural factors. These estimates should be based on large samples and, of course, the density and distance units of measurement must be the same. The use of the second, third, . . . nth nearest neighbours, discussed by Morisita (1954) and Thompson (1956), enables not only more accurate density determinations to be made, but also the dispersion pattern to be detected over a larger area. However, as Waloff and Blackith (1962) pointed out, competition effects detectable between nearest neighbours may be masked at greater distances, because of heterogeneity of the habitat. Thompson (1956) shows that the statistic: 2p DÂ x 2n

(2.56)

where D = density and x = distance, is distributed as a c2 with 2Nn degrees of freedom (N = number of observations). This value may be calculated and compared with the expected value under the hypothesis of randomness (the c2 value for 2N is found from tables). A probability of c2 greater than 0.95 indicates significant aggregation, whilst a probability of less than 0.05 indicates significant regularity. A simpler but — according to Thompson (1956) — less reliable comparison is given by the proportionality constants in the equation: x 2n =

p D

(2.57)

where p = the proportionality constant and the other symbols are as above. For a random distribution for n = 1 (i.e. the nearest neighbour), as indicated above, p = 0.25; for n = 2, p = 0.87; for n = 3, p = 0.97; for n = 4, p = 1.05. 2.5.3.2 Closest-individual or distance method In this method, a point is selected at random and the distance between this and the nth-nearest individual measured. The main theoretical difficulty is the possibility of the random point being closer to the boundary than to an individual. Practically, it is more time-consuming than the nearest neighbour, because of the need to find the third-nearest or more distant individual. However, in spite of these problems, because of the difficulties of true random selection of an animal, referred to above (difficulties that apply less to botanical studies), this method is to be preferred. The method has also been derived, apparently independently, by Keuls et al. (1963) and used for population estimations of the snail Limnaea (= Galba) truncatula. It depends on the population being randomly distributed, although, if the discontinuities are small, the errors that are made will tend to cancel each other out. The basic equation is: D=

r -1 1 ◊ p an2

(2.58)

where r = the rank of the individual in distance from the randomly selected point, e.g. for the nearest neighbour r = 1, for the second nearest 2, a = the distance

THE SAMPLING PROGRAMME

53

between the randomly selected point and the individual, and other symbols are as above. This basic equation is used in the ordered distance estimators in Chapter 9 to estimate density. Various indices of non-randomness have been derived from equation 2.58 (for a fuller discussion, see Pielou, 1969; Seber, 1982). Two are as follows. 1 Pielou’s index:

_

Ia = a 2i Dp

(2.59)

which requires a knowledge of the density per unit area (D), as well as the mean distance from a random point to the closest individual (ai), Ia = 1 for random distributions, >1 for contagious, and