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Unformatted text preview: 5 are worth a
K 5.
;‘ 3 Population Genetics The famous geneticist Theodosius Dobzhansky once wrote that
nothing in biology makes sense except in the light of evolution.
This is certainly true about many of the ecological questions of
interest to us. Life history patterns, for example, are shaped by
selection. The ages at which individuals have offspring, when
they disperse, and even when they die are the results of selection.
We need to develop some concepts about the genetic properties
of populations to help us understand questions in evolutionary
ecology. In turn, we will consider the issue of variation within a
species — a strictly genetic question — and then turn to questions
at the interface between ecology and genetics. 3.1 Genetic questions What properties of populations can be classiﬁed as genetic in na
ture? Perhaps the most important aspect concerns variation among
individuals. One of the primary questions of population genet—
ics is to determine the reasons for the maintenance of polymer—
pbz‘smsMwo or more different types Within a single population.
Until the 19603, the variants studied by geneticists were almost all 42 3. Population Genetics What other examples of visible
polymorphisms can you think of? The technique of gel electrophoresis is
restricted to looking at soluble proteins that
catalyze reactions. Visible polymorphisms: they could be detected by looking at the
organism in question. One striking example of a visible polymorphism is the exis—
tence of two sexes in most higher organisms. There is clearly a
mechanism for the maintenance of the polymorphism. If there is
only one male in a population and many females, the number
of offspring for the male will clearly be higher then the average
number of offspring for the females. Thus, any males produced
will also have more offspring than any females produced. Conse—
quently, the fraction of males in a population will tend to increase
over time. This argument also works in reverse if there is only one
female to start with and many males. In general, if a sex becomes
rare it is then more ‘valuable’. There are a number of other classic cases of visible polymor
phism. One is the maintenance of dark and light forms in the
moth Biston betularia and the change in the frequency of these
forms following an increase in soot from industrial pollution —
the case of industrial melam'sm. Another wellknown example is
the maintenance of the gene responsible for sicklecell anemia
in humans, a disease in which the red blood cells are deformed
and less efﬁcient in carrying oxygen. We will explore this case
in the problems at the end of the chapter. A third example is the
snail Cepaea nemomlis, whose numerous shell color and banding
polymorphisms have been the subject of extensive study. Until the 19605 many geneticists thought genetic variants were
relatively rare. Then, beginning with the classic work of Lewontin
and Hubby (1966), the technique of gel electrophoresis emerged
as a method to look for polymorphisms that did not have obvious
macroscopic visual effects. Classic gel electrophoresis begins by
placing soluble proteins in a gel made of starch. The gel is then
subjected to an electric current, and different variants of the same
protein may move at different rates, depending on the properties
of the amino acids on the outside of the protein. Slices of the
gel are treated with the substrates and cofactors catalyzed by a
particular enzyme (protein) and a dye that reacts with the end
product of the reaction. Proteins migrate under the inﬂuence of
the electric ﬁeld and appear as bands on the gel. Bands which
travel different distances are different alleles. (Some amino acid FIGURE 3.1.Scl
Each numberc
moves in the g
chemical that'
which travel a
respectively, h
arbitrarily. substitutio
tion rates
of populal
of extensi‘ The qur
tion of the
variation r
tion (diffe
is the maj
phism. Ar
of differei
pled with
the major During
look for p
Techniqur
even the l
electrophr
direct exa
1995). Th
responsib The trai fa Iation Genetics
king at the __is the exis
" is clearly a
'e . If there is
l the number
, the average
'5 produced
, ced. Conse—
3‘ to increase
, is only one
i x becomes Le polymor
rrns in the
icy of these
pollution —
l example is
‘x 11 anemia
7 deformed
‘_  this case ple is the d banding
dy. ‘
‘. 'nts were
.I‘ Lewontin
g emerged
 e obvious
It begins by
gel is then the same
3. properties es of the
A, ed by a
‘i the end r uence of
Y. ds which
’ ino acid 3.1 Genetic questions FIGURE 3.1 . Schematic diagram illustrating gel electrophoresis as a tool forﬁnding genotypes.
Each number corresponds to an individual. Afterthe application of an electric current, a protein
moves in the gel, and then the substrates of a reaction catalyzed by that protein as well as a
chemical that will react with the end product are placed on the gel. The alleles of the protein,
which travel at different rates, show up as bands on the gel. Thus individuals 1 through 5,
respectively, have the genotypes BC, BB, CC, AB, and AC. The names of the alleles are chosen
arbitrarily. substitutions which do not result in proteins with different migra
tion rates under electrophoresis will not be detected.) The result
of population surveys using gel electrophoresis was the discovery
of extensive polymorphisms in almost all species examined. The question of interest to geneticists became the determina
tion of the forces responsible for the maintenance of the extensive
variation revealed by electrophoresis. One possibility is that selec—
tion (differential reproduction or survival of different genotypes)
is the major force responsible for the maintenance of polymor—
phism. Another possibility is that dnft (changes in the frequency
of different genotypes caused by random sampling effects) cou
pled with mutation (the chance production of new alleles) are
the major forces responsible for polymorphism. During the past decade the techniques used by geneticists to
look for polymorphisms have become even more sophisticated.
Techniques from molecular biology have been used to show that
even the high estimates of levels of polymorphism revealed by gel
electrophoresis were lower than those determined by direct or in
direct examination of DNA sequences (e.g., Begun and Aquadro,
1995). The major goal remains the determination of the forces
responsible for the maintenance of polymorphism. The traits examined through electrophoresis or DNA sequence 43 Genetic techniques may help ecologists to
determine rates of exchange of breeding
individuals between different populations. 44 What other quantitative traits can you think
of? 3. Population Genetics Frequency 0 50 1 00 1 50 200 Ratio of wing vein length x 100 FIGURE 3.2. Distribution of a quantitative trait, the ratio of wing vein lengths in Drosophi/a
melanogaster (data from Scharloo et al., 1967). The frequency distribution is very roughly
normal. techniques may not be the ones that are important to an ecolo—
gist, in contrast to a trait such as the size of an individual. A very
different kind of question emerges when focusing on quantita—
tive characters — the kind that can be measured, such as height,
weight, or the ratio of sizes of different body parts (Figure 3.2).
Here the issue is why is every individual not the same — presum
ably optimal  size? Although some variation results from envi
ronmental factors, there is also underlying genetic variability in
virtually all cases. A related issue is how fast such a quantitative
trait evolves in response to selection. 3.2 Evolutionary and ecological questions There are several areas where the interface between ecology and
genetics has produced fruitful insights. One involves the use of
optimization concepts in ecology. Many ecologists studying the
behavior of organisms assume that the behavior is ‘optimal’. The
natural question is whether it is reasonable to expect that the outcome of
is the evolz
takes place theory — hc
food items.
For ecolc 0 Under
traits ‘ 0 Why
optim o If the
envirr
spon:
evolv Other topi:
elude coev
and parasi
the very di 3.3 Or. We will no
to all thee
complex, subject of :‘
that may 5
our under Hardy—ii At the tin
factors re:
understoo
generatior
ents. Greg
heritance Lulation Genetics 3 ll ‘ in Drosophila
is very roughly 7? an ecolo al. A very
I quantita
as height,
igure 3.2).
L' — presum m envi
‘ ,. 'ability in antitative r is and
e e use of
‘ ying the
WI 1’. The that the 3.3 Onelocus model without selection outcome of evolution is optimal behavior? One speciﬁc example
is the evolution of life histories — when and how reproduction
takes place. Another general area of study is optimal foraging
theory — how a foraging animal searches for patchily distributed
food items. For ecological questions such as these we may ask: 0 Under what circumstances will natural selection produce
traits that are optimal? 0 Why are all organisms within the population not of the
optimal type: why is variability maintained? 0 If the environment is Changing, or the organism is in a novel
environment, one would like to know how fast is the re
sponse to selection. How quickly does the population
evolve toward the optimum? Other topics at the interface between ecology and evolution in
clude coevolution, the joint evolution of predator and prey, host
and parasitoid, host and pathogen, or plant and pollinator, and
the very difﬁcult issue of the process of speciation. 3.3 Onelocus model without selection We will not be able to produce models that will guide our answers
to all these questions. The models involved can become very
complex, and many of the issues we have raised are still the
subject of intense study. Instead, we will study very simple models
that may suggest some simple answers. These models will guide
our understanding of more complex cases. Hardy— Wemberg law At the time of Darwin, the midto—late nineteenth century, the
factors responsible for the maintenance of variability were not
understood. It was unclear why variability was not reduced each
generation by the offspring looking like the ‘average’ of the par
ents. Gregor Mendel, who deduced the particulate nature of in
heritance through his extensive experiments with peas, provided 45 We will look later at comparisons between
plants that reproduce many times and
those that reproduce once and die. Response to changing environments is a
question of immense current interest. 46 We start with the simplest case, a single
locus with two alleles. We do” not claim that
this is realistic. Think ofhow each assumption might be
violated. As we formulate the model,
consider how the model might have to be
changed to accommodate changes in the
assumptions. There are twice as many alleles as
individuals because we are thinking of a
diploid organism. 3. Population Genetics the mechanism — heritable genes —— that explained how variabil
ity could be maintained. Although Mendel was a contemporary
of Darwin, his work was not well known until the early twen
tieth century. Also, in the early twentieth century, G.H. Hardy,
a British mathematician known for his contributions to number
theory and analysis, showed mathematically that in the absence
of other forces variability is preserved, as we now demonstrate. Consider in a diploid organism a single locus with just two
alleles, A and a. Thus there are three different possible genotypes
in the population: AA, Aa, and am. We will make the following
assumptions. 0 Mating is random among individuals in the population. 0 There is no selection: the probability of mating and survival
is independent of the genotype. 0 Generations are nonoverlapping.
c There is no immigration or emigration. o The population is so large that we can ignore stochastic
effects and consider only the frequencies (fractions) of dif
ferent genotypes. 0 There are no mutations at this locus. What happens to variability in the population at this locus with
these assumptions? Let pAA be the frequency of AA individuals,
1),,“ be the frequency of Ad individuals, and pm be the frequency
of aa individuals. Let the frequency of A alleles be p. If there
are N individuals and 2N alleles, then the number of A alleles is
2NpAA + NpAa. The frequency of A alleles is P=——————=PAA+— (3.1) The frequency of a alleles is 6] = 1 *‘P = 1% +Paa (3.2) To determine what happens after one generation of mating,
we consider all possible matings, their frequency, and the possi
ble offspring and their frequency. These are listed in Table 3.1. 3.3 One—locusr TABLE 3.1. Matin
fraction of offspri
by multiplying tl Summing l
eration’, w Similar rea and Thus, after
frequencie We drav
First, after
pletely de " ation Genetics variabil
' temporary arly twen— .H. Hardy, to number
:1 e absence '  genomes following lation. ‘~ (1 survival stochastic of dif .§
a locus with
' individuals,
_ frequency
fp. If there
A alleles is (3.1) (3.2) 1‘ of mating,
' the possi
‘Table 3.1. 3.3 Onelocus model without selection 47
TABLE 3.1. Matings, frequencies, and offspring in a onelocus, twoallele model. The three 'offspring fraction’ columns are the
fraction of offspring of the mating that are of the given genotype. The ’contribution to next generation' columns are obtained
by multiplying the fraction of offspring of each genotype by the frequency of the mating.
mating offspring fraction contribution to next generation
mating frequency AA Aa aa AA Aa cm
I!“ pi) “‘
1/2 1/2 In“
AA x a“ nun 2mm _
p2)
n 1/2 1/2 In
nun
Summing up the entries in the column ‘contribution to next gen—
eration’, we ﬁnd the frequency of AA the in next generation:
pAA =17)?“ +PAAPAa +pfla/4 (33)
= (1m tom/2)2 (3.4)
= p2. (3.5)
Similar reasoning leads to the conclusion that
p1“, =pﬁa/4 +pAapaa +122“ (3.6)
= (pita +paa/2)2 (3.7)
= q2 (38)
and
pﬁia = 1 _p2 _ qz Equation (3.10) iollowsfromthe
= zpq. (3.10) observation thatl = p + q implies that 1(p+q)ZP"+21>q+q3 Thus, after one generation, no matter what the initial genotypic
frequencies were, the genotypic frequencies are given by (3.13) sum to 1.
PM =p2 (3.11)
pAa = 21261 (3.12)
pad = ‘12 (3.13) We draw several very important conclusions from this result.
First, after one generation the genotype frequencies are com
pletely determined by the initial allele frequencies. This also Note that the frequencies in (3.1 1) through 48 3. Population Genetics Measuring selection directly can be very
difﬁcult because very small differences at
the individual level in survivorship probability or reproduction can have large effects at the population level. means that we can express our models in terms of the single
quantity, p, the frequency of A, which is a great simpliﬁcation.
Second, the allele frequencies and genotype frequencies remain
constant from generation to generation after the ﬁrst generation;
genetic variability is not eliminated. Third, any deviation from
the genotype frequencies (3.13), which are called the Hardy—
Weinberg frequencia to honor the mathematicians who derived
the equations, must result from violation of one of our assump
tions. Unfortunately, in practice it is very difﬁcult to detect statisti
cally deviations from Hardy—Weinberg frequencies, because none
of the available statistical tests are very sensitive. 3.4 Onelocus model with selection At this point we can begin exploring the consequences of chang
ing any of the assumptions we have made in our initial model.
Our initial model predicts that variability would be maintained,
but does not provide any reason why a particular allele frequency
should be found. Also, some variability will clearly be lost because
of random sampling, so we need to look at forces that can truly
maintain variability. We thus turn to an examination of the role of
selection. In this context, we are also motivated by our interest in
understanding the dynamics of traits of ecological interest. We thus add to the simple one—locus model the complicating
factor of selection and ask: 0 How strong must selection be to produce an observed
change in allele frequencies? We will use the answer to this
question to consider the case of ‘industrial melanism’, the
rapid increase in the frequency of dark forms of a forest
moth following the presence of soot on trees. 0 Under what conditions will a polymorphism be maintained?
We will examine the case of sickle—cell anemia in the prob
lems. Fitness To answer
tion. The l
hind offsp
survival, ai
on the ph
viva] prob
whereby ti Viability . In additior
cumstance
model inc
change thi
in ﬁtnesse
pendent o
in the met The moi
we can a
Weinberg
of mating
is requiret
relative at
by the rel
tions to fr ' Iation Genetics f the single
I» pliﬁcation.
Cies remain
generation;
iation from
the Hardy—
ho derived
,ur assump
stect statisti
r‘cause none is of chang
‘,__'tial model.
‘ aintained,
r frequency
> St because
3 t can truly
/ the role of
V interest in
rest. mplicating ‘ 3 observed
n er to this
“ nism’, the
of a forest i 7! . intained?
the prob— 3.4 Onelocus model with selection Juveniles
Reproduction Differential
by random surVIval
mating
Adults FIGURE 3.3. Life cycle used in the model Fitness To answer these questions, we must talk about ﬁtness and selec—
tion. The ﬁtness of an individual is its probability of leaving be
hind offspring as determined through differential reproduction,
survival, and possibly representation among gametes, depending
on the phenotype. We will temporarily equate ﬁtness with sur
vival probability, or viability. By selection we mean the process
whereby the more ﬁt individuals are chosen. Viability selection In addition to the assumptions made under Hardy—Weinberg cir—
cumstances, we will make further assumptions to obtain a simple
model incorporating selection. We assume that viabilities do not
change through time. We also ignore any frequency dependence in ﬁtnesses; ﬁtnesses of each genotype are assumed to be inde pendent of the current makeup of the population. The two stages
in the model are illustrated in Figure 3.3. The model is described in Table 3.2. At the initial juvenile stage
we can assume that the frequencies are given by the Hardy—
Weinberg proportions, because the juveniles resulted from a cycle
of mating and we have already shown that only one generation
is required to establish Hardy—Weinberg proportions. To ﬁnd the
relative adult proportions, we multiply the juvenile frequencies
by the relative survival rates. To convert these relative propor—
tions to frequencies, we divide by their sum, the (relative) mean 49 Can you think of cases in which
frequencydependent selection is likely? You can add the entries in the last row of
Table 3.2 to see that they sum to i. 50 Both by guessing, and by following similar
steps, ﬁnd the equation for q’. 3. Population Genetics TABLE 3.2. The onelocus, two—allele model with selection.
—
— Juvenile relative survival rates wAA
relative adult frequencies 2
adult frequencies prAA/E ﬁtness: E = 1?sz + quwAa + qzwaa. (5.14) We then use these results to determine the frequency of A alleles
in adults, which is the same as the frequency of A alleles in the
juveniles of the next generation, p’ . From the last row of Table 3.2
we see that p’ = p14,, wig/2 (3.15)
= prAA/w + (1/2)2pqua/E (3.16)
= PCPwAA: qua) (5.17)
w
= ‘13, (3.18)
w where in the last line we have deﬁned the mean ﬁtness of allele
A to be WA = PwAA + qua. (3.19) Does this make sense? If we know that an individual has a single
A allele, then with probability p its other allele is A, while with
probability q its other allele is 61. Thus the mean relative ﬁtness of
an individual carrying an A allele is prA + qua. From equation
(3.18) we see that allele A increases in frequency if the mean
ﬁtness of individuals carrying allele A is greater than the mean
ﬁtness of the population. Before using this model to examine some of the questions
posed earlier, we phrase the model in terms of a different set
of parameters that simplify the algebra and make our conclusions 3.4 One—locus more app: At this po
parameter and that Equilibria
We are n(
posed ear]
The notio
the text. A
constant ti Let Ap l
to general Denote th
change in
so set Ap if tion Genetics M
q wad
waa/ w ‘1/ (3.14) 6fA alleles
les in the
4: Table 3.2 (3.15)
(3.16) (3.17)
(3.18) l of allele (3.19) a single hile with
f ﬁtness of equation
the mean
:the mean 4 uestions
'l Serent set
so nclusions 3.4 Onelocus model with selection 51 more apparent. Let wAA = 1 — s (3.20)
10/151 = wad = 1 — t. (3.22) At this point, the sign of s and t is arbitrary. With this choice of Canyougiveaheuristicdiscussionofthe ' 7
parameters, we see that mean'”9°f””d"
Remember to usethe fact that p + q = 1. wA=p(1—s)+q=p+q—ps=1—ps (3.23)
wa=P+q(1—t)=p+q—qt=1—qt (3.24)
and that Remember that
1=(p+q)2=p2+2pq+qz. w =p2(1 — s) + 2M + 42(1 — t) (3.25) =p2 + M + 612 —p25 — qzt (3.26) = 1 —pzs — qzt (3.27) =p(1 —ps) + q(1 — qt). (3.28) Equilibria We are now ready to begin to answer the ﬁrst of the questions
posed earlier: when is a polymorphism maintained at equilibrium?
The notion of an equilibrium is central to much of the rest of
the text. An equilibrium is a value for a variable that can remain
constant through time. Let Ap be the change in p, the frequency of A, from generation to generation. Then Compare the development hereto the graphical approach used in Figures 3.4 AP =p/ _p = p% _p. through3Jbelow. Denote the equilibrium value of p by At an equilibrium, the
change in allele frequencies from generation to generation is zero, so set Ap = 0:
_ [31,014 _ A
0 — —ZU_ p (3.30)
13(1 —ﬁs) A
= _——————A ""‘ u 1
1_pzs_é2t p (33)
15(1 —135) —[3(1 —j)zs — qzt) A = e —p. (3.32) We get this pair because if the righthand
side of (3.34) is zero, one or the other of the
factors must be. We are trying to divide by the factor 1 — p. 3. Population Genetics Box 3.1. Finding the equilibrium ofa discrete time model with a single variable. To ﬁnd the equilibrium of the discrete time model p’ =f(p)
write the change in p, 8p, as
6p =f(p) —p. Set Sp to be zero, and ﬁnd the equilibrium, [3, as the solution
of the resulting equation: 0 =f(13) I3. We continue by setting the numerator of (3.32) to zero: 0 =13(1 —13$) —13(1 #325 — 812!) (3.33)
=ﬁ[1 —ﬁs — (1 ~fazs — 21%)]. (3.34)
Thus either
.5 = 0 (3.35)
01‘
1 —13s — (1 —132s — 8ft) = 0. (3.36) It makes sense that [a = 0 is an equilibrium, because if there
are no A alleles in one generation and there is no mutation or
immigration (as we have assumed), there will be no A alleles the
next generation. Similar reasoning suggests that f) = 1 must be an
equilibrium as well and thus 1 — 15 must be a factor of (3.36). We
therefore notice that (using & = 1 — ﬁ) 0 = 1 —135 — (1 12325 — 3ft) (3.37)
= 1 —135 — 1 +1325 + (1 —;3)% (3.38)
= — 135 +325 + (1 —13)2t (3.39)
= (1 —13)[#135 + (1 — 13):] = 0. (3.40) We conclude that f) = 0, or 15 = 1 or that 4» + (1 — 13): = 0. (3.41) 3.4 Onelocus Solving fo We ﬁnally Note that
equilibriur
which cle
both posit
tbe hetero. Stability of We have
three equi
frequencic
But, what
that is not
on appro:
times in tl
We ﬁrst
happens i
number oi
rate or a s
at the cas
equation 1 We are
that if p =
approxim; ’ ‘rpulation Genetics 3  1'0: (3.33)
(3.34) (3.35) (3.36) 74 use if there
" mutation or
A alleles the
._ 1 must be an
of (3.36). We (3. 37)
(3.38)
(3.39)
(3.40) (3.41) 3.4 One—locus model with selection Solving for f), we ﬁnd: .135 _‘ﬁt + t = 0 (3.42)
t =[Js +1’3t. (3.43) We ﬁnally conclude that 13 = L (3.44) s+t‘ Note that for (3.44) to represent a polymorphic equilibrium (an
equilibrium with both alleles present) we must have 0 < f) < 1,
which clearly requires that either 5 and t are both negative or
both positive. Yhus a polymorphic equilibrium is possible only if
the heterozygote is the most ﬁt or the least ﬁt genotype. Stability of equilibria We have discovered that the onelocus model can have up to
three equilibria. We now know what the outcome is if the allele
frequencies are exactly zero, or exactly one, or exactly t/(s + t).
But, what happens if the population starts at an allele frequency
that is not one of the equilibria? We will use an argument, based
on approximations near an equilibrium, that we repeat several
times in this text. We ﬁrst start near the 13 = 0 equilibrium. We ﬁrst look at what
happens if we start with no A alleles and then introduce a small
number of A alleles, as would happen if there was a low mutation
rate or a small number of immigrants with A. We thus are looking
at the case where p is very small. We begin by writing out the
equation for p’ in terms of p only, eliminating q. / = p(1 _p5)
p ——1 _p25 _ qzt (3.45)
_ p —p2s
————1 _p25 _ (1 _p)2t (3.46)
_ 2
= p p S (5.47) 1—p25—t+2pt—p2t' We are ready to use our assumption that p is very small. Notice
that ifp = 10—2, then p2 = 10—4, so ifp is very small we can
approximate the numerator by [2 because the other term will be 53 Thus this model can have either two or
three possible equilibria. What does (3.44) imply aboutf; ifs > 0
and t < 0? We begin by going through a series of steps
that in factjustify the simpler procedure we
will use in general. The equilibrium [3 = 0 corresponds to a
population with noA alleles. 54 3. P0pulation Genetics We are ignoring the special case in which t
is precisely 1. Give a deﬁnition of an unstable equilibrium. We are still justifying, and not simply
describing, the procedure we will use in
practice to ﬁnd stabiliw. much smaller. Similarly, in the denominator the term 1 —t will be
much larger than any of the other terms, all of which contain p.
Thus, we approximate (3.47) as L 1 _ t. (3.48) p’ %
Do you recognize this equation? It is identical to the equation for
discrete time exponential growth which we studied earlier. Thus,
we identify two distinct cases: 1 o If t > 0, then 1—; > 1 so p increases when it is small. 0 If t < 0 then % < 1, so p decreases when it is small. Does this conclusion make sense? If p is small and A is rare,
there will be so few AA individuals that we can ignore them.
If the heterozygote Aa is more ﬁt than the homozygote aa, we
would expect that A would increase in frequency. A more ﬁt
heterozygote corresponds to t > 0. We can use a similar argument to show that if q is small, or p
is near 1, then: 0 If s > 0, then q increases (p decreases).
o If s < 0, then q decreases (p increases). We have just determined when each of the equilibria f) = 0 and
f) = 1 is stable. An equilibrium is stable if a population starting
near the equilibrium approaches the equilibrium. The notion of
stability is very important, because we expect to ﬁnd populations
near stable equilibria and do not expect to ﬁnd populations near unstable equilibria because only stable equilibria are approached
as time increases. A simpler way of computing stability We would not want to go through all these steps every time we
want to compute stability. Luckily, there is a different way of view
ing the calculation we have just done. Think of p’ as a function
ofp, FTP) If we wish to approximate 19’ near the equilibrium [3 = 1, we can
use a Taylor series (Box 3.2). The ‘variable’ in the Taylor series 3.4 Onelocu When
equilil: will be p We knov
just this (
approxir. We subs ...
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at University of California, Berkeley.
 Fall '10
 WayneM.Getz

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