Data-driven Modelling of Structured Populations: A Practical Guide to the Integral Projection Model

339 Pages·2016·7.348 MB·English

by  Stephen P. Ellner, Dylan Z. Childs, Mark Rees (auth.)

Lecture Notes on Mathematical Modelling
in the Life Sciences
Stephen P. Ellner
Dylan Z. Childs
Mark Rees
Data-driven
Modelling of
Structured
Populations
A Practical Guide to the Integral
Projection Model
Lecture Notes on Mathematical Modelling
in the Life Sciences
SeriesEditors:
MichaelMackey
AngelaStevens
Moreinformationaboutthisseriesathttp://www.springer.com/series/10049

Stephen P. Ellner • Dylan Z. Childs (cid:129) Mark Rees
Data-driven Modelling
of Structured Populations
A Practical Guide to the Integral Projection
Model
123
StephenP.Ellner DylanZ.Childs
EcologyandEvolutionaryBiology AnimalandPlantSciences
CornellUniversity,CorsonHall UniversityofSheffield
Ithaca,NY,USA Yorkshire,UK
MarkRees
AnimalandPlantSciences
UniversityofSheffield
Yorkshire,UK
ISSN2193-4789 ISSN2193-4797 (electronic)
LectureNotesonMathematicalModellingintheLifeSciences
ISBN978-3-319-28891-8 ISBN978-3-319-28893-2 (eBook)
DOI10.1007/978-3-319-28893-2
LibraryofCongressControlNumber:2016933253
MathematicsSubjectClassification(2010):92D25;92D40;92C80;92-02;62P10;97M60
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Preface
Over the past 30 years, one of the most rapidly expanding areas in ecology
has been the development of structured population models, where individuals
have attributes such as size, age, spatial location, sex, or disease status that
affect their interactions and fate. These models can be roughly divided into
conceptual models where the effects of biological structure are explored in the
simplestpossiblesettingtoexposegeneralprinciplesanddata-drivenmodelsin
which the parameters and functions are estimated from observations of actual
individuals, allowing inferences about population behavior. By far the most
popular data-driven models are matrix projection models, which assume that
the population can be divided into discrete classes. Much of the success of
matrix models is a consequence of Hal Caswell writing a detailed “how to”
book (Caswell 2001, 1st edition 1989) in which he described, with appropriate
Matlab code, how to build and use matrix models.
TheIntegralProjectionModel(IPM),introducedbyEasterlingetal.(2000),
is a generalization of matrix models to cover populations where individuals are
classified by continuous traits. An IPM makes full use of hard-won data on the
precise state of individuals, without an increase in the number of parameters
to estimate; in fact it often needs fewer parameters, especially when the model
includesenvironmental variability. TheaimofthisbookistodoforIPMswhat
Caswell did for matrix models. We want this book to be useful and accessible
to ecologists and evolutionary biologists interested in developing data-driven
models for animal and plant populations. IPMs can seem hard as they involve
integrals, and the link between data and model, although straightforward, can
seem abstract. We hope to demystify IPMs so that they can become the model
of choice for populations structured by continuously varying attributes. To do
this, we structure the book around real examples of increasing complexity and
showhowthelifecycleofthestudyorganismnaturallyleadstotheappropriate
statistical analysis, which in turn leads to the IPM itself. We also provide com-
plete, self-contained R code (R Core Team 2015) to replicate all of the figures
and calculations in the book at https://github.com/ipmbook/first-edition.
v
vi Preface
Wesetouttosummarizethestateoftheart,takingadvantageofbookformat
to start from the beginning, and to provide more details, more examples, more
practical advice, and (we hope) clearer explanations. But we could not always
resist the impulse to go beyond the current literature. Much of Chapters 3
and9isnew.Chapter4takesanewapproachtoeigenvaluesensitivityanalysis.
Chapter 10 introduces a general way of putting demographic stochasticity into
an IPM and some ideas about fitting IPMs to mark-recapture data that we
hope might spur more applications to animal populations (or better yet, spur
some statisticians to bemoan our naivet´e and come up with something better).
And while we were writing, the current literature often went beyond us, so we
apologize to everyone whose new papers we ignored in the push to finish this
book.
We also apologize to beginning IPM users and to advanced users. Advanced
users will find that we start at the beginning, with things they already know.
But we take anewapproach tosome of the firststeps, sowe hope that you will
read the whole book. Beginners will find that we don’t stop until we hit the
limitsofwhatweknow,soweurgeyoutonot readthewholebookuntilyou’ve
gotten some hands-on experience with IPMs. And we hope that you won’t give
up completely when you hit something confusing. The start of the next section
or chapter will probably be written for you, unless it starts with a warning to
the contrary.
It has been just over three years since we started preparing this book. We
have been fortunate to receive guidance and support from many friends and
colleagues duringthis time. Wehave also beenvery fortunate to work with and
learnfrommanystudents,postdocs,andothercollaborators.Theliststartswith
Philip Dixon (who got this started nearly 20 years ago by posing the question,
histogram:kernel density estimate = matrix projection model:what?, and help-
ingtoanswerit)–anditcontinuestoexpand.Thosewhohaveshapedourthink-
ing over the years include Peter Adler, Ottar Bjørnstad, Ben Bolker, Yvonne
Buckley, Hal Caswell, Peter Chesson, Margaret Cochran, Evan Cooch, David
Coomes, Tim Coulson, Mick Crawley, Johan Dahlgren, Michael Easterling,
Roberto Salguero-G´omez, Edgar Gonza´lez, Peter Grubb, Nelson Hairston Jr.,
Elze Hesse, Giles Hooker, Eelke Jongejans, Dave Kelly, Patrick Kuss, Jonathan
Levine,SvataLouda,MarcMangel,JessMetcalf,SeanMcMahon,CoryMerow,
Heinz Mu¨eller-Sch¨arer, Arpat Ozgul, Satu Ramula, Richard Rebarber, Karen
Rose, Sebastian Schreiber, Kat Shea, Andy Sheppard, Robin Snyder, Britta
Teller, Shripad Tuljapurkar, Lindsay Turnbull, Larry Venable, and really each
and every one of our students and postdocs.
Two anonymous reviewers of some early chapter drafts provided advice and
much needed encouragement to carry on developing the text. A further three
reviewersofthenearlycompletemanuscriptofferedyetmoreencouragement,as
wellassomevaluableandconstructivesuggestions.AchiDosanjh,oureditorat
Springer, has shown great patience throughout the process of taking the book
from inception to publication. Parts of the book are based on our published
papers, and we thank the reviewers and editors who read our manuscripts and
helped us improve them.
Preface vii
A number of funding agencies provided direct and indirect support for our
researchonIPMsoverthelast15years,includingtheNaturalEnvironmentRe-
searchCouncil(DZC,MR),theLeverhulmeTrust(DZC,MR),theUSNational
Science Foundation (SPE), and the US National Institutes of Health through
the program on Ecology and Evolution of Infectious Diseases (SPE).
Finally, we’d like to thank our families for their enduring support and
patience.
Ithaca, NY, USA Stephen P. Ellner
Sheffield, Yorkshire, UK Dylan Z. Childs
Sheffield, Yorkshire, UK Mark Rees

Contents
1 Introduction................................................ 1
1.1 Linking individuals, traits, and population dynamics .......... 1
1.2 Survey of research applications ............................. 2
1.3 About this book.......................................... 3
1.3.1 Mathematical prerequisites .......................... 4
1.3.2 Statistical prerequisites and data requirements ......... 5
1.3.3 Programming prerequisites .......................... 5
1.4 Notation and nomenclature ................................ 6
2 Simple Deterministic IPM .................................. 9
2.1 The individual-level state variable .......................... 9
2.2 Key assumptions and model structure....................... 10
2.3 From life cycle to model: specifying a simple IPM ............ 12
2.3.1 Changes........................................... 15
2.4 Numerical implementation................................. 17
2.5 Case study 1A: A monocarpic perennial ..................... 19
2.5.1 Summary of the demography ........................ 19
2.5.2 Individual-based model (IBM) ....................... 21
2.5.3 Demographic analysis using lm and glm ............... 22
2.5.4 Implementing the IPM.............................. 24
2.5.5 Basic analysis: projection and asymptotic behavior ..... 26
2.5.6 Always quantify your uncertainty! .................... 30
2.6 Case study 2A: Ungulate .................................. 32
2.6.1 Summary of the demography ........................ 33
2.6.2 Individual-based model ............................. 34
2.6.3 Demographic analysis............................... 35
2.6.4 Implementing the IPM.............................. 36
2.6.5 Basic analysis...................................... 39
2.7 Model diagnostics ........................................ 41
2.7.1 Model structure.................................... 42
2.7.2 Demographic rate models............................ 42
ix