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1990 | Book

Introduction Read chapter Read first chapter

Population Dynamics in Variable Environments

Author: Shripad Tuljapurkar

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Biomathematics

Included in: Professional Book Archive

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About this book

Demography relates observable facts about individuals to the dynamics of populations. If the dynamics are linear and do not change over time, the classical theory of Lotka (1907) and Leslie (1945) is the central tool of demography. This book addresses the situation when the assumption of constancy is dropped. In many practical situations, a population will display unpredictable variation over time in its vital rates, which must then be described in statistical terms. Most of this book is concerned with the theory of populations which are subject to random temporal changes in their vital rates, although other kinds of variation (e. g. , cyclical) are also dealt with. The central questions are: how does temporal variation work its way into a population's future, and how does it affect our interpretation of a population's past. The results here are directed at demographers of humans and at popula­ tion biologists. The uneven mathematical level is dictated by the material, but the book should be accessible to readers interested in population the­ ory. (Readers looking for background or prerequisites will find much of it in Hal Caswell's Matrix population models: construction, analysis, and in­ terpretation (Sinauer 1989) ). This book is in essence a progress report and is deliberately brief; I hope that it is not mystifying. I have not attempted to be complete about either the history or the subject, although most sig­ nificant results and methods are presented.

Table of Contents

Frontmatter
1. Introduction
Abstract
Demography relates observable facts about individuals to the dynamics of populations. If the dynamics axe linear and do not change over time, the classical theory of Lotka (1907) and Leslie (1945) is the central tool of demography. This book addresses the situation when the assumption of constancy is dropped. In many practical situations, a population will display unpredictable variation over time in its vital rates, which must then be described in statistical terms. Most of this book is concerned with the theory of populations which are subject to random temporal changes in their vital rates, although other kinds of variation (e.g., cyclical) are also dealt with. The central questions are: how does temporal variation work its way into a population’s future, and how does it affect our interpretation of a population’s past.
Shripad Tuljapurkar
2. Beginnings: Classical Theory
Abstract
Most of this book deals with demography in discrete time, i.e., time is divided into successive intervals of some equal length τ, and the times τ, 2τ, etc. are labeled as discrete times 1, 2, etc. Demographers usually drop the τ for convenience, as will I, but note that this practice obscures the dimensions (i.e., units) of all the objects in our equations. This chapter summarizes demographic theory in discrete time with constant rates, emphasizing terminology and results which I later use freely without further explanation. For details consult Keyfitz (1968) or the more compact account of Pollard (1973). Coale’s (1972) book uses continuous time but is valuable.
Shripad Tuljapurkar
3. Deterministic Temporal Variation
Abstract
I now consider what happens when the fixed matrix b of (2.1.2) is replaced by a deterministically varying sequence of matrices. This is a long-standing problem with Norton (1928), Coale (1957), and Lopez (1961) being the classical contributions. Golubitsky et al. (1977), Hajnal (1976), Kim and Sykes (1976), Seneta (1981), Cohen (1979b), Tuljapurkar (1984) and Kim (1987) are more recent explorations. This work is a prerequisite to the study of random rates. I first consider general variation, and then cyclical variation.
Shripad Tuljapurkar
4. Random Rates: General Theory
Abstract
Demographic theory with random vital rates is built on powerful general properties of random matrix products. This chapter sets out the kinds of random models I want to analyze, and summarizes general random matrix properties. Later chapters consider applications and questions which require a more concrete study of particular models.
Shripad Tuljapurkar
5. Examples
Abstract
There is increasing interest in the impact of environmental variability on vital rates. I summarize below examples in which such variation has actually- been measured. The examples include plants and animals and point to a growing recognition that such data are needed and useful. However, most of the data are limited in their sampling both of environments and of the transition structure between environments.
Shripad Tuljapurkar
6. ESS and Allele Invasion
Abstract
Population biologists have long been concerned with the evolution of demographic vital rates, and so with the analysis of combined genetic-demographic models. One approach to such models is to study ESS (evolutionary stable state) criteria (e.g., Charlesworth 1980): consider a genetic model in which two alleles at a single diploid locus affect the life history phenotypes in a randomly mating population. Suppose the population is initially homozygous for one allele and introduce the other allele at low frequency. Then ask: what conditions determine the invasion (increase in frequency) of the rare allele? The answer identifies the natural “fitness” measure for a discussion of the evolution of life histories. The analysis for sexually reproducing populations is different from that in which clonal reproduction is also possible, and I treat these separately below.
Shripad Tuljapurkar
7. Moments of the Population Vector
Abstract
The basic population equation (4.1.1) is linear and this linearity ought to be of some use. This chapter shows that for both the I.I.D. and the (finite) Markov models, the linearity allows us to compute moments of the population vector.
Shripad Tuljapurkar
8. Random Survival or Fertility: Exact Results
Abstract
A model which ought to be simple is one with 2 age classes and random survival or fertility. In this chapter we will see that this is only sometimes true depending on the question one asks. The examples should breathe some life into the general theory.
Shripad Tuljapurkar
9. Age Structure: Bounds, Growth, Convergence
Abstract
The analysis of age structure remains of central interest to many demographers and I present here some useful results: bounds on age structure in the presence of random rates, a relatively simple formula for growth rate a, and an analysis of periodic matrices.
Shripad Tuljapurkar
10. Synergy, Constraints, Convexity
Abstract
This book deals with many of the differences between the multidimensional dynamics of structured populations and those of scalar growth models. It is important to identify those features of the structured case which differ markedly from the scalar case. The first two sections below show that the differences can be quite considerable. Section 1 discusses the effect of autocorrelation. Section 2 presents an example where serially uncorrelated random variation raises the growth rate of a population above its possible deterministic growth rate. The third section summarizes a potentially useful result of Cohen concerning the parametric sensitivity of stochastic rates. The last section shows how strong constraints on population vital rates can lead to “scalar” behavior.
Shripad Tuljapurkar
11. Sensitivity Analysis of Growth Rate
Abstract
A central question in demography is: how do vital rates interact to determine growth rate and population structure? The answer is a bit complicated even in the deterministic case, because there is no analytical formula giving r in terms of vital rates. Ecologists and demographers have resorted to studying how r changes when vital rates change (Lewontin 1965, Keyfitz 1968, 1977, Caswell 1978, Arthur 1980). This amounts to studying the derivatives of r, and is called sensitivity analysis (Caswell 1978). The stochastic analog is the sensitivity analysis of the long-run growth rate a. In this chapter I begin with a quick summary of sensitivity analysis for r, and then set out the sensitivity analysis for a.
Shripad Tuljapurkar
12. Growth Rates for Small Noise
Abstract
The long-run growth rate a is central to questions of evolution (Chapter 6), prediction (Chapter 14) and extinction. However, it is only useful if we can describe how vital rates and uncertainty determine a. This is difficult because there is no general formula to compute a for arbitrary vital rates and variability. In addition, the exactly known cases of a (Chapter 8) do not generalize; worse, they reveal singular behavior near parameter limits where demographic ergodicity is lost. One useful and general approach is to develop a systematic approximation to a when the magnitude of random variation is small. This was done by Tuljapurkar (1982b) and the results have since been applied to a number of ecological and demographic problems. The method and some extensions are presented below.
Shripad Tuljapurkar
13. Population Structure for Small Noise
Abstract
This chapter extends the expansion method of Chapter 12 to the population structure vector Y t and the reproductive value vector V t As a byproduct of this analysis we get information on the growth rate of population over time, and on the serial correlation structure of populations over time in a varying environment. We learn how the history of environmental perturbations is filtered by population response. The first section below deals with the method itself while later sections consider implications of the results.
Shripad Tuljapurkar
14. Population Projection
Abstract
Projection (or forecasting) is the first of the applications of the preceding theory to be discussed in this book. In discussing applications I aim to highlight the central issues and the most promising approaches for dealing with them. I will however be very selective.
Shripad Tuljapurkar
15. Life History and Iteroparity
Abstract
Fisher (1930) posed the central question about life histories, asking how the apportionment of reproduction over life might have evolved. Cole (1954) framed the style of many recent studies in comparing the evolutionary advantage of semelparity (reproducing once) and iteroparity (reproducing more than once). Since then there has been considerable work on general features of life history evolution (e.g., Williams 1966, the review by Stearns 1976, Begon, Harper and Townsend 1986), and on features specific to certain species or genera (e.g., Denno and Dingle 1981, Jackson, Buss and Cook 1985). It is clear from Lewontin (1965) that classical demographic arguments can only account for some of the Life historical patterns in nature, and later workers have tried to incorporate factors outside the classical framework. One of these is an environment which produces random variations in vital rates and thereby generates selection pressures on life histories.
Shripad Tuljapurkar
16. Life History Evolution: Delayed Flowering
Abstract
Chapter 5 mentioned the work of Klinkhammer and de Jong on the advantages of delayed flowering in biennials. The species under consideration (such as Cirsium vulgare) reproduce by flowering only once and then die. The strict biennial habit involves flowering in the second year of life, but delayed flowering of up to several years is commonly observed. Klinkhammer and de Jong (1983) argued that delayed flowering is akin to iteroparity and evolved as a response to variable reproductive success; they used a computer simulation model. Roerdink (1987, 1989) analyzed their model using the theory of stochastic demography. I discuss below his work and some related questions; this analysis complements the more general discussion of the preceding chapter.
Shripad Tuljapurkar
Backmatter
Metadata
Title
Population Dynamics in Variable Environments
Author
Shripad Tuljapurkar
Copyright Year
1990
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-51652-8
Print ISBN
978-3-540-52482-3
DOI
https://doi.org/10.1007/978-3-642-51652-8