The Theoretical Biologist’s Toolbox ppt

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The Theoretical Biologist’s Toolbox
Quantitative Methods for Ecology and Evolutionary Biology
Mathematical modeling is widely used in ecology and evolutionary biology
and it is a topic that many biologists find difficult to grasp. In this new
textbook Marc Mangel provides a no-nonsense introduction to the skills
needed to understand the principles of theoretical and mathematical bio-
logy. Fundamental theories and applications are introduced using numerous
examples from current biological research, complete with illustrations to
highlight key points. Exercises are also included throughout the text to show
how theory can be applied and to test knowledge gained so far. Suitable for
advanced undergraduate or introductory graduate courses in theoretical and
mathematical biology, this book forms an essential resource for anyone
wanting to gain an understanding of theoretical ecology and evolution.
MARC MANGEL is Professor of Mathematical Biology and Fellow of
Stevenson College at the University of California, Santa Cruz campus.

The Theoretical Biologist’s Toolbox
Quantitative Methods for
Ecology and Evolutionary Biology
Marc Mangel
Department of Applied Mathematics
and Statistics
University of California, Santa Cruz
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
First published in print format
isbn-13 978-0-521-83045-4

isbn-13 978-0-521-53748-3
isbn-13 978-0-511-24512-1
© Marc Mangel 2006
2006
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/9780521830454
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
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Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
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erback
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erback
eBook (EBL)

eBook (EBL)
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To all of my teachers, but especially Susan Mangel.

Contents
Preface page ix
Permissions xiii
1 Four examples and a metaphor 1
2 Topics from ordinary and partial differential
equations 20
3 Probability and some statistics 80
4 The evolutionary ecology of parasitoids 133
5 The population biology of disease 168
6 An introduction to some of the problems of
sustainable fisheries 210
7 The basics of stochastic population dynamics 248
8 Applications of stochastic population dynamics
to ecology, evolution, and biodemography 285
References 323
Index 369
vii

Preface: Bill Mote, Youngblood Hawke,
and Mel Brooks
I conceived of the courses that led to this book on sabbatical in
1999–2000, during my time as the Mote Eminent Scholar at Florida
State University and the Mote Marine Laboratory (a chair generously
funded by William R. Mote, who was a good friend of science). While at
FSU, I worked on a problem of life histories in fluctuating environments
with Joe Travis and we needed to construct log-normal random vari-

ables of specified means and variances. I did the calculation during my
time spent at Mote Marine Laboratory in Sarasota and, while doing the
calculation, realized that although this was something pretty easy and
important in ecology and evolutionary biology, it was also something
difficult to find in the standard textbooks on probability or statistics.
It was then that I decided to offer a six-quarter graduate sequence in
quantitative methods, starting the following fall. I advertised the course
initially as ‘‘Quantitative tricks that I’ve learned which can help you’’
but mainly as ‘‘The Voyage of Quantitative Methods,’’ ‘‘The Voyage
Continues,’’ etc. This book is the result of that course.
There is an approximate ‘‘Part I’’ and ‘‘Part II’’ structure. In the first
three chapters, I develop some basic ideas about modeling (Chapter 1),
differential equations (Chapter 2), and probability (Chapter 3). The
remainder of the book involves the particular applications that inter-
ested me and the students at the time of the course: the evolutionary
ecology of parasitoids (Chapter 4), the population biology of disease
(Chapter 5), some problems of sustainable fisheries (Chapter 6), and the
basics and application of stochastic population theory in ecology, evo-
lutionary biology and biodemography (Chapters 7 and 8).
Herman Wouk’s character Youngblood Hawke (Wouk 1962) bursts
on the writing scene and produces masterful stories until he literally has
nothing left to tell and burns himself out. The stories were somewhere
between the ether and the inside of his head and he had to get them out.
Much the same is true for music. Bill Monroe (Smith 2000) and Bob
Dylan (Sounes 2001) reported that their songs were already present,
either in the air or in their heads and that they could not rest until the
songs were on paper. Mozart said that he was more transcribing music
that was in his head than composing it. In other words, they all had a
ix
story to tell and could not rest until it was told. Mel Brooks, the

American director and producer, once wrote ‘‘I do what I do because I
have to get it out. I’m just lucky it wasn’t an urge to be a pickpocket.’’
I too have a story to get out, but mine is about theoretical biology, and
once I began writing this book, I could not rest until it was down on paper.
Unlike a novel, however, you’ll not likely read this book in a weekend or
before bed. But I hope that you will read it. Indeed, it took me two years
of once a week meetings plus one quarter of twice a week meetings with
classes to tell the story (in Chapter 1, I offer some guidelines on how to
use the book), so I expect that this volume will be a long-term companion
rather than a quick read. And I hope that you will make it so. Like my
other books (Mangel 1985, Mangel and Clark 1988, Hilborn and Mangel
1997, Clark and Mangel 2000), my goal is to bring people – keen
undergraduates, graduate students, post-docs, and perhaps even a faculty
colleague or two – to a skill level in theoretical biology where they will
be able to read the primary literature and conduct their own research. I do
this by developing tools and showing how they can be used. Suzanne
Alonzo, a student of Bob Warner’s, post-doc with me and now on the
faculty at Yale University, once told me that she carried Mangel and
Clark (1988) everywhere she went for the first two years of graduate
school. In large part, I write this book for the future Suzannes.
Before writing this story, I told most of it as a six quarter graduate
seminar on quantitative methods in ecology and evolutionary biology.
These students, much like the reader for whom I write, were keen to
learn quantitative methods and wanted to get to the heart of the matter –
applying such methods to interesting questions in ecology and evolu-
tionary biology – as quickly as possible. I promised the students that if
they stuck with it, they would be able to read and understand almost
anything in the literature of theoretical biology. And a number of them
did stick through it: Katriona Dlugosch, Will Satterthwaite, Angie
Shelton, Chris Wilcox, and Nick Wolf (who, although not a student

earned a special certificate of quantitude). Other students were able to
attend only part of the series: Nick Bader, Joan Brunkard, Ammon Corl,
Eric Danner, EJ Dick, Bret Eldred, Samantha Forde, Cindy Hartway,
Cynthia Hays, Becky Hufft, Teresa Ish, Rachel Johnson, Matt
Kauffman, Suzanne Langridge, Doug Plante, Jacob Pollock, and Amy
Ritter. Faculty and NMFS/SCL colleagues Brent Haddad, Karen Holl,
Alec MacCall, Ingrid Parker, and Steve Ralston attended part of the
series too (Brent made five of the six terms!). To everyone, I am very
thankful for quizzical looks and good questions that helped me to clarify
the exposition of generally difficult material.
Over the years, theoretical biology has taken various hits (see, for
example, Lander (2004)), but writing at the turn of the millennium,
x Preface: Bill Mote, Youngblood Hawke, and Mel Brooks
Sidney Brenner (Brenner 1999) said that there is simply no better
description and we should use it. Today, of course, computational
biology is much in vogue (I sometimes succumb to calling myself a
computational biologist, rather than a theoretical or mathematical biol-
ogist) and usually refers to bioinformatics, genomics, etc. Although
these are not the motivational material for this book, readers interested
in such subjects will profit from reading it. The power of mathematical
methods is that they let us approach apparently disparate problems with
the same kind of machinery, and many of the tools for ecology and
evolutionary biology are the same ones as for bioinformatics, genomics,
and systems biology.
I have tried to make this book fun to read, motivated by Mike
Rosenzweig’s writing in his wonderful book on species diversity
(Rosenzweig 1995). There he asserted – and I concur – that because a
book deals with a scientific topic in a technical (rather than popular)
way, it does not have to be thick and hard to read (not everyone agrees
with this, by the way). I have also tried to make it relatively short, by

pointing out connections to the literature, rather than going into more
detail on additional topics. I apologize to colleagues whose work should
have been listed in the Connections section at the end of each chapter,
but is not.
For the use of various photos, I thank Luke Baton, Paulette
Bierzychudek, Kathy Beverton, Leon Blaustein, Ian Fleming, James
Gathany, Peter Hudson, Jay Rosenheim, Bob Lalonde, and Lisa
Ranford-Cartwright. Their contributions make the book both more
interesting to read and more fun to look at. Permissions to reprint figures
were kindly granted by a number of presses and individuals; thank you.
Nicole Rager, a graduate of the Science Illustration Program at UC
Santa Cruz and now at the NSF, helped with many of the figures, and
Katy Doctor, now in graduate school at the University of Washington,
aided in preparation of the final draft, particularly with the bibliography
and key words for indexing.
Alan Crowden commissioned this book for Cambridge University
Press. His continued enthusiasm for the project helped spur me on. For
comments on the entire manuscript, I thank Emma A
˚
dahl, Anders
Brodin, Tracy S. Feldman, Helen Ivarsson, Lena Ma
˚
nsson, Jacob
Johansson, Niclas Jonzen, Herbie Lee, Jo¨rgen Ripa, Joshua Uebelherr,
and Eric Ward. For comments on particular chapters, I thank Per
Lundberg and Kate Siegfried (Chapter 1), Leah Johnson (Chapter 2),
Dan Merl (Chapter 3), Nick Wolf (Chapter 4), Hamish McCallum,
Aand Patil, Andi Stephens (Chapter 5), Yasmin Lucero (Chapter 6),
and Steve Munch (Chapters 7 and 8). The members of my research
group (Kate, Leah, Dan, Nick, Anand, Andi, Yasmin, and Steve)

Preface: Bill Mote, Youngblood Hawke, and Mel Brooks xi
undertook to check all of the equations and do all of the exercises, thus
finding bloopers of various sizes, which I have corrected. Beverley
Lawrence is the best copy-editor with whom I have ever worked; she
deserves great thanks for helping to clarify matters in a number of
places. I shall miss her early morning email messages.
In our kind of science, it is generally difficult to separate graduate
instruction and research, since every time one returns to old material,
one sees it in new ways. I thank the National Science Foundation,
National Marine Fisheries Service, and US Department of
Agriculture, which together have continuously supported my research
efforts in a 26 year career at the University of California, which is a
great place to work.
At the end of The Glory (Wouk 1994), the fifth of five novels about
his generation of destruction and resurgence, Herman Wouk wrote
‘‘The task is done, and I turn with a lightened spirit to fresh beckoning
tasks’’ (p. 685). I feel much the same way.
Have a good voyage.
xii Preface: Bill Mote, Youngblood Hawke, and Mel Brooks
Permissions
Figure 2.1. Reprinted from Washburn, A. R. (1981). Search and
Detection. Military Applications Section, Operations Research Society
of America, Arlington, VA., with permission of the author.
Figure 2.21a. Reprinted from Ecology, volume 71, W. H. Settle and
L. T. Wilson, Invasion by the variegated leafhopper and biotic interactions:
parasitism, competition, and apparent competition, pp. 1461–1470.
Copyright 1990, with permission of the Ecological Society of America.
Figure 3.12. Reprinted from Statistics, third edition, by David Freeman,
Robert Pisani and Roger Purves. Copyright 1998, 1991, 1991, 1978 by
W. W. Norton & Company, Inc. Used by permission of W. W. Norton &

Company, Inc.
Figure 4.11. Reprinted from May, Robert M., Stability and Complexity
in Model Ecosystems. Copyright 1973 Princeton University Press, 2001
renewed PUP. Reprinted by permission of Princeton University Press.
Figure 4.15. Reprinted from Theoretical Population Biology, volume 42,
M. Mangel and B. D. Roitberg, Behavioral stabilization of host-parasitoid
population dynamics, pp. 308–320, Figure 3 (p. 318). Copyright 1992,
with permission from Elsevier.
Figure 5.3. Reprinted from Proceedings of the Royal Society of
London, Series A, volume 115, W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, pp. 700–721,
Figure 1. Copyright 1927, with permission of The Royal Society.
Figure 5.9. Reprinted from Ecology Letters, volume 4, J. C. Koella
and O. Restif, Coevolution of parasite virulence and host life history,
pp. 207–214, Figure 2 (p. 209). Copyright 2001, with permission
Blackwell Publishing.
Figure 5.12. From Infectious Diseases of Humans: Dynamics and
Control by R. M. Anderson and R. M. May, Figure 14.25 (p. 408).
Copyright 1991 Oxford University Press. By permission of Oxford
University Press.
xiii
Figure 8.3. Robert MacArthur, The Theory of Island Biogeography.
Copyright 1967 Princeton University Press, 1995 renewed PUP.
Reprinted by permission of Princeton University Press.
Figure 8.10. Reprinted from Ecological Monographs, volume 70,
J. J. Ande rson, A vitality-based model relating stressors and environ-
mental properties to organismal survival, pp. 445–470, Figure 5 (p. 455)
and Figure 14 (p. 461). Copyright 2000, with permission of the
Ecological Society of America.
xiv Permissions

Chapter 1
Four examples and a metaphor
Robert Peters (Peters 1991) – who (like Robert MacArthur) tragically
died much too young – told us that theory is going beyond the data.
I thoroughly subscribe to this definition, and it shades my perspective
on theoretical biology (Figure 1.1). That is, theoretical biology begins
with the natural world, which we want to understand. By thinking about
observations of the world, we conceive an idea about how it works. This
is theory, and may already lead to predictions, which can then flow back
into our observations of the world. Theory can be formalized using
mathematical models that describe appropriate variables and processes.
The analysis of such models then provides another level of predictions
which we take back to the world (from which new observations may
flow). In some cases, analysis may be insufficient and we implement the
models using computers through programming (software engineering).
These programs may then provide another level of prediction, which
can flow back to the models or to the natural world. Thus, in biology
there can be many kinds of theory. Indeed, without a doubt the greatest
theoretician of biology was Charles Darwin, who went beyond the data
by amassing an enormous amount of information on artificial selection
and then using it to make inferences about natural selection. (Second
place could be disputed, but I vote for Francis Crick.) Does one have to
be a great naturalist to be a theoretical biologist? No, but the more you
know about nature – broadly defined (my friend Tim Moerland at
Florida State University talks with his students about the ecology of
the cell (Moerland 1995)) – the better off you’ll be. (There are some
people who will say that the converse is true, and I expect that they
won’t like this bo ok.) The same is true, of course, for being able to
1
develop models and implementing them on the computer (although, I

will tell you flat out right now that I am not a very good programmer –
just sufficient to get the job done). This book is about the middle of
those three boxes in Figure 1.1 and the objective here is to get you to be
good at converting an idea to a model and analyzing the model (we will
discuss below what it means to be good at this, in the same way as what
it means to be good at opera).
On January 15, 2003, just as I started to write this book, I attended a
celebration in honor of the 80th birthday of Professor Joseph B. Keller.
Keller is one of the premier applied mathematicians of the twentieth
century. I first met him in the early 1970s, when I was a graduate
student. At that time, among other things, he was working on mathe-
matics applied to sports (see, for example, Keller (1974)). Joe is fond of
saying that when mathematics interacts with science, the interaction is
fruitful if mathematics gives something to science and the science gives
something to mathematics in return. In the case of sports, he said that
what mathematics gained was the concept of the warm-up. As with
athletics, before embarking on sustained and difficult mathematical
exercise, it is wise to warm-up with easier things. Most of this chapter
is warm-up. We shall consider four examples, arising in behavioral and
evolutionary ecology, that use algebra, plane geometry, calculus, and a
tiny bit of advanced calculus. After that, we will turn to two metaphors
about this material, and how it can be learned and used.
Foraging in patchy environments
Some classic results in behavioral ecology (Stephens and Krebs 1986,
Mangel and Clark 1988, Clark and Mangel 2000) are obtained in the
Natural world:
Observations
An idea of how the world works:
Theory and predictions
Variables, processes:

Mathematical models
Analysis of the models:
A second level of prediction
Implementation of
the models:
Software engineering
A third level of
prediction
Figure 1.1. Theoretical biology
begins with the natural world,
which we want to understand.
By thinking about observations
of the world, we begin to
conceive an idea about how it
works. This is theory, and may
already lead to predictions,
which can then flow back into
our observations of the world.
The idea about how the world
works can also be formalized
using mathematical models
that describe appropriate
variables and processes. The
analysis of such models then
provides another level of
predictions which we can take
back to the world (from which
new observations may flow).
In some cases, analysis may
be insufficient and we choose

to implement our models
using computers through
programming (software
engineering). These programs
then provide another level of
prediction, which can also flow
back to the models or to the
natural world.
2 Four examples and a metaphor
study of organisms foraging for food in a patchy environment
(Figure 1.2). In one extreme, the food might be distributed as individual
items (e.g. worms or nuts) spread over the foraging habitat. In another,
the food might be concentrated in patches, with no food between the
patches. We begin with the former case.
The two prey diet choice problem (algebra)
We begin by assuming that there are only two kinds of prey items (as
you will see, the ideas are easily general ized), which are indexed by
i ¼1, 2. These prey are characterized by the net energy gain E
i
from
consuming a single prey item of type i, the time h
i
that it takes to handle
(capture and consume) a single prey item of type i, and the rate l
i
at
which prey items of type i are encountered. The profitability of a single
prey item is E
i
/h

i
since it measures the rate at which energy is accumu-
lated when a single prey item is consumed; we will assume that prey
(a)
(b)
(c)
Figure 1.2. Two stars of foraging experiments are (a) the great tit, Parus major, and (b) the common starling Sturnus
vulgaris (compliments of Alex Kacelnik, University of Oxford). (c) Foraging seabirds on New Brighton Beach,
California, face diet choice and patch leaving problems.
Foraging in patchy environments 3
type 1 is more profitable than prey type 2. Consider a long period of
time T in which the only thing that the forager does is look for prey
items. We ask: what is the best way to consume prey? Since I know the
answer that is coming, we will consider only two cases (but you might
want to think about alternatives as you read along). Either the forager
eats whatever it encounters (is said to generalize) or it only eats prey
type 1, rejecting prey type 2 whenever this type is encountered (is said
to specialize). Since the flow of energy to organisms is a fundamental
biological consideration, we will assume that the overall rate of energy
acquisition is a proxy for Darwinian fitness (i.e. a proxy for the long
term number of descendants).
In such a case, the total time period can be d ivided into time spent
searching, S, and time spent handling prey, H. We begin by calculating
the rate of energy acquisition when the forager specializes. In search
time S, the number of prey items encountered will be l
1
S and the time
required to handle these prey items is H ¼h
1
(l

1
S ). According to our
assumption, the only things that the forager does is search and handle
prey items, so that T ¼S þH or
T ¼ S þ h
1
l
1
S ¼ Sð1 þ l
1
h
1
Þ (1:1)
We now solve this equation for the time spent searching, as a
fraction of the total time available and obtain
S ¼
T
1 þ l
1
h
1
(1:2)
Since the number of prey items encountered is l
1
S and each item
provides net energy E
1
, the total energy from specializing is E
1
l

1
S, and
the rate of acquisition of energy will be the total accumulated energy
divided by T. Thus, the rate of gain of energy from specializing is
R
s
¼
E
1
l
1
1 þ h
1
l
1
(1:3)
An aside: the importance of exercises
Consistent with the notion of mathematics in sport, you are developing a
set of skills by reading this book. The only way to get better at skills is
by practice. Throughout the book, I give exercises – these are basically
steps of analysis that I leave for you to do, rather than doing them here.
You should do them. As you will see when reading this book, there is
hardly ever a case in which I write ‘‘it can be shown’’ – the point of this
material is to learn how to show it. So, take the exercises as they come –
in general they should require no more than a few sheets of paper – and
really make an effort to do them. To give you an idea of the difficulty of
4 Four examples and a metaphor
exercises, I parenthetically indicate whether they are easy (E), of med-
ium difficulty (M), or hard (H).
Exercise 1.1 (E)

Repeat the process that we followed above, for the case in which the forager
generalizes and thus eats either prey item upon encounter. Show that the rate of
flow of energy when generalizing is
R
g
¼
E
1
l
1
þ E
2
l
2
1 þ h
1
l
1
þ h
2
l
2
(1:4)
We are now in a position to predict the best option: the forager is
predicted to specialize when the flow of energy from specializing i s greater
than the flow of ene rgy from g eneralizing. This will occur when R
s
> R
g
.

Exercise 1.2 (E)
Show that R
s
> R
g
implies that
l
1
>
E
2
E
1
h
2
À E
2
h
1
(1:5)
Equation (1.5) defines a ‘‘switching value’’ for the encounter rate
with the more profitable prey item, since as l
1
increases from below to
above this value, the behavior switches from generalizing to speciali-
zing. Equation (1.5) has two important implications. First, we predict
that the foraging behavior is ‘‘knife-edge’’ – that there will be no partial
preferences. (To some extent, this is a result of the assumptions. So if
you are uncomfortable with this conclusion, repeat the analysis thus far
in which the forager chooses prey type 2 a certain fraction of the time, p,

upon encounter and compute the rate R
p
associated with this assumption.)
Second, the behavior is determined solely by the encounter rate with the
more profitable prey item since the encounter rate with the less profitable
prey item does not appear in the expression for the switching value.
Neither of these could have been predicted a priori.
Over the years, there have been many tests of this model, and much
disagreement about what these tests mean (more on that below). My
opinion is that the model is an excellent starting point, given the simple
assumptions (more on these below, too).
The marginal value theorem (plane geometry)
We now turn to the second foraging model, in which the world is assumed
to consist of a large number of identical and exhaustible patches contain-
ing only one kind of food with the same travel time between them
Foraging in patchy environments 5
Travel time
(a)
τ
25
20
0
0
0.02
0.04
Rate of gain, R(t)
0.06
0.08
0.1
(c)

510
Residence time, t
15
0–5
0
0.2
0.4
Gain
0.6
0.8
1
(d)
510
Residence time
Optimal residence time
15
0
5
10
15
20
25
0
0.2
0.4
Gain, G(t)
0.6
0.8
1
(b)

Residence time, t
Figure 1.3. (a) A schematic of the situation for which the marginal value theorem applies. Patches of food
(represented here in metaphor by filled or empty patches) are exhaustible (but there is a very large number of them)
and separated by travel time . (b) An example of a gain curve (here I used the function G(t) ¼t/(t þ3), and (c) the
resulting rate of gain of energy from this gain curve when the travel time  ¼3. (d) The marginal value construction
using a tangent line.
6 Four examples and a metaphor
(Figure 1.3a). The question is different: the choice that the forager faces is
how long to stay in the patch. We will call this the patch residence time,
and denote it by t. The energetic value of food removed by the forager
when the residence time is t is denoted by G(t). Clearly G(0) ¼0(since
nothing can be gained when no time is spent in the patch). Since the patch
is exhaustible, G(t) must plateau as t increases. Time for a pause.
Exercise 1.3 (E)
One of the biggest difficulties in this kind of work is getting intuition about
functional forms of equations for use in models and learning how to pick them
appropriately. Colin Clark and I talk about this a bit in our book (Clark and
Mangel 2000). Two possible forms for the gain function are G(t) ¼at/(b þt) and
G(t) ¼at
2
/(b þt
2
). Take some time before reading on and either sketch these
functions or pick values for a and b and graph them. Think about what the
differences in the shapes mean. Also note that I used the same constants (a and
b) in the expressions, but they clearly must have different meanings. Think
about this and remember that we will be measuring gain in energy units (e.g.
kilocalories) and time in some natural unit (e.g. minutes). What does this imply
for the units of a and b, in each expression?
Back to work. Suppose that the travel time between the patches

is . The problem that the forager faces is the choice of residence in the
patch – how long to stay (alternatively, should I stay or should I go
now?). To predict the patch residence time, we proceed as follows.
Envision a foraging cycle that consists of arrival at a patch, resi-
dence (and foraging) for time t and then travel to the next patch, after
which the process begins again. The total time associated with one
feeding cycle is thus t þ and the gain from that cycle is G(t), so that
the rate of gain is R(t) ¼G(t)/(t þ). In Figure 1.3, I also show an
example of a gain function (panel b) and the rate of gain function
(panel c). Because the gain function reaches a plateau, the rate of g ain
has a peak. For residence times to the left of the peak, the forager is
leaving too soon and for residen ce times to the right of the peak the
forager is remaining too long to optimize the rate of gain of energy.
The questio n is then: how do we find the location of the peak, given
the gain function and a travel time? One could, of course, recognize that
R(t) is a function of time, depending upon the constant  and use
calculus to find the residence time that maximizes R(t), but I promised
plane geometry in this warm-up. We now proceed to repeat a remark-
able construction done by Eric Charnov (Charnov 1976). We begin by
recognizing that R(t) can be written as
RðtÞ¼
GðtÞ
t þ 
¼
GðtÞÀ0
t ÀðÀÞ
(1:6)
Foraging in patchy environments 7
and that the right hand side can be interpreted as the slope of the line that
joins the point (t, G(t)) on the gain curve with the point (À, 0) on the

abscissa (x-axis). In general (Figure 1.3d), the line between (À, 0) and
the curve will intersect the curve twic e, but as the slope of the line
increases the points of intersection come closer together, until they meld
when the line is tangent to the curve. From this point of tangency, we
can read down the optimal residence time. Charnov called this the
marginal value theorem, because of analogies in economics. It allows
us to predict residence times in a wide variety of situations (see the
Connections at the end of this chapter for more details).
Egg size in Atlantic salmon and parent–offspring
conflict (calculus)
We now come to an example of great generality – predicting the size of
propagules of reproducing individuals – done in the context of a specific
system, the Atlantic salmon Salmo salar L. (Einum and Fleming 2000).
As with most but not all fish, female Atlantic salmon lay eggs and the
resources they deposit in an egg will support the offspring in the initial
period after hatching, as it develops the skills needed for feeding itself
(Figure 1.4). In general, larger eggs will improve the chances of off-
spring survival, but at a somewhat decreasing effect. We will let x
denote the mass of a single egg and S(x) the survival of an offspring
through the critical period of time (Einum and Fleming used both 28 and
107 days with similar results) when egg mass is x. Einum and Fleming
chose to model S(x)by
SðxÞ¼1 À
x
min
x

a
(1:7)
where x

min
¼0.0676 g and a ¼1.5066 are parameters fit to the data.
We will define c ¼(x
min
)
a
so that S(x) ¼1 Àcx
Àa
, understanding that
S(x) ¼0 for values of x less than the minimum size. This function is
shown in Figure 1.5a; it is an increasing function of egg mass, but has a
decreasing slope. Even so, from the offspring perspective, larger eggs
are better.
However, the perspective of the mother is different because she has
a finite amount of gonads to convert into eggs (in the experiments of
Einum and Fleming, the average female gonadal mass was 450 g).
Given gonada l mass g, a mother who produces eggs of mass x will
make g/x eggs, so that her reproductive success (defined as the expected
number of eggs surviving the critical period) will be
Rðg; xÞ¼
g
x
SðxÞ¼
g
x
ð1 À cx
Àa
Þ (1:8)
8 Four examples and a metaphor
and we can find the optimal egg size by setting the derivative of R(g, x)

with respect to x equal to 0 and solving for x.
Exercise 1.4 (M)
Show that the optimal egg size based on Eq. (1.8)isx
opt
¼fcða þ 1Þg
1=a
and
for the values from Einum and Fleming that this is 0.1244 g. For comparison, the
observed egg size in their experiments was about 0.12 g.
(c)
(b)(a)
Figure 1.4. (a) Eggs, (b) a nest,
and (c) a juvenile Atlantic
salmon – stars of the
computation of Einum and
Fleming on optimal egg size.
Photos complements of Ian
Fleming and Neil Metcalfe.
Egg size in Atlantic salmon and parent–offspring conflict (calculus) 9