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1991 | Book

Introduction Read chapter Read first chapter

Estimation of Mortality Rates in Stage-Structured Population

Authors: Simon N. Wood, Roger M. Nisbet

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Biomathematics

Included in: Professional Book Archive

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
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 matheamtically complete system which can be solved.
Simon N. Wood, Roger M. Nisbet
Chapter 2. Mortality Estimation Schemes Related to Stage Structured Population Models
Abstract
The formalism developed in recent years for modelling stage structured populations would appear to provide a rigorous and appealing basis from which to approach mortality estimation (van Straalen 1986). This formalism is applicable to models of organisms whose life history is made up of a sequence of clearly identifiable stages, of known but not necessarily equal duration (Gurney, Nisbet and Lawton 1983; Gurney, Blythe and Nisbet 1986). In their simplest form, these models assume that all individuals within a stage have the same vital rates (in particular per capita death rate) at a given time, and that stage duration is time-independent. The population of the jth stage, ηj(t), at time t changes through recruitment of new individuals to the stage, maturation from the stage to its successor, and death. Mathematically this implies that the stage population obeys the differential equation
$$ d\eta (t)/dt = {R_j}(t) - {M_j}(t) - {\mu_j}(t){\eta_j}(t) $$
(2.1.1)
where Rj(t) and Mj(t) are respectively recruitment rate to, and maturation rate from, stage j at time t, and μj(t) is per capita death rate in the stage. The maturation rate is related to the recruitment rate to, or the age distribution within, the stage at previous times.
Simon N. Wood, Roger M. Nisbet
Chapter 3. Cubic Splines and Histosplines
Abstract
In chapter 2 we provided an explanation for the failure of many methods intended to find time- and age-dependent mortality rates from stage structured population data. It was also suggested that an effective method of mortality estimation would be a more local method than those so far suggested. In chapter 4 we present such a method based on surface fitting with cubic spline functions. The method requires some standard techniques for fitting spline functions to noisy point data, which are outlined briefly in section 3.2. Chapter 4 also requires some extensions of spline theory to cover noisy aggregate data and covariance, which are the subjects of sections 3.3 and 3.4 respectively. Some of the development in section 3.3 is new, and section 3.5 briefly discusses some possible areas of application.
Simon N. Wood, Roger M. Nisbet
Chapter 4. Population Surfaces: A New Method of Mortality Estimation
Abstract
The approach to mortality estimation in chapter 2 was based on the formalism of stage structure models (see for example Gurney et. al. 1983), but it was shown that estimators based on such models are prone to instability when applied to less than perfect data. Therefore we now consider a less restrictive approach based directly on the von Foerster equation (see for example Nisbet and Gurney 1982, chapter 3); our aim is to produce an estimate of the age-specific death rate which is consistent with stage abundance data from a population, but which displays stability with respect to observational errors. The end point of our work is a rather elaborate recipe, detailed in section 4.8.
Simon N. Wood, Roger M. Nisbet
Chapter 5. Tests of the New Method
Abstract
In chapter 4 a new mortality rate estimation scheme was developed. This new method will now be compared to methods proposed by Parslow, Sonntag and Matthews (1979), Manly (1987) and Hay, Evans and Gamble (1988) using simulated data which for most of the comparisons, we constructed in a way that would, a priori favour the “riva” method. The two variants of the new method of chapter 4 will then be compared in a similar way.
Simon N. Wood, Roger M. Nisbet
Chapter 6. Loch Ewe Copepods: Some Speculation
Abstract
As noted in the Introduction to this monograph, our interest in problems of mortality estimation derived from a number of studies of marine copepods. In particular, we were interested in results from a series of mesocosm experiments performed by members of the DAFS Marine lab, Aberdeen in the 1970s and early 1980s. Interactions between the component populations of complex marine communities were studied by enclosing these communities in large plastic bags suspended in the Loch. In this way systems could be studied which were of far greater complexity than those which might feasibly be reconstituted in an aquarium and the enclosed systems were subjected to a physical environment similar to that found in the rest of the Loch. At the same time the system was closed making accurate sampling of species and environment easy in comparison with the open sea. For a survey of the mesocosm approach see Grice and Reeve (1982).
S. J. Hay
Chapter 7. Discussion
Abstract
In the preceding chapters we identified instabilities which largely explain the problems that bedevil mortality estimation methods for stage structured populations. We then showed how these problems can be overcome using the biologically sensible assumption that mortality rate varies slowly over the development time of a cohort. The proposed method fits stage structured population data with a solution of the McKendrick-Von Foerster equation based on spline functions. On the basis of computer simulation studies this method appears to offer substantially better results than ‘systems identification’ or the method of Manly (1987) and preliminary results with real data are encouraging.
Simon N. Wood, Roger M. Nisbet
Backmatter
Metadata
Title
Estimation of Mortality Rates in Stage-Structured Population
Authors
Simon N. Wood
Roger M. Nisbet
Copyright Year
1991
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-49979-1
Print ISBN
978-3-540-53979-7
DOI
https://doi.org/10.1007/978-3-642-49979-1