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Lecture Notes in Biomathematics 90
Managing Editor:
S. A. Levin
Editorial Board:
Ch. DeLisi, M. Feldman, J. B. Keller, M. Kimura
R. May, J. D. Murray, G. F. Oster, A. S. Perelson
L. A. Segel
Simon N. Wood Roger M. Nisbet
Estimation of Mortality Rates
in Stage-Structured Population
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
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Authors
Simon N. Wood
Centre for Population Biology
Ascot Berks, SL5 7QY, U.K.
Roger M. Nisbet
University of Strathclyde
Department of Statistics and Modelling Science
Glasgow, GI1XH, U.K.
Mathematics Subject Classification (1980): 92D40, 92D25
ISBN 978-3-540-53979-7 ISBN 978-3-642-49979-1 (eBook)
DOl 10.1007/978-3-642-49979-1
This work is subject to copyright. All rights are reserved, whether the whole or part of
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act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991
2146/3i40-54321O -Printed on acid-free paper
PREFACE
The stated aims of the Lecture Notes in Biomathematics allow for work that
is "unfinished or tentative". This volume is offered in that spirit. The problem
addressed is one of the classics of statistical ecology, the estimation of
mortality rates from stage-frequency data, but in tackling it we found ourselves
making use of ideas and techniques very different from those we expected to
use, and in which we had no previous experience. Specifically we drifted
towards consideration of some rather specific curve and surface fitting and
smoothing techniques. We think we have made some progress (otherwise why
publish?), but are acutely aware of the conceptual and statistical clumsiness of
parts of the work. Readers with sufficient expertise to be offended should
regard the monograph as a challenge to do better.
The central theme in this book is a somewhat complex algorithm for
mortality estimation (detailed at the end of Chapter 4). Because of its
complexity, the job of implementing the method is intimidating. Any reader
interested in using the methods may obtain copies of our code as follows:
Intelligible Structured Code
1. Hutchinson and deHoog's algorithm for fitting smoothing splines by cross
validation
2. Cubic covariant area-approximating splines
3. Cubic interpolating splines
4. Cubic area matching splines
5. Hyman's algorithm for monotonic interpolation based on cubic splines.
Prototype User-Hostile Code
6. Positive constrained interpolation
7. Positive constrained area matching
8. The "full method" from chapter 4
9. The "simpler" method from chapter 4.
In addition we can supply executable copies of the 3-D plotter used to
produce the figures. This will screen dump to an Epson-FX compatible printer.
Items 1-9 above are available only as Turbo Pascal (version 5.0 or 5.5)
source code. The problems in translating to other dialects of Pascal are minor
except for items 8 and 9. The code is supplied free and without any
guarantees(!). However we request payment of US $20 or 10 sterling to cover
disks, postage and hassle. Requests for code should be addressed to R.M.
Nisbet, Department of Statistics and Modelling Science, University of
Strathclyde, Glasgow G1 1XH, Scotland.
Many people contributed time and ideas to this project. Bill Gurney and
Steve Blythe collaborated in previous work based on stage structure models;
much of their insight and creativity carries over to the present work. Ann
Jones, Andre de Roos and Hans Metz contributed ideas on instability and
ill-conditioning that significantly influenced chapter 2. We thank Peter Maas
for computational advice and for a beer mat rediscovery of quadratic
histosplines! We thank John Gamble and Steve Hay (who is co-author of
chapter 6) for access to unpublished data on Loch Ewe copepods, and for
remaining supportive during a very long period over which we made little
visible progress. Comments by Hal Caswell, Saran Twombly and Nelson
Hairston at a Conference on "Future Directions in Zooplankton Biology" in
Michigan (October 1989) helped place the work in a broader perspective. We
thank Jerome Casas, Steve Hay, Joe Horwood, Hans Metz and Charles Taylor
for valuable comments at different stages of the manuscript preparation. We
particularly appreciated a thorough reading by Hans Metz who checked much
of the algebra, though taking our lead from modem terrorists we claim
responsibility for the remaining errors. Kay Johnston battled with illegible
handwriting and with text from two different word processors used on several,
subtly different computers to produce the final manuscript.
Most of the work was carried out at the University of Strathclyde and we
thank our colleagues in the Department of Physics and Applied Physics and in
the Department of Statistics and Modelling Science for their interest. The work
was completed while SW was employed first at the MAFF Fisheries
Laboratory, Lowestoft, and latterly at the Centre for Population Biology,
Imperial College. Financial support came from the UK Natural Environment
Research Council through grant GR 5210 and a studentship held by SW.
Simon Wood University of Strathclyde
Roger Nisbet October 1990
CONTENTS
CHAPTER 1: INTRODUCTION 1
1 .1 Inverse problems in population ecology 1
1.2 Copepod mortality rate estimation 2
1.3 The theoretical problems of mortality estimation 4
1.4 Existing methods for mortality estimation 4
1.5 This monograph 8
CHAPTER 2: MORTALITY ESTIMATION SCHEMES RELATED TO
ST AGE STRUCTURED POPULATION MODELS 9
2.1 Introduction 9
2.2 Preliminaries: mortality estimation for an unstructured
population with known birth rate 10
2.3 Mortality estimation for a single stage of known duration 15
2.4 Instabilities associated with mortality estimators 16
2.5 Discussion 21
CHAPTER 3: CUBIC SPLINES AND HISTOSPLINES 23
3.1 Introduction 23
3.2 A brief guide to splines 23
3.3 Cubic splines for exact or noisy histogram data 30
3.4 Choosing the smoothing parameter with covariant error
terms 33
3.5 Two examples of the application of cubic area splines to
age-structured population data 34
CHAPTER 4: POPULATION SURFACES: A NEW METHOD OF
MORT ALITY ESTIMATION 40
4.1 The structured population model 40
4.2 Preliminary estimation of population surface f(a,t) 41
4.3 Characteristics of f(a,t) 44
4.4 Error estimates 47
4.5 What to do about adults 49
4.6 A simpler method 49
4.7 Messy practicalities 49
4.8 Summary 52
CHAPTER 5: TESTS OF THE NEW METHOD 54
5.1 Parslow and Sonntag's Lag-Manly model method 54
5.2 The method of Hay, Evans and Gamble 60
5.3 Manly's (1987) method 66
5.4 Comparison and testing of the sophisticated and
simplified versions of the new method 68
5.5 Discussion 76
CHAPTER 6: LOCH EWE COPEPODS: SOME SPECULATION 77
(Written with S.l. Hay)
6.1 The Loch Ewe mesocosm experiments 77
6.2 Stage durations 78
6.3 Modelling sampling error 83
6.4 Death rate patterns for the different copepod species
in bag C2: comparison and speculation 87
CHAPTER 7: DISCUSSION 92
7.1 What we've done 92
7.2 Experimental suggestions 93
7.3 Production estimation 94
7.4 Methodological improvements 94
7.5 The take home message 95
APPENDIX 96
REFERENCES 97
CHAPTER 1
INTRODUCfION
1.1 Inverse problems in population ecology
A major aim of population ecology is to explain observed patterns of plant
and animal abundance. At the single species level, one approach is to
construct mathematical models of abundance whose parameters are estimated in
a manner quasi-independent of the data which it is hoped to explain.
Disagreement between such a model and observed population data implies that
the model is false, agreement that it may be correct. Mathematically, this
approach to explaining abundance yields a 'direct' problem: the model and its
parameter estimates constitute a mathematically complete system which can be
solved.
The direct approach has been used successfully by many workers to interpret
data on laboratory populations, and has potential as a tool for linking
laboratory and field data (e.g. Nisbet et ai., 1989). However there are many
difficulties, of which those associated with mortality estimation are possibly the
most fundamental. Plants and animals die from a worryingly diverse set of
causes; if we have to try and explain a set of population data by hypothesising
too many (poorly understood) causes of mortality, then there is a good chance
of obtaining some very trendy dynamics without very much insight.
One way of by-passing the difficulties of the direct approach is to try and
solve an 'inverse problem' . Given some fairly general modelling assumptions
can we extract patterns of recruitment and mortality most likely to explain a
given set of population data? Unlike the direct problem, such problems are
usually underconstrained.
Ecological inverse problems answer different questions to those requiring the
direct approach. A mortality pattern derived by solving an inverse problem
may "explain" a set of population data in the sense of enabling a
reconstruction of the original data which is not too offensive to a friendly
statistician, but on its own it says nothing about the physiological and
environmental factors which are responsible for this mortality. However there
are many situations (e.g. production calculations) in which ecologists want a
quantitative characterisation of mortality rates which is independent of any
assumption about the root causes. Such information may of course be used
subsequently to identify possible causes, for example by relating zooplankton
death rates to growth rates of larval fish.
Some inverse problems are hopelessly underdetermined. For example, any
population trajectory can be produced by an infinite family of time-dependent
per capita birth and death rates. Mathematically this can be seen by
considering a population, 1)(t), described by
