Chapter 3. Coevolution of polygenic traits
Biological Motivation
In the previous chapter we considered the process of coevolution for those cases where a single
gene of major effect mediated the outcome of interactions. In many other well-studied cases, however,
interactions are more likely mediated by quantitative traits. Take, for example, the interaction between
the seed boring weevil, Curculio camelliae, and its host plant the Japanese camellia, Camellia japonica.
Female weevils attempt to deposit their eggs into the fruit of the Japanese camellia by first boring
through the protective pericarp of the fruit with their rostrum. If the female weevil is successful, she lays
an egg (EGGS?) into the fruit which then provides food for the developing larvae (REFS). Thus, this
interaction has the potential to exert strong reciprocal selection on weevil and plant. Studies exploring
this interaction over its native range have identified substantial variation in rostrum length and pericarp
thickness among populations (REFS), with some populations showing extremely exaggerated trait values
(REFS). Another interesting observation made by these studies is that weevil rostrum length and
camellia pericarp thickness are well-matched in only a subset of populations (REFS). Because similar
patterns have been demonstrated within a wide range of systems (REFS), this chapter will focus on
evaluating the extent to which simple mathematical models of coevolution based on quantitative traits
can explain these commonly observed empirical patterns.
Key Questions:



When should coevolution cause arms races and escalation?
What factors determine which species wins?
Should we expect coevolution to cause traits of interacting species to match?
Building a Model of Weevil-Camellia Coevolution
Although the genetic basis of rostrum length in the weevil and pericarp thickness in its host
plant is unknown, the continuous nature of phenotypic variation in these traits is incompatible with a
single gene of major effect. Instead, the continuous nature of phenotypic variation suggests these may
be quantitative traits, controlled by the action of many genes of small effect. Although we could, in
principle, begin to develop a coevolutionary model which attempted to track the frequencies of all
possible genotypes, such a model would be ridiculously complicated. For instance, even if as few as five
diploid loci determine rostrum length and pericarp thickness, we would need 35 = 243 equations for
each species even if only two alleles segregated at each locus! Obviously we will need a simpler
approach if we are to gain any insight into the process of coevolution for such genetically complex traits.
An obvious avenue to explore in such a case is the theoretical framework provided by
quantitative genetics. The key assumption made by the quantitative genetic approach is that
phenotypes are determined by the action of very many genes, each of which has only a small effect on
the phenotype {Lynch, 1998 #1188;Turelli, 1994 #255}. Under such conditions, it is often reasonable to
assume that the distribution of phenotypes is Gaussian and can thus be described by only its mean and
variance {Lande, 1976 #500;Lande, 1979 #499;Turelli, 1994 #255}. Equally important, as long as
selection is sufficiently weak for mutation to replace the genetic variation it erodes, the variance may
remain approximately constant, at least over relatively short evolutionary timescales. Remarkably, when
these conditions are met, we can predict evolutionary change in the mean phenotype of a population, 𝑧̅,
using the classic equation:
̅
1 𝜕𝑊
𝜕𝑧̅
∆𝑧̅ = 𝐺𝑧 𝑊
̅
(1)
̅ is the mean fitness of the population, and 𝑧̅ is the
where 𝐺𝑧 is the additive genetic variance for trait z, 𝑊
average value of the trait, z, within the population {Lande, 1976 #500}. Because the quantitative genetic
framework assumes that additive genetic variance 𝐺𝑧 is a fixed parameter, using equation (1) to predict
̅ , be calculated. Thus, the key to
evolutionary change requires only that the population mean fitness, 𝑊
making coevolutionary predictions using quantitative genetics is to develop expressions for population
mean fitness in each of the interacting species.
Although myriad factors undoubtedly determine the fitness of weevils and camellias in the real
world, our goal here is not to be comprehensive, but rather to be insightful. With this goal in mind, we
can focus on writing down the simplest possible expressions for fitness in weevil and Camellia that
capture the essence of their interaction. The first step in writing down these fitness expressions is to
predict the outcome of an encounter between a weevil with rostrum length x and a camellia with
pericarp thickness y. Fortunately, elegant experimental studies by Toju and Sota {, 2006 #1457} have
quantified how weevil and camellia traits interact to determine the outcome of the interaction.
Specifically, this work shows that the probability a weevil succeeds in boring through the protective
pericarp of the plant fruit and depositing an egg depends on the thickness of the pericarp relative to the
length of the weevil’s rostrum {Toju, 2006 #1457}. If the weevil’s rostrum is too short to penetrate the
plant’s pericarp, oviposition fails; if the weevil’s rostrum is long enough to penetrate the plant’s
pericarp, oviposition succeeds. Thus, the probability that a weevil with rostrum length x succeeds in an
encounter with a camellia with pericarp thickness y could be reasonably modeled using the following
function:
1
𝑃(𝑥, 𝑦) = 1+𝐸𝑥𝑝[𝛼(𝑥−𝑦)]
(2)
where the parameter α determines how sensitive weevil oviposition success is to differences in pericarp
and rostrum thickness (Figure 1). Many other interactions, such as those between toxic newts and garter
snakes {Brodie, 1999 #104} or between parsnips and parsnip webworms {Berenbaum, 1998 #73} also
appear to be well described by a phenotype differences function like (2).
Although equation (2) captures an important component of fitness — the probability that a
weevil succeeds or fails to deposit eggs in the camellia —the impact weevil success (or failure) has on
the fitness of weevil and camellia is also critical. This additional component of fitness can be easily
modeled by assuming that successful weevil oviposition increases weevil fitness by some amount sX but
reduces camellia fitness by some amount sY. Putting these two fitness components together yields
expressions for the fitness consequences of an encounter between an individual weevil with rostrum
length x and an individual camellia with pericarp thickness y:
𝑊𝑋 = 1 + 𝑠𝑋 𝑃(𝑥, 𝑦)
(3a)
𝑊𝑌 = 1 − 𝑠𝑌 𝑃(𝑥, 𝑦)
(3b)
We now have simple expressions for the fitness outcome of encounters between individual weevils and
camellias at our disposal. What we really need in order to capitalize on the classical framework of
quantitative genetics, however, is expressions for the population mean fitness of both weevil and
camellia.
Population mean fitness is nothing more than the average or expected value of fitness within a
given population and can thus be calculated by integrating over all possible values of fitness multiplied
by the frequency with which they occur (i.e., a weighted average). As long as we are willing to assume
weevils and camellias encounter one another at random, calculating population mean fitness is
particularly straightforward and can be accomplished in two discrete steps. First, calculate the expected
fitness of particular phenotypes within the focal species by integrating over the phenotype distribution
of the interacting species:
𝐸[𝑊𝑋 (𝑥)] = ∫ 𝑊𝑋 𝜑𝑦 𝑑𝑦 = 1 + 𝑠𝑋 ∫ 𝑃(𝑥, 𝑦)𝜑(𝑦)𝑑𝑦
(4a)
𝐸[𝑊𝑌 (𝑦)] = ∫ 𝑊𝑌 𝜑𝑥 𝑑𝑥 = 1 − 𝑠𝑌 ∫ 𝑃(𝑥, 𝑦)𝜑(𝑥)𝑑𝑥
(4b)
where 𝜑(𝑥) is the frequency of weevils with phenotype x and 𝜑(𝑦) is the frequency of camellias with
phenotype y. Next, calculate the expected value of (4) over the phenotype distribution of the focal
species:
̅𝑋 = ∫ 𝐸[𝑊𝑋 (𝑥)]𝜑𝑥 𝑑𝑥 = 1 + 𝑠𝑋 ∫ ∫ 𝑃(𝑥, 𝑦)𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
(5a)
̅𝑌 = ∫ 𝐸[𝑊𝑌 (𝑦)]𝜑𝑦 𝑑𝑦 = 1 − 𝑠𝑌 ∫ ∫ 𝑃(𝑥, 𝑦)𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
(5b)
Although equations (5) look a bit frightening, remember that the double integral in each is nothing more
than two weighted averages: one taken over the phenotype distribution of the interacting species and
the other over the phenotype distribution of the focal species.
Our challenge now, is to evaluate the integrals in (5). Remarkably, this integration is not as
challenging as you might think, and does not even require reaching for a calculus text. As long as we are
willing to make a simplifying assumption, the integrals boil down to nothing more than an application of
basic statistical rules for taking expectations of random variables. One way to make progress, and the
one I personally favor, is to assume the probability of a successful interaction, 𝑃(𝑥, 𝑦), does not depend
too strongly on the phenotypes of the interacting individuals. This, of course, begs the obvious question:
what do you mean by “too strongly”? Although this question can be answered in several ways, the
essence of all answers is that it must be possible to approximate the logistic function (2) with the linear
function:
1
𝛼
𝑃(𝑥, 𝑦) ≈ 2 − 4 (𝑥 − 𝑦)
(6)
while maintaining a sufficient level of accuracy. Formally, the linear approximation (6) can be derived by
taking a first order Taylor Expansion of 𝑃(𝑥, 𝑦) around the point α = 0. This approximation ignores all
terms of order α2 and higher which will be negligible for small values of α (Figure 2; change function).
Figure 2 makes clear that this approximation is quite accurate as long as α is small and the phenotypes
of the two interacting individuals are not too far apart. Remembering the latter is very important when
analyzing coevolutionary models which may generate sustained evolutionary change and drive the traits
of the interacting species outside the range over which the approximation is valid!
Our task now is to use the linear approximation (6) to facilitate evaluation of the integrals in (5).
Substituting (6) into (5) shows that the integral can now be written as:
̅𝑋 ≈ 1 + 𝑠𝑋 ∫ ∫ (1 − 𝛼 (𝑥 − 𝑦)) 𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
2
4
(7a)
̅𝑌 ≈ 1 − 𝑠𝑌 ∫ ∫ (1 − 𝛼 (𝑥 − 𝑦)) 𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
2
4
(7b)
where each integral is simply taking the expectation of the approximate fitness function over one of the
random variables (either x or y). Taking expectations with respect to the non-focal species yields:
̅𝑋 ≈ 1 + 𝑠𝑋 ∫ (1 − 𝛼 (𝑥 − 𝑦̅)) 𝜑(𝑥)𝑑𝑥
𝑊
2
4
(8a)
̅𝑌 ≈ 1 − 𝑠𝑌 ∫ (1 − 𝛼 (𝑥̅ − 𝑦)) 𝜑(𝑦)𝑑𝑦
𝑊
(8b)
2
4
where overbars indicate expectations. Next, taking expectations over the focal species yields:
̅𝑋 ≈ 1 + 𝑠𝑋 (1 − 𝛼 (𝑥̅ − 𝑦̅))
𝑊
2
4
(9a)
̅𝑌 ≈ 1 − 𝑠𝑌 (1 − 𝛼 (𝑥̅ − 𝑦̅))
𝑊
2
4
(9b)
with overbars again indicating expectations. Voila! Doing nothing more than taking statistical
expectations has allowed us to derive approximate expressions for mean fitness in the interacting
species.
Before proceeding to use mean fitness to predict coevolutionary change, it is worth taking a
moment to think a bit about what these simple expressions tell us about the way in which weevil and
camellia influence each other’s population mean fitness. Although crude, studying population mean
fitness provides a tie to population growth and insight into how the phenotype distributions of the
interacting species might impact their ecology. Two realizations are key: First, the population mean
fitness of weevil and camellia depends on the extent to which their population mean egg colorations
match. The closer the average egg coloration of the two species, the lower the fitness of the warbler and
the greater the fitness of the cuckoo. This result is quite intuitive and suggests that close phenotype
matching may have negative impacts on warbler population size but positive impacts on cuckoo
population size.
With expressions for population mean fitness of weevil and camellia in hand, we can now make
the final step toward predicting the dynamics of coevolution between these species. Specifically, we
need only substitute (10) into the classical quantitative genetics expression (1), evaluate the partial
derivative, and ignore all higher order terms in α, which are negligible under our assumption that α is
small. Making these steps yields the following expressions for coevolutionary change in population
mean values of weevil rostrum length, x, and camellia pericarp thickness, y:
∆𝑥̅ ≈ 𝑆𝑋 𝐺𝑋
(10a)
∆𝑦̅ ≈ 𝑆𝑌 𝐺𝑌
(10b)
𝑠
𝑋
where 𝑆𝑋 = 𝛼 2(2+𝑠
𝑋)
𝑠
𝑌
and 𝑆𝑌 = 𝛼 2(2−𝑠
.
)
𝑌
Analyzing the Model
We are now in a position to study the process of coevolution between weevil rostrum length
and camellia pericarp thickness. Because it is the difference between traits which matters for the
outcome of interactions, we define the new variable 𝛿 = 𝑦̅ − 𝑥̅ as we did in the previous section,
yielding a very simple expression for the change in δ which occurs over a single generation:
∆𝛿 = ∆𝑦̅ − ∆𝑥̅ = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 = 𝐾
(11)
Because this recursion equation is so very simple, we can derive a complete solution for the entire timecourse of coevolution by simply solving it directly:
𝛿1 = 𝛿0 + 𝐾
𝛿2 = 𝛿1 + 𝐾 = 𝛿0 + 2𝐾
𝛿3 = 𝛿2 + 𝐾 = 𝛿0 + 3𝐾
⋮
𝛿𝑡 = 𝛿0 + 𝐾𝑡 = 𝛿0 + (𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 )𝑡
(12)
Inspection of (12) shows that coevolutionary dynamics are even simpler than they were for the case of
the cuckoo and warbler we studied previously, although they continue to depend solely on the quantity,
𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 . If this quantity is positive, the mean pericarp thickness of the camellia population is
increasing more rapidly than is the mean rostrum length of the weevil population and the weevil is
losing the coevolutionary race (Figure 3a). In contrast, if this quantity is negative, the mean rostrum
length of the weevil population is increasing more rapidly than is the mean pericarp thickness of the
camellia population and the camellia is losing the evolutionary race (Figure 3b).
Specifically, equation (12) shows that no matter what the value of K, trait values increase in
perpetuity for both camellia and weevil; there is no possibility for decreases in trait values, nor is there
any possibility of a coevolutionary equilibrium. Instead, our model predicts that camellia and weevil will
become inextricably linked in a perpetual coevolutionary arms-race with the winner being the species
which can increase its trait values most rapidly.
Answers to Key Questions:
When should coevolution cause arms races and escalation?
Our analysis of coevolution inspired by the interaction between the seed boring weevil C.
camelliae and its host plant C. japonica suggests that coevolution should invariably lead to a
coevolutionary arms race and perpetual increases in rostrum length and pericarp thickness (Figure X).
The rate at which this arms race proceeds and traits become exaggerated depends on the strength of
reciprocal selection and the amount of additive genetic variance for the traits.
What factors determine which species wins?
The results of our simple analysis suggest that the species which wins the coevolutionary arms
race is the one with the largest response to reciprocal selection, measured by the quantity, 𝑆𝐺. Thus,
there are two factors that need to be considered before predicting which species will win: additive
genetic variance and the
Should we expect coevolution to cause traits of interacting species to match?
Although there is some intuitive reason to expect traits of interacting species will tend to evolve
well-matched phenotypes, our results do not support this idea for interactions like those between C.
camelliae and C. japonica. Instead, our results suggest that the traits of the interacting species will be
ever increasing, with a general trend toward being more dissimilar over time.
New Questions Arising:
Our simple model of weevil camellia coevolution provides us with valuable insights into the
dynamics and outcome of coevolution mediated by quantitative traits. We now have some idea which
biological conditions favor perpetual coevolutionary arms-races and determine which species is likely to
ultimately win. We have also learned that there may be little reason to expect interactions such as those
between weevils and camellias to result in closely matched traits. At the same time, however, our model
analysis raises several important new questions:



What keeps traits from zooming off to infinity?
Do our results hold for antagonistic interactions with a different mechanistic basis?
Do our results extend to other types of ecological interactions?
In the next three sections, we will develop generalizations of our simple model which allow us to answer
these questions and gain further insight into the process of coevolution mediated by quantitative traits.
Generalizations
Generalization 1: What keeps traits from zooming off to infinity?
As you read through the analysis of the previous sections and studied Figure 3, you likely noticed
that trait values were unbounded and could increase or decrease in perpetuity. Almost surely, this
prediction struck you as unrealistic and made you suspect something important was missing from our
model. One of the most obvious omissions is, of course, selective constraint. For instance, producing a
very thick protective pericarp or a freakishly long rostrum might very well be energetically expensive
and result in reduced fitness. Under such conditions, we might expect stabilizing selection to act on the
traits mediating the interaction favoring intermediate levels of pericarp thickness and rostrum length. In
this section, we integrate selective constraints into the model of weevil-Camellia coevolution developed
in the previous section and explore how the inclusion of constraint alters our expectations for
coevolution.
Stabilizing selection can be readily incorporated into a quantitative genetic model by assuming
the fitness of an individual decreases as its phenotype becomes more distant from some optimum value,
θ. Often, it is assumed that stabilizing selection is of a Gaussian form such that fitness is given by:
𝑊 = exp[−𝛾(𝑧 − 𝜃)2 ]
(13)
where 𝛾 measures the strength of stabilizing selection, z is the phenotype of an individual, and θ is the
phenotypic optimum (Figure 6). If we assume that stabilizing selection and species interactions influence
fitness independently, individual fitness can be found by multiplying the fitness consequences of
constraint (19) by the fitness consequences of biotic interactions (3). For our scenario of coevolution
between the weevil and Camellia, this yields the following expressions for the fitness consequences of
an encounter between an individual weevil with rostrum length x and an individual Camellia with
pericarp thickness y:
𝑊𝑋 = exp[−𝛾𝑋 (𝑧𝑋 − 𝜃𝑋 )2 ](1 + 𝑠𝑋 𝑃(𝑥, 𝑦))
(14a)
𝑊𝑌 = exp[−𝛾𝑌 (𝑧𝑌 − 𝜃𝑌 )2 ](1 − 𝑠𝑌 𝑃(𝑥, 𝑦))
(14b)
As was the case in the absence of constraint, our goal now is to calculate the population mean fitness of
both weevil and Camellia so that coevolutionary change can be predicted using the classical quantitative
genetics equation (1).
Just as we did previously for our model without constraint, we can calculate population mean
fitness by integrating expressions (20) twice: once over the frequency distribution of phenotypes in the
interacting species, and once over the frequency distribution of phenotypes in the focal species. Our
approach to evaluating these integrals will rely on our previous assumption that α is small and a new
assumption that 𝛾 is also small. Biologically, these assumptions imply that the outcome of interactions
between individuals does not depend too strongly on their phenotypes, and that stabilizing selection is
relatively weak. Mathematically, these assumptions allow us to approximate (20) by a first order Taylor
expansion near the points α = 0 and 𝛾 = 0. The result of this mathematical manipulation is the following
approximate expressions for population mean fitness in weevil and Camellia:
̅𝑋 ≈ 1 + 𝑠𝑋 + ∫ ∫ (𝛼𝑠𝑋 (𝑥 − 𝑦) − 2𝛾𝑋 (2 + 𝑠𝑋 )(𝑥 − 𝜃𝑋 )2 ) 𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
2
4
=1+
𝑠𝑋 𝛼𝑠𝑋
𝑠𝑋
(𝑥̅ − 𝑦̅) − 𝛾𝑋 (1 + ) ((𝑥̅ − 𝜃𝑋 )2 + 𝑉𝑋 )
+
2
4
2
̅𝑌 ≈ 1 − 𝑠𝑌 + ∫ ∫ (𝛼𝑠𝑌 (𝑦 − 𝑥) − 2𝛾𝑌 (2 − 𝑠𝑌 )(𝑦 − 𝜃𝑌 )2 ) 𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
2
4
=1−
(15a)
(15b)
𝑠𝑌 𝛼𝑠𝑌
𝑠𝑌
(𝑥̅ − 𝑦̅) − 𝛾𝑋 (1 − ) ((𝑦̅ − 𝜃𝑌 )2 + 𝑉𝑌 )
−
2
4
2
∆𝑥̅ ≈ 𝑆𝑋 𝐺𝑋 + 2𝛾𝑋 𝐺𝑋 (𝜃𝑋 − 𝑥̅ )
(16a)
∆𝑦̅ ≈ 𝑆𝑌 𝐺𝑌 + 2𝛾𝑌 𝐺𝑌 (𝜃𝑌 − 𝑦̅)
(16b)
where 𝑆𝑋 = 𝛼
𝑠𝑋
2(2+𝑠𝑋 )
and 𝑆𝑌 = 𝛼
𝑠𝑌
2(2−𝑠𝑌 )
.
Although equations (22) are more complicated than those we studied in the absence of
constraint, it remains possible to solve them directly and thus characterize the full time course of
coevolution. In the presence of constraint, however, we cannot reduce this pair of equations to a single
recursion for the variable 𝛿 = 𝑦̅ − 𝑥̅ as we did previously. Instead, we must study how the mean
phenotype itself evolves in each species, which results in the following solutions for the population
mean phenotypes of the coevolving species as a function of time, t:
𝑥̅ (𝑡) =
𝑦̅(𝑡) =
𝑆𝑋 +2𝛾𝑋 𝜃𝑋 −(1−2𝛾𝑋 𝐺𝑋 )𝑡 (𝑆𝑋 +2𝛾𝑋 (𝜃𝑋 −𝑥̅0 ))
2𝛾𝑋
𝑆𝑌 +2𝛾𝑌 𝜃𝑌 −(1−2𝛾𝑌 𝐺𝑌 )𝑡 (𝑆𝑌 +2𝛾𝑌 (𝜃𝑌 −𝑦̅0 ))
2𝛾𝑌
(17a)
(17b)
where 𝑥̅0 and 𝑦̅0 are the initial population mean phenotypes for weevil rostrum length and Camellia
pericarp thickness, respectively. Inspection of equation (23) reveals that the coevolutionary dynamics of
this system are very simple. Specifically, as evolutionary time progresses (i.e., increasing t), the term
(1 − 2𝛾𝐺)𝑡 becomes smaller and smaller, ultimately approaching zero as 𝑡 → ∞. This occurs because
the product 2𝛾𝐺 must be less than one due to our assumption of weak stabilizing selection, such that
the quantity (1 − 2𝛾𝐺) is always between 0 and 1 and thus decays when raised to a high power, t.
Consequently, as time proceeds, population mean phenotypes of the two species inevitably evolve to a
unique equilibrium:
𝑆
𝑥̂ = 𝜃𝑋 + 2𝛾𝑋
(18a)
𝑋
𝑆
𝑦̂ = 𝜃𝑌 + 2𝛾𝑌
(18b)
𝑌
At this equilibrium, each species exceeds the optimal phenotype favored by stabilizing selection by an
amount which depends on the strength of coevolutionary selection, 𝑆, relative to the strength of
stabilizing selection, 𝛾.
The equilibrium (24) provides us with the opportunity to understand how selective constraints
influence the dynamics and outcome of coevolution between weevil and Camellia. Perhaps the single
most important insight is simply that an equilibrium is now possible, and, in fact, inevitable. This is in
stark contrast to our expectations for the coevolutionary process in the absence of constraint, which led
us to believe that Camellia and Weevil would become locked in a coevolutionary arms race where trait
means escalate in perpetuity. Furthermore, whereas this perpetual arms-race was inevitably won by the
species with the greatest response to selection, we now see that the outcome of the coevolutionary
process depends on relative levels of constraint in the interacting species. Specifically, at equilibrium,
the phenotypic distance between mean rostrum length in weevils and mean pericarp thickness in
Camellias, 𝛿 = 𝑦̅ − 𝑥̅ , is given by:
𝑆
𝑆
𝛿̂ = 𝜃𝑌 − 𝜃𝑋 + 2𝛾𝑌 − 2𝛾𝑋
𝑌
𝑋
(19)
This quantity will be largest when the selective constraints acting on weevil rostrums (𝜃𝑌 , 𝛾𝑌 ) are
smaller than the selective constraints acting on Camellia pericarps (𝜃𝑋 , 𝛾𝑋 ), and the strength of
coevolutionary selection acting on weevil rostrum length (𝑆𝑌 ) exceeds that acting on Camellia pericarp
thickness (𝑆𝑋 ). The relationship between selective constraints, coevolutionary selection, and the
equilibrium difference between weevil and Camellia traits is illustrated in Figure 7. Although we have
explored only how constraints influence coevolution mediated by a mechanism of phenotype
differences, constraints also play an important role in interactions mediated by phenotype matching {,
#946;Gavrilets, 1997 #77}.
RESULT 25 HAS IMPORTANT IMPLICATIONS FOR THE GEOGRAPHIC STRUCTURE OF
INTERACTIONS WHICH MAY SHED SOME LIGHT ON AVAILABLE DATA. FOR INSTANCE, IN THE CAMELLIA
SYSTEM, A GMTC IS PRESENT. 25 SHOWS THIS COULD EASILY RESULT FROM SPATIALLY VARIABLE
ABIOTIC OPTIMA, OR COSTS…
Generalization 2: Interactions with a different mechanistic basis
the interaction between the nest parasitic cuckoo Cuculus canorus and its reed warbler host
Acrocephalus scirpaceus appears to depend upon egg coloration in both cuckoo and host {Aviles, 2012
#2666}. One of the most interesting aspects of this interaction is the large amount of spatial variation
which exists among populations for rates of parasitism, rates of parasite egg rejection, and the degree to
which the egg coloration of the parasite matches that of its host (Aviles et al. 2012).
the coloration of cuckoo and warbler eggs match. If the cuckoo lays eggs which closely match
the coloration of the warbler host, the warbler may fail to recognize and reject these eggs and raise
them as its own. Alternatively, if the cuckoo lays eggs which differ appreciably in coloration from those
of the warbler, the warbler may recognize them and eject them from the nest. Thus the probability that
a cuckoo with egg coloration x succeeds when it encounters a warbler with egg coloration y could be
reasonably modeled using the following function:
𝑃(𝑥, 𝑦) = 𝐸𝑥𝑝[−𝛼(𝑥 − 𝑦)2 ]
(20)
where the parameter α determines how perceptive the warbler is to differences in egg coloration
(FIGURE 1).
Although equation (2) captures an important component of fitness — the probability that a
cuckoo succeeds or fails to have its eggs reared by the warbler —the impact cuckoo success (or failure)
has on the fitness of cuckoo and warbler is also critical. This additional component of fitness can be
easily modeled by assuming that successful cuckoo parasitism increases cuckoo fitness by some amount
sX but reduces warbler fitness by some amount sY. Putting these two fitness components together yields
expressions for the fitness consequences of an encounter between an individual cuckoo with egg
coloration x and an individual warbler with egg coloration y:
𝑊𝑋 = 1 + 𝑠𝑋 𝑃(𝑥, 𝑦)
(21a)
𝑊𝑌 = 1 − 𝑠𝑌 𝑃(𝑥, 𝑦)
(21b)
We now have simple expressions for the fitness outcome of encounters between individual cuckoos and
warblers at our disposal. What we really need in order to capitalize on the classical framework of
quantitative genetics, however, is expressions for the population mean fitness of both cuckoo and
warbler.
Population mean fitness is nothing more than the average or expected value of fitness within a
given population and can thus be calculated by integrating over all possible values of fitness multiplied
by the frequency with which they occur (i.e., a weighted average). As long as we are willing to assume
cuckoos and warblers encounter one another at random with respect to egg coloration, calculating
population mean fitness is particularly straightforward and can be accomplished in two discrete steps.
First, calculate the expected fitness of particular phenotypes within the focal species by integrating over
the phenotype distribution of the interacting species:
𝐸[𝑊𝑋 (𝑥)] = ∫ 𝑊𝑋 𝜑𝑦 𝑑𝑦 = 1 + 𝑠𝑋 ∫ 𝑃(𝑥, 𝑦)𝜑(𝑦)𝑑𝑦
(22a)
𝐸[𝑊𝑌 (𝑦)] = ∫ 𝑊𝑌 𝜑𝑥 𝑑𝑥 = 1 − 𝑠𝑌 ∫ 𝑃(𝑥, 𝑦)𝜑(𝑥)𝑑𝑥
(22b)
where 𝜑(𝑥) is the frequency of cuckoos with phenotype x and 𝜑(𝑦) is the frequency of warblers with
phenotype y. Next, calculate the expected value of (4) over the phenotype distribution of the focal
species:
̅𝑋 = ∫ 𝐸[𝑊𝑋 (𝑥)]𝜑𝑥 𝑑𝑥 = 1 + 𝑠𝑋 ∫ ∫ 𝑃(𝑥, 𝑦)𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
(23a)
̅𝑌 = ∫ 𝐸[𝑊𝑌 (𝑦)]𝜑𝑦 𝑑𝑦 = 1 − 𝑠𝑌 ∫ ∫ 𝑃(𝑥, 𝑦)𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
(23b)
Although equations (5) look a bit frightening, remember that the double integral in each is nothing more
than two weighted averages: one taken over the phenotype distribution of the interacting species and
the other over the phenotype distribution of the focal species.
Our challenge now, is to evaluate the integrals in (5). Remarkably, this integration is not as
challenging as you might think, and does not even require reaching for a calculus text. As long as we are
willing to make a simplifying assumption, the integrals boil down to nothing more than an application of
basic statistical rules for taking expectations of random variables. One way to make progress, and the
one I personally favor, is to assume the probability of a successful interaction, 𝑃(𝑥, 𝑦), does not depend
too strongly on the phenotypes of the interacting individuals. This, of course, begs the obvious question:
what do you mean by “too strongly”? Although this question can be answered in several ways, the
essence of all answers is that it must be possible to approximate the Gaussian function (2) with the
quadratic function:
𝑃(𝑥, 𝑦) ≈ 1 − 𝛼(𝑥 − 𝑦)2
(24)
while maintaining a sufficient level of accuracy. Formally, the quadratic approximation (6) can be
derived by taking a first order Taylor Expansion of 𝑃(𝑥, 𝑦) around the point α = 0. This approximation
ignores all terms of order α2 and higher which will be negligible for small values of α (Figure 2). Figure 2
makes clear that this approximation is quite accurate as long as α is small and the phenotypes of the two
interacting individuals are not too far apart. Remembering the latter is very important when analyzing
coevolutionary models which may generate sustained evolutionary change and drive the traits of the
interacting species outside the range over which the approximation is valid!
Our task now is to use the quadratic approximation (6) to facilitate evaluation of the integrals in
(5). Substituting (6) into (5) shows that the integral can now be written as:
̅𝑋 ≈ 1 + 𝑠𝑋 ∫ ∫(1 − 𝛼(𝑥 2 − 2𝑥𝑦 + 𝑦 2 ))𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
(25a)
̅𝑌 ≈ 1 − 𝑠𝑌 ∫ ∫(1 − 𝛼(𝑥 2 − 2𝑥𝑦 + 𝑦 2 ))𝜑(𝑥)𝜑(𝑦)𝑑𝑥 𝑑𝑦
𝑊
(25b)
where each integral is simply taking the expectation of the approximate fitness function over one of the
random variables (either x or y). Taking expectations with respect to the non-focal species yields:
̅̅̅2 − 2𝑥̅ 𝑦 + 𝑦 2 )) 𝜑(𝑥)𝑑𝑥
̅𝑋 ≈ 1 + 𝑠𝑋 ∫ (1 − 𝛼(𝑥
𝑊
(25a)
̅𝑌 ≈ 1 − 𝑠𝑌 ∫ (1 − 𝛼(𝑥 2 − 2𝑥𝑦̅ + ̅̅̅
𝑊
𝑦 2 )) 𝜑(𝑦)𝑑𝑦
(25b)
where overbars indicate expectations. Next, taking expectations over the focal species yields:
̅̅̅2 − 2𝑥̅ 𝑦̅ + ̅̅̅
̅𝑋 ≈ 1 + 𝑠𝑋 (1 − 𝛼(𝑥
𝑊
𝑦 2 ))
(26a)
̅̅̅2 − 2𝑥̅ 𝑦̅ + ̅̅̅
̅𝑌 ≈ 1 − 𝑠𝑌 (1 − 𝛼(𝑥
𝑊
𝑦 2 ))
(26b)
with overbars again indicating expectations. The last step in calculating mean fitness is to remember a
basic identity from statistics: 𝑉[𝑥] = 𝐸[𝑥 2 ] − 𝐸[𝑥]2 . Using this identity to replace the terms ̅̅̅
𝑥 2 and ̅̅̅
𝑦2
and performing a small bit of algebra yields the following more insightful and useful expressions for
mean fitness: MANY FOUND THE STRETCH ABOVE CONFUSING…
̅𝑋 ≈ 1 + 𝑠𝑋 (1 − 𝛼(𝑥̅ − 𝑦̅)2 − 𝛼(𝑉𝑥 + 𝑉𝑦 ))
𝑊
(27a)
̅𝑌 ≈ 1 − 𝑠𝑌 (1 − 𝛼(𝑥̅ − 𝑦̅)2 − 𝛼(𝑉𝑥 + 𝑉𝑦 ))
𝑊
(27b)
where Vx and Vy measure the phenotypic variance for trait x and trait y, respectively.
Before proceeding to use mean fitness to predict coevolutionary change, it is worth taking a
moment to think a bit about what these simple expressions tell us about the way in which cuckoo and
warbler influence each other’s population mean fitness. Although crude, studying population mean
fitness provides a tie to population growth and insight into how the phenotype distributions of the
interacting species might impact their ecology. Two realizations are key: First, the population mean
fitness of cuckoo and warbler depends on the extent to which their population mean egg colorations
match. The closer the average egg coloration of the two species, the lower the fitness of the warbler and
the greater the fitness of the cuckoo. This result is quite intuitive and suggests that close phenotype
matching may have negative impacts on warbler population size but positive impacts on cuckoo
population size. Second, and much less intuitively, the population mean fitness of cuckoo and warbler
depends on levels of phenotypic variation in egg coloration in the two species. The more phenotypically
variable the egg coloration of the two species, the greater the population mean fitness of the warbler
and the lesser the population mean fitness of the cuckoo. The reason for this interesting result is a
fundamental difference in the form of selection coevolution imposes on the two species. Specifically,
when coevolutionary interactions are mediated by a mechanism of phenotype matching like that which
defines the interaction between cuckoo and warbler, the species which does not want to match (in this
case the warbler) experiences disruptive selection whereas the species which does want to match (in
this case the cuckoo) experiences stabilizing selection {Nuismer, 2005 #1454;Kopp, 2006 #1643}.
Consequently, phenotypic variance is “good” for the species which does not want to match (warbler)
but “bad” for the species which wants to match (cuckoo). This qualitative difference in the form of
selection acting on the two species has interesting consequences for levels of standing genetic variation
in the two species {Nuismer, 2005 #1454;Kopp, 2006 #1643}, patterns of diversification across space
{Yoder, 2010 #2309}, and potentially even the propensity for sympatric speciation (REFS).
With expressions for population mean fitness of cuckoo and warbler in hand, we can now make
the final step toward predicting the dynamics of coevolution between these species. Specifically, we
need only substitute (10) into the classical quantitative genetics expression (1), evaluate the partial
derivative, and ignore all higher order terms in α, which are negligible under our assumption that α is
small. The result is a set of very simple equations predicting how cuckoo and warbler egg coloration will
coevolve:
∆𝑥̅ ≈ 𝑆𝑋 𝐺𝑋 (𝑦̅ − 𝑥̅ )
(28a)
∆𝑦̅ ≈ 𝑆𝑌 𝐺𝑌 (𝑦̅ − 𝑥̅ )
(28b)
𝑠
𝑠
𝑌
Where 𝑆𝑋 = 2𝛼 1+𝑠𝑋 and 𝑆𝑌 = 2𝛼 1−𝑠
measure the strength of selection for and against egg coloration
𝑋
𝑌
matching in cuckoo and warbler, respectively, and GX and GY measure the additive genetic variance for
cuckoo and warbler egg coloration, respectively.
The pair of recursion equations defined by (11) are much simpler than those which we tackled in
the previous chapter for scenarios where coevolution was mediated by a single gene of major effect.
The equations are so simple, in fact, that it is possible to reduce them to a single equation which allows
us to answer the biological questions posed at the beginning of this chapter. Specifically, if our goal is to
understand the degree to which egg coloration of cuckoo and warbler matches, we need only study how
the difference between the mean phenotype of the two species itself evolves. Defining a new variable
𝛿 = 𝑦̅ − 𝑥̅ and replacing all occurrences of 𝑦̅ − 𝑥̅ in (11) with δ, yields an expression for the change in δ
which occurs over a single generation:
∆𝛿 = ∆𝑦̅ − ∆𝑥̅ = 𝑆𝑌 𝐺𝑌 (𝛿) − 𝑆𝑋 𝐺𝑋 (𝛿) = (𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 )𝛿 = 𝐾𝛿
(29)
Because this recursion equation is so simple, we have the luxury of doing something we can almost
never do in coevolutionary theory: deriving a general time dependent solution. This general solution can
be found by following the logical progression of 𝛿𝑡 where the subscript t indicates time, or the current
generation:
𝛿1 = 𝛿0 + 𝐾𝛿0 = (1 + 𝐾)𝛿0
𝛿2 = 𝛿1 + 𝐾𝛿1 = 𝛿0 + 𝐾𝛿0 + 𝐾(𝛿0 + 𝐾𝛿0 ) = (1 + 𝐾)2 𝛿0
𝛿3 = 𝛿2 + 𝐾𝛿2 = 𝛿0 + 𝐾𝛿0 + 𝐾(𝛿0 + 𝐾𝛿0 ) + 𝐾(𝛿0 + 𝐾𝛿0 + 𝐾(𝛿0 + 𝐾𝛿0 )) = (1 + 𝐾)3 𝛿0
⋮
𝛿𝑡 = (1 + 𝐾)𝑡 𝛿0 = (1 + 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 )𝑡 𝛿0
(30)
Ailene points out K is not obviously bounded on -1<K<inf
Inspection of (13) shows that coevolutionary dynamics are extremely simple, with the outcome
depending only on the sign of the quantity 𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 , which must take values -1 < K < 1 because
of our earlier assumption that α is small and G not too large. If this quantity is positive, the absolute
distance between the population mean phenotypes of the two species increases over time, indicating
that the warbler is winning the coevolutionary race and leaving the cuckoo behind (Figure 3). Unless the
cuckoo is able to switch hosts, such a scenario would ultimately result in the extinction of the cuckoo
since it would no longer be able to have its young reared by the warbler. In contrast, if the quantity K is
negative, the absolute distance between average cuckoo and warbler egg coloration decreases over
time until an equilibrium is reached where δ = 0. At this equilibrium, the cuckoo perfectly matches the
coloration of its warbler host (Figure 3), such that the warbler suffers a chronic negative fitness impact
from cuckoo parasitism.
The simple analysis in the previous paragraph suggests the outcome of coevolution between
cuckoo and warbler rests entirely on the sign of the quantity 𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 . What is it about this
quantity that makes it so important to the outcome of coevolution? Biologically speaking, this quantity is
of central importance because it measures the relative rate of evolution in the two species for any
specific distance, δ, between their mean phenotypes. For example, if the current difference in egg
coloration between cuckoo and warbler is 6nm, the response to selection for the warbler is 6nm(𝑆𝑌 𝐺𝑌 )
and the response to selection in the cuckoo is 6nm(𝑆𝑋 𝐺𝑋 ). Consequently, if 𝑆𝑌 𝐺𝑌 exceeds 𝑆𝑋 𝐺𝑋 the
warbler’s response to selection is greater than the cuckoo’s and the warbler will pull ahead in the
coevolutionary race (Figure 3a). If, the converse is true, however, the cuckoo’s response to selection
exceeds that of the warbler and it will capture the warbler in phenotype space (Figure 3b).
Based on this analysis of our model we can begin to answer the questions we posed at the
beginning of this chapter using what we have learned from our mathematical analysis.
Generalization #3: Other forms of ecological interaction
There is no doubt that antagonistic interactions such as those we have considered between
cuckoos and warblers or weevils and camellias are taxonomically diverse and ecologically important. At
the same time, however, antagonism is just one form of ecological interaction. For instance, competitive
interactions among species are thought to structure biological communities, shape probabilities of
invasion, and influence species ranges. Similarly, mutualistic interactions play a key role in the function
of biological communities, facilitating seed-dispersal, pollination, and nitrogen fixation. Clearly, any
theory of coevolution must accommodate these diverse and ecologically important forms of interaction.
Although developing a general mathematical treatment for interactions of any ecological form is beyond
the scope of this chapter, the general approach and several key insights can be illustrated by developing
and analyzing a model for a specific biological interaction.
Seed dispersal mutualisms are widespread and of significant ecological importance. For instance
dispersal of the wild nutmeg, Virola sebifera, by chestnut mandibled toucans, Ramphastos swainsonii,
plays an important role in nutmeg recruitment and range expansion within the Amazon. At the same
time, nutmeg provides an important food source for toucan populations within this region {Howe, 1981
#2678;Howe, 1981 #2679}. Assuming the rate at which successful consumption and dispersal occur
depends on the size of nutmeg seeds and the depth of toucan beaks, how might these traits coevolve
within this mutualistic interaction?
Within many seed-dispersal mutualisms, rates of successful consumption and dispersal increase
as beak depth increases relative to seed size (REF). Consequently, an appropriate functional form for the
interaction between nutmeg seed size, x, and toucan bill depth, y, would be very similar to the function
we used previously to study coevolution between weevils and camellias:
1
𝑃(𝑥, 𝑦) = 1+𝐸𝑥𝑝[𝛼(𝑥−𝑦)]
(31)
where 𝛼 now measures the extent to which differences in beak depth and seed width influence the
probability of consumption and dispersal (FIGURE 8). When a toucan succeeds in consuming and
dispersing a nutmeg seed, the mutualistic nature of the interaction suggests that both individuals
experience increased fitness:
𝑊𝑋 = 1 + 𝑠𝑋 𝑃(𝑥, 𝑦)
(32a)
𝑊𝑌 = 1 + 𝑠𝑌 𝑃(𝑥, 𝑦)
(32b)
where 𝑠𝑋 and 𝑠𝑌 are the fitness benefits to a nutmeg of being dispersed, and to a toucan of consuming a
nutmeg seed, respectively.
Although the interaction between nutmeg and toucan shapes the fitness of the interacting
species, we might expect that other forces contribute to fitness and constrain evolution in response to
species interactions. For instance, it seems likely that nutmeg seeds which are very small or very large
would have lower fitness for reasons extrinsic to the species interaction (e.g., life-history trade-offs,
minimal provisioning, etc.). Similarly, it seems likely that very small or very large beak sizes would be
detrimental to Toucan fitness for reasons unrelated to interactions with wild nutmeg (e.g., flight
performance, ability to utilize alternative food sources, sexual selection). Consequently, it seems
prudent to include stabilizing selection acting on the traits mediating the interaction as we did in the
previous section such that the fitness consequences of an encounter between an individual nutmeg with
seed size x and an individual Toucan with beak depth y are given by:
𝑊𝑋 = exp[−𝛾𝑋 (𝑧𝑋 − 𝜃𝑋 )2 ](1 + 𝑠𝑋 𝑃(𝑥, 𝑦))
(33a)
𝑊𝑌 = exp[−𝛾𝑌 (𝑧𝑌 − 𝜃𝑌 )2 ](1 + 𝑠𝑌 𝑃(𝑥, 𝑦))
(33b)
where 𝛾𝑋 and 𝛾𝑌 are the strengths of stabilizing selection acting on seed size and beak depth
respectively.
Taking a step back and looking at equations (28), it becomes apparent that they are structurally
identical to the equations we used to study the interaction between weevil and Camellia. The only
difference being that now both species benefit from the interaction such that the sign preceding the
parameter 𝑆𝑋 in (28a) is now positive rather than negative. Consequently, coevolution between nutmeg
and Toucan must be described by an identical set of recursion equations differing only in the sign of the
compound parameter, 𝑆𝑋 :
∆𝑥̅ ≈ −𝑆𝑋 𝐺𝑋 + 2𝛾𝑋 𝐺𝑋 (𝜃𝑋 − 𝑥̅ )
(34a)
∆𝑦̅ ≈ 𝑆𝑌 𝐺𝑌 + 2𝛾𝑌 𝐺𝑌 (𝜃𝑌 − 𝑦̅)
𝑠
(34b)
𝑠
𝑋
where 𝑆𝑋 = 𝛼 2(2+𝑠
𝑋)
𝑌
and 𝑆𝑌 = 𝛼 2(2+𝑠
. As was the case for equations (22) describing
)
𝑌
coevolutionary change between Weevil and Camellia, solving these recursion equations directly shows
that coevolution between nutmeg and toucan always leads to a unique equilibrium where population
mean seed size and beak depth are given by:
𝑆
𝑥̂ = 𝜃𝑋 − 2𝛾𝑋
𝑋
𝑆
𝑦̂ = 𝜃𝑌 + 2𝛾𝑌
𝑌
(35a)
(35b)
As expected, these expressions differ from the equilibrium solutions for antagonistic interactions only in
the sign of the parameter, 𝑆𝑋 , which is now negative rather than positive.
Our equilibrium solutions for population mean trait values in nutmeg and toucan provide two
insights into mutualistic coevolution. First, we expect coevolution to drive toucan beak depths to values
greater than the optimal value favored by stabilizing selection and nutmeg seed sizes to values smaller
than those favored by stabilizing selection. This result makes sense given our original assumption that
toucans with greater beak depth can consume a greater range of nutmeg seed sizes and nutmegs with
smaller seeds can be consumed and dispersed by a greater range of toucan beak sizes (i.e., Figure 8).
Second, in contrast to antagonistic interactions, both nutmeg and toucan evolve in a way that increases
the rate of successful interactions between species (FIGURE 9). The reason for this result, which is
actually quite general across mutualistic interactions, is that the fitness interests of both species are
aligned such that both evolve to enhance the efficacy of the interaction.
Conclusions and Synthesis
HERE PRESENT A GENERALIZED MODEL WITH TAYLOR SERIES GENERAL FUNCTIONS…
References
Figure Legends
Figure 1. The interaction function 𝑃(𝑥, 𝑦) for three different values of the parameter α. Larger values of
α cause the probability of successful nest parasitism to fall off more rapidly with differences in cuckoo
and warbler egg coloration.
Figure 2. The exact interaction function 𝑃(𝑥, 𝑦) (solid line), and its quadratic approximation (dashed
line), for three different values of the parameter α. As the value of α increases, the range of phenotypes
over which the quadratic approximation remains accurate decreases.
Figure 3. The coevolution of the difference between population mean egg coloration in warbler and
cuckoo, 𝛿 = 𝑦̅ − 𝑥̅ , over 200 generations. The solid line shows a case where the composite parameter
𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 > 0, indicating the warbler has a greater response to selection. In this case, the
mean egg coloration of the warbler population increases more rapidly than that of the cuckoo
population. The dashed line shows a case where the composite parameter 𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 < 0,
indicating the cuckoo has a greater response to selection. In this case, the egg coloration of the cuckoo
increases more rapidly than that of the warbler population, resulting in a matching equilibrium where
the egg coloration of the two species is identical.
Figure 4. The interaction function 𝑃(𝑥, 𝑦) for three different values of the parameter α. Larger values of
α cause the probability of successful weevil oviposition to decline more rapidly as pericarp thickness
increases relative to rostrum length.
Figure 5. The coevolution of the difference between population mean pericarp thickness in the Camellia
and rostrum length in the weevil, 𝛿 = 𝑦̅ − 𝑥̅ , over 100 generations. The solid line shows a case where
the composite parameter 𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 > 0, indicating the Camellia has a greater response to
selection. In this case, pericarp thickness increases more rapidly than rostrum length. The dashed line
shows a case where the composite parameter 𝐾 = 𝑆𝑌 𝐺𝑌 − 𝑆𝑋 𝐺𝑋 < 0, indicating the weevil has a
greater response to selection. In this case, rostrum length increases more rapidly than pericarp
thickness. In both cases, coevolution generates sustained arms races where mean rostrum length and
pericarp thickness increase in perpetuity.
Figure 6. Gaussian stabilizing selection for three different values of the parameter γ. In all cases, the
optimal phenotype is θ = 0.
Figure 7. Coevolutionary dynamics of population mean trait values (left hand panels) and corresponding
rates of interspecific interaction (right hand panels). In panels a and b, the weevil experiences only very
weak selective constraints acting on rostrum length (γX = 0.01), whereas the Camellia pericarp is strongly
constrained (γY = 0.06). As a consequence, coevolutionary selection is able to significantly exaggerate
rostrum length but not pericarp thickness, thus increasing the average rate of parasitism by the weevil
over time. In panels c and d, weevil rostrum length is strongly constrained (γX = 0.06) whereas Camellia
pericarp experiences only weak constraints (γY = 0.01). As a consequence, coevolutionary selection is
able to significantly exaggerate pericarp thickness but not rostrum length, thus decreasing the average
rate of parasitism by the weevil over time.
Figure 8. The interaction function 𝑃(𝑥, 𝑦) for three different values of the parameter α. Larger values of
α cause the probability of successful consumption and dispersal of nutmeg seeds by the toucan to
decline more rapidly as seed size increases relative to beak depth.
Figure 9. Coevolutionary dynamics of population mean trait values (left hand panels) and corresponding
rates of interspecific interaction (right hand panels). In panels a and b, the nutmeg experiences only very
weak selective constraints acting on seed size (γX = 0.01), whereas Toucan beak depth is strongly
constrained (γY = 0.06). As a consequence, coevolutionary selection is able to significantly reduce seed
size but not beak depth. In panels c and d, nutmeg seed size is strongly constrained (γX = 0.06) whereas
Toucan bill depth experiences only weak constraints (γY = 0.01). As a consequence, coevolutionary
selection is able to significantly exaggerate bill depth but not significantly reduce seed size. In both
cases, the rate at which toucans consume and disperse nutmeg seeds by increases over time because
coevolutionary selection drives the mean phenotypes of both species in a direction which promotes the
interaction.