Adaptive Dynamics: Branching Phenomena and
the Canonical Equation
Carl Boettiger
May 8, 2007

Contents
1 Introduction 1
1.1 From Population Genetics to Adaptive Dynamics . . . . . . . . . 1
1.2 Fundamentals in Adaptive Dynamics . . . . . . . . . . . . . . . . 2
2 The Canonical Equation 6
2.1 The First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The Second Moment . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Singular Strategies 11
4 Applications to a Branching Equilibrium Example 16
4.1 Generalizing a Common Branching Model . . . . . . . . . . . . . 16
4.2 Predicting Dynamics in Evolutionary Branching . . . . . . . . . 18
4.3 A Simulation-based Test of Theoretical Predictions . . . . . . . . 20
4.3.1 Confirming Branching . . . . . . . . . . . . . . . . . . . . 20
4.3.2 The Instability of Branching . . . . . . . . . . . . . . . . 21
4.3.3 Evaluating the Canonical Equation and its Variance . . . 22
5 Conclusions 24
A Branching Criterion for Generalized Logistic Equation 27
B Subsequent Branching 30
C Simulation Code 33
1

Abstract
In this paper we provide an introduction to the field of Adaptive Dynamics.
We present derivations for two of the fundamental components of the theory:
“canonical equation” and the classification of singular strategies. We supple-
ment the existing theory with a derivation of the variance associated with the
canonical equation. We then consider a common ecological model (an instance
of the logistic equation) that has been used to explore branching in the con-
text of Adaptive Dynamics. We show that the branching properties of this
model are maintained in a much more general form of which the familiar ex-
ample is a particular instance. We then determine the expected evolutionary
trajectory of a population in this model using the canonical equation, and find
the associated variance of this trajectory. We also examine the stability of the
dimorphic, branched population. Having determined each of these components
analytically, we then confirm these predictions by implementing a model using
an individual-based stochastic simulation.

Chapter 1
Introduction
1.1 From Population Genetics to Adaptive Dy-
namics
The Modern Synthesis in the first half of the twentieth century served to unify
Darwin’s evolution with Mendel’s genetics, providing a mechanism for natural
selection to operate. The field emerging from this unification became known
as population genetics, which experienced great success in several areas, such
as an understanding under simple models how even very weak forces of natural
selection could determine the evolution of large populations. However, the treat-
ment of population genetics failed to account for ecological aspects of evolution,
and consequently cannot provide an understanding of ecological aspects such as
density and frequency dependent selection and their role in long-term evolution.
Such considerations were first addressed through the techniques of mathemat-
ical game theory beginning in the 1970s [DOE Report, 2005]. Through this
work the concept of an Evolutionarily Stable Strategy, ESS, (originally called
an Evolutionarily Unbeatable Strategy, EUS) was developed [Metz et al., 1996].
These game-theoretical techniques did not enable the classification of the dy-
namics of such points. The existence of such ESS did not guarantee that a
population could ever reach such a strategy. This treatment has become the
domain of a set of methods collectively known as Adaptive Dynamics [Geritz
et al., 1998,Dieckmann & Doebeli, 1999,Dieckmann & Law, 1996]. In this paper
we provide an introduction to these techniques and illustrate their application
to explore the phenomenon of branching in evolutionary populations, a question
which serves as a paradigm with which to address one of the most interesting
and challenging questions in long-term evolution: that of speciation.
Adaptive Dynamics offers an essentially analytic approach to the questions it
seeks to address. Like most good theories, it makes certain simplifying assump-
tions and then employs a set of mathematical tools to explore the relevant ques-
tions. In population genetics, the assumptions simplify the ecology drastically,
allowing for an explicit treatment of the genetics. Adaptive Dynamics works in
1

the other regime; trading genetic detail for ecological complexity [Dieckmann &
Doebeli, 2005]. As such it is not a replacement for but rather an extension to
the existing field. Such distinctions have been at the source of much recent con-
troversy regarding the significance and role of Adaptive Dynamics. In light of
what has largely been a controversy of misunderstanding [Kisdi & Gyllenberg,
2005], it is important to bear several distinctions in mind. The predictions made
by theory are only valid insomuch as the assumptions are satisfied, consequently
we will state these assumptions as explicitly as possible. Further, one must keep
the distinction between the set of tools and methods that comprise Adaptive
Dynamics theory from the applications to particular ecological models. In this
vein we begin we begin by developing the abstract theory and follow with an
example of its application. Finally, we employ the common practice of com-
puter simulation to visualize the ecological model and the predictions made in
our applications section. The practice has also contributed to some confusion
in the literature, where such visualizations have been mistaken as the primary
evidence of a particular result [Waxman & Gavrilets, 2005,Kisdi & Gyllenberg,
2005].
1.2 Fundamentals in Adaptive Dynamics
Our mission is to understand the evolution of a population in some ecologi-
cal setting. This requires a model both for the ecology (population structure,
growth, competition, and so forth) and a model for the evolution. Our objective
is not to develop the most appropriate model for any of these aspects, but rather
to develop a theoretical framework and set of techniques that can be applied to
any particular model one might wish to investigate. Bearing this in mind, we
construct a notation with which to address these questions, and then develop
the tools we will later use to investigate a particular model of interest. First,
the terminology and notation.
By population we imagine a group of individuals who can all be identified
as having the same value for a particular trait, x.1 Borrowing from the game-
theory literature, we will consider this trait as the strategy employed by this
population refer to them as x-strategists. This population may not be alone,
but can be competing (or merely co-existing) with n other populations. We
can then represent the strategies of each of these populations as elements of a
vector, X = {Xi}ni=1. Similarly, allow the vector N = {Ni}
n
i=1 to represent the
number of individuals practicing the corresponding strategy Xi. Finally, let us
place our populations in some environment, E( ~N, ~X, t)that depends not only
on external factors which may vary in time but on the strategies in play and
the numbers of individuals practicing them.
1We imagine this trait as some continuous variable x ∈ R that characterizes the population.
There is no reason that this value must be positive; for instance the variable of interest may be
the logarithm of the length. It is also possible to treat the trait values as functionals instead
of real numbers, employing the calculus of variations. For simplicity we will continue to treat
these values as real numbers.
2

Our initial question is how each strategy performs: that is, how does its pop-
ulation grow or shrink? We expect its average rate of growth to be proportional
to its current size, that is,
dNi
dt
= s(Xi,E(N,X, t)) ·Ni (1.1)
Here s(Xi,E) is the average rate of growth, which will depend on the strat-
egy of the population in question Xi = x and also on environmental factors,
E(N,X, t), which of course include the effects of other populations. This results
in the population growing (or shrinking) exponentially at rate %(x,E). The ex-
ponential growth of Ni cannot be maintained indefinitely. As the population
grows, it modifies the environment in a way which will cause its growth rate to
deteriorate, reflecting limited resources, population density or other dependence
that E(N,X, t) might have on N. This causes the population to reach some
equilibrium level which we denote as Ni = N∗i . Clearly this will depend on the
current environment which includes the traits and sizes of all other populations
in the community. In the case of a monomorphic community (one where only a
single strategy is being practiced) this population is obviously determined only
by that strategy and the fixed environmental conditions. Under these conditions
we will refer to the steady-state population size as being the carrying capacity,
K(x). In multi-strategy communities the steady-state populations can often
be expressed as function of this inherent carrying capacity of the strategy in
question and the influence of the other populations. We mention this for later
reference, for now we continue to treat the general case of possibly polymorphic
populations. Our first simplifying assumption is in regard to these equilibrium
population sizes Nx(x):
Assumption 1: Fluctuations in the steady state population size, Nx(x) are
negligible with respect to the dynamics. For instance, we won’t worry about
a population going extinct due to a chance fluctuation of low births and high
deaths in one generation.
Note that the quantity s(x,E) gives the expected exponential growth rate of
a population, providing us with a precise and meaningful definition of the fitness
of the species bearing trait x in the given environment E. Our next assumption
deals with this quality:
Assumption 2: s(x,E) is a continuously smooth function of the x. Equiv-
alently, small changes in phenotype x result in correspondingly small changes in
the fitness s. This biological generality of this assumption has been challenged,
see [Barton & Polechova, 2005].
We now build a mutation-based model of evolution upon this framework. Let
µ represent the mutation probability per birth. In general this could depend on
the trait value x, but for simplicity we will take this as constant for all possible
trait values. The expected number of mutations per generation in a population
Ni is then µ · b(x) · Ni, where b(x) is the birth rate per individual. When
a mutant occurs, it carries a new trait y chosen from a normal distribution
centered around a mean of x with variance σ2µ. Here we introduce two critical
assumptions of Adaptive Dynamics theory.
3

Assumption 3: µ ·max(N)  1. This assumption allows for the separation
of timescales for evolutionary and ecological processes. Through this assumption
we can assume that the mutant will occur in a population that has already
equilibrated to steady state, Ex. This steady state will depend not only on x,
the trait value of that population, but on all other populations in the community
(which may also need to be at equilibrium in order that population with trait
x is at equilibrium).
Assumption 4: Mutants are nearly clonal offspring. This assumption to-
gether with assumption 1 will allow an approximation to s which we shall see
shortly.
We are now prepared to consider what happens when a mutant first occurs
in a population. Since individuals with the mutant trait are initially rare,their
effect on the environment can be neglected. By assumption 3, the ecological
dynamics of the population have equilibrated. From these two observations,
we can write down the invasion fitness (growth rate) of the rare mutant y
in a population of x as s(y,Ex). To distinguish this as the invasion fitness,
where y is rare, this is generally written in the literature as sx(y), which is
read as the invasion fitness of y in a resident population of trait x. Here the
dependence not only on the trait value from which the mutant originated (x)
but on equilibrium environment Ex is implied by the subscript x. Ex of course
depends on all other populations as well as a-biotic factors, allowing this more
compact notion to remain completely general. Though we will not continue to
write the Ex explicitly, we do not intend to suggest that we have abandoned a
general treatment where the invasion fitness depends on environmental factors
and any other resident populations.
By assumptions 1 and 3 we can approximate sx(y) by Taylor expanding
around y = x,
sx(y) ≈ sx(x) +D(x) · [y − x] + . . . (1.2)
Where
D(x) ≡
[
∂sx(y)
∂y
]
y=x
(1.3)
By definition sx(x) = 0. The mutant y can invade successfully when sx(y) >
0. Hence if the invasion fitness gradient D(x) is negative, only mutants with
y < x can invade, while if the gradient is positive, only those with y > x will
invade. As the population with trait y grows, we transition into an environment
Ey where x is now rare. For nonzero D(x) we will find sy(x) < 0 and hence
the mutant completely replaces the parent population2. This results in the
population climbing the gradient through successive mutations. Our first task
2This follows from assumptions 2 and 4. Because y is sufficiently similar to x (by assump-
tion 4) the invasion fitness of x when rare will be close to that of the invasion fitness of y (by
assumption 2). Being similar, the sign of the gradients D(x) will not change once x becomes
rare in a population of y; that is when x and y are swapped in these equations. However
the term y − x will change sign, therefore sy(x) < 0 as claimed. It is important to bear in
mind that this is not the case in the region of a singular point where D(x) = 0. For further
discussion see [Geritz et al., 2002]
4

will be to describe the deterministic mean path of this stochastic trajectory.
This is accomplished through the so-called canonical equation of Dieckmann
and Law [Dieckmann & Law, 1996]. In addition to their classic derivation we
will present a calculation of the variance associated with this mean trajectory.3
This trajectory may take the population to a value for Xi where D(Xi) = 0,
and it becomes impossible to predict its evolution from the fitness gradient. We
denote such a strategy Xi = x∗, which is termed an Evolutionarily Singular
Strategy, or singular point. [Geritz et al., 1998] We first derive the expected
path to a singular strategy, and then consider their classification.
3I am unaware of such a calculation having yet appeared in the literature.
5

Chapter 2
The Canonical Equation
Our task here is to derive an equation giving the expected trajectory of an
evolving population starting with a trait value x0 6= x∗ in some environment
E. This is given by the canonical equation of Adaptive Dynamics, and was
first derived in this context by Ulf Dieckmann and Richard Law [Dieckmann
& Law, 1996]. Rather than direct the reader to this literature, we present
the derivation as simply as possible employing the more familiar notation of
S.A.H. Geritz [Geritz et al., 1998] which we have been following thus far. The
canonical equation gives the first moment (the mean trajectory) for this process.
We follow up this classic result with a derivation of the second moment, allowing
us to predict the variance in the trajectory of an evolving population from the
mean trajectory given by the canonical equation.
2.1 The First Moment
We begin by observing that the trajectory of a population through the space
of possible traits over time is Markovian, depending only on the current trait
value of the population at time t. Consequently we can describe the evolution
for the probability of the population having a particular trait value x at time t
by the following master equation:1
d
dt
P (x, t) =
∫
dy [w(x|y) · P (y, t)− w(y|x) · P (x, t)] (2.1)
Where P (x, t) represents the probability of having trait x at time t while
w(y|x) gives the probability per unit time of making the transition x → y2. This
transition probability consists of two events: (1) a mutant with trait y occurs
in the population, with probability M, and (2) the mutant survives accidental
1The reader is directed to any text on stochastic processes for a discussion and derivation
of this class of equations, e.g. van Kampen [van Kampen, 1981].
2Perhaps the most familiar example of these transition probabilities is that given by Fermi’s
Golden Rule, describing transitions of electrons between excited states of an atom.
6

extinction (drift), with probability D. Since these processes are independent,
the probability that both occur is simply their product:
w(y|x) = M(y, x)D(y, x) (2.2)
The probability that a mutation enters a population is given by
M(y, x) = µ · b(x) ·Nx(x) ·M(x, y − x) (2.3)
Where b(x) is the mean birth rate per individual, µ the mutation rate per
birth, Nx(x) the equilibrium population size for a population with trait x, and
M(x, y−x) is the distribution from which the mutant trait is drawn. Meanwhile,
the probability of surviving drift given the mean individual birth rate b and
mean death rate d for the mutant y is given by a classical result from branching
process theory:
D(y, x) =
{
b(y,x)−d(y,x)
b(y,x) d(y, x) < b(y, x)
0 d(y, x) ≥ b(y, x)
(2.4)
Where we use the notation of the dependencies, b(y, x) to imply that this
is the birth rate of a mutant with trait y in a population of x. The case of
D = 0 being trivial, we will focus only on the case where d(y, x) < b(y, x).
The derivation of this result can be found in any text on branching processes,
e.g. [Feller, 1968]. Following equation (2.2) and simplifying, we have
w(y|x) = µ ·Nx(x) · b(x) ·M(x, y − x) · [b(y, x)− d(y, x)]/b(y, x) (2.5)
We can simplify this further by means of a convenient approximation. Taylor
expanding D(y, x) about y = x, noting that b(x, x)− d(x, x) = 0:
D(y, x) = [b(y, x)− d(y, x)]/b(y, x) ≈
∂y[b(y, x)− d(y, x)]y=x
b(x, x)
· [y − x] (2.6)
Where b(x, x) = b(x), the birthrate of an x strategist in a population of x
being by definition b(x). The quantity b(y, x) − d(y, x) is simply the expected
growth rate, our familar sx(y). Recall that from equation (2.4) that D ≥ 0, and
consequently we have the condition that ∂ysx(y)
∣
∣
y=x
and the quantity [y − x]
must have the same sign. Not coincidentially, this is identically the condition
for invasion we determined in discussing the invasion fitness gradient D(x) in
equation (1.3). We can then express the transition probability as:
w(y|x) = [y − x] · µ ·Nx(x) ·M(x, y − x) · ∂ysx(y)
∣
∣
y=x
(2.7)
We are now ready to consider the dynamics of the expected trait value:
d
dt
〈x〉(t) =
∫
dx · x ·
d
dt
P (x, t) (2.8)
7

Using the master equation to replace ddtP (x, t) and performing a change of
variables we find
d
dt
〈x〉(t) =
∫
dx
∫
dy · [y − x]w(y|x)P (x, t) (2.9)
defining the kth jump moment [van Kampen, 1981]3 as
ak(x) =
∫
dy · [y − x]kw(y|x) (2.10)
We then have
d
dt
〈x〉(t) =
∫
dx · a1(x)P (x, t) = 〈a1(x)〉(t) ≈ a1(〈x〉(t)) (2.11)
Which tells us that the quantity we seek is just the expectation value of the
first jump moment, 〈a1(x)〉(t). If a1(x) is linear, we could take 〈a1(x)〉 = a1(〈x〉).
In general this will not be true, and we would instead write
〈a1(x)〉 ≈ a1(〈x〉) + 12 〈(x− 〈x〉)
2〉a′′1(〈x〉) + . . . (2.12)
Hence the evolution of 〈x〉 is not determined only by 〈x〉 but is also influ-
enced by its fluctuations. The statement that we will ignore the fluctuations
is exactly what we do when we consider any macroscopic quantity that results
from stochastic fluctuations, such as chemical rate constants or Ohm’s law. We
desire to describe the world in terms of these macroscopic quantities alone,
when a precise description should depend not only on these values but on their
fluctuations. The assumption that the macroscopic quantity is by itself a mean-
ingful description of the phenomenon (without reference to its variance) is the
assumption that the the second term in this approximation is negligible. N.G.
van Kampen describes this as the macroscopic approximation. With regard to
the process considered here, Dieckmann and Law simply refer to van Kampen
for a justification of this approximation for nonlinear a1. While this approach
will provide an accurate description of the mean path, it gives no such jus-
tification as to whether the mean path is a meaningful macroscopic quantity
analogous to the resistance of a wire. With this precaution, we continue:
Using equation (2.7) in equation (2.10) we have:
d
dt
〈x〉(t) =
∫
dy · [y − x]2µNx(x)M(x, y − x)∂ysx(y)
∣
∣
y=x
(2.13)
We now change our integration variable to ∆x ≡ y − x. Since we have
the condition on (2.4) that [y − x] is either always positive or always negative,
we must restrict our range of integration to half of the real line. Since M is
3van Kampen introduces this term for this concept in place of an earlier term, “derivative
moment,” which never became established. Unfortunately, van Kampen’s term does not seem
to have established itself much beyond Dieckmann and Law. Nevertheless, we shall see that
the concept is quite powerful.
8

symmetric and the only function depending on ∆x, we can equivalently integrate
over all of R and introduce a factor of 1/2,
d
dt
〈x〉(t) =
1
2
· µ ·Nx(x) · ∂ysx(y)
∣
∣
y=x
·
∫
d∆x ·∆x2M(x, y − x) (2.14)
The remaining integral we recognize simply as the variance in M , the distri-
bution from which the mutational trait is drawn. Denoting this integral simply
σ2µ we recover the canonical equation:
d
dt
〈x〉(t) =
1
2
· µ · σ2µ ·Nx(x) · ∂ysx(y)
∣
∣
y=x
(2.15)
2.2 The Second Moment
Along these same lines, it is possible to derive the variance expected for this
mean path. Analogously to before, we have the identity
d
dt
〈x2〉(t) =
∫
dx
∫
dy · [y2 − x2]w(y|x)P (x, t)
=
∫
dx
∫
dy ·
[
[y − x]2 + 2x[y − x]
]
w(y|x)P (x, t)
= 〈a2(x)〉(t) + 2〈xa1(x)〉(t)
Recall the variance is given by σ2(t) = 〈x2〉(t)−〈x〉2(t). Differentiating with
respect to time, we have
dσ2
dt
= 〈a2(x)〉(t) + 2〈[x− 〈x〉(t)]a1(x)〉(t) (2.16)
Then with the assumption of linearity for both jump moments and expanding
a1 around x = 〈x〉, we can write:
dσ2
dt
= a2(〈x〉(t))(t) + 2σ
2a′1(〈x〉(t)) (2.17)
This gives us the variance associated with the canonical equation. Clearly
the second jump moment is always positive. Meanwhile in order for equa-
tion (2.15) to approach equilibrium a′1 must be negative near the equilibrium
point. Consequently we expect σ2 will converge towards a steady state of
σ2 =
a2
2|a′1|
(2.18)
Hence the variance will grow initially but does not continue to diverge, in-
stead settling in towards a maximum value. This provides a valuable measure
of the accuracy of the canonical equation for any particular instance. Further,
9

this allows us to add a correction to the canonical equation itself. So far we have
written the canonical equation as 〈x˙〉 = a1(〈x〉). Under the expansion in (2.12)
we could instead write
〈x˙〉 = a1(〈x〉) + 12σ
2a′′1(〈x〉) (2.19)
This is currently rather abstract and will hopefully become clearer when we
compute these quantities for a particular model of interest in a later section.
10

Chapter 3
Singular Strategies
Through our derivation of the canonical equation we have reached an expres-
sion describing the average trajectory of a population across a fitness gradient.
When a population has reached the singular strategy, x∗, the gradient D(x)
can no longer tell us about the dynamics; we must consider higher terms in
our expansion of the invasion fitness, equation (1.2). We will show that any
singular point can be divided into one of eight categories on the basis of its
second derivatives. The discussion here is intended to be self contained but not
expansive. For a more thorough treatment see [Geritz et al., 1998].
Our first consideration is to establish whether or not the singular point
is evolutionarily stable. To determine if a mutant can invade a population
with strategy x∗ (that is, s(y, x∗) > 0) we simply continue our expansion in
equation (1.2) to second order:
sx(y) ≈ sx(x) + ·[y − x]
[
∂sx(y)
∂y
]
y=x=x∗
+ [y − x]2
[
∂2sx(y)
∂y2
]
y=x=x∗
(3.1)
Since the first time vanishes by definition as before, and the gradient is zero
by definition at x∗, we are left with the following condition for mutant of some
nearby strategy y to be able to invade a population with strategy x∗:
B ≡
[
∂2sx(y)
∂y2
]
y=x=x∗
> 0 ⇔ x∗ is invasible (3.2)
Considering the sign of the second derivative offers another way to think
about the question. For B < 0, the invasion fitness landscape sx(y) has a local
maximum with respect to y at the critical point, (x∗, x∗). Since s(x∗, x∗) zero
be definition, this means that all nearby values for y have negative invasion
fitness, s(y, x∗) < 0, and hence none of hem can invade x∗. This means x∗ is
uninvasible, and consequently acts as an evolutionary trap: any population with
strategy x∗ will never be ousted by its mutants. This does not tell us anything
about the dynamics in general however. Whether a population with a strategy
11

initially different from x∗ could ever reach the strategy x∗ is an independent
question. This is one of the key distinctions that Adaptive Dynamics adds to
the game-theoretical concept of evolutionarily stable. We will see that even if
x∗ is uninvasible, it may be impossible for a population to reach the strategy
x∗ by evolution.
For instance, we must first ask: can a rare mutant with strategy y = x∗
invade a nearby population with strategy x? Mathematically, to invade x∗
needs a positive invasion fitness, s(y = x∗, x) > 0. This time we expand around
the second variable, taking the resident population x near the fixed value of
y = x∗,
sx(y = x
∗) ≈ sx∗(x
∗) + (x− x∗)
[
∂sx(y = x∗)
∂x
]
x∗=x
+
(x− x∗)2
2
[
∂2sx(x∗)
∂x2
]
x=x∗
(3.3)
The first term sx∗(x∗) and the first derivative term are again zero for the
same reasons. Again we are left with the sign of the second derivative giving
us the sign of the invasion fitness sx(y = x∗). A positive invasion fitness for
rare x∗ corresponds with the invasion fitness landscape having a local minimum
with respect to x. Recalling that sx∗(x∗) is zero, we see that for any x near x∗,
the strategy has a positive invasion fitness and hence when rare. Our condition
that x∗ strategists can invade is thus
A ≡
[
∂2sx(y)
∂x2
]
y=x=x∗
> 0 ⇔ x∗ can invade (3.4)
Note that this condition is completely independent of our first condition.
One could have a singular point for which any population beginning with the
value x∗ would be stable, but even populations beginning very close to the point
could never reach. Even if we know that x∗ is uninvasible (when established)
and can invade nearby populations (when rare), we still know nothing about its
global stability; that is, whether a population with a strategy initially far from
x∗ will converge towards it or away from it. Since the population moves along
the fitness gradient, this requires that sx(y) > 0 for y nearer x∗ than x and
conversely, sx(y) < 0 for |x∗− y| > |x∗−x|. Since D(x) changes sign at x∗, this
requires that D(x) be a decreasing function of x, that is,
dD(x)
dx
=
∂2sx(y)
∂x∂y
+
∂2sx(y)
∂y2
< 0 (3.5)
Where we have expanded the full derivative of D(x) (from equation (1.3))
in terms of the second partials. Recall that since the average rate of growth
for a steady state population must be zero, s(y = x, x) = 0. Consequently, any
derivatives of sx(y) along the line y = x direction must be zero. Expanding the
second derivative along this line we find:
∂2sx(y)
∂x2
+ 2
∂2sx(y)
∂x∂y
+
∂2sx(y)
∂y2
= 0 (3.6)
12

Which we use to replace the mixed partial in equation (3.5), giving us the
condition
∂2sx(y)
∂x2
−
∂2sx(y)
∂y2
> 0 ⇔ x∗ is convergence stable (3.7)
Following our definitions of A and B (equations (3.4) and (3.2)) we can
write the convergence stability condition simply as A−B > 0. Finally, we have
observed that whether x∗ can invade when rare is independent of whether x∗
can be invaded when common. Hence around the critical point it is possible for
both s(y, x∗) > 0 (x∗ invasible by nearby mutants) and s(x∗, x) > 0 (x∗ can
invade nearby populations). In this case neither can replace the other, and the
population must become dimorphic. In general this requires the invasibilty to
appear the same when we swap y and x, that is, reflect across the line y = x
in the invasion landscape. Hence the strategies to be mutually invasible sx(y)
must have a minimum in the direction perpendicular to y = x,
∂2sx(y)
∂x2
− 2
∂2sx(y)
∂x∂y
+
∂2sx(y)
∂y2
> 0 (3.8)
Using this to replace the mixed partial in equation (3.5), we have the con-
dition
∂2sx(y)
∂x2
+
∂2sx(y)
∂y2
> 0 (3.9)
Which is simply the condition A+B > 0 for polymorphisms to be protected.
Clearly our last two conditions are not independent, but amount to the question
of whether |A| − |B| > 0. Any singular point could have any combination for
the sign of A, B, and |A| − |B|, making for a total of eight unique types of
critical points. It is possible to make a simplified sketch of these invasion fitness
landscapes as follows. Imagine that we have a topographic map with values of
x along the abissca, values of y along the ordinate, and the corresponding value
sx(y) as the height of that point on the map. Clearly along the line y = x the
height is zero, or sea-level. Rather than worry about the exact value of sx(y)
everywhere, we simply color in the areas on this map lying above sea-level. This
is called a pairwise invasibilty plot, and are discussed in greater detail in [Geritz
et al., 1998]. See figure 3.1 and figure 3.2 as examples.
The critical point in which each of these signs is positive is particularly inter-
esting. Such a point is convergence stable, invasible, able to invade, and permits
polymorphisms. Any point satisfying these criteria is known as a branching
point, since a monomorphic population beginning anywhere in the trait space
will necessarily branch into populations with different traits. Our study of
diversification in asexual populations will require a dynamics satisfying these
criteria for branching. Having characterized the behavior of the singular points
by means of these four properties, we turn our attention to describing the dy-
namics of a population evolving away from a critical point; that is, when the
selection gradient is nonzero.
13

Figure 3.1: Sample pairwise invasibilty plot (PIP). A population must start
on the line y = x at equilibrium. It can then take a small mutational step
along the vertical (along the ordinate) direction into any positive region. These
mutants then successfully invade, carrying the population horizontally (along
the abissca) back to the line y = x. This PIP exhibits a singular point at
x = x∗ and represents an uninvasible (evolutionarily stable) strategy. Figure
reproduced from [Geritz et al., 1998].
14

Figure 3.2: Eight types of singular points. The singular point (x∗, x∗) is unin-
vasible if a vertical line through the point lies entirely in the positive (shaded)
region, (a, b, g, h). The singular point can invade when rare if a horizontal line
through the point lies entirely in the positive regions, (a, b, c, d). The singular
point exhibits protected polymorphisms if the line perpendicular to y = x lies in
positive regions, (h, a, b, c). Overlapping positive regions obtained by reflect-
ing the image along the line y = x determines which polymorphisms that are
protected. Lastly, if the positive region lies above the line y = x for values left
of the singular point and below the region for values right of the point, then the
point is convergence stable (b, c, d, e). Figure reproduced from [Geritz et al.,
1998].
15

Chapter 4
Applications to a Branching
Equilibrium Example
4.1 Generalizing a Common Branching Model
We now turn to consider a particular example which will serve to illustrate
these techniques and also allow us to explore the phenomenon of branching
populations. Dieckmann and Doebelli consider a classic form for ecological
dynamics that exhibits branching in the paper on speciation [Dieckmann &
Doebeli, 1999]. We demonstrate that the properties of the model they consider
are in fact preserved by a more general class of such models. This derivation
serves as an illustration of the methods outlined in the previous section and also
as a demonstration that these models feature the branching property we wish
to explore. We write down our general form for these models as:
dN(x, t)
dt
= [b(x,N)− d(x,N)]N(x, t) (4.1)
Where b represents a generalized birth rate and d a generalized death rate.
The motivation for this formulation is then clear: the population is expected
to grow exponentially at the net rate at which individuals enter the population
(births minus deaths). These rates are allowed to depend on the particular trait
value (the so-called strategy, x) of the population in question. We assume the
following general forms for birth and death as linear functions of the population:
b(x, t) := ν(x)− β(x)N(x, t), d(x, t) := ρ(x) + α(x)N(x, t) (4.2)
Clearly, the steady state occurs when b(x) = d(x). That is, when
ν(x)− βNx(x) = ρ(x) + αNx(x) (4.3)
Where Nx indicates the steady state population as before, which we will
identify as the carrying capacity K(x) of some fixed environment. Solving for
this we find:
16

Nx =
ν(x)− ρ(x)
α(x) + β(x)
= K(x) (4.4)
We will assume that this carrying capacity is also a Gaussian, offering the
maximum capacity K0 for some optimal trait x∗.
K(x) = K0e
−(x−x∗)2/(2σ2k) (4.5)
The only necessary features for K(x) in our derivation will be K(x∗) = K0,
while K ′(x)|x=x∗ = 0 and K ′′(x)|x=x∗ < 0. We now consider the per capita
growth rate of the rare mutant1. We will assume that the rare mutant competes
against a discounted population size, weighted by how similar the mutant is to
the population it invades. This weighting we will denote by C(x− y) indicating
that it depends only on how different two populations in the community are,
not on the values themselves. This C(x − y) is called the competition kernel,
and is conventionally taken to be a simple Gaussian centered at the trait itself.
sx(y) = [ν(y)− ρ(y)]− [α(y) + β(y)]N(x)C(x− y) (4.6)
Where C(x− y) has the explicit form:
C(x− y) = exp(−(x− y)2/(2σ2c )) (4.7)
As with K(x), we will only be interested in the values this function and its
derivatives takes at the critical point, not the actual form. Given equation (4.6),
we are ready to apply the machinery we have developed into identifying and
classifying the equilibrium point. These derivations are carried out explicitly
in the appendix, and the main results quoted here. Not surprisingly, we find
the equilibrium condition to be x∗. We find the condition for x∗ to be able to
invade as:
∂2xsx(y)
∣
∣
y=x=x∗
=
K0[α(x) + β(x)]
σ2k
+
ν(x)− ρ(x)
σ2c
∣
∣
∣
x=x∗
(4.8)
Which is clearly always greater than 0 (given ∀x ν(x) − ρ(x) > 0), hence
x∗ can always invade. Our next condition is that x∗ is invasible,
ν(x)− ρ(x)
σ2c
−
K0[α(x) + β(x)]
σ2k
∣
∣
∣
x=x∗
(4.9)
Which is only positive for σk > σc, making this a necessary requirement
for branching to occur. Our condition for protected polymorphisms is similarly
always satisfied, as the sum of these is always positive:
[
∂2xsx(y) + ∂
2
ysx(y)
]
x=x∗
=
2[ν(x)− ρ(x)]
σ2c
∣
∣
∣
x=x∗
(4.10)
1were the mutant not rare, the second term would be [αy + β(y)]N(x)C(x − y) + N(y),
but since the mutant is rare N(y) ≈ 0.
17

Finally, our condition that x∗ is convergence stable, requires the difference
to be always positive:
[
∂2xsx(y) + ∂
2
ysx(y)
]
x=x∗
=
2K0[α(x) + β(x)]
σ2k
(4.11)
This leaves us with the sole condition σk > σc for x∗ to be a branching point.
If the inequality is reversed, a population will converge to x∗, rare mutants with
the strategy x∗ will then emerge and successfully invade, and the population
will remain stably at x∗ despite invasion attempts by nearby mutants. Note
that only each of these steps are satisfied by independent conditions, and only
the last does not hold in the case of branching.
4.2 Predicting Dynamics in Evolutionary Branch-
ing
Having shown that we obtain a branching point for the general form of the logis-
tic equation given in (4.1), we will now assume an explicit form so that we can
explore the dynamics more closely. Matching the landmark paper on evolution-
ary branching in Adaptive Dynamics by Dieckmann and Doebelli [Dieckmann
& Doebeli, 1999], we take the the following assignments for equations (4.2):
ν(x) = r, β = 0, ρ(x) = 0, and α = rK(x) we can write our invasion fitness
(equation (4.6)) as:
sx(y) = r −
r
∑n
i N(xi)C(xi − y)
K(y)
(4.12)
We assume the same form for K(x) and C(y − x) as specified in equa-
tions (4.5) and (4.7). As this is of the general form discussed above, we have
already shown that this model will have a branching point at x∗. Having deter-
mined that our species will branch, we now ask what happens next? We assume
that the branching initially results in two populations of equal sizes symmet-
rically spaced about x∗ (which for convenience is taken to be at x∗ = 0), and
applying the techniques we have discussed, (see appendix (B) for the calcula-
tions) we find that two new single points exist, symmetrically spaced at:
x = ±
[
σ2c
2
log
(
2σ2k − σ
2
c
σ2c
)] 1
2
(4.13)
Similarly we can evaluate the stability of these points. Somewhat surpris-
ingly2 Taking the second derivative of the invasion fitness (4.12) and evaluating
at the branching points above we find (see appendix (B)):
−2σ4k + 4σ
2
kx
2 + 2σ2c (σ
2
k − x
2)
σ2cσ
2
k(σ
2
c − 2σ
2
k)
(4.14)
2Dieckmann and Doebelli give no indication that these points are unstable, suggesting that
this simply results in two new species [Dieckmann & Doebeli, 1999].
18

This term is always positive for σk > σc, even when σk gets arbitrarily close
to σc (see appendix (B)). Consequently this point is always unstable and result
in subsequent branching. This prediction is explored in more depth through
simulation.
Meanwhile, we can also derive the expected trajectory for a monomorphic
population starting far from the critical point using the canonical equation,
(2.15). We evaluate the fitness gradient,
∂ysx(y)
∣
∣
y=x
=
−r[x− x∗]
σ2k
(4.15)
Further, recall that for a monomorphic population the equilibrium popu-
lation size is the carrying capacity associated with that trait, Nx(x) = K(x).
Hence we have
d
dt
〈x〉(t) = a1(〈x〉) = −
1
2
rµσ2µK(x)
−[x− x∗]
σ2k
(4.16)
As the carrying capacity depends on x, we will integrate this numerically to
obtain the expected trajectory.
Similarly, we can find the variance associated with this on the basis of equa-
tion (2.17). If we first calculate the second jump moment using the same ap-
proximation as before,
〈a2〉(t) =
∫
dy · [y − x]3µNx(x)M(x, y − x)∂ysx(y)
∣
∣
y=x
(4.17)
We immediately see that for our Gaussian M that this vanishes. Since the
steady state variance depend on this term, we need to revisit the approximation
we made in equation (2.6), carrying it out to higher order than we had to merely
to obtain the canonical equation originally. In doing so we will make use of the
fact that in our model birth rates are constant. We have also already substituted
in for the invasion fitness, b(y, x)− d(y, x) = sx(y).
D(y, x) ≈
∂y[sx(y)]y=x
b(x, x)
[y − x] +
∂2y [sx(y)]y=x
b(x, x)
[y − x]2 (4.18)
Note that this will not actually change the first jump moment, since we see
by equation (2.10) that the new term will result in being the third moment of
M(x, y − x) in a1, which vanishes as we have just seen. Meanwhile, by equa-
tion (2.10) we now obtain the fourth moment of M(x, y − x) in our expression
for a2,
a2 = µK(x)∂
2
y [sx(y)]y=x
∫
d∆x ·∆x4M(x,∆x) (4.19)
19

Evaluating the integral and the derivatives we recover3
a2 = 3rµσ
4
µK(x)
[
1
σ2c
−
1
σ2k
]
(4.20)
Returning to our first jump moment and taking the derivative with respect
to x we have,
d
dx
a1(x) =
1
2
rµσ2µ
σ2k
[
[x− x∗]2
σ2k
− 1
]
K(x) (4.21)
Putting this into equation (2.17) gives us our differential equation for the
variance as a function of time.
We now turn from our analytical treatment of this well-known ecological
model to computer simulation in order to confirm our predictions and further
explore the dynamics.
4.3 A Simulation-based Test of Theoretical Pre-
dictions
We design an individual-based simulation for the model given by equation (4.12).
We employ an event driven approach which offers an exact simulation for stochas-
tic processes known as Gillespie’s algorithm [Gillespie, 1977]. Using this model
we can simulate equation (4.12) and evaluate the predictions we derived from
it. Parameter values for the simulations are given in table 4.1. Code is provided
in Appendix C for reference.
4.3.1 Confirming Branching
Our model provides a clear example of an evolutionary branching point for
σk > σc, as seen in simulations such as figure 4.1. Our analysis predicts that
a monomorphic population starting some distance from the singular point will
converge towards the singular point while remaining essentially monomorphic
(consisting of only a growing mutant population and the dying resident pop-
ulation) until it reaches the vicinity of the singular point. At this point, the
further mutations lead to the population branching into two distinct popula-
tions, as seen in figure 4.1(a).
It is possible to confirm that the model is in the regime specified by as-
sumption 3 (that the mutation rate is low enough to allow for equilibration)
3A close look at this equation reveals that it becomes negative for values of x larger than
some value (for which their is no simple analytical representation but can easily be determined
numerically). This is contrary to the definition of a2 given by equation 2.10 and consequently
must be an artifact introduced by the approximations made. Simply expanding (2.6) to
higher orders is however insufficient to guarantee that a2 is always positive for this particular
formulation of sx(y). We do not have space to explore this issue further within the confines
of this paper, and it is thus left as an observation open to further investigation. We note that
this effectively gives a parameter range of starting conditions for which this expression works.
20

Parameter symbol value
Mutation rate per birth µ 0.001 *, 0.0001 **
Variance in mutation values σµ 1/20
Variance for Carrying Capacity, K(x) σk 1
Variance of Competition kernel, C(x− y) σc .4
Maximum of Carrying Capacity K0 500 *, 5000 **
Value of singular point strategy x∗ 0
Starting population trait x0 1, 0, .9 †
Table 4.1: * Values for figure 4.1(a) and 4.2. ** Values for figure 4.1(b). †
Values for figure 4.1(a), 4.1(b), and 4.2 respectively. All other values universal
to all simulations.
by observing that populations do indeed grow to their equilibrium sizes be-
fore mutants from that population start to grow into new populations. When
this assumption is violated, the population need not converge while remaining
monomorphic. Instead, when a resident population gives rise to mutants, those
mutants generate further mutants before the resident population can go extinct.
In this regime populations still converge to the singular point and branch, but
always with this higher level polymorphisms due to the multiple mutants. As
this lies outside the realm of adaptive dynamics proper we merely provide this
observation as a caution regarding the simulation parameters employed.
In our calculations we demonstrated that though this initial branching re-
sults in two distinct populations spaced some distance from the equilibrium
population, these points are in fact unstable. We further explore this instability
by running the simulations long enough to see the repeated branching.
4.3.2 The Instability of Branching
The instability of the initial branching event is somewhat surprising in light of
the treatment this particular ecological model receives within the literature. The
simulations suggest that it is not merely this first branching that is unstable,
but that these points result in further divisions. It is unclear that this should
have any direct connection to period doubling bifurcations of chaotic systems.
For instance, the subsequent branching of the original two branches need not
occur at the same time. Not only is this a consequence of the stochastic nature
of the simulation, but that the branching of one effects the other, changing the
position of the singular point.
Equation 4.13 predicts the location of the first branching, given that the
branching occurs symmetrically. While this condition need not be satisfied,
nevertheless our expression usually approximates well the location of the first
subsequent branching. We do not yet have sufficent data to determine if this
branching process continues or stablizes after a finite number of branchings.
These features can be seen in figure 4.1.
21

(a) (b)
Figure 4.1: Population branching: Figures show time on the abissca and trait
value on the ordinate. Population size is indicated by the size of the dot. Partic-
ular parameter values used in each simulation are given at the top of each graph.
(a) A population starting initially at some distance away from the branching
point converges towards it and then begins to branch. (b) A population un-
dergoes successive branching. The blue curve is the weighted mean trait value.
Note that for the parameters illustrated here, the symmetric critical points,
equation (4.13), occur at ±0.44, roughly agreeing with the onset of the first
subsequent branching.
4.3.3 Evaluating the Canonical Equation and its Variance
Integrating the canonical equation for this system, equation (4.16), we can pre-
dict the mean trajectory of a monomorphic population starting some distance
from the singular point. In order to compare our prediction with the data,
we must first collect enough samples to have a reliable measure of the actual
mean path. Additionally we can integrate the variance equation (2.17), using
the jump moments we calculated in (4.20) and (4.21). As seen in figure 4.2, the
match of both the canonical equation and the varaince to the simulation data is
rather good. The correction that we calculated to the canonical equation given
by equation 2.19, accounting for the variance dependence of the mean, is also
plotted, and shows a small but noticible improvement in predicting the mean
path. The simulation is run in a regime such that most trajectories will not
acually reach the branching point and begin to branch, since this behaviour is
no longer described by the canonical equation.
22

Figure 4.2: Confirming the canonical equation and its associated variance rela-
tion: The simulation is run for a fixed number of events with a single population
starting at a set distance (.9) from the singular point. The plot shows time on
the abissca and trait value on the ordinate, as in figure 4.1. The blue curves show
the weighted mean trait values for each run. The red curve gives the mean path
(average of the blue curves). Dashed pink curves are give the standard deviation
of the paths. The black curve is the canonical equation, numerically integrated,
which lies very close to the mean path. The yellow curve is the canonical equa-
tion with the correction discussed in the text, which offers a small but noticable
improvement relative to the fit of the original canonical equation of Dieckmann
and Law (black curve), [Dieckmann & Law, 1996]. The solid green curves give
the standard deviation calculated in the text (integrated numerically), which
also appears to agree reasonably well with the actual variance in these 50 runs.
23

Chapter 5
Conclusions
In this paper we have introduced the basic concepts of Adaptive Dynamics
theory and derived its fundamental results: the canonical equation and the
classification of singular points. In addition, we have added a derivation of
the variance associated with the canonical equation, which we showed can also
be used to provide a correction to the canonical equation of the mean. We
showed that the properties of a model for a branching singular point that is
commonly used in the literature hold for a more generalized form. Using the
common model as a particular instance of such a branching singular point, we
applied the canonical equation and the associated variance equation to this
model when a monomorphic population begins far from the critical point. We
confirmed these predictions through stochastic simulation. We also perform
stability analysis to the model where two populations have branched off from
the critical point, determining that these branches are unstable. The stochastic
simulation confirm that following the initial branching event, the populations
continue to branch.
Several interesting possibilities remain for further exploration. The limit of
continual branching could be explored further through simulation, and one may
also be able to analytically prove no stable configuration of finite n branches
exists. Using the same model for σk < σc, the singular point becomes evolu-
tionarily stable, and remains able to invade, remains convergence stable and
still features protected polymorphism. The dynamics associated with polymor-
phisms in this case would be rather different than the branching case, and may
be interesting to explore further. One could also explore the effects of using
functions other than Gaussians that still statisfy the criteria for branching out-
lined in the generalization section. Though the timescale of the initial relaxation
to the equilibrium point for a monomorphic population is given by the canonical
equation, no similar characterization exists for the timescale of the branching
events. Presumably this could be explored analytically in a similar vein, and it
could certainly explored in further simulation. Similarly, it may be interesting to
study the canonical equation’s relaxation prediction in the event of polymorphic
populations.
24

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26

Appendix A
Branching Criterion for
Generalized Logistic
Equation
We begin by determining the critical point of the growth rate given by equa-
tion (4.6). Taking the first derivative with respect to y and evaluating at x and
setting equal to 0,
∂sx(y)
∂y
∣
∣
∣
y=x
= ∂y[ν(y)− ρ(y)]− ∂y[α(y) + β(y)]K(x)C(x− y)
− [α(y) + β(y)]K(x)∂yC(x− y)
∣
∣
∣
y=x
= 0 (A.1)
Since the derivative is evaluated at y = x, we see immediately that C(x−y) =
1 and ∂yC(x− y) = 0, while we can simply write ν, ρ, α and β as functions of
x and take the partial with respect to x instead, leaving,
∂x[ν(x)− ρ(x)]−K(x)∂x[α(x) + β(x)] = 0 (A.2)
Solving this expression for K(x) and setting it equal to the condition we
found in equation (4.4)
K(x) =
∂x[ν(x)− ρ(x)]
∂x[α(x) + β(x)]
=
ν(x)− ρ(x)
α(x) + β(x)
(A.3)
Which gives us the condition,
∂x[ν(x)− ρ(x)] · [α(x) + β(x)]− [ν(x)− ρ(x)] · ∂x[α(x)− β(x)] = 0 (A.4)
We then observe that since the derivative of K in equation (4.4) is
∂xK(x) =
∂x[ν(x)− ρ(x)] · [α(x) + β(x)]− [ν(x)− ρ(x)] · ∂x[α(x)− β(x)]
[α(x) + β(x)]2
(A.5)
27

That equation (A.4) guarantees that ∂xK(x) = 0. Using our expression for
K(x), (equation (4.5)),
∂xK(x) =
(x−x∗)
σ2k
e(−(x−x
∗)2/2σ2k) (A.6)
We find that this gives us the condition that y = x = x∗ is indeed the critical
point, as expected.
We now employ the machinery of Adaptive Dynamics by taking second
derivatives of sy(x) to determine the nature of the critical point. For simplicity,
we make the following assignments: r(x) = ν(x)−ρ(x) and φ(x) = α(x)+β(x).
For the same reason, we’ll use primes to denote x derivatives and assume that
r and φ are always functions of x without explicitly writing out the argument.
We first take a look at the derivatives of K which will help us simplify our
expressions. From equations (4.4) and (4.5) we have for the first derivative:
K ′ =
r′
φ
−
rφ′
φ2
= 0 (A.7)
and for the second derivative:
∂2xK(x) = −
2r′φ′
φ2
+
2rφ′2
φ3
+
r′′
φ
−
rφ′′
φ2
=
−K0
σ2k
(A.8)
Observe that multiplying both sides by 2φ′/φ we have:
2r′φ′
φ2
−
2rφ′2
φ3
= 0 (A.9)
Which lets us simplify our expression for the second derivative of K to:
r′′
φ
−
rφ′′
φ2
=
−K0
σ2k
(A.10)
We are now ready to consider the second derivative conditions. First, we
look at the condition that x∗ can invade, which requires the second x derivative
to be positive:
∂2xsx(y)
∣
∣
y=x=x∗
=
2r′φ′
φ
− r′′ +
r
σ2c
−
2rφ′2
φ2
+
rφ′′
φ
∣
∣
x=x∗
(A.11)
Multiplying equation (A.9) by φ we realize that two of the terms cancel
immediately, leaving two terms that are simply −φ ·K ′′ (equation (A.10)). This
lets us rewrite our condition as
∂2xsx(y)
∣
∣
y=x=x∗
=
K0φ
σ2k
+
r
σ2c
∣
∣
∣
x=x∗
(A.12)
As long as r > 0 and φ > 0, we have expression (A.12) which is always
positive, satisfying our first condition. Our next condition is that x∗ is invasible,
which is satisfied if the following is positive:
28

∂2ysx(y)
∣
∣
y=x=x∗
= r′′ −
rφ′′
φ
+
r
σ2c
∣
∣
∣
x=x∗
(A.13)
Recognizing two of the terms as being equal to φK ′′ (equation (A.10)) we
can simplify our equation (A.13) for the x∗ invasibilty criterion to:
r
σ2c
−
K0φ
σ2k
∣
∣
∣
x=x∗
(A.14)
This equation is not trivially satisfied, but requires r(x∗)σ2k > K0φ(x
∗)σ2c .
Dividing over the φ term, we have
r(x∗)
φ(x∗)
σ2k > K0σ
2
c (A.15)
Recalling our first condition for K, equation (4.4) we see that r(x∗)/φ(x∗) =
K(x∗) = K0, and we can simplify the condition for x∗ to be invasible:
σk > σc (A.16)
Pressing on, the protected polymorphisms condition is:
[
∂2xsx(y) + ∂
2
ysx(y)
]
x=x∗
=
2r′φ′
φ
+
2r
σ2c
−
2rφ′2
φ2
∣
∣
∣
x=x∗
(A.17)
From the condition imposed by φ ·K ′ this simplifies immediately to
[
∂2xsx(y) + ∂
2
ysx(y)
]
x=x∗
=
2r
σ2c
∣
∣
∣
x=x∗
(A.18)
Which is clearly always positive as desired for branching. Finally, the con-
dition for x∗ to be an attractor (convergence stable) is:
[
∂2xsx(y) + ∂
2
ysx(y)
]
x=x∗
=
2r′φ′
φ
+
2rφ′′
φ
− 2r′′ −
2rφ′2
φ2
∣
∣
∣
x=x∗
(A.19)
Canceling the same terms from the K ′ condition we’re left with
=
2rφ′′
φ
− 2r′′
∣
∣
∣
x=x∗
(A.20)
Which we recognize as −φ/2 times K ′′. Substituting gives
=
2φK0
σ2k
(A.21)
Which again is positive, satisfying the convergence stability criterion. Hence
our sole condition for this generalized logistic equation to have a branching point
at x∗ is σk > σc. If this is not satisfied, x∗ still features protected polymor-
phisms, convergence stability, and can invade when rare, but can never then be
invaded, hence it is an evolutionarily stable strategy.
29

Appendix B
Subsequent Branching
Here we use the particular form of the Logistic equation implemented in the
simulation rather than the general form originally presented in the text. Our
population will have the following dynamics:
S(y, x) = r − rKx[C(x, y) + C(x,−y)]/K(y) (B.1)
We take the derivative with respect to y
∂S(y, x)
∂y
∣
∣
∣
y=x
=
r[KxC(−x, x) +KxC(x, x)]K ′(x)
K(x)2
−
r[Kx∂yC(−x, y)|y=x +Kx∂yC(x, y)|y=x)
K(x)
(B.2)
This simplifies under the following observations. We note that Kx ≡ K(x),
C(x, x) = 1, ∂yC(x, y)|y=x = 0, ∂yC(−x, y)|y=x = (−2x/σ2c )C(−x, x), and
K ′[x] = (−x/σ2k)K(x) Then we have
−x
σ2k
r[C(−x, x) + 1]− r
[
−2x
σ2c
C(−x, x)
]
(B.3)
Setting equal to zero we can immediately cancel r and x and solve for
C(−x, x):
C(−x, x) =
σ2c
2σ2k − σ
2
c
(B.4)
We note that the left-hand side is greater than zero for σk > σc, the regime
in which we are interested. Recall that since the form of C is fixed, this is a
condition on the particular allowable values of x. Plugging in for C(−x, x) we
find the equilibrium point x = a is given by
a = ±
[
σ2c
2
log
(
2σ2k − σ
2
c
σ2c
)] 1
2
(B.5)
30

Having found the stable points, we evaluate the second derivative of S(y, x)
with respect to y. If this is negative this equilibrium point x = a is not invasible
from rare mutants.
∂2S(y, x)
∂y2
∣
∣
∣
y=x
=
1
K(x)2
[r[
C(−x, x)[−2K ′(x)2 +K(x)K ′′(x)]
+ C(x, x)[−2K ′(x)2 +K(x)K ′′(x)]
−K(x) · −2K ′(x)[∂yC(−x, y)|y=x + ∂yC(x, y)|y=x]
−K(x)2[∂2yC(−x, y)|y=x + ∂
2
yC(x, y)|y=x]]] (B.6)
Taking the same conditions as before, and also the following conditions for
second derivatives:
K ′′(x) =
(
x2
σ4k
− 1σ2k
)
K(x)
∂2yC(x, y)|y=x = −1/σ
2
c
∂2yC(−x, y)|y=x =
(
4x2
σ4c
− 1σ2c
)
C(−x, x)
We then have:
∂2S(y, x)
∂y2
∣
∣
∣
y=x
=
1
K(x)2
[
r
[
[
C(−x, x) + 1
][
−2
(
−x
σ2k
)2
K(x)2 +
(
x2
σ4k
−
1
σ2k
)
K(x)2
]
+ 2
(
−x
σ2k
)
K(x)2
(
−2x
σ2c
)
C(−x, x)
−K(x)2
(
4x2
σ4c
−
1
σ2c
)
C(−x, x)−K(x)2
(
−1
σ2c
)]]
(B.7)
Simplifying, we have
= rC(−x, x)
[
−2
x2
σ4k
+
x2
σ4k
−
1
σ2k
+
4x2
σ2kσ
2
c
−
4x2
σ4c
+
1
σ2c
]
+ r
[
−2
x2
σ4k
+
x2
σ4k
−
1
σ2k
+
1
σ2c
]
(B.8)
Recalling our condition for C(−x, x) in equation (B.4) and simplifying we
can rewrite this as:
−2σ4k + 4σ
2
kx
2 + 2σ2c (σ
2
k − x
2)
σ2cσ
2
k(σ
2
c − 2σ
2
k)
(B.9)
31

Since the denominator is negative for σk > σc, we need the numerator to
be positive for the point a to be stable. Using our solution for x = a given in
equation (B.5), we find that this amounts to the condition:
(
2
σ2k
σ2c
− 1
)
log
(
2
σ2k
σ2c
− 1
)
+ 2
(
σ2k
σ2c
−
σ4k
σ4c
)
(B.10)
This term is always negative for σk > σc, hence this point is always unstable.
It is a straight forward exercise to show that the other conditions are always
satisfied.
32

Appendix C
Simulation Code
Simulations were carried out in MATLAB R©7.1.
33



  


 
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