Integrated Population Models: Theory and Ecological Applications with R and JAGS

By Michael Schaub and Marc Kéry

Integrated Population Models: Theory and Ecological Applications with R and JAGS is the first book on integrated population models, which constitute a powerful framework for combining multiple data sets from the population and the individual levels to estimate demographic parameters, and population size and trends. These models identify drivers of population dynamics and forecast the composition and trajectory of a population.

Written by two population ecologists with expertise on integrated population modeling, this book provides a comprehensive synthesis of the relevant theory of integrated population models with an extensive overview of practical applications, using Bayesian methods by means of case studies. The book contains fully-documented, complete code for fitting all models in the free software, R and JAGS. It also includes all required code for pre- and post-model-fitting analysis.

Integrated Population Models is an invaluable reference for researchers and practitioners involved in population analysis, and for graduate-level students in ecology, conservation biology, wildlife management, and related fields. The text is ideal for self-study and advanced graduate-level courses.

• Offers practical and accessible ecological applications of IPMs (integrated population models)
• Provides full documentation of analyzed code in the Bayesian framework
• Written and structured for an easy approach to the subject, especially for non-statisticians

• Language: English
• Publisher: Academic Press
• Release date: Nov 12, 2021
• ISBN: 9780128209158

Michael Schaub

Michael Schaub is the Head of the Ecology Department at the Swiss Ornithological Institute and a courtesy Professor at the University of Bern. His research interests include population dynamics, capture-recapture models, integrated population models, and migratory birds. He has coauthored approximately 130 peer-reviewed journal publications and the book Bayesian Population Analysis using WinBUGS.

Book preview

Integrated Population Models

Theory and Ecological Applications with R and JAGS

Michael Schaub

Swiss Ornithological Institute, Sempach, Switzerland

Marc Kéry

Swiss Ornithological Institute, Sempach, Switzerland

Table of Contents

Cover image

Title page

Copyright

Foreword

Preface

Acknowledgments

Chapter 1. Introduction

1.1. Population Modeling in Population Ecology and Management

1.2. The Two-Step Approach to Population Modeling

1.3. Integrated Population Models

1.4. Developing Integrated Population Models with the BUGS Language

1.5. This Book

Part 1. Theory of Integrated Population Models

Chapter 2. Bayesian Statistical Modeling Using JAGS

2.1. Introduction

2.2. Parametric Statistical Modeling

2.3. Maximum Likelihood Estimation in a Nutshell

2.4. Bayesian Inference

2.5. Bayesian Computation

2.6. BUGS Software: WinBUGS, OpenBUGS, JAGS, and NIMBLE

2.7. Using JAGS to Fit Simple Statistical Models from R: Generalized Linear and Generalized Linear Mixed Models

2.8. Fitting General Integrated Models in JAGS

2.9. Why We Have Become Bayesians…

2.10. Summary and Outlook

2.11. Exercises

Chapter 3. Introduction to Stage-Structured Population Models

3.1. Introduction

3.2. Age- and Stage-Structured Population Models

3.3. Classical Analysis of a Matrix Population Model

3.4. Analysis of Matrix Population Models With Markov Chain Monte Carlo Software

3.5. Summary and Outlook

3.6. Exercises

Chapter 4. Components of Integrated Population Models

4.1. Introduction

4.2. Overview of the Key Types of Data and Associated Models That Go Into an IPM

4.3. Models for Population Size Surveys

4.4. Models for Productivity Surveys

4.5. Models for Survival Surveys

4.6. Introduction to Spatial Capture-Recapture Modeling

4.7. Summary and Outlook

4.8. Exercises

Chapter 5. Introduction to Integrated Population Models

5.1. Introduction

5.2. Feeding Demographic Data into the Analysis of a Matrix Population Model

5.3. Our First Integrated Population Model

5.4. The Three-Step Approach to Integrated Population Modeling

5.5. Simulation Assessment of a Simple Integrated Population Model

5.6. Outlook and Summary

5.7. Exercises

Chapter 6. Benefits of Integrated Population Modeling

6.1. Introduction

6.2. Parameter Estimates With Increased Precision

6.3. Estimation of Demographic Parameters for Which There Are No Explicit Data

6.4. Estimation of Process Correlation Among Demographic Parameters

6.5. Estimation of Population Structure

6.6. Flexibility

6.7. Summary and Outlook

6.8. Exercises

Chapter 7. Assessment of Integrated Population Models

7.1. Introduction

7.2. Assumptions of Integrated Population Models

7.3. Under- and Overfitting

7.4. Effects of a Misspecified Observation Model

7.5. Outlook and Summary

7.6. Exercises

Chapter 8. Integrated Population Models With Density Dependence

8.1. Introduction

8.2. Density Dependence in Red-Backed Shrikes

8.3. Advantages of IPMs for the Study of Density Dependence

8.4. Summary and Outlook

8.5. Exercises

Chapter 9. Retrospective Population Analyses

9.1. Introduction

9.2. Correlations Between Demographic Rates and Population Growth

9.3. Life-Table Response Experiments

9.4. Transient Life-Table Response Experiments

9.5. Summary and Outlook

9.6. Exercises

Chapter 10. Population Viability Analysis

10.1. Introduction

10.2. Challenges for Demographic Population Viability Analysis

10.3. Bayesian Population Viability Analysis

10.4. Use of an Integrated Population Model in Population Viability Analysis

10.5. A Population Viability Analysis for Simulated Woodchat Shrike Data

10.6. Population Viability Analysis of a Population With Immigration

10.7. Summary and Outlook

10.8. Exercises

Part 2. Integrated Population Models in Practice

Chapter 11. Woodchat Shrike

11.1. Introduction

11.2. Data Sets

11.3. Population Model

11.4. Component Data Likelihoods

11.5. The Integrated Population Model

11.6. Results

11.7. More Parsimonious Models

11.8. Discussion

Chapter 12. Peregrine Falcon

12.1. Introduction

12.2. Data Sets

12.3. Population Model

12.4. Component Data Likelihoods

12.5. The Integrated Population Model

12.6. Results

12.7. Discussion

Chapter 13. Horseshoe Bat

13.1. Introduction

13.2. Data Sets

13.3. Population Model

13.4. Single Data Likelihoods

13.5. The Integrated Population Models

13.6. Results

13.7. Prior Sensitivity Analysis

13.8. Discussion

Chapter 14. Hoopoe

14.1. Introduction

14.2. Data Sets

14.3. Population Model

14.4. Component Data Likelihoods

14.5. Integrated Population Model

14.6. Results

14.7. Discussion

Chapter 15. Black Grouse

15.1. Introduction

15.2. Data Sets

15.3. Population Model

15.4. Component Data Likelihoods

15.5. Integrated Population Model

15.6. Results

15.7. Discussion

Chapter 16. Barn Swallow

16.1. Introduction

16.2. Data Sets

16.3. Population Model

16.4. Component Data Likelihoods

16.5. The Integrated Population Model

16.6. Results

16.7. Discussion

Chapter 17. Elk

17.1. Introduction

17.2. Elk in Idaho

17.3. Population Model

17.4. Component Data Likelihoods

17.5. The Integrated Population Model

17.6. Results on Elk Population Dynamics

17.7. Prior Sensitivity Analysis

17.8. Specification of the Survival Process with Hazard Rates

17.9. Discussion

Chapter 18. Cormorant

18.1. Introduction

18.2. Data Sets

18.3. Population Model

18.4. Component Data Likelihoods

18.5. The Integrated Population Model

18.6. Results

18.7. Discussion

Chapter 19. Gray Catbird

19.1. Introduction

19.2. Data Sets

19.3. Population Model

19.4. Component Data Likelihoods

19.5. The Integrated Population Model

19.6. Results

19.7. Discussion

Chapter 20. Kestrel

20.1. Introduction

20.2. Data Sets

20.3. Population Model

20.4. Component Data Likelihoods

20.5. The Integrated Population Model

20.6. Results

20.7. Discussion

Chapter 21. Black Bear

21.1. Introduction

21.2. Data Sets

21.3. Population Model

21.4. Component Data Likelihoods

21.5. The Integrated Population Model

21.6. Results

21.7. Discussion

Chapter 22. Conclusions

22.1. Fitting Integrated Population Models: A Steep Mountain … but One That's Really Worth the Climb!

22.2. Should We Always Integrate?

22.3. The Great Importance of Long-Term Ecological Research

22.4. Future Directions in Integrated Population Modeling

22.5. Concluding Remarks

References

Author Index

Subject Index

Copyright

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Foreword

Population biology is a substantial discipline composed of two important subdisciplines, population genetics and population ecology. Population genetics and closely related fields ultimately provide a bridge to molecular and organismal processes by revealing the mechanisms responsible for genetic differences among individuals within and between populations. Population ecology, on the other hand, is the study of the mechanisms responsible for changes in the distribution and abundance of individuals over space and time. These evolutionary and ecological topics are in fact interrelated. To paraphrase an eloquent statement by G. Evelyn Hutchinson, the individuals of a population acting within an ecological theatre can affect the forces of natural selection for individuals and the evolution of the population (the company of actors). These eco-evolutionary dynamics affect lower organizational levels of biology and are essential ingredients that influence higher levels of organization such as biological communities and ecosystems.

At first glance, the primary parameters affecting the distribution and abundance of individuals over space and time are so simple that a troglodyte could be a population ecologist. In fact, when I attempt to explain my work as a wildlife population ecologist to family and friends, their responses are usually something along the lines of Oh, you count animals for a living. Sigh—a stark reminder to work on my elevator talk! I say this because when expressed as rates, the abundance of individuals at a particular location x and point in time t (N x,t ) is simply a function of the preceding local b irth rate, the rate at which individuals i mmigrate into the location from elsewhere, the local d eath rate, and the rate at which individuals e migrate out of the location to other locales, which collectively act on local abundance at the previous point in time t −1: N x,t   =   N x,t −1   +   ( b x,t −1   +   i x,t −1   −   d x,t −1   −   e x,t −1)N x,t −1. When this simple discrete-time equation is rearranged, the balance of these bide demographic parameters defines the geometric rate of population growth from one time step to the next: N x,t /N x,t −1   =     =   1   +   b x,t −1   +   i x,t −1   −   d x,t −1   −   e x,t −1. However, the dynamics of some (e.g., birth-flow) populations are better described in continuous time, and the demographic parameters and abundance state variable might importantly be structured by age, developmental stage, size, phenotype, or even genotype. These states of individuals might be measured continuously or with discrete categories, and in the real world, all demographic parameters are influenced by a large suite of biotic, abiotic, and anthropogenic forces. Too many, in fact, to estimate everything, and we must always deal with multiple uncertainties when making inferences about population dynamics. Alas, the job of a population ecologist is extremely challenging!

Population ecology is a mature subdiscipline of biology with a rich theory set and empirical findings to back it up. Nevertheless, there are many unknowns in population ecology and perhaps an equally great number of misunderstandings. Given that community and even ecosystem-level processes hinge on population-level processes, this should be concerning. Can we pretend to seek truths about these higher levels of biological organization by throwing demographic mechanisms under the rug and instead chase patterns in the outcomes at these higher levels? I think not. Can we work toward decreasing uncertainty and enhancing our understanding of demography to achieve a more mechanistic understanding of how multiple levels of biological organization are connected? I think so.

If we desire less uncertainty and greater understanding, population ecology must spread its wings as a quantitative force of nature while not forgetting its roots in natural history. How can we achieve this in such challenging times when all we want is to daydream of a better tomorrow? Well, I invite you to daydream of a magical place near the foot of the Swiss Alps—a small village along a glistening lake—and if you wish, grab a virtual bottle of local Riesling and a brick of virtual cheese to enjoy while viewing wintering red-crested pochards or after a long summer hike in meadows of wildflowers. Here you will find two gentlemen who work at the Schweizerische Vogelwarte, seemingly ordinary in every way, except they are most generous with their time and possess a genius for teaching the masses how to become better quantitative population ecologists. Following on the heels of their highly acclaimed book, Bayesian Population Analysis Using WinBUGS: A Hierarchical Perspective, Michael Schaub and Marc Kéry bring to us this blockbuster, Integrated Population Models: Theory and Ecological Applications Using R and JAGS.

Integrated population models (IPMs) are statistical models of population dynamics as opposed to mathematical models. Rather than estimating each bide parameter separately and then plugging them into my simple equation above to project abundance from a starting value, IPMs integrate available time-series data for demographic parameters and abundances. That is to say, IPMs combine data for both sides of the equation. This approach is much more efficient and seamless for estimation and modeling, with many advantages over traditional piecemeal approaches to population modeling. In practice, IPMs are often applied to structured populations, such as those structured by discrete ages or developmental stages, and are intricately connected to matrix population models. By providing estimates of the population structure and how it changes over time (and space), IPMs are ideal for empirically examining transient and stochastic dynamics. Moreover, IPMs are used to conduct prospective and retrospective perturbation analyses, scenario projections, and population forecasting; identify metapopulation sources and sinks, density dependence, natural selection, and eco-evolutionary dynamics; and inform harvest management, habitat management, and spatial conservation prioritization. They've even been extended to include demographic interactions among multiple species and to model community dynamics. The other IPM commonly used to model continuously structured populations is the integral projection model and can be implemented as an integrated population model, thus forming an IIPM (or IPM²). The possibilities of IPMs are limited only by our creativity and computing skills.

Michael Schaub and Marc Kéry's new book will help you expand your population modeling creativity and computing skills to transform this creativity into science. Building on Bayesian Population Analysis, which provided in-depth coverage of Bayesian demographic models that can be used as IPM submodels (e.g., mixed, state-space, capture-mark-recapture, and occupancy models), Integrated Population Models provides broad coverage of the underlying theory behind IPMs, including the connection to matrix population models, and a range of formal analyses that can be performed with IPMs. The book then covers a wide array of IPMs applied to real-world data, such that we can get a feel for the art of modifying an IPM to address particular objectives, available types of data, and any shortcomings of the data (e.g., missing data). With complete sets of how-to computer code provided from start to finish, the reader is never left hanging. The work of Michael and Marc, and the generosity of their time, helped reinvigorate my career, for which I will be forever thankful. I hope you will find this new book equally exciting and helpful in your career.

David N. Koons

Colorado State University

Fort Collins, Colorado

January 2021

Preface

Demographic models to understand population changes and the mechanisms underlying them and to predict these changes into the future represent a key analytical technology in an extensive range of biological sciences, including ecology, conservation biology, wildlife biology, and wildlife management. Often, multiple sources of information are available about the demography of a study species and the changes in the population size over time. A smart approach then is to exploit all the available information in a single statistical model. The joint analysis of different data sets that are informative about the demography and population dynamics of a species is the topic of integrated population models (IPMs) and of this book.

IPMs were first developed in fisheries science about 40 years ago (Fournier and Archibald, 1982) and were then introduced in terrestrial ecology in a seminal Biometrics paper by Besbeas et al. (2002). Since then, IPMs have become a very active area of research in statistical ecology and have been employed in an ever-increasing number of application papers at the interface of ecology/wildlife management and statistical science (Zipkin and Saunders, 2018). However, from the beginning, the uptake of IPMs by practitioners has been slow, despite a seminal paper in 2004 by Brooks et al. (2004) that demonstrated the first Bayesian implementation of an IPM in the widely used BUGS software. This opened up the possibility for ecologists to fit their own IPMs, something also documented by the first paper on the topic by one of us (Schaub et al., 2007). Over the last 10 years, there has been vigorous growth in IPM applications, most of them implemented in some variant of BUGS software (for an earlier review, see Schaub and Abadi, 2011; and more recently, Zipkin and Saunders, 2018). Nevertheless, it must have proven difficult or even impossible for most ecologists to translate the often highly technical descriptions of an IPM in the primary literature, such as in Biometrics, Ecology, or Methods in Ecology and Evolution, into an actual analysis in R or BUGS for their own data. We believe this is so even nowadays when code for analyses is increasingly published in electronic appendices to published papers. Our book provides you with that translation for a large number of case studies, and it attempts a comprehensive synthesis of the current state of research on IPMs. We honestly hope it will empower you to apply IPMs that interest you to data sets from your study.

In addition, owing to the vigorous growth of the IPM field, it has become difficult to obtain an overview of the burgeoning number of different classes, methods, and goals of IPM-based analyses. Despite several published reviews (Schaub and Abadi, 2011; Maunder and Punt, 2013; Zipkin and Saunders, 2018), to date, a comprehensive synthesis of IPMs does not exist, and our book aims to fill that gap.

This book gives broad coverage of the theory of this powerful class of models and therefore should appeal to statisticians and quantitative ecologists who wish to obtain an overview of this field. But at the same time, it is an applied book that aims to empower ecologists, wildlife managers, and similar professionals to fit their own IPMs with confidence. Therefore, an important part of the book consists of almost a dozen fully worked and complete example analyses presented in a richly commented tutorial style. Thus, we emphasize learning by doing and we invite you to work through fully documented example analyses. From teaching our workshops and our own progress in understanding statistical models, including IPMs, it is our experience that this is by far the best way for ecologists to learn and understand a new technique or statistical model. Our collection of fully worked examples will first help you to understand a model and then serve as a template for your own analyses, which you can start from our code and then modify to your needs. This, by the way, is exactly how most statistical analyses are conducted in practice anyway. We very rarely write code for a model from scratch, but rather, we typically start from a template for a similar model that we may have written earlier, was given to us by a colleague, or we took from the appendix of a paper, and we then adapt the code for our case. Our book also shows how IPMs can be used to understand past and predict future population dynamics, thus synthesizing various important topics in population ecology. More specifically, our book has the following emphases:

• It covers the theory of IPMs and shows how data sampled at the population level (e.g., population count data) and those from the individual level (e.g., capture-recapture data) can be jointly analyzed in an IPM to make better inferences about population processes and more accurately predict the future trajectory of a population.

• Matrix population models are used often in population ecological studies to compute life history summaries, carry out retrospective population analyses, or analyze population viability. The classic manner of conducting such analyses is by a two-step procedure: first, estimates of various demographic parameters such as survival are obtained from different data sets in separate analyses, and second, these parameter estimates are fed into a matrix population model. This two-step procedure is clumsy and can be statistically inefficient. This book explains the basics of classic matrix population models and then shows how the matrix population model in an IPM is seamlessly embedded into an estimation framework. That is, in an IPM, the population analysis and parameter estimation steps of traditional applications of matrix population models are combined in a single analysis. We demonstrate the multiple benefits that result.

• The book serves as a manual for practicing population analysts and a graduate-level text for students in ecology, conservation biology, wildlife management, and related fields. Specifically, the book makes the fitting of IPMs easily accessible to practitioners and graduate students with an accessible and not very math-heavy style of presentation and many tutorial-style illustrations of the analyses possible with an IPM in a wide range of case studies. These fully worked examples serve as a template to start your own analyses and can be modified for your own needs. Thus, our book builds a bridge between the typically far more technical descriptions of IPMs in the primary literature and the practitioners who want to use them to answer questions of scientific interest or management importance. It contains fully documented, cookbook-style analyses using R and the Bayesian modeling software JAGS and thus is a textbook on IPMs fit with R and JAGS software. So far, no canned software exists to fit IPMs (and quite likely there never will be), and over the last 15 years, software that adopts the BUGS language such as WinBUGS, JAGS and NIMBLE has become the standard manner in which IPMs are fit.

Michael has gained much experience with IPMs over the last 15 years in countless IPM applications and simulation studies, resulting in more than 20 papers on the topic thus far. Marc has a great deal of experience with Bayesian statistical modeling. Together, we wrote a successful earlier book, Bayesian Population Analysis (Kéry and Schaub, 2012), that is similar in style and concept. Therefore, we have strived to replicate that book’s successful recipe in our new book on the topic of IPMs. In preparing to write this book, we taught more than 10 workshops on IPMs with a total audience of about 400 people. The experience we gained during those workshops, and the questions and comments we received, were essential contributions to this book. Although we hope this book will also be of interest to statisticians who want to learn about IPMs and population modeling in general, it was primarily written by ecologists for ecologists. Perhaps this is a natural result, but we have written the book we would like to have had when we started with the topic of integrated population modeling.

Who Should Read This Book?

We hope that any quantitative population ecologist who conducts empirical studies on population dynamics can greatly benefit from reading this book because it will foster a better understanding of a whole range of analytical methods for estimating demographic rates and population sizes. Even if you do not actually carry out an IPM, we think there is much to be learned from our book that will benefit your demographic analyses of a natural population.

As a professor in ecological research or related disciplines, including natural resource management, you will have to choose a book on which to base your class on modern population modeling using matrix population models and related approaches. You may have heard that in the new wave of these models, the matrix modeling part and the parameter estimation part can be unified into a single framework and that the result is called an IPM. You will have questions such as What is a comprehensive yet easy to understand text that presents both the theory and the practical application of these powerful models, ideally with open-source software such as R, in a way that my students, often lacking adequate mathematical background and with limited software experience, can still understand? We believe that our book fully addresses your concerns.

Finally, as a wildlife or fisheries manager, you may be working at a national park, government agency, or a similar place where you are in charge of monitoring and modeling natural resources. Perhaps you need to know, "How many individuals from my population can be harvested in a sustainable way?, Which is the best strategy to reverse the negative population trend of an endangered species into a positive one?, What is the best eradication strategy for an invasive weed?, How long is a small population likely to survive? or What is the main demographic driver for the decline of a given species?" In all cases, our book will allow you to confidently apply an IPM to your data to answer these and related questions.

In particular, we hope that our book and the techniques we cover will receive a much wider uptake than hitherto in the management of hunted species, i.e., game management. Whenever we are in the United States to teach, we are impressed by the quantitative rigor of wildlife management on both the monitoring and the management side. For instance, it appears to us that in the U.S. population assessments without rigorous incorporation of false-negative errors (i.e., those arising from imperfect detection) are rare and that decision analyses in the management of hunted species are typically based on a formal population viability analysis. Also, adaptive harvest management is much more widely conducted in the U.S. We have no doubt that our book will be widely useful in such an environment where formal, quantitative population assessments are de rigueur, but we sincerely hope the ideas about quantitative population analysis that our book espouses will become much more widely adopted in similar circles, especially in Europe, including Switzerland.

Conventions in This Book

The book is composed of two parts, which start after the introductory Chapter 1. Part 1 covers the theory of IPMs, including required knowledge about Bayesian inference, Bayesian computing, matrix population models, and many examples of what we call the ingredient models of an IPM. We introduce IPMs in two conceptually slightly different ways in the hope that you find at least one of them appealing and insightful. We highlight the benefits of IPMs but also stress their limitations and discuss the possible consequences when model assumptions are not sufficiently well met. Most of the time, we develop and use an IPM for our model species—the beautiful woodchat shrike, which also adorns the cover of our book. We think it is helpful to use the same study system consistently when explaining different topics because attention on a specific topic becomes more focused. We show how using an IPM can result in a better understanding of population dynamics and can be invaluable for more accurate population forecasting. Part 2 of the book is a rich collection of case studies, most of which have already been published by our colleagues or by us. Finally, we have a concluding chapter where we attempt a brief synthesis and offer our view about the likely future developments of the IPM enterprise.

In its style and concept, this book is similar to other books written by Marc (Kéry, 2010; Kéry and Royle, 2016, 2021) and especially our Bayesian Population Analysis book (Kéry and Schaub, 2012). The key style concepts are the following:

• We provide code for all analyses and figures on our book web page (see below). However, all the essential code is also printed in the book.

• We have a clear and consistent layout for all analyses, and we use fixed-width font for all code (i.e., this is how we write R and BUGS code). In addition, all complete BUGS code appears in a light-grey box (see below).

• We use the free software R and JAGS for all data preparation, model fitting, and output processing.

• All statistical models are specified using the powerful BUGS language, used also by other Bayesian software including NIMBLE, OpenBUGS, and WinBUGS. Hence, alternative software could be used for model fitting with hardly any modifications of the code.

• We often show how to simulate data to foster a clearer understanding of the analyzing model. Indeed, we claim that if you can simulate data under a certain model, you likely understand that model.

• We aim for light and engaging language.

We use models a great deal in this book, and there are different kinds. First, we have mathematical models such as matrix population models (Chapter 3). They do not attempt to estimate quantities but simply produce output based on some input parameters and calculations, often in a what-if spirit. Many statistical models are available for estimating parameters based on your data. Such models are, for example, a Poisson regression model (Chapter 2), a state-space, a Cormack-Jolly-Seber, occupancy, or an N-mixture model (Chapter 4). An IPM also belongs in that category, but it is composed of multiple, connected statistical models. We often refer to the latter as submodels or component models of an IPM, and related to that, we may write of single-data likelihood or component data likelihood. Finally, we also use model to describe the spatial or temporal patterns in a parameter. For example, for survival in a capture-recapture model, we may adopt a model with temporal random effects or a linear trend. Hence, the meaning of the word model can be manifold, but we think it should become clear from the context.

Computing

Throughout this book, we use R (R Core Team, 2018) and JAGS (Plummer, 2003) for data management, model fitting, and output processing, including the drawing of figures. JAGS uses the simple but very powerful BUGS language, which is also implemented in the programs WinBUGS (Lunn et al., 2013), OpenBUGS (Thomas et al., 2006), and NIMBLE (de Valpine et al., 2017; de Valpine et al., 2020a; 2020b), among others. Thus, when we refer to the BUGS language, we do not denote any particular software, as it could be any of those mentioned here.

To run the code in our book, you need to have a recent version of R and JAGS installed on your computer (we have used R 4.0 and JAGS 4.3). Both programs can be downloaded for free: JAGS, including its manual, from https://sourceforge.net/projects/mcmc-jags/, and R from https://www.r-project.org/. In R, you also need to install several packages. Apart from the R package that accompanies this book (IPMbook; see below), the most important is the interface between R and JAGS, and in this book, we use the package jagsUI (Kellner, 2019). Other interface packages, such as runjags, R2jags, rjags, or dclone, could also be used. These packages may differ slightly in how MCMC options are defined and in the format of the output they produce. Therefore, if you want to use one of these other interface packages for R and JAGS, you obviously would have to modify some of our code in the book, although any code that defines IPMs for JAGS will remain exactly the same.

All the code we provide must be executed in R. Some of that code will write a plain text file containing a description of the IPM or another statistical model defined in the BUGS language. JAGS uses this file to fit the model. You have to be very clear about which part of the code belongs to R and which is for JAGS. Fortunately, the syntax of both programs is very similar, and at first sight, they may even look identical. To make it absolutely clear which is which, we have added a light grey layer underneath all code belonging to the model definition file for JAGS. All the rest, in Courier New font, is R code (though note that in very rare cases we show a small number of lines of BUGS code to demonstrate something specific, and they do not have the grey layer). The text file for JAGS is created with the R function cat. So whenever there is JAGS code for a complete model, you will see the following construction:

In this example, R writes the code within the grey layer into a text file with the name model1.txt. It is saved into the R working directory, which is the folder in which R is active and that you can identify using getwd(). You can change the R working directory with the setwd function in R. Especially when you start with JAGS, it can be insightful to search for the text file produced and have a look at it. In addition, when JAGS reports an error, it will indicate a line number in the model. You can open the text file to see which part of the model the error message refers to.

These model files always have the name modelX.txt, where X is an increasing integer that starts anew at 1 in every chapter. The output from JAGS is always stored in R objects with a name such as outX, where X again starts anew at 1 in each chapter. Sometimes we fit the same model to different data sets, in which case the numbers between model and output files may no longer correspond in later parts of a chapter. Moreover, we sometimes fit additional models using code you will find on the book web page, but that is not printed in the book. These model and output files are included in the numbering. Therefore, it is for instance possible to generate an output file out9 from the model model7.txt.

Throughout the book, we use several other R packages for data manipulation, to produce plots, or for further analyses. Here is the complete list of packages (in alphabetic order): AHMbook, denstrip, grid, ggplot2, IPMbook, jagsUI, lme4, maptool, MCMCglmm, plotrix, plyr, raster, RColorBrewer, rgdal, rgeos, scales, sp, wiqid.

The IPMbook package

We are indebted to Mike Meredith for creating and maintaining the R package IPMbook that accompanies this book and that you can download from CRAN. It contains various useful R functions that we have developed to simulate data, format data in various ways, and provide initial values for model fitting (initial values can be a critical part of modeling with JAGS). It also contains all the data sets used in the book, including those for the case studies in Part 2. To get an overview of the functions and data sets provided in the package, you can use this:

Book Web Page

Additional material for the book is provided on the book web page at www.vogelwarte.ch/ipm. Besides a description of the book, you will find a link to a GitHub page maintained by Mike Meredith that provides the full code. This is handy because you can download the code in your favorite R interface and because the code is complete, while in the book only parts of code (the important ones of course) are printed. Hence, you should be able to reproduce every analysis presented and draw any graph shown in the book. The solutions to the exercises can also be found on the web page.

An important landing page for this and several related books by some colleagues and us is www.hierarchicalmodels.com. It contains additional resources and information on hierarchical modeling in ecology, discussion groups and help forums, workshops, and some particularly relevant R packages on CRAN and GitHub repositories. In particular, we would like to draw your attention to the Google group email list, where questions about topics covered in our book can be asked: https://groups.google.com/g/hmecology.

We truly hope you will find our book useful for the fitting of IPMs and to enhance your understanding of these complex hierarchical models. We hope that applying IPMs will lead to better inferences that are useful in management and conservation. Indeed, we consider one of our major contributions to the mitigation of the biodiversity crisis to be the provision of analytical knowledge and help, such that an increasing number of ecologists and wildlife managers can apply complex statistical models to better analyze their population data and make better decisions about managed populations. For us, putting all this material together has been a great learning experience, but it is clear that our understanding of IPMs is far from complete. We therefore also hope that our book will inspire much future research on the fascinating topic of integrated population models.

Michael and Marc

December 2020

Acknowledgments

We thank Panagiotis (Takis) Besbeas, who, along with colleagues, wrote the foundational paper on integrated population models (IPMs) back in 2002 and who has continued to conduct important research on this class of models in the frequentist mode of inference ever since. We also thank Steve Brooks and his colleagues for the first translation of an IPM into the BUGS language in 2004. That translation made these models practically accessible to a much larger audience, including ourselves.

Our colleague Mike Meredith read and checked the code throughout the entire book and suggested corrections and ways to make things better, more efficient, less clumsy, or correct. He checked and improved all R functions and the structure of the data sets, maintains a GitHub site hosting the IPM book code and most of all, he created the R package that accompanies this book, called IPMbook. We are indebted to him for his huge contribution to this project and for his friendship, his patience, and the incredible swiftness with which he always dealt with all our queries.

We warmly thank the following people who read and commented on parts of the book: Ismael Verrastro Brack, Thomas Bregnballe, Richard Chandler, Dilsad Dagtekin, Dan Gibson, J. Nicholas Hendershot, Joseph Hightower, Cat Horswill, Pepe Jimenez, Dave Koons, Omar Lenzi, Wendy Leuenberger, Dan Linden, Mark Maunder, Rachel McCrea, Mike Meredith, Helen Moor, Josh Nightingale, Joseph Hightower, Floriane Plard, Rob Robinson, Andy Royle, Sarah Saunders, Eric Stolen, Cat Sun, Chris Sutherland, Gesa von Hirschheydt, Greg Wann, Amanda Warlick, Matt Weldy, Kristopher Winiarski and Qing Zhao. In particular, Thomas Riecke read and commented on most chapters, and his input was vital. We are grateful to Dave Koons for writing his inspiring foreword.

We also thank the persons who have made data sets available that we could use as well as offer to the world for free use: Farshid Ahrestani, Raphaël Arlettaz, Gaby Banderet (†), Radames Bionda, Thomas Bregnballe, Bob Clark, Martin Grüebler, Hans Jakober (†), Ulrich Köppen, René-Jean Monnerat, Luca Rotelli, Wolfgang Stauber (†), and Bruno Ullrich. Next, we are very grateful to the photographers for their great pictures, and in particular, that amazing young artist, Helene Rimbach, who created the fantastic drawing of the woodchat shrike for the book cover.

We are also grateful to the many people who have participated in our workshops. Our interactions with them helped improve the material compiled in this book, and in many cases, helped us come up with some content in the first place.

Our employer, the Swiss Ornithological Institute (www.vogelwarte.ch), deserves our gratitude for giving us the freedom to write this book and for the liberty in our research to pursue what we thought was important, fascinating, and beautiful.

Finally, we thank our families, particularly our wives Christine and Susana and our children Lilly, Lukas, and Gabriel, for their support, love, and patience while we wrote the book.

Special Thanks by Michael

I thank my good old friend Marc for suggesting to write this book during our unforgettable BPA workshop in Esporles in September 2012, and for going along this way together. It has been a real pleasure. I’m also very grateful to Olivier Gimenez, who introduced me to the wonderful world of IPMs 15 years ago. Finally, I would like to thank Fitsum Abadi, Todd Arnold, Rémi Fay, David Fletcher, Dave Koons, Thomas Riecke, Simone Tenan, Mitch Weegman and Floriane Plard for various inspiring collaborations on different aspects of IPMs.

Special Thanks by Marc

I would like to thank Miguel for his friendship and help over the years and for inviting me to become a coauthor of this book, which was a wonderful learning experience for me. In addition, I feel forever indebted to my friend and colleague Andy Royle, to my mentor Jim Nichols, and to Jim Hines for so much inspiration and support over so many years. Without them and the fantastic time I spent at the legendary Patuxent Wildlife Research Center in Laurel, Maryland, I would never have become the statistical ecologist I am now. Finally, it is my great pleasure to acknowledge Andrés Calamaro for so much joy and inspiration over the years, right from the start 20 years ago, when all this hierarchical modeling began for me.

Literature Cited

Besbeas P, Freeman S.N, Morgan B.J.T, Catchpole E.A. Integrating mark-recapture-recovery and census data to estimate animal abundance and demographic parameters.  Biometrics . 2002;58:540–547.

Brooks S.P, King R, Morgan B.J.T. A Bayesian approach to combining animal abundance and demographic data.  Anim. Biodiv. Cons.  2004;27.1:515–529.

de Valpine P, Paciorek C.J, Turek D, Michaud N, Anderson-Bergman C, Obermeyer F, Wehrhahn Cortes C, Rodriguez A, Temple Lang D, Paganin S.  NIMBLE User Manual  R package manual version 0.10.1. 2020.

de Valpine P, Paciorek C.J, Turek D, Michaud N, Anderson-Bergman C, Obermeyer F, Wehrhahn Cortes C, Rodriguez A, Temple Lang D, Paganin S.  NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling  R package version 0.10.1. 2020.

de Valpine P, Turek D, Paciorek C.J, Anderson-Bergman C, Lang D.T, Bodik R. Programming with models: writing statistical algorithms for general model structures with NIMBLE.  J. Comput. Graph. Stat.  2017;26:403–413.

Fournier D, Archibald C.P. A general theory for analyzing catch at age data.  Can. J. Fish. Aquat. Sci.  1982;39:1195–1207.

Kellner K.F.  Package ‘jagsUI’: A Wrapper Around ‘rjags’ to Streamline ‘JAGS’ Analyses . 2019 https://cran.r-project.org/web/packages/jagsUI/index.html.

Kéry M.  Introduction to WinBUGS for Ecologists – A Bayesian Approach to Regression, ANOVA, Mixed Models and Related Analyses . Burlington: Academic Press; 2010.

Kéry M, Royle J.A.  Applied Hierarchical Modeling in Ecology – Analysis of Distribution, Abundance and Species Richness in R and BUGS .  Volume 1 Prelude and Static Models . London: Elsevier/Academic Press; 2016.

Kéry M, Royle J.A.  Applied Hierarchical Modeling in Ecology – Analysis of Distribution, Abundance and Species Richness in R and BUGS .  Volume 2 Dynamic and Advanced Models . London: Elsevier/Academic Press; 2021.

Kéry M, Schaub M.  Bayesian Population Analysis Using WinBUGS – A Hierarchical Perspective . Boston: Academic Press; 2012.

Lunn D, Jackson C, Best N, Thomas A, Spiegelhalter D.  The BUGS Book. A practical introduction to Bayesian Analysis . Boca Raton: CRC Press; 2013.

Maunder M.N, Punt A.E. A review of integrated analysis in fisheries stock assessment.  Fisheries Research . 2013;142:61–74.

Plummer M.  JAGS: A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling . Wien, Vienna, Austria: Technische Universität; 2003.

R Core Team.  R: A Language and Environment for Statistical Computing . Vienna: R Foundation for Statistical Computing; 2018.

Schaub M, Abadi F. Integrated population models: a novel analysis framework for deeper insights into population dynamics.  J. Ornithol.  2011;152:227–237.

Schaub M, Gimenez O, Sierro A, Arlettaz R. Use of integrated modeling to enhance estimates of population dynamics obtained from limited data.  Cons. Biol.  2007;21:945–955.

Thomas A, O’Hara R.B, Ligges U, Sturtz S. Making BUGS open.  R news . 2006;6:12–17.

Zipkin E.F, Saunders S.P. Synthesizing multiple data types for biological conservation using integrated population models.  Biol. Cons.  2018;217:240–250.

Chapter 1: Introduction

Abstract

Knowing how an animal or plant population works is of paramount importance in many fields including population ecology, conservation, and wildlife management. In the typical case, this requires estimation of demographic rates and population size based on multiple and disparate data sets. The traditional approach to population analysis is usually a two-step approach—first the parameters are estimated, and second, the estimates are used to parameterize a mathematical population model, often a matrix population model, for making inferences about population dynamics. This approach can be inefficient, because information about demographic processes contained in the timeseries of population size data is not exploited. The two-step approach may also be inaccurate because error propagation is typically not done correctly. Integrated population models jointly analyze individual- and population-level data in a single statistical model to allow the use of available information and to achieve comprehensive error propagation throughout the model. The resulting models are fairly complex hierarchical models, but owing to the flexible, powerful, and most of all, highly accessible BUGS language, these models are relatively easy to code, even for nonstatisticians. Here we introduce integrated population models, provide a brief overview of the contents of the book, and make suggestions about how the book can be used most efficiently for teaching and self-study.

Keywords

BUGS; Conservation; Data simulation; Error propagation; Parameter estimation; Population ecology; Population management

1.1 Population Modeling in Population Ecology and Management

1.2 The Two-Step Approach to Population Modeling

1.3 Integrated Population Models

1.4 Developing Integrated Population Models with the BUGS Language

1.5 This Book

1.5.1 Why This Book?

1.5.2 Structure and Overview of This Book

1.5.3 The Importance of Simulation

1.5.4 Use of This Book in Courses and for Teaching

1.1. Population Modeling in Population Ecology and Management

The number of individuals in study species changes in both time and space, and these changes are broadly known as population dynamics. The description of these changes and the demographic and ecological mechanisms that cause them are important for many scientific questions in population ecology, conservation biology, evolutionary ecology, and disease ecology as well as in applied fields such as fisheries, pest control, and general conservation practice (Caughley, 1994; Norris, 2004; Conroy and Carroll, 2009; Mills, 2013).

In an ideal world, when attacking almost any scientific question or decision problem in population dynamics, we would have perfect knowledge about the state and location of all the individuals of a study species at a certain point in time and then be able to follow them through time. We can thus define the state s of an individual at time t and location i as the fundamental information unit in demography (see also Section 4.2), where the individual state denotes features such as sex, age, size, or a measure of quality. Individuals may survive, may disperse to another place, may reproduce, and will eventually die. Survival, movement/dispersal, and reproduction are the individual demographic processes, and they may vary by state and location and also over time. Over time, demographic processes result in changes in the spatial distribution of individuals, their state frequencies, and temporal changes in their numbers. We could depict these individuals on a map and draw such a map at different points in time. We then would see points appearing, moving, and disappearing again (Fig. 1.1). In other words, when we follow the population dynamics processes over time, we will see points that blink on and off, move, and spawn new points. Mathematically then, this becomes all about describing spatio-temporal point patterns (Illian et al., 2008; Baddeley et al., 2016; Wikle et al., 2019).

In such a simple model of a dynamic population of points, the goal of a population dynamics study can be defined as the description of the blinking, movement, and spawning behavior of these points and the attempt to understand the processes that govern these behaviors. Which demographic process is responsible for points occurring more densely in some areas than in others; or becoming scarcer over time in some areas while becoming more locally abundant in others? Perhaps we want to predict (i.e., forecast) how that cloud of blinking points looks in the future. Will the density of these points increase or decrease? What happens if we eliminate some points—will this affect the long-term dynamics of the remaining points? These questions could be translated into the real world: What are the demographic drivers of spatial and spatio-temporal variations in abundance? What are the demographic causes of an observed population change, such as a decline, over time? How large is the extinction probability of a study population? How many individuals can be removed (i.e., killed, harvested) from the population without putting its long-term viability at risk?

In practice, addressing these and many related questions requires simplifications. That is, it requires abstract models, because not having them results in no model, no understanding, as Keyfitz and Caswell (2005) succinctly and delightfully put it. The first and typically necessary simplification is the definition of a certain space wherein we study the individuals; we define an area and then define the individuals within its boundaries as our study population (Fig. 1.1). These individuals are the target of our inference. Rarely are the boundaries of a study area set such that all individuals of a species are included, i.e., the world population of the study species. Nearly always, a study area includes only a small fraction of the world population of a species (for a rare exception, see Sillett et al., 2012). Boundaries may be defined based on biological considerations, e.g., when the spatial distribution of the individuals is patchy, boundaries may be chosen such that one or several distinct patches are included. Boundaries can also be defined based on practical reasoning or by what we want to know, e.g., whether we are interested in population dynamics on small or large spatial scales. It is also possible to define several such areas, resulting in multiple studied populations, or perhaps colonies for colonial species such as many seabirds. Again, these areas may be defined based on biological reasoning such that, for instance, all occupied patches are defined. Alternatively, they may be defined more arbitrarily or as part of a spatially replicated study design.

Figure 1.1 Conceptual outline of a population dynamics study over a 2-year period in a perfect world depicted as part of a continuous spatio-temporal point pattern (perfect world refers to perfect observability of the process). Growth of the population is governed by the demographic processes of survival, reproduction, and natal dispersal. In year 1, mature adults are shown as black dots (left). In year 2, surviving individuals are shown as black symbols, while individuals that have died since year 1 are denoted by gray symbols (right). Circles are individuals that were already mature in year 1, and triangles are individuals born in year 1. Gray lines show natal dispersal from the birth location to the breeding location. The beige quadrat defines the area of the study population, the boundaries of which are arbitrary to some degree. Dispersal results in immigration and emigration with respect to the defined population.

If we are truly honest, we must admit that in practice, population definitions are often extremely arbitrary in terms of their location, and especially, their extent. We most often study a species where it is numerous (because we need a large sample size in our statistical analyses) and where it is easily accessible. The extent of what we call a population is most typically defined by the area wherein all the individuals live that our PhD students can get their hands on during one field season. Hence, the size of a population is most often a function of the logistical resources available to a researcher.

Whichever way we define a spatial area to contain the individuals that we study, it will affect our view of the fundamental point pattern and will help us understand it. By referring to a spatial area, we add one level to the basic level of the individual point in the pattern—the population level, i.e., the whole set of points as defined by our choice of study area. Recognizing the population level allows us to summarize the set of individuals; for example, we can now count the number of individuals and characterize this population by its size (N). The change in population size over time, then, is the most basic, and simplest, description of population dynamics. Properties of individuals can also be summarized by quantities such as the sex ratio, the age distribution of the individuals, or the distribution of any other classification chosen for them.

The outcomes of the individual demographic processes (survival, dispersal, reproduction) are never perfectly predictable and in this sense are stochastic. That means that we need to describe them as realizations from a chance process, which in practice is a chosen statistical distribution (see Chapter 2). For example, the survival of an individual is akin to the flipping of a loaded coin, and the load of the coin (affecting its propensity, or probability, to produce a head or a tail) will depend on characteristics of the individual as well as on time and space. That is, the outcome of the coin flip is a chance process, and we want to estimate the probability governing it from some observations of that chance process. Both dispersal and reproduction can similarly be viewed as the outcomes of chance processes, and the parameters that govern their average behavior may similarly depend on features of the individual as well as on time and location. Estimation of these demographic parameters may require scaling-up to the population level. Scaling-up from the individual to population level has especially strong effects on the way we perceive dispersal, i.e., the movement into or out of a chosen study area. Individual dispersal movements are characterized at the population level by their outcomes with respect to the defined area—a movement starts from inside or outside the area and ends inside or outside. A movement from inside to outside is called an emigration event, whereas a movement from outside to inside is called an immigration event.

A further possible simplification of the general spatio-temporal point pattern is in reference to time. In practice, the numbers and locations of individuals changes continuously over time, but in practice we are often interested in a characterization of the population at specific points in time, e.g., during the reproductive season, when individuals are more tied to certain places and may be easier to study. Therefore, we often discretize time, which also renders simpler models. To understand population dynamics, it is often sufficient to describe a population only at specific points in time.

Finally, we may make simplifications regarding demographic processes that in reality will hardly ever be identical for two individuals and are unlikely to be identical over time and in space. In our simplifications of the demographic processes, i.e., in our models, we typically make a number of constancy assumptions such that demographic processes are assumed to be identical for all individuals in the same state, to not vary in space (or in only a restricted way, e.g., as explained by a linear model of some covariate), to be constant over time, or to vary only in some restricted way (e.g., as a time trend).

Thus, at a fundamental level we have a population concept as a spatio-temporal point pattern that we could describe at the individual level where the demographic processes actually operate. The results of these processes are the appearance, movement, and disappearance of individuals over time and space and their branching into new individuals (i.e., reproduction). In addition, there is the population level, which is artificial to some degree but of paramount importance for our understanding and for a parsimonious description of the population. The population level is typically our starting point for thinking about models to describe and understand population dynamics.

The most fundamental models of population dynamics, however, are individual-based (Huston et al., 1988; Fahse et al., 1998), which describe the demographic performance of every individual. They do not attempt to describe effects at the population level; rather, the dynamics at the population level are an emergent property that results from processes acting at the individual level. That is, the number of individuals or their age distribution in a certain area is obtained by simple summarization, and we can define space any which way we like. These models closely resemble the fundamental point-process-based view of a population described earlier. A drawback of these models is their great computational burden, as modeling each individual separately is quite expensive as well as a conceptual burden because their descriptions are mathematically complex.

A second class of population dynamics models explicitly includes both individual- and population-level data. A spatial unit is defined, and the number of individuals and its change over time is modeled within this spatial unit that defines the population. Individuals are characterized by age or stage, and the age- or stage-specific numbers of individuals are important quantities in these models. Information about the location of individuals is reduced to whether or not they are inside or outside the limits of the population. These models are the celebrated stage-structured or matrix population models (Caswell, 2001). A further development of this class of models recognizes that individual heterogeneity in demographic performance is often substantial even among individuals within an age group or stage in a matrix population model. In these models, heterogeneity is defined in a continuous rather than categorical manner, and demographic rates may be specified that depend, in parametric form, on continuous individual traits such as size. Models that include these features are called integral projection models (Easterling et al., 2000; Coulson, 2012).

A third class of models considers only the population level and therefore does not include any demographic parameters that describe individual processes. A change in the number of individuals in a population is modeled as a function of the population growth rate only. These models are much more phenomenological and are known as time-series analyses in population dynamics (Dennis et al., 1991; Dennis and Taper, 1994; Dennis et al., 2006; Knape et al., 2013).

The use of these models may answer different and multiple questions relevant to a fundamental understanding of population dynamics, conservation, and management. Depending on the class of model used, we can address various questions. With individual-based and matrix population models, we arguably come closer to a mechanistic understanding of a population, i.e., at the level of the actual demographic processes that always operate at the individual level. But the costs of these models include increased complexity and greater demands on the data. In contrast, time-series models of population abundance represent a more phenomenological description of population dynamics but are less complex, and the data they require are usually cheaper in the sense that they are more widely available or logistically easier to obtain.

1.2. The Two-Step Approach to Population Modeling

Population models contain parameters such as survival probability, productivity, or the population growth rate. In practice, these parameters are virtually never known. Sometimes we may have information from previous studies or the experience of experts, but in most cases when we study population dynamics, these parameters must be estimated from data (which must be collected in the first place). Estimation means the analysis of data from (ideally) a random sample of individuals in the population by means of a statistical model, which is an idealized description of the processes that are thought to have produced the data. Often these models must account for imperfect detection, i.e., for the fact that an individual can be alive and inside a study area but not be detected. Failure to accommodate this type of measurement error in models of animal populations can have severe consequences when inferences are made based on those models. This recognition has given rise to the development of a large branch of ecological statistics that we collectively call capture-recapture models (Seber, 1982; Borchers et al., 2002; Williams et al., 2002; Royle and Dorazio 2008; King et al. 2010; Royle et al., 2014; Buckland et al., 2015; McCrea and Morgan, 2015; Seber and Schofield, 2019). Such models allow us to correct for this particular measurement error in our inferences about temporal processes such as survival or dispersal that require us to follow recognizable individuals over time; they also correct for the measurement error that arises when we estimate the locations of individuals. Of course, formal estimation also means that we obtain a measure of uncertainty associated with a parameter estimate, which depends on sample size, noise in the system, and model complexity, among other factors.

The estimated parameters can then be plugged into a population model, most commonly a matrix population model, which allows the calculation of quantities such as the population growth rate or growth rate sensitivities (Caswell, 2001). Hence, inferences about population dynamics are obtained by a two-step approach in which the first step is parameter estimation, and the second step is the use of the parameter estimates obtained in step 1 as inputs in a population model.

Data sampling in population studies usually includes both individual- and population-level data. That is, typically longitudinal data on productivity and marked or otherwise recognizable individuals are sampled in addition to counts of the population, where no individuality is usually distinguished. While parameters estimated from the individual-level data are used to parameterize a population model, the data from the population level can be used to evaluate the population model developed, i.e., to gauge whether the predicted and observed population dynamics agree.

Such piecemeal analyses are not wrong, but usually they ignore the sampling (co-)variation of the estimators of the demographic rates. They are also statistically inefficient because information about individual-level processes contained in the population-level data is not exploited. In addition, the estimation uncertainty coming from the first step of the