Contents

Preface

Acknowledgements

Companion website

1 Introduction: the role of science in conservation

Evolution of conservation science

Conservation advocacy versus science

Good science is cost effective

Conservation under fire

Structure of the book

Part I Basic concepts in scientific investigations for conservation 7

2 Using models in conservation biology

Types of models

Basic principles of modeling

3 Models of population dynamics

Balance equation of population growth

Density-independent (geometric) population growth

Density-dependent population growth

Populations: age-structured growth

Stochastic (random) effects on population growth

Spatial structure and population growth

Models for two or more species

4 Applying population models to conservation

Harvest models

Small population models

Modeling genetic effects

Including habitat or other factors in models

Use of “canned” modeling software

5 Basics of study design and analysis

Statistical models

Sampling designs

Comparison of time and space (monitoring)

Accounting for incomplete detection

Summary

Part II Conservation studies and monitoring programs

6 General principles of estimation

Census versus estimate

Indices

General relationship between counts and abundance or occupancy

Count data as indices

“Closed” versus “open” estimation

Summary

7 Occupancy (presence–absence) analysis

Design and analysis of occupancy studies

Occupancy estimation

Application and programs

Summary

8 Sample counts for abundance estimation

Complete detection assumed

Incomplete detection assumed

Study design

9 Distance sampling for estimating density and abundance

Line transect sampling

Point transect sampling

10 Capture–mark–recapture studies for estimating abundance and density

Basics of capture–mark–recapture

Two-sample (Lincoln–Petersen)

Multiple sample CMR methods (Schnabel census)

Removal sampling

Estimation of density with CMR

Other methods

Summary

11 Estimation of survival from radiotelemetry, nesting success studies, and age distributions

Basic estimation problem

“Known fate” radiotelemetry studies

Nesting success studies

Other approaches to estimating reproduction rates

Analysis of age frequency data

Modeling variation in survival and reproduction rates

Summary

12 Markߝrecapture for estimating survival, recruitment, abundance, and movement rates

Basic estimation problem

Survival estimation with tag (ring, band) recovery data

Survival estimation with capture–recapture (CJS models)

Survival, abundance, and recruitment estimation with Jolly–Seber

Combined open and closed models: the Robust Design

Estimation of movement rates from recovery and capture–recapture data

Summary

13 Analysis of habitat

Sample units and scale

Assessing habitat from Design I studies

Assessing habitat using radiotagged animals from Design II, III, and IV studies

Sampling in radiotelemetry studies

Statistical methods for assessing habitat use from radiotagged animals

Future directions

14 Estimation of species richness and other community parameters

Detection and communities

Estimation of species richness

Community richness using occupancy models

Estimating parameters of community dynamics

Summary

Part III Integrating modeling and monitoring for conservation

15 Elements of conservation decision making

Elements of decision making

Steps in decision making

Example – reserve construction

Example – habitat management for two species

Example – sustainable harvest

Summary

16 Accounting for uncertainty in conservation decisions

Sources of uncertainty

Probability as a measure of uncertainty

Expected values of outcomes

Using probability and expected value in decision making

Summary

17 Learning and adaptive management

Prediction and monitoring

Constructing the models

Updating belief: Bayes’ Theorem

Adaptive management

The value of information

Summary

18 Case study: decision modeling and adaptive management for declining grassland birds in the southeastern USA

Elements of a BQI decision model

Dealing with uncertainty

Monitoring of response following management

Adaptive management

Summary

19 Summary and recommendations

Use of models

Proper design of field studies

Proper analysis of data

Systematic approaches to decision making

Literature cited

Glossary

Appendix A: Statistical and modeling programs available on the worldwide web

Appendix B: Other internet resources

Appendix C: Modeling and statistical notation

Appendix D: Key to abundance and parameter estimation

Index

Wiley End User License Agreement

An electronic companion to the book is available a webpage maintained at the University of Georgia, accessible through

www.blackwellpublishing.com/conroy

See p. x for further details.

This edition first published 2009

© 2009 by Michael J. Conroy and John P. Carroll

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Library of Congress Cataloguing-in-Publication Data
Conroy, Michael J. (Michael James), 1952-
Quantitative conservation of vertebrates / Michael J. Conroy and John P. Carroll.
p. cm.
Includes bibliographical references.
ISBN: 978-1-4051-9098-5 (hardback : alk. paper)
ISBN: 978-1-4051-8228-7 (paperback : alk. paper)
1. Vertebrate populations–Mathematical models. 2. Conservation biology–Mathematical models. I. Carroll, John P. II. Title.
QL607.6.C66 2009
639.901′5118–dc22
2008029041

A catalogue record for this book is available from the British Library.

Preface

This book is intended for use by field biologists and others, including future field biologists who might be in a university course, engaged in the day-to-day study and conservation of vertebrate animals. Our goal is that conservation biologists use this book as a (with apologies to our colleague Evan Cooch) “gentle introduction” to the field of quantitative ecology. We hope to convince readers that the methods and approaches within are not the domain of mathematicians, statisticians, and computer programers, but in fact are essential tools to doing the job of conservation in the twenty-first century.

We intend this book to be used. Read it, mark it up, and take it into the field. Consult it before collecting field data, to motivate and design your monitoring and research programs (Part I), and afterwards to properly analyze and interpret the data that (hopefully) have now been properly gathered (Part II). Especially in Part II we hope that this introduction gives field biologists the basic tools and confidence to tackle the specialized books covering many of the field techniques that we cite in this book.

We draw particular attention to Part III of the book, which deals with structured decision making and adaptive management. We would actually encourage readers to skip ahead to this section and get to grips with the essential ideas of applying models to conservation decisions. Then return to Parts I and II for the details of how to build models and collect and analyze field data. This will help keep the focus where we want it – on the practical application of these methods to solving real conservation problems.

As noted on page ii, we have provided a website for the book. The website will be updated (and corrected) periodically as new developments occur and as (inevitably) mistakes are found. The website is essential resource for the book, so we strongly encourage readers to use them to repeat for themselves the analyses performed in the book, and as a template for performing analyses of their own data.

Finally, this book is not intended as a substitute for other, more comprehensive books, notably Williams et al. (2002). To keep this book to a reasonable length, and in order to remain accessible to a less mathematically oriented audience, we chose not to cover all the methods available, and have not provided the sort of depth that more advanced references such as Williams et al. (2002) provide. So, for example, we have only briefly described such important approaches as the Pradel temporal symmetry model, Barker/Burnham joint recapture–recover models, and multi-state models (Chapter 12), but have instead provided a context in which readers might assess whether these models could be useful in their applications, and have pointed the reader to the appropriate in-depth background and relevant software. Likewise, we have given only a “barebones” treatment of the Robust Design (Chapter 12), but we have provided an example that should give readers a good idea of just how powerful an approach this is. We have, rather, emphasized practical applications and examples – which is why we again exhort readers to use the website to full advantage.

Acknowledgments

This book was made possible by the support and help of many people. M.J.C. thanks Liz, Mary,and Laura for putting up with another book project; his current and past graduate students for keeping him relatively honest; and colleagues and occasional partners in crime including Richard Barker, Bob Cooper, Chris Fonnesbeck, Clint Moore, Jim Nichols, Bill Palmer, Jim Peterson, Jon Runge, Juan Carlos Senar, Jeff Thompson, Ken Williams, and of course his coauthor John Carroll, to name a few. J.P.C. thanks Eileen, Caitlin, and Sean for their patience and understanding especially during periods of frustration and chaos. Also, to his partner in this venture, Mike Conroy, who provided lots of interesting times during our tenure teaching Applied Population Dynamics together and then the Galliformes Shortcourse. J.P.C. particularly thanks the ‘UGA Gamebird Posse,’ especially Brant Faircloth, Theron Terhune, and Jeff Thompson. We all upgraded our quantitative skills together, although J.P.C. now lags well behind the rest. It was 5 or 6 years of APD students who put the authors on the course for writing this book and they deserve our thanks. As a student once wrote in a course evaluation, ‘Dr. Conroy needs to learn how to dumb down and Dr. Carroll needs to get on Ritalin.’ This pretty much sums up our working relationship. Cheers!

Companion website

1

Introduction: the role of science in conservation

The impetus for this book began as the result of a rather fortunate convergence of the careers of the authors at the University of Georgia. Although we were educated in traditional wildlife management programs during the 1970s and 1980s, we both developed an interest in what is now better defined as conservation biology. Interestingly, we underwent an evolution in our thinking, leading to similar ideas relative to what we perceived as weaknesses in our own profession and to how the creation of conservation biology as a profession, while addressing some of these weaknesses, fell short in many areas. We have also become increasingly involved in international issues in wildlife conservation, leading to further career intersections with other collaborators. Indeed, we have discovered that our interest in mixing conservation and science transcends political boundaries and sub-disciplines.

Evolution of conservation science

The integration of science and conservation of wildlife has quite a long history and is found in many forms. Game management in Europe and North America is based on the fundamentals of agricultural management and animal husbandry. This form of conservation biology is essentially the treatment of stocks of wild animals as domestic livestock and has evolved over hundreds of years. In both North America and Europe, wildlife management as a profession developed over much of the twentieth century following a somewhat parallel course that focused on particular species or groups of species and their management. The resulting body of literature and understanding of the population dynamics of those species and their management is enormous and some of the best information is available on vertebrates.

A second development occurred during the latter part of the twentieth century as interest and concern for the diversity of wildlife in mainly tropical parts of the world moved to the forefront. Scientists who worked predominantly in the area of ecology theory began several attempts to integrate ecology as a science with biological conservation. Driven in large part by North American and Australian scientists and coming to fruition by the late 1980s, we see the wholesale movement of scientists who traditionally dealt with empirical questions in ecology adopt an additional strategy concerned with the conservation of biological diversity.

The above developments resulted in several scientific disciplines, each with different strengths, converging to form scientific conservation biology. We believe that each discipline brings different strengths to conservation science. For example, wildlife management in North America has an excellent track record of applying scientific research to management and policy making. By contrast, the discipline of conservation biology has generally excelled at integrating ecological principles and conservation. The third important component here is the popularization of conservation among the general public which has resulted in an enormous influence of popular culture and activism on conservation and biodiversity management.

These developments then leave us with two scientific disciplines – wildlife (and/or game) management and its sister profession conservation biology. These disciplines can aptly be described by the general heading applied ecology, and are driven in part by non-scientific goals. This creates an interesting and sometimes complex series of relationships that can affect the ability of professional “applied ecologists” to strive toward their scientific objectives of obtaining reliable knowledge. We encounter several issues that are critically important at this juncture. First, as with any applied or endpoint-driven research, we must be particularly careful that our research does not simply become a series of self-fulfilling prophecies. Just as in theoretical–ecological research, our preconceptions about how systems operate must not cloud our ability to undertake objective research. In many ways the goal objectivity is easier to attain in theoretical research, because the results of theoretical¹ research might only involve individual egos and career development, rather than ecological systems and biodiversity that we as individuals and conservation biologists hold dear to us. Over the course of history in scientific endeavors someone who develops some “new” theory would be under some pressure to defend the theory and other scientists might strive to find evidence to falsify it. These traditional scientific tensions are also important in applied research; however, there are now the added pressures created by outside forces from those having a stake in the outcome of research. This is because conservation scientists operate within a socio-economic-political “real world” that includes other values and tradeoffs. Even with a sympathetic public, conservation scientists and managers must act responsibly to allow policy makers to make the best decisions possible, often with limited resources and competing demands.

Conservation advocacy versus science

We distinguish between conservation advocacy – where conservationists become directly involved in promoting policies relative to biological diversity – and conservation science – which uses science to help society make more informed decisions. The latter is the target of this book. We believe that by adopting a scientific approach, not only is science better served, but also in the long-term conservation will be better served. The task is to simultaneously increase our understanding of systems in a dynamic world and to provide decision makers with the necessary information. This is why we believe that modeling approaches and adaptive management are critical components of the conservation research “system.”

In this book we will often use models to summarize how we think a particular population or other resource behaves and might respond to various factors, including management. Our models will involve a combination of (i) our understanding of basic biological processes, and (ii) data available to support the various model components (Figure 1.1, from Williams et al. 2002, p. 112). Typically we will need to increase each of these two factors, as represented by the x and y axes of the graph (Figure 1.1). On the x axis, we wish to move to the right, which represents increasing understanding of the mechanisms driving a particular system. On the y axis, we move vertically as we increase the amount of data available on the system or parts of it. The box itself represents general areas of data quantity and system understanding. Ideally we should be in Box 3, where we have good amounts of empirical data and also good understanding of how particular systems operate. This is why we can use our understanding of gravity, material properties, and other components of physics to build bridges – which generally do not fail. Although this works well in many of the physical sciences, the complexity of combining biological and social/political issues makes this a difficult direction to move in for conservation questions or issues. More likely we are working in Boxes 2 or 4, where we lack data, knowledge, or both. In fact, in dealing with some conservation questions and species outside of the charismatic megafauna or well-studied game species, we are often starting in Box 4 with the proverbial “blank slate.” Making matters worse is the fact that biologists are often in the position where policy and/or management recommendations are expected after a single inadequately funded study. Although never ideal, this is political reality. Beginning in Box 2 is slightly better because we can use our base knowledge of similar systems to hopefully give us a stronger starting point. In both of these scenarios (Boxes 2 and 4) modeling approaches combined with good data collection will be useful. In wildlife conservation and tradition game or wildlife management in North America and Europe we may also be operating in Box 1. We may have high-quality and long-term biological data, but in many cases our understanding of the mechanisms driving systems for issues we are interested in tackling is still lacking. The integration of science in conservation management will ultimately provide the foundation for more informed conservation decisions and management.

Good science is cost effective

One of the common issues in conservation biology is the problem of inadequate funding. Outside of a few areas of conservation that can garner large amounts of money, most conservation biologists are faced with enormous questions and tasks, but limited time and financial resources. Even in programs that are relatively well funded, such as game management in North America and Europe, the scale at which many biological questions should be addressed and the resources that are available often are quite disparate. Conservation research competes for limited funding with conservation implementation (“management”). This means that if funding for research increases then funding for management decreases and vice versa. The only way to “win” at this zero-sum game is to improve efficiency. As we will argue in a number of places in this book, bad research is almost worse than no research. It can lead to wrong conclusions and wrong management.

In this book we argue that poorly designed conservation research projects also steal resources from conservation. That is, spending money on bad science not only wastes that money, but also takes money from good science and good management. Thus good science combined with improved efficiency will yield better conservation.

Conservation under fire

More and more frequently we are faced with skeptical and even hostile groups, who demand that conservationists “prove” claims of adverse impacts of desired development, ecological benefits of restrictions of forest logging or other resource consumption, proposed reserve systems, or declaring a species endangered, to name a few. While it can never be possible to “prove” (in the logical sense) such assertions, it is possible to collect and analyze data in such a way that the evidence so provided is repeatable and defensible. Conversely, data collected or analyzed in an unscientific way lead to conclusions that, while perhaps intuitively reasonable, are not repeatable, and will not stand up to scientific scrutiny. Increasingly the opponents of conservation projects are technically informed, and will eagerly reveal conclusions made by the conservation community that are based on flawed approaches. Here we emphasize that ethical, scientific conservation includes the honest reporting of study and data flaws, so that results may be appropriately interpreted. Improper reporting of results, especially to exaggerate certitude of estimates or effects, is both unethical and, in the long term, counter-productive, because when (inevitably) discovered the resulting loss of credibility can be devastating (Beier et al. 2006; Conroy et al. 2006).

Structure of the book

We envision this book as a practical and hands-on resource for field biologists. This book should be analogous to the field identification guides. It is the book you take with you and use all the time, but is not the one where you go to obtain the in-depth theory or mathematical derivations. We hope it complements some recent volumes, such as Williams et al. (2002), in assisting practitioners and students. We also envision the book being used in short courses for field conservationists. In fact, the impetus for this book came as a result of the participation of J.P.C. and M.J.C. in development of a week-long short course following the main conference in each of the last three International Galliformes Symposia.

Part I covers mainly the background we believe all biologists should review when presented with a conservation problem or question and asked to develop a research program. Chapter 2 provides some basic concepts in modeling. This is not “ugly” and complex modeling that most field biologists fear, but practical modeling that assists us in problem solving. Chapter 3 is a review and application of some basic population models. Chapter 4 deals with the issues of applying models to conservation questions. Chapter 5 provides a basic review of study design. Again, this part of the book is setting the stage for couching conservation questions in a way that makes our research more scientifically sound, economically efficient, and defendable.

Part II moves on to those topics of most importance to field biologists in collecting appropriate data in answering conservation questions. In Chapter 6, we begin with the general principles of estimation. Chapter 7 is a basic overview of occupancy studies. We believe that occupancy research is underutilized, but will eventually be viewed as one of the most important techniques in conservation. Chapter 8 covers the estimation of abundance from sample counts, and introduced the importance issue of incomplete detection, a recurrent theme through the book. Chapters 9 covers the basic principles of distance sampling, including line transect and point counts (the latter are also now called point transects). Chapter 10 provides background on mark-recapture (re-sighting) and mark-removal sampling in abundance estimation. Chapter 11 focuses on the estimation of demographic rates using data from radio-telemetry, nesting success, and age distributions. Chapter 12 expands on the issues of demographic parameters by incorporating some aspects of Chapters 10 and 11. Chapter 13 deals with the issue of habitat use and selection. Finally, in Chapter 14 we touch on some sampling and estimation issues for wildlife communities.

In Part III we begin to apply modeling and estimation tools to conservation decision making. In Chapter 15 we describe how conservation goals can be combined with predictive models and used as tools for decision making. In Chapter 16 we deal with issues of uncertainty in research and conservation decision making. We remind readers that in the real world we are faced with profound uncertainties, in part because nature cannot be controlled, but also because of our incomplete understanding of how ecological systems work. This leads on to Chapter 17, in which we show how monitoring information can be integrated into decision making, leading to adaptive management. In Chapter 18 we illustrate many of the principles of the book via an example of conservation of grassland birds in North America. Chapter 19 provides a short summary of the book.

We also provide several appendices that we hope readers will find useful. Because many readers will be familiar with some but not all the terminology we use, in the Glossary we provide a comprehensive list of terms. See p. ii of this book for numerical examples in electronic form with a detailed accompanying narrative. In Appendices A and B we provide links to sites where software and other resources can be obtained, much of it at no cost. In Appendix C we provide a comprehensive explanation and cross-referencing for modeling and statistical notation. Finally, in Appendix D we provide a dichotomous key for abundance and parameter estimation that can be used to assist in identifying appropriate estimation techniques, in much the same way that taxonomic keys are used to aid in animal or plant identification.

We especially hope that the chapters in this book give field conservationists the courage to tackle some new ways of viewing conservation problems. This is where we believe this book is most useful – in taking the fear out of quantitative and modeling approaches to conservation, and making field conservationists realize they are not “black boxes” that are to be relegated to “systems ecologists” locked away in an office somewhere.

¹Interestingly, theoretical biologists now find that outside influences are very much invading their realm, including recent debates involving religious organizations in the USA and other countries over evolutionary theory and natural selection.

Part I

Basic concepts in scientific investigations for conservation

2

Using models in conservation biology

The word “model” tends to strike fear in the hearts of conservation biologists, who think of models as devices that can only be constructed (and understood) by quantitative specialists – and definitely not by field biologists. In fact, we will now argue that models are part of everyday life, and are used (consciously or not) by virtually everyone to make daily decisions. This is a critical starting point – we all use models all the time, so why are we so afraid of them when applied to conservation issues?

In this chapter, we first endeavor to de-mystify models, with some very simple examples, in which modeling is stripped to its essential elements. We then apply simple graphical and numerical models – most of which can be built using spreadsheets – to investigate population growth (exponential and logistic). More complicated models incorporating age structure and random variability are also considered. We apply simple models to two of the most important problems in conservation, namely that of harvesting (and “overharvest”) and conservation of “small” populations. We extend modeling to populations with spatial structure, critical to the important concepts of source–sink dynamics, habitat fragmentation, and population isolation, and to simple competition and predator–prey models.

We later spend some time discussing the use of “canned” models for investigating population dynamics, particularly population viability analysis (PVA). At this point we strongly warn readers about the dangers of naïve use of these packages, which often rely on numerous (and often not testable) assumptions. These warnings also apply to all statistical packages – generally no matter what you put in the front end of the program it will generate an “answer.” In addition, “canned” models are often designed to fit a wide range of systems – therefore many of these might not be appropriate for specific applications, creating additional problems. We are proponents of simpler, but more transparent, models that will be more informative about the system, and will be easier to evaluate with data. We provide several real examples of population modeling, both with user-developed models and “canned” approaches, to illustrate these ideas.

Types of models

What is a model? Quite simply, a model is an abstraction of something real. Even though models are mysterious and frightening to many, in fact we use them in everyday life. A map is a model – basically a scaled-down, two-dimensional drawing of a part of the Earth’s surface. It is not the Earth itself, nor is it even necessarily very realistic – it simply needs to get us from point A to point B and may do so in a rather artificial way. A very simple sketch of roads from your house to a friend’s house can be crude but extremely useful in meeting your objective of getting to your friend’s house safely. This just emphasizes that there are many kinds of models, and they don’t all live in computers or in lists of complicated equations.

The following is a range of models that the reader should recognize and use, some on a daily basis:

1. Conceptual models are really just ideas about how a system looks, works, or interacts with other systems. Such a model may reside completely in the brain of the person thinking about it – the “conceptualizer,” be represented in flow diagrams, or be formalized mathematically.

2. Physical models are physical representations of a system that work in some way that is analogous to how the system of interest works. For example, Pearson (1960) developed a sort of pinball machine (for those younger than the authors, these are old-fashioned mechanical video games) as an analogy for how birth and death processes work in populations: new balls released represented births, and balls dropping through holes deaths.

3. Graphical models are represented by anything we might show in graph form. For example, plots of average temperature versus rainfall might be very useful in predicting regions of drought.

4. Analytical models turn our ideas into a series of mathematical equations which may then be converted into computer code.

5. Numericalmodels report the quantitative outcome, often of a number of pieced-together predictions calculated by hand, in a spreadsheet, or in a computer program.

6. Empirical or statistical models use data in order to estimate parameters, and then test predictions and other hypotheses, using sample data. It is this type of model and the previous two that often scare and excite conservation biologists and comprise the heart of many useful conservation programs.

It should be clear to the astute reader that the above classification of models is artificial, and that there is much overlap among the categories. At the very least, it is easy to see how development of one type of model can easily lead to another. For example, one might have a purely theoretical idea of a relationship between a biological response and an environmental predictor (conceptual model), which might then be plotted (graphical model). One might even wish to make up some values for the coefficients of the model and generate some predictions (numerical model), or explore the general behavior of the model under a wide range of possible parameter values (analytical model). As good field biologists, however, we should not be happy until we have collected some data, estimated parameters, and tested some predictions of alternative models (statistical models). Closing the loop, the statistical models may reinforce the original conceptual model, or they may challenge it, leading to a revised conceptual model, more graphing, analysis, and prediction, and so on (Figure 2.1).

Basic principles of modeling

Perhaps the second biggest mistake conservation biologists make about modeling – next to fearing and avoiding the topic – is embarking on a modeling exercise in a haphazard and arbitrary way. Modeling can be very useful, but only if done so methodically, with a purpose in mind, and remembering George Box’s admonition that “all models are wrong, but some may still be useful” (Box 1979).

Defining the objective

As with any approach, it is important to keep the objective of modeling firmly in mind. This will determine such things as the scale or detail of the modeling; what sort of system features need to be included and which do not; and whether the model is purely a “thinking exercise” versus an actual predictive tool. For example, if our objective is to manage a population for sustainable harvest, we should be mainly interested in modeling aspects of population dynamics as they relate to the impacts of harvest. Inclusion of details about individual animal behavior, resource selection, and other features, while interesting, are probably not germane to the issue of sustainable harvest, and should not be included. In fact, these latter components might serve to make understanding the dynamics of harvest much more difficult to tease out of the system, by creating large amounts of noise in our model. Conversely, genetic composition of a population may be critical to a decision we are about to make about conservation. In this case, modeling focused only on abundance and other population-level parameters will be insufficiently detailed for our needs.

Defining parameters, variables, and functional relationships

Most models will involve a variety of inputs, outputs, and functional relationships that specify the biological features we are trying to mimic. Features like population size, age structure, and habitat condition that tend to vary over time and space are typically referred to as state variables (or simply “variables”). Constants that control the rate of change, or otherwise express relationships between variables, are parameters. Examples of parameters include survival and birth rates. Two simple examples illustrate these terms, and the idea of simple model building. First, suppose that we wish to translate the prediction “as amount of habitat increases the abundance of our species of interest increases linearly.” In this example, abundance and habitat are state variables, which we might label as Y (abundance) and X (amount of habitat), respectively. The hypothetical increase in abundance with increasing habitat occurs at a rate b, so our model is simply:

(Something to think about – we could have included a Y -intercept term, rather than the assumed value of “0” in our equation, but any positive value would have resulted in a prediction for positive abundance in the absence of any habitat! Some more observant readers might catch us here and point out that there could be a negative Y intercept. This suggests that there is some level of habitat (X intercept) for which the population drops to “0”.) The actual value of b could be chosen by us arbitrarily; such as a value approximately based on first principles of biology, or estimated statistically (e.g., using linear regression methods).

To take another example, suppose we wish to predict population change over time. Our state variable, abundance, now should be indexed to time (t); we will represent abundance at time t as N (t). A simple model of population growth (which we will explore further in the next section) supposes that birth (b) and death rates (d) are constant over time, leading (in the absence of immigration or emigration) to constant growth r = b – d. These are the model parameters (in this case r alone suffices). Our basic dynamic model is

simply, that next year’s abundance [N (t + 1)] is equal to this year’s abundance times a growth multiplier. Actually, we need one more feature to complete our model: a place for the population to start. This initial condition is the population’s abundance at some initial time t = 0, N(0). Given this value, and a value for r, we have a working model. For example, if we take r = 0.05 and N(0) = 100 we have:

and so on. The result is that a small amount of information about a population can allow us to create a simple and possibly useful model in describing the abundance of some species.

Discrete or continuous models: stochastic or deterministic

There are many mathematical ways to construct models, and the above simple examples represent two types of choices for model construction. If our models are dynamic (meaning that the population or other state changes over time), then we must decide whether to represent time in discrete terms as above, or in continuous form. For most of the modeling in this book we favor discrete-time models, because many of the animals we deal with reproduce seasonally, so discrete time seems appropriate. Also, many population surveys are conducted annually at specific dates, making it easier to think about population change as occurring over intervals [t, t + 1] rather than instantaneously. Finally, the mathematics of continuous-time models can be more difficult, involving differential calculus; discrete-time models are represented by difference equations, somewhat easier to grasp, and also easier to translate into spreadsheets or other computer code. However, continuous-time models are important in much of ecology, so advanced readers are encouraged to study them further in references such as Williams et al. (2002).

Conservation modelers must also decide whether to build models that are deterministic – that is, contain no random elements – or stochastic, containing random elements (and thus predictions that vary from one run to the next). There are advantages and disadvantages of each type of model. It is often a good idea to start with deterministic models and focus on the mathematical behavior of the model, such as equilibrium and sensitivity analysis, without the added distraction of random effects. For example, the relative impact of one parameter versus another on the outcome from your modeling will be much clearer without random variation included. Once the mathematics of the model are well understood then random effects are often added to create additional model realism (since real populations are subject to many different types of random influences!) In the next section we build both deterministic and stochastic population models.

Verification and validation

Hopefully, any model we create is at least a plausible, if not necessarily correct, version of reality. However, it is very important to perform “reality checks” on models that we have built. Model verification is essentially a check to see that the model produces results similar to what we intended and (if data are involved) matches well the data used to build the model. At this step, we may discover that the model produces results that are nonsensical or biologically counterintuitive (e.g., negative population values, or survival rates > 1.0) or that otherwise make no sense. This is the time to discover why these aberrant results are occurring – and to fix them. Of course, just because the model produces results that look right does not mean that they are right. This is where model validation comes in. True validation involves the comparison of model predictions to data collected independently of the model construction (the data were not used to build the model). This is a very strong test of the model, and one that is unfortunately rarely done. We will return to model validation and ways to improve models through time later in this chapter.

Model behavior and sensitivity analysis

It is often very important to have an idea of how models behave over ranges of parameter values, or (for dynamic models) how they perform over time. Model behavior is especially relevant to conservation issues such as sustainable harvest and viability analysis, since we are usually interested in how real populations will behave over long time horizons, or in response to management actions. For dynamic models, it is especially important to determine whether the system will reach equilibrium, and, if so, where it (or they) exists. An equilibrium is simply a state at which the system no longer changes over time; for abundance that is where N (t + 1) = N (t). For many models, it is also important to determine model stability, essentially the tendency of a model to return to an equilibrium following a perturbation.

It is also important to evaluate the model’s sensitivity to variations in parameter values, for a number of reasons. First, some parameters may be controllable by management, and therefore knowledge of how these parameters influence the system (e.g., equilibrium abundance) can be very important to conservation and/or management. Second, most parameters will not be known with certainty, and a knowledge of how much it matters that parameter values are possibly varying from true values can focus priorities on improving these values.

In this chapter we should have accomplished two important things about models. First, and most importantly, we should have shown the reader that models are not to be feared and should be fairly useful to those of us doing conservation science. The second goal is to provide the lead-in for Chapter 3 where we expand our understanding of models to some practical examples.