A Course in Mathematical and Statistical Ecology

296 Pages·2001·6.25 MB·English

by  Anil Gore, Sharayu Paranjpe (auth.)

A COURSE I:" MATIIF.~t\T1CAL A:"JD STATISTICAL ECOLOGY
THEORY AND DECISION LIBRARY
General Editors: W.Leinfellner (Vienna) and G. Eberlein (Munieh)
SeriesA:Philosophy and Methodologyofthe Social Seiences
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SERIES B:MATHEMATICALAND STATISTICALMETHODS
VOLUME42
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Thetitlespublishedinthisseriesare listed at theendofthis volume.
A COURSE IN MATHEMATICAL
AND STATISTICAL ECOLOGY
by
ANILGORE
and
SHARAYU PARANJPE
Department 0/Statistics,
University0/Pune,
Pune,India
,
•
Springer-Science+Business Media, B.Y.
ISBN978-90-481-5616-0 ISBN978-94-015-9811-8(eBook)
DOI 10.1007/978-94-015-9811-8
Printedonacid-free paper
All RightsReserved
©2001 SpringerScience+BusinessMediaDordrecht
OriginallypublishedbyKluwerAcademicPublishersin200I.
Softcoverreprintofthehardcover 1stedition2001
No part ofthematerial protectedby this copyrightnotice may be reproduced or
utilized inany form orbyany means,electronicor mechanical,
includingphotocopying,recordingorby any information storageand
retrievalsystem, withoutwritten permission from the copyrightowner.
Table of Contents
PREFACE " . ix
1 INTRODUCTION 1
1.1 Background ... 1
1.2 Modeling In Ecology 3
1.3 Scope . 4
2 SINGLE SPECIES POPULATIONS 9
2.1 Introduction .. ... 9
2.2 Linear Growth . . , 9
2.3 Exponential Growth 10
2.4 Sigmoidal Growth . 12
2.4.1 The Logistic Equation 12
2.4.2 The Gompertz Curve 15
*
2.4.3 An Alternative Derivation Of The Logistic Equation 15
*
2.4.4 Scramble And Contest . . . . . 16
2.4.5 Logistic Model In Discrete Time 19
*
2.4.6 Models With Time Lags . . . . 22
2.4.7 The Allee Effect 25
2.5 Populations With Age Structure (Discrete Time) 25
2.5.1 Leslie Matrix Model . . . . . 26
2.5.2 Stable Age Distribution . . . 28
*
2.5.3 Density Dependent Model . 31
*
2.5.4 Some Other Variants . . . . 32
2.6 Populations With Age Structure (Continuous Time) 33
2.6.1 Lotka's Estimates Of Overall Growth Rates 33
2.6.2 Life Tables 36
2.7 Summarizing Survivorship Data . 38
2.7.1 Exponential Distribution 39
2.7.2 Weibull Distribution 41
2.7.3 Bath Tub Models . 41
2.8 Stochastic Models .. . .. 43
2.8.1 Pure Birth Process . 43
2.8.2 Pure Death Process 46
2.8.3 Simple Birth And Death Process 47
VI
2.8.4 Estimation Of Parameters . 53
2.9 Exercises . 55
3 POPULATIONS OF TWO INTERACTING SPECIES 63
3.1 Introduction.. .... ....... 63
3.2 Competition 63
3.2.1 Lotka - Volterra Equations 64
3.2.2 *Some Variants . . 70
3.3 Symbiosis . . . . . . . . . . . 71
3.4 Predation And Parasitism . . 73
3.4.1 Lotka-Volterra Model 73
3.4.2 Model Diagnostics Using Community Matrix 79
3.4.3 Model With Carrying Capacity . . . . . . . 80
3.4.4 *Functional Response .. .. . . . . . . . . 82
3.4.5 * Model Incorporating Functional Response 84
3.4.6 * Nicholson-Bailey Model . . . . . . . . . . 86
3.4.7 * Nicholson-Bailey Model: Type 2 Functional Response 90
3.5 Exercises 92
4 ESTIMATION OF ABUNDANCE 95
4.1 Introduction ... .. .. . ... .. 95
4.2 Nearest Neighbor Distance Methods 95
4.2.1 The Basic Estimator . . . . . 95
4.2.2 Batcheler's Correction . . . . 98
4.2.3 Extension To r-th Nearest Individual. 99
4.2.4 *Cost Efficiency 99
4.2.5 *Aggregated Forest. . . . . . . . . . . 100
4.2.6 *Treating Trees As Circ1es Instead Of Points 101
4.2.7 T2_ Sampling 103
4.2.8 Estimation Of Prey Density From Predator Behavior
Using Nearest Individual Distance . . . 104
4.3 Line Transect Sampling And Related Methods 108
4.3.1 The Basic Approach . . . . . . . . . 108
4.3.2 The Exponential Detection Function 110
4.3.3 Line Intersect Sampling . . . . . . . 112
4.3.4 Other Methods Based On Detection 114
4.3.5 Insights Of Practitioners . 116
4.4 Capture - Recapture Methods . 117
4.4.1 Closed Populations 118
4.4.2 Some Variants 121
4.4.3 Open Population With Single Release And Multiple
Recaptures .. . . . . . . . . . . . . . . . . . . . .. 122
vii
4.4.4 Open Population With Multiple Releases And Single
Recapture . . . . . . . . . . . . . . . . . . . . . . . . 124
4.4.5 Open Population With Multiple Releases And Re-
captures . . . . . . . . . . . . . 125
4.5 Fish stock Assessment . . . . . . . . . . . . . . . . . . 127
4.5.1 Estimating Pattern Of Growth . . . . . . . . . 128
4.5.2 Modal Progression And Bhattacharya Method 131
4.5.3 Estimation Of Natural And Fishing Mortalities 133
4.5.4 Virtual Population Analysis . . . . . . . 138
4.6 Indirect Methods of Estimation . . . . . . . . . 139
4.6.1 Estimation Using Counts Of Dung Piles 139
4.6.2 Tiger Count Using Pug-marks . .. . 140
4.6.3 Lion Identification By Whisker Marks 146
4.6.4 Waterhole Census 147
4.7 Exercises . . . . . . . . . . . . . . . . . 148
5 BIODIVERSITY 153
5.1 Introduction . 153
5.1.1 Species Abundance Distributions 154
5.1.2 Negative Binomial Distribution 155
5.1.3 Logarithmie Series Distribution 157
5.1.4 Log normal distribution . 158
5.2 Diversity . 159
5.2.1 The Concept Of Diversity 159
5.2.2 Simpson's Index . . . .. 160
5.2.3 Shannon - Wiener Index . 160
5.2.4 Diversity as average rarity . 165
5.2.5 Measurement of evenness . 167
5.3 Effort Needed to Measure Biodiversity 169
5.4 Measurement of Species Riehness . . . 171
5.5 Situation Specific Diversity Measures . 172
5.5.1 Diversity Across Geographie Levels. 172
5.5.2 Diversity At Taxonomie Levels .. . 174
5.5.3 Diversity Indiees For Host Parasite System 175
5.5.4 Index Incorporating Interspecies Differences 176
5.6 Other Aspects of Biodiversity 176
5.7 Conservation Priority 177
5.8 Exercises . 179
6 HARVESTING BIOLOGICAL POPULATIONS 185
6.1 Introäuction . 185
6.2 Surplus Yield Approach . . . . . . . . . . . 186
6.2.1 Maximum Sustainable Yield (MSY) 186
Vlll
6.2.2 Bionomie Equilibrium . . . . . . . . . . . . . 190
6.2.3 Tragedy of Commons 192
6.2.4 Optimal Harvesting Policy for a Sole Owner . 193
6.2.5 Beverton-Holt Model: 195
6.2.6 Thomson and Bell's Method. . . . . . . . 197
6.2.7 Optimal Harvesting in Primitive Societies 198
6.3 Harvesting Under Matrix Model 201
6.3.1 Doubleday's Approach 201
6.3.2 Usher's Approach. 205
6.4 Exercises 208
7 OPTIMAL DECISION MODELS IN ANIMAL BEHAV-
IOR SYSTEMS 211
7.1 Introduction........ 211
7.2 Optimal Foraging Models 212
7.2.1 Diet Choice Model 212
7.2.2 Diet Choiee With Constraints on Intake of Toxins 215
7.2.3 * Problem of Recognition 218
7.2.4 Patch Residence Time . 221
7.2.5 Central Place Foraging . 223
7.2.6 Risk Sensitive choice . . 224
7.2.7 *Vigilance........ 225
7.3 Models for Reproductive Traits in Animals and Plants 229
7.3.1 Clutch Size Model . . . . . . 230
7.3.2 *Egglaying 230
7.3.3 Ovule Number Optimization 235
7.3.4 Seed Number Distribution. . 238
7.3.5 Seed Size Optimization '" 240
7.3.6 Flower Number Optimization 241
7.4 Contests and game theory ..... 242
7.4.1 Two Person Game . . . . . . 243
7.4.2 Evolutionarily Stable Strategy 247
7.4.3 Games With More Than Two Pure Strategies 252
7.4.4 A Waiting Game 257
7.5 Exercises......................... 259
REFERENCES 266
GLOSSARY 281
Index 285
PREFACE
As the world enters the new millennium,mankindfaces a series ofnewprob
lems, many of them created by man himself. These include overpopulation,
air and water pollution, global warming, accumulation of greenhouse gases,
darnage to the ozone layer and loss of biodiversity. Perhaps these problems
were around even earlier in an incipient stage, but they have now assumed
global proportions and are uppermost in the minds of all. A natural con
sequence is enhanced interest in sciences connected with these problems.
Ecology is a field that is immensely useful in understanding many of them.
In the seventies, nature conservation became a concern of wide sections
of society, well beyond the small group ofexpert ecologists. Species extinc
tion and depletion of biological resources were seen as major threats to
human welfare. It was therefore natural for scientists from different disci
plines to seek reasons behind these developments. We were no exceptions
and when opportunity to interact with ecologists as statistreal consultants
came, we found ourselves reading more and more of ecology and evolution
ary biology.
Several yearsago weproposed startingofan electiveone semester course
on statistical ecology for graduate students ofstatistics of Pune University.
Such a course has now been taught for about ten years, Material for the
course was borrowed from various texts, monographs and journals. It was
felt that synthesizing all this material into a cohesive textbook may help
both the teacher and the taught. Present book is a result of our endeavors
in this direction.
Looking back,it is clear that Pielou (1977) provided the basicframework
of our course but others like Seber (1973), Maynard Smith (1974,1982),
Stephens and Krebs (1986) were also used extensively. We found an emerg
ing perspective which combined studyofhypothetical and simplified ecosys
tems (rnathematical ecology) with study of statistical tools to interpret
actual field and laboratory observations (statistical ecology). Our expecta
tion is that the present book will prepare the reader to react to issues in
quantitative ecology in a broadly informed manner,
Primary audience of the book is graduate or senior undergraduate stu
dents in mathematics and statistics. They willbeable to seehow basic tools
in their disciplines can be employed to elucidate seemingly intricate issues
in ecology. The description of mathematical or statistical methods given is
usually informal and sketchy. Ecological aspects are discussed just enough
to motivate the particular quantitative technique. While it is not necessary
to know ecological phenomena in all details, a broad perspective is most
useful which is why we recommend concurrent reading ofsome ecology text
such as Collinvaux (1986). Our book should also be useful to quantitative
ecologists and managers of natural resources. We hope they will feel con-