Chapter 4. Coevolution and Population Dynamics
Biological Motivation
The previous chapters considered the process of coevolution without touching on potential
feedbacks between demography and evolution. For those systems where population sizes are regulated
more by abiotic conditions than by interactions with other species, this may be a good approximation. In
other systems, however, we know that species interactions drive demographic change, creating the
possibility for interesting feedbacks between coevolution and population ecology (Burdon and Thrall
2000; Yoshida et al. 2003; Thrall et al. 2012). One of the best examples of such feedbacks is presented by
the interactions between wild flax and flax rust we first studied in Chapter 2. As we already learned, this
interaction is thought to involve a gene-for-gene interaction and thus has the potential for coevolution.
What we left out when we introduced this system in Chapter 2, however, is the fact that population
sizes of both rust and flax fluctuate substantially over time, with much of this demographic change
driven by the interaction itself (Burdon and Thrall 1999). Moreover, recent work demonstrates that
population fluctuations in this system are tied to changes in gene frequencies (Thrall et al. 2012),
suggesting that we may need to study both demography and evolution simultaneously if we are to
develop a complete understanding of the coevolutionary process.
Our goal in this chapter is to revisit some of the simple coevolutionary models we introduced in
Chapters 2 and 3 and explore when and how their predictions change when we allow species
interactions to drive both demographic and evolutionary change.
Key Questions:



Can coevolution shape population dynamics?
Do population dynamics influence the coevolutionary process?
Will coevolution ever drive a species to extinction?
Building a model of coevolution between wild Flax and Flax-Rust
Our general approach to integrating demography and coevolution into a model will be to first
develop an ecological model of the interaction between flax and flax rust. Once we understand the basic
behavior of this purely ecological model, we will add genetic variation to the interacting species and
allow this genetic variation to influence key parameters of the ecological model. In this way, we will
allow coevolution and ecology to intermingle.
Preliminaries — developing the ecological context for coevolution
In order to begin, we must formulate a model that describes the ecology of the interaction
between these species. Although we could do this by developing a novel ecological model that captures
the potentially important nuances of this system, we will instead take an easier (and hopefully more
general) approach that capitalizes on existing ecological theory. Specifically, we will assume the ecology
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
of the interaction between Flax and Flax Rust is adequately described by the (slightly modified) LotkaVolterra model of predator-prey interactions:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝛼𝜉𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌 .
𝛼𝛽𝑁𝑋 𝑁𝑌
(1a)
(1b)
This system of ordinary differential equations describes how the population size of Flax (𝑁𝑋 ) and Flax
Rust (𝑁𝑌 ) change over time in response to Flax population growth (r), density dependence in the Flax
population (k), the probability that an encounter between Flax and Flax Rust leads to infection (𝛼), the
rate of death or virulence of infected Flax individuals (𝛽), the rate at which successful rust infections
churn out new rust spores (𝜉), and the rates at which free living rust spores die off (d). The key
assumption of the model is, much like the coevolutionary models we studied previously, that individuals
of the two species encounter one another at random, and that population sizes are sufficiently large
that random effects (e.g., demographic stochasticity) can be safely ignored.
Unlike the discrete time recursion equations we studied in the previous two chapters, these
continuous time differential equations are most appropriate for systems lacking discrete divisions
between generations. Put differently, our continuous time model works best when all events are always
occurring at some small rate (e.g., birth, death, interactions). Although many of the analytical tools we
use to study continuous time systems are identical to those we employed in the previous chapters (e.g.,
solving for equilibria), others differ slightly (e.g., local stability analyses). An ancillary goal of this chapter
is to introduce you to these subtle differences in analysis of continuous time systems.
Now that we have specified a simple ecological model, what can we learn about the population
dynamics of this interaction? Hopefully, we will be able to answer some simple questions such as when
both species are likely to coexist, when one or the other will go extinct, and when we might expect
fluctuations in population size like those well-documented in the Flax, Flax-Rust system. Although the
most pleasing way to answer these questions would be to solve the system of differential equations (1)
directly, finding such a solution is well beyond the mathematical scope of this book. Fortunately, with a
little creativity, we can still answer these questions and satisfy our curiosity using analyses of equilibria
and their local stability.
The easiest place to begin when analyzing such a system of equations is to solve for the
equilibria. Even though we are now working with differential equations rather than the discrete time
difference equations we explored in earlier chapters, the procedure for identifying equilibria is identical:
simply set the left hand side of the equations equal to zero and solve the resulting system of equations:
0 = 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝛼𝛽𝑁𝑋 𝑁𝑌
(2a)
0 = 𝛼𝜉𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌 .
(2b)
Using only relatively simple algebra it is possible to show that there are three possible equilibrium
solutions:
2
𝑁𝑋 = 0 𝑎𝑛𝑑 𝑁𝑌 = 0
(3a)
𝑁𝑋 = 𝑘 𝑎𝑛𝑑 𝑁𝑌 = 0
(3b)
𝑁𝑋 =
𝑑
𝛼𝜉
𝑎𝑛𝑑 𝑁𝑌 =
𝑟(𝛼𝑘𝜉−𝑑)
.
𝛼 2 𝑘𝜉𝛽
(3c)
The first solution is rather boring and identifies the trivial case where both species have gone extinct.
The second solution is a bit more interesting and corresponds to a scenario where the rust has gone
extinct and the host is at its carrying capacity, k. In contrast, the third solution reveals a case of
significantly more interest where both rust and flax potentially coexist. I say potentially here because
(3c) only predicts biological relevant positive population abundances for a subset of parameter
conditions. Our goal now is to identify when these abundances are positive such that equilibrium (3c)
exists. In this case, these existence conditions can be identified by simple inspection of (3c), revealing
that 𝑁𝑋 is always positive and that 𝑁𝑌 is positive anytime:
𝛼𝜉𝑘 > 𝑑
(4)
This condition makes sense because it tells us that the rust can only persist if it can turn host individuals
into new rust individual (𝛼𝜉𝑘) faster than rust individuals die (d). Put differently, this condition shows
that the rust can persist only when its per capita growth rate is greater than zero when interacting with
a host population at its carrying capacity (k).
We now know that there are three possible equilibrium states for our ecological model,
corresponding to extinction of both species, extinction of only the rust, and coexistence of both rust and
flax. Next we need to identify the conditions leading the system toward each of these possible
equilibria. To answer this question, we must employ local stability analyses. Just as we did for the
system of discrete time equations describing changes in allele frequencies (Chapter 2), the first step is to
create a Jacobian Matrix. For the system of ordinary differential equations (2), this Jacobian Matrix is
given by:
𝑟−
𝐽=[
2𝑟𝑁𝑋
𝑘
− 𝛼𝛽𝑁𝑌
𝛼𝜉𝑁𝑌
−𝛼𝛽𝑁𝑋
𝛼𝜉𝑁𝑋 − 𝑑
].
(5)
Next, just as we did in Chapter 2 we calculate the eigenvalues of the Jacobian Matrix and evaluate them
at each of the equilibria (3). The eigenvalues for each of the equilibria are:
𝜆1 = 𝑟 𝑎𝑛𝑑 𝜆2 = −𝑑
(6a)
𝜆1 = −𝑟 𝑎𝑛𝑑 𝜆2 = 𝛼𝜉𝑘 − 𝑑
(6b)
𝜆1 =
−𝑟𝑑+√𝑟√𝑑√𝑟𝑑+4𝛼𝑑𝑘𝜉−4𝛼2 𝜉 2 𝑘 2
2𝛼𝑘𝜉
𝑎𝑛𝑑 𝜆2 =
−𝑟𝑑−√𝑟√𝑑√𝑟𝑑+4𝛼𝑑𝑘𝜉−4𝛼2 𝜉 2 𝑘 2
2𝛼𝑘𝜉
respectively.
3
(6c)
Now that we have the eigenvalues in hand, we just need to use them to understand the ecology
of the interaction between flax and flax rust. Let’s start with the simplest and most transparent results.
First, the equilibrium where both species are extinct is never locally stable because one of its
eigenvalues is always positive and real. This makes perfect biological sense since we would imagine that
the Flax population would always increase from rarity in the absence of any infection by Flax Rust or
intraspecific competition. Second, the equilibrium where Flax is present but the rust extinct can be
locally stable, but only if the rust per capita growth rate when rare is negative. By noticing that this
condition is intimately related to the condition we identified for the existence of the coexistence
equilibrium, a pleasing connection emerges. Specifically, when extinction of the rust is locally stable, the
coexistence equilibrium does not exist; when the coexistence equilibrium exists, rust extinction is locally
unstable. So far so good! We can move on to trying to figure out when the rust and flax populations will
be drawn to the third equilibrium, and what their ecological dynamics will be like near this equilibrium.
Right off the bat, we can draw one conclusion of biological importance: if the quantity under the
more complicated root (𝑟𝑑 + 4𝛼𝑑𝑘𝜉 − 4𝛼 2 𝜉 2 𝑘 2 ) is negative, the eigenvalues will have an imaginary
component and the species abundances will cycle inwards toward the equilibrium. The reason for this is
that in continuous time systems, stability is determined by only the real part of the eigenvalue and if the
quantity under the more complicated root is negative, the real part of both eigenvalues must be
negative (Appendix 1; local stability analysis). What if instead, however, the quantity under the more
complicated root is positive? In this case, we know that Flax and Flax-Rust populations will not cycle, but
whether the equilibrium is stable or not depends on the value of the eigenvalues that will now be
entirely real. Together, these considerations and some simple algebra allow us to summarize when
various sorts of ecological outcomes are expected to occur (Table 1). By solving the system of ordinary
differential equations (2) numerically, we can also visualize ecological dynamics for various parameter
combinations (Figure 1). In the next section, we integrate genetic variation for the ecological parameters
of our model and investigate how the potential for coevolution impacts the ecology of the interaction.
Integrating coevolution
Our goal now is to take the ecological template we developed in the previous section and add to
it the potential for coevolution. Although there are many ways in which this could be done, we will
assume as we did in Chapter 2, that the probability with which infection occurs in a random encounter
between Flax and Flax-rust, 𝛼, depends on the genotype of each individual at a single, haploid, locus.
Specifically, we will assume that Flax population has two possible alleles, R and r with abundances 𝑁𝑋,𝑅
and 𝑁𝑋,𝑟 , respectively, and the rust population has two possible alleles, V and v with abundances 𝑁𝑌,𝑉
and 𝑁𝑌,𝑣 , respectively. With these assumptions, we can use the ecological model (1) to specify a system
of four ordinary differential equations describing how the abundance of each allele in each species
changes over time:
𝑑𝑁𝑋,𝑅
𝑑𝑡
𝑑𝑁𝑋,𝑟
𝑑𝑡
= 𝑟𝑁𝑋,𝑅 (1 −
= 𝑟𝑁𝑋,𝑟 (1 −
𝑁𝑋,𝑅 +𝑁𝑋,𝑟
𝑘
𝑁𝑋,𝑅 +𝑁𝑋,𝑟
𝑘
) − 𝛼𝑅,𝑉 𝛽𝑁𝑋,𝑅 𝑁𝑌,𝑉 − 𝛼𝑅,𝑣 𝛽𝑁𝑋,𝑅 𝑁𝑌,𝑣
(7a)
) − 𝛼𝑟,𝑉 𝛽𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝛼𝑟,𝑣 𝛽𝑁𝑋,𝑟 𝑁𝑌,𝑣
(7b)
4
𝑑𝑁𝑌,𝑉
𝑑𝑡
𝑑𝑁𝑌,𝑣
𝑑𝑡
= 𝛼𝑅,𝑉 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑉 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝑑𝑁𝑌,𝑉
(7c)
= 𝛼𝑅,𝑣 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑣 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑣 − 𝑑𝑁𝑌,𝑣
(7d)
where the term 𝛼𝑖,𝑗 indicates the probability of infection in an encounter between a flax with genotype i
and a rust with genotype j. Together, equations (7) are sufficient to predict the ecological and
coevolutionary dynamics of the interaction between Flax and Flax-Rust.
Analyzing the Model
With a lovely set of differential equations in hand, the temptation is strong to jump in and start
analyzing! In this case, however, we can make much more progress and enrich our biological
understanding by being patient and first making a change of variables. The change of variables I
recommend in this case is one that shifts the focus from the raw genotypic abundances tracked by
equations (7) to the total abundance of Flax and Flax rust and the genotype frequencies within each. The
reason this change of variables is so powerful, is that it effectively disentangles (to the extent possible)
ecological and coevolutionary dynamics. This clarifies the way in which feedbacks between demography
and evolution occur, and allows us to easily identify cases in which coevolution and demography are
effectively independent. The first step in making this change of variables is to define new variables
corresponding to the total population abundance of Flax and Flax-Rust:
𝑁𝑋 = 𝑁𝑋,𝑅 + 𝑁𝑋,𝑟
(8a)
𝑁𝑌 = 𝑁𝑌,𝑉 + 𝑁𝑌,𝑣
(8b)
and allele frequency within Flax and Flax-Rust:
𝑝𝑋 =
𝑝𝑌 =
𝑁𝑋,𝑅
(8c)
𝑁𝑋,𝑅 +𝑁𝑋,𝑟
𝑁𝑌,𝑉
(8d)
𝑁𝑌,𝑉 +𝑁𝑌,𝑣
where 𝑝𝑋 is the frequency of the R allele in the Flax and 𝑝𝑌 is the frequency of the V allele in the FlaxRust. The next step in accomplishing our change of variables is to apply the chain rule from calculus:
𝑑𝑁𝑋
𝑑𝑡
= 𝜕𝑁 𝑋
𝜕𝑁
𝑑𝑁𝑌
𝑑𝑡
= 𝜕𝑁 𝑌
𝑑𝑝𝑋
𝑑𝑡
= 𝜕𝑁 𝑋
𝑑𝑝𝑌
𝑑𝑡
= 𝜕𝑁 𝑌
𝑋,𝑅
𝜕𝑁
𝑌,𝑉
𝜕𝑝
𝑋,𝑅
𝜕𝑝
𝑌,𝑉
𝑑𝑁𝑋,𝑅
𝑑𝑡
𝑑𝑁𝑌,𝑉
𝑑𝑡
𝑑𝑁𝑋,𝑅
𝑑𝑡
𝑑𝑁𝑌,𝑉
𝑑𝑡
𝑑𝑁𝑋,𝑟
𝜕𝑁
+ 𝜕𝑁 𝑋
𝑋,𝑟
𝑑𝑁𝑌,𝑣
𝜕𝑁
+ 𝜕𝑁 𝑌
𝑑𝑁𝑋,𝑟
𝜕𝑝
𝑋,𝑟
𝜕𝑝
+ 𝜕𝑁 𝑌
𝑌,𝑣
(9b)
𝑑𝑡
𝑌,𝑣
+ 𝜕𝑁 𝑋
(9a)
𝑑𝑡
(9c)
𝑑𝑡
𝑑𝑁𝑌,𝑣
(9d)
𝑑𝑡
5
Carrying out the derivatives specified by the chain rule (9) leads to the following system of differential
equations describing how total population abundances of Flax and Rust change over time (ecology) and
how the frequencies of Flax and Flax Rust genotypes change over time (evolution):
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
(10b)
𝑑𝑝𝑋
𝑑𝑡
= −𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 (𝑝𝑌 (𝛼𝑅,𝑉 − 𝛼𝑟,𝑉 ) + 𝑞𝑌 (𝛼𝑅,𝑣 − 𝛼𝑟,𝑣 ))
(10c)
𝑑𝑝𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 (𝑝𝑋 (𝛼𝑅,𝑉 − 𝛼𝑅,𝑣 ) + 𝑞𝑋 (𝛼𝑟,𝑉 − 𝛼𝑟,𝑣 ))
(10d)
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(10a)
where 𝛼̅ = 𝛼𝑅,𝑉 𝑝𝑋 𝑝𝑌 + 𝛼𝑅,𝑣 𝑝𝑋 𝑞𝑌 + 𝛼𝑟,𝑉 𝑞𝑋 𝑝𝑌 + 𝛼𝑟,𝑣 𝑞𝑋 𝑞𝑌 and measures the average infectivity of the
pathogen population. Re-written in this way, the biology hidden within the equations becomes
transparent and we can immediately learn much about the interplay between ecology and evolution.
Perhaps the single most important thing we can distill from equations (10) without performing
any real mathematical analysis is that demography impacts coevolution through its influence on the
strength of coevolutionary selection. This can be clearly seen by noticing that population sizes appear in
the evolutionary equations (10c-d) only as a multiplier. Thus, as we might expect intuitively, the greater
the population size of the interacting species, the greater its impact on focal species evolution. Equally
intuitive is the way in which coevolution impacts demography. Specifically, if coevolution increases the
average rate of infection, as might be the case if the rust is winning the coevolutionary race, 𝛼̅ becomes
larger, and it becomes increasingly likely that the parasite population will avoid extinction and coexist
with the Flax. To see this, simply substitute the quantity 𝛼̅ in for 𝛼 within Table 1 and ask what happens
as this parameter increases. In contrast, if coevolution reduces the average rate of infection, as might be
the case if the host is winning a coevolutionary race, 𝛼̅ becomes smaller and extinction of the rust
becomes increasingly likely. Of course, what we really need to know in order to use these general
insights to make predictions in the Flax and Flax-Rust system is how, exactly, the quantity 𝛼̅ changes in
reponse to coevolution. To make this more precise sort of prediction, we must specify the genetic basis
of the interaction between Flax and Flax Rust further.
We learned in Chapter 2 that available evidence suggests the interaction between Flax and FlaxRust is mediated by a gene-for-gene interaction. If we replace the arbitrary 𝛼 parameters in (10) with
their values specified by a gene-for-gene interaction (𝛼𝑅,𝑉 = 1, 𝛼𝑅,𝑣 = 0, 𝛼𝑟,𝑉 = 1, 𝛼𝑟,𝑣 = 1), we arrive
at the following set of equations describing demographic and coevolutionary change in Flax and Flax
Rust:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(11a)
(11b)
6
𝑑𝑝𝑋
𝑑𝑡
= 𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 𝑞𝑌
(11c)
𝑑𝑝𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 𝑝𝑋
(11d)
where, 𝛼̅ = 1 − 𝑝𝑋 𝑞𝑌 .
These equations show us something remarkable: if you look back to Chapter 2 where we first modeled
this system ignoring demographic change, you will see that the equations predicting coevolution there
(5) are virtually identical to those describing coevolution here (11c,d). How cool is that? The only impact
population sizes have in this case is to modulate the strength of coevolutionary selection. More
importantly, because population sizes are always positive numbers, equations (11c,d) predict exactly the
same coevolutionary outcome as we expected when we ignored demography: fixation of the virulent
allele in the Rust resulting in complete infectivity at the population level (Figure 2). The only impact
ecology has on coevolution is to cause fluctuations in the rate of coevolution as it proceeds toward
fixation of the virulent allele. In short, for this particular example, coevolution is quite insensitive to
population ecology and changes in demography.
Although demography has little impact on coevolution in this scenario, coevolution can have
large – although transient – consequences for ecology. Specifically if we imagine both Flax and Flax Rust
are initially fixed for the susceptible allele (r) and the avirulent allele (v), the population would begin in a
state of high infectivity where 𝛼̅=1. Now, imagine a new mutant resistant allele (R) appears within the
host population and begins to sweep through the population. Until a corresponding mutation to the
virulent (V) allele arises within the Rust population, average infectivity will fall, reducing the per capita
growth rate of the Rust population, potentially driving it toward extinction. Once a mutation to the
virulent allele occurs within the Rust population, however, we expect the average rate of infection to
begin to increase, ultimately returning to 𝛼̅=1 once the virulent allele has spread to fixation. Thus, over
the course of the coevolutionary process, population sizes of Flax and Flax Rust may rise and fall as new
mutations arise and increase in frequency, only to be counteracted by new mutations in the interacting
species (Figure 2).
Answers to Key Questions:
Can coevolution shape population dynamics?
Absolutely. Our general results show that coevolution can change the average rate of ecological
interaction that can have important implications for species persistence and population dynamics. The
extent to which this occurs, however, depends on the particular type of coevolution and its
consequences for rates of interaction. For instance, for the gene-for-gene interaction thought to
mediate interactions between Flax and Flax Rust, changes in population dynamics driven by coevolution
are expected to be only transient.
Do population dynamics influence the coevolutionary process?
7
Not much. Our analyses revealed that the way in which population sizes impact coevolution is
by modulating the strength of coevolutionary selection. When population sizes are large, encounters
occur frequently, and coevolutionary selection is strong. When population sizes are small, encounters
become less frequent and coevolutionary selection weaker. As a consequence, for the gene-for-gene
model thought to mediate interactions between Flax and Flax Rust, demography has no impact on the
outcome of coevolution, causing only short-term fluctuations in the rate of coevolution.
Will coevolution ever drive a species to extinction?
No, with caveats. For the gene-for-gene model underlying Flax Flax-Rust coevolution we
considered, coevolution cannot lead to extinction of either species. However, if both species are initially
fixed for the susceptible (r) and avirulent (v) alleles and a new mutation to the resistant allele (R) arises
within the Flax population and spreads to a high frequency prior to occurrence and spread of a virulent
(V) allele in the rust population, the rust population could be driven to very low abundances. Because
our model is deterministic, however, this will never result in the extinction of the rust population which
will ultimately rebound once a mutation to the virulent allele occurs and spreads through the
population. It is easy to see, however, how coevolution could lead to extinction in a more realistic model
that integrated finite population sizes and the nuances of demographic and genetic stochasticity.
New Questions Arising:
Our simple model has yielded interesting conclusions and predictions about the interplay
between coevolution and ecology. These predictions, however, may rest on the specific assumptions we
made as we developed and analyzed our model, raising several important questions:



Would including costs of resistance and virulence alter our conclusions?
Do our general conclusions hold for other forms of genetic interaction?
Are our conclusions applicable to coevolution mediated by quantitative traits?
In the next three sections, we will generalize our simple model in ways that allow us to answer these
questions.
Generalizations
Generalization 1: Integrating costs of resistance and virulence
We learned in Chapter 2 that integrating costs of resistance and virulence strongly influences
the dynamics of gene-for-gene coevolution, creating the potential for genetic polymorphism and cyclical
dynamics. It also seems likely that costs of resistance and virulence could increase the scope for
feedbacks between ecology and evolution because the benefits of carrying a resistant or virulent allele
depends on the abundance of pathogens and hosts whereas costs of carrying these alleles may be static
in many cases. Thus, we might expect changes in population size to now have the potential to shift the
outcome of coevolution rather than simply adjust its rate.
8
Costs could be integrated into our model in many ways. For instance, it might be the case that
the resistant R allele lowers the competitive ability of its carrier through costs of expression or costs
associated with altering cell-surface receptors in a way that impacts functions other than species
interactions (Tian et al. 2003). In such a case, it would be appropriate to assume Flax individuals carrying
the R gene had a reduced carrying capacity. Alternatively, carrying the resistant R allele might have no
impact on competitive ability but instead reduce growth rate. Because this latter case is mathematically
more straightforward, we will assume that it is in growth rate that costs of carrying the resistant (R)
allele are manifested. Similarly, there are multiple ways to integrate costs of carrying the virulent (V)
allele. We will, however, focus on the simplest case here as well, where the death rate (d) of individuals
carrying the virulent (V) allele is greater than that of individuals carrying the avirulent (v) allele. With
these assumptions, we can re-write our coevolutionary model in the following way:
𝑑𝑁𝑋,𝑅
𝑑𝑡
𝑑𝑁𝑋,𝑟
𝑑𝑡
𝑑𝑁𝑌,𝑉
𝑑𝑡
𝑑𝑁𝑌,𝑣
𝑑𝑡
= 𝑟𝑅 𝑁𝑋,𝑅 (1 −
= 𝑟𝑟 𝑁𝑋,𝑟 (1 −
𝑁𝑋,𝑅 +𝑁𝑋,𝑟
𝑘
𝑁𝑋,𝑅 +𝑁𝑋,𝑟
𝑘
) − 𝛼𝑅,𝑉 𝛽𝑁𝑋,𝑅 𝑁𝑌,𝑉 − 𝛼𝑅,𝑣 𝛽𝑁𝑋,𝑅 𝑁𝑌,𝑣
) − 𝛼𝑟,𝑉 𝛽𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝛼𝑟,𝑣 𝛽𝑁𝑋,𝑟 𝑁𝑌,𝑣
(12a)
(12b)
= 𝛼𝑅,𝑉 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑉 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝑑𝑉 𝑁𝑌,𝑉
(12c)
= 𝛼𝑅,𝑣 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑣 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑣 − 𝑑𝑣 𝑁𝑌,𝑣
(12d)
where 𝑟𝑖 is the growth rate of a Flax individual carrying allele i and 𝑑𝑖 is the death rate of a rust
individual carrying allele i. Although we could analyze this system of equations directly, we will again
employ the change of variables to population sizes and allele frequencies that we used in the previous
section.
After applying the chain rule (9) to equations (12) using the new variables defined by (8) we
arrive at the following system of differential equations for the specific case of the gene-for-gene model:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟̅ 𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝛼̅𝜉𝑁𝑋 𝑁𝑌 − 𝑑̅𝑁𝑌
𝑑𝑝𝑋
𝑑𝑡
= 𝑝𝑋 𝑞𝑋 (𝛽𝑁𝑌 𝑞𝑌 − 𝜏𝑅 (1 −
𝑑𝑝𝑌
𝑑𝑡
= 𝑝𝑌 𝑞𝑌 (𝜉𝑁𝑋 𝑝𝑋 − 𝜏𝑉 )
𝛼̅𝛽𝑁𝑋 𝑁𝑌
(13a)
(13b)
𝑁𝑋
))
𝑘
(13c)
(13d)
where 𝛼̅ = 1 − 𝑝𝑋 𝑞𝑌 , 𝑟̅ = 𝑟𝑅 𝑝𝑋 + 𝑟𝑟 𝑞𝑋 , 𝑑̅ = 𝑑𝑉 𝑝𝑌 + 𝑑𝑣 𝑞𝑌 , 𝜏𝑅 = 𝑟𝑟 − 𝑟𝑅 , 𝜏𝑉 = 𝑑𝑉 − 𝑑𝑣 . The most
important thing to take away from these equations is that while costs of carrying the resistant and
virulent alleles remain constant, the benefits accruing form carrying these alleles depend on the
population size of the interacting species. This asymmetry significantly increases the scope for ecology
and evolution to interact as changes in population size shift the balance between costs and benefits of
9
carrying resistant and virulent alleles. Going beyond this general and somewhat vague statement
requires that we employ a more formal analysis to address a particular question.
Although there are many questions we could use our model to answer, one particularly
interesting possibility is to ask when the rust can invade the host population. To answer this question,
we need to identify those conditions where equilibria corresponding to rust extinction are
simultaneously unstable. The first step, then, is to identify equilibrium solutions to equations (13). We
can do this in the usual way by setting their left hand sides of the equations to zero and solving for the
variables. Because we are interested only in identifying the cases where the rust is absent from the
population or locally extinct, we can focus on the subset of equilibria where the population size of the
rust is zero (Table 2). If we can show that conditions exist where all five of these equilibria are
simultaneously unstable, we will have successfully identified conditions guaranteeing a persistent
(endemic) rust population.
To analyze the local stability of these equilibria, we follow the usual steps: 1) Make a Jacobian
Matrix, 2) Find the eigenvalues of this Jacobian Matrix, and 3) Evaluate the eigenvalues at the
equilibrium of interest. The only substantive difference from the stability analyses we have previously
performed is that we now have a system of four dynamical equations. As such, our Jacobian Matrix will
now have four rows and four columns (whereas before it had only two), and when we find the
eigenvalues of this Jacobian Matrix, there will be four rather than two. Carrying out the requisite
calculations using Mathematica allows us to identify the conditions under which the equilibria are
locally unstable (Table 2). Looking carefully at this collection of stability conditions, and remembering
that costly virulence implies 𝑑𝐵 > 𝑑𝑏 , reveals that the only way persistence of the rust population can
be guaranteed is if 𝑑𝐵 < 𝜉𝑘. Biologically, this result makes perfect sense: as long as the virulent allele is
able to increase in abundance when introduced into a Flax population at its carrying capacity (k), the
rust population is bounded away from extinction. Interestingly, the frequency of the resistant Flax allele
(R) is irrelevant, as are all parameters within the Flax population other than the carrying capacity. What
this result represents, then, is a demographic constraint on coevolution: unless this condition holds and
the Rust is able to invade, the populations cannot even begin to engage one another in coevolution.
We now know when the rust is able to invade the flax population but have no idea what
happens once the rust becomes established. To gain insight into the coevolutionary dynamics and
demographic changes that ensue, we would need to identify equilibria where both Flax and Rust coexist
and then analyze the local stability of these equilibria. Because such an analysis is quite complex
(although possible), we are not going to pursue this here. Instead, we will simply take a look at what
happens as we reduce the death rate of the virulent allele using numerical solutions to equations (13). In
short, as we decrease the death rate for this virulent allele in the Rust, coevolution begins to engage
between rust and flax and cyclical dynamics, accompanied by potentially quite dramatic population
cycles ensue (Figure 3). Comparing the coevolutionary dynamics in these figures with those we saw in
Chapter 2 for the case where demography played no role, appears to support our basic insight that for
the gene-for-gene model with costs, demography and coevolution interact to generate novel dynamical
regimes.
10
Generalization 2: Matching alleles interaction
Our model of coevolution between Flax and Flax-Rust indicates that demography and
coevolution interact in an interesting way only when costs of resistance and virulence are present in a
gene-for-gene model. Would this insight hold for other types of interactions not mediated by a gene-forgene interaction? For instance, would interactions such as that between Daphnia magna and Pasteuria
ramosa that are thought to be mediated by a matching alleles model (Luijckx et al. 2013) also require
costs in order for interesting interactions between demography and coevolution to occur? In order to
answer this question, we can simply integrate the genetic assumptions of the matching alleles model
into the Lotka-Volterra model we developed previously in this chapter.
As we did in Chapter 2, we will assume that both the host, D. Magna, and parasite, P. Ramosa,
have two possible alleles at a single haploid locus. Specifically, we will assume the D. Magna population
has two possible alleles, A and a with frequencies pX and (1-pX) and the P. Ramosa population also has
two possible alleles, B and b with frequencies pY and (1-pY). With these assumptions, we can follow the
exact same steps we did for the gene-for-gene model to develop the following set of differential
equations describing change in population densities and allele frequencies over time:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
(14b)
𝑑𝑝𝑋
𝑑𝑡
= −𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 (𝑝𝑌 (𝛼𝐴,𝐵 − 𝛼𝑎,𝐵 ) + 𝑞𝑌 (𝛼𝐴,𝑏 − 𝛼𝑎,𝑏 ))
(14c)
𝑑𝑝𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 (𝑝𝑋 (𝛼𝐴,𝐵 − 𝛼𝐴,𝑏 ) + 𝑞𝑋 (𝛼𝑎,𝐵 − 𝛼𝑎,𝑏 ))
(14d)
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(14a)
If we now specify the values of 𝛼 corresponding to a matching alleles model, we can simplify (14) to the
following set of differential equations:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
(15b)
𝑑𝑝𝑋
𝑑𝑡
= 𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 (1 − 2𝑝𝑌 )
(15c)
𝑑𝑝𝑌
𝑑𝑡
= −𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 (1 − 2𝑝𝑋 )
(15d)
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(15a)
where 𝛼̅ = 𝑝𝑋 𝑝𝑌 + 𝑞𝑋 𝑞𝑌 . We now have a system of equations that describe how population sizes and
allele frequencies of D. magna and P. ramosa change over time.
Much as we saw with the gene-for-gene model (without costs), coevolution appears to be
relatively insensitive to demography. To see this, compare equations (15c and 15d) to equations (6) in
Chapter 2 and notice that the only difference is that here, the strength of coevolutionary selection is
11
modulated by population sizes. This shows that — all else being equal — evolutionary change within
each species is most rapid when the interacting species is abundant. It also shows, however, that
including demography is unlikely to have major qualitative impacts on the coevolutionary process since
it adjusts only the strength of coevolutionary selection. Thus, it seems reasonable to assume coevolution
will cause allele frequency cycles, just as it did in the absence of demography (Chapter 2). How will
these allele frequency cycles impact the demography of the interaction? Inspecting equations 15a and
15b shows that allele frequency oscillations have the potential to strongly impact the ecology of the
interaction through their impact on average rates of infection within the Daphnia population (𝛼̅). Would
this then make cyclical ecological dynamics more common than they would be in the absence of
coevolution? To answer this question, we turn to a more formal analysis of equilibria.
In the preceding paragraph we made some logical predictions about the dynamics and outcome
of the interaction between Daphnia and P ramose based on previous analyses (e.g., Chapter 2) and
inspection of the general structure of equations (15). Although these predictions may well be sound, it is
always a good idea to complement mathematical intuition with formal analysis. Specifically, we can
formalize our understanding of this system by identifying equilibria and evaluating the conditions under
which they exist and are locally stable. Since this whole exercise should be becoming old hat by now, I
will leave the details to the accompanying Mathematica notebooks and jump ahead to the most
interesting and relevant results. Specifically, the results derived in the accompanying Mathematica
notebooks show that there is a single biologically relevant equilibrium where Daphnia and P. Ramosa
coexist and are genetically polymorphic. At this equilibrium, the population abundances of the two
species are:
𝑁𝑋 =
2𝑑
𝜉
𝑎𝑛𝑑 𝑁𝑌 =
2𝑟(−2𝑑+𝑘𝜉)
𝑘𝛽𝜉
(16a)
and their allele frequencies are:
1
1
𝑝𝑋 = 2 𝑎𝑛𝑑 𝑝𝑌 = 2.
(16b)
Inspecting equations (16) immediately leads us to several important conclusions. First, the population
abundances of the two species at this equilibrium are identical to that expected from our purely
ecological model if we replace α with the value of 𝛼̅ near this equilibrium (). Second, the allele
frequencies of the species are exactly those we expect from the matching alleles model in the absence
of demography. Thus, at least with respect to the equilibrium, no interesting interaction between
ecology and coevolution exists. But what happens if we perturb this system? Would we then see an
interesting interaction between coevolution and demography emerge? To answer this question, we
need to analyze the local stability of the equilibrium.
Following the standard protocol of creating a Jacobian Matrix and then calculating its
eigenvalues at the equilibrium of interest leads to the following set of four eigenvalues:
𝜆1 = −
√𝑑√𝑟√2𝑑−𝑘𝜉
(17a)
√𝑘√𝜉
12
𝜆2 =
√𝑑√𝑟√2𝑑−𝑘𝜉
𝜆3 = −
𝜆4 =
(17b)
√𝑘√𝜉
𝑑𝑟+√𝑑𝑟(−𝑘 2 𝜉 2 +𝑑(𝑟+2𝑘𝜉))
𝑘𝜉
(17c)
−𝑑𝑟+√𝑑𝑟(−𝑘 2 𝜉 2 +𝑑(𝑟+2𝑘𝜉))
𝑘𝜉
(17d)
Because the equilibrium (16) exists only when 𝑘𝜉 > 2𝑑, the first two of these eigenvalues are always
purely imaginary and thus predict cyclical dynamics. This pair of eigenvalues corresponds to the
coevolutionary dynamics of allele frequencies (see Mathematica notebook). In contrast, the second two
eigenvalues always have a negative real component such that the equilibrium is locally stable and may
also have an imaginary component indicating a cyclical approach to this equilibrium. This pair of
eigenvalues corresponds to the demographic dynamics of the interacting species (see Mathematica
notebook). Interestingly, the latter pair of eigenvalues are identical to those characterizing the local
stability of the purely ecological model if we replace (𝛼) with the value of (𝛼̅) near this equilibrium.
Together, these results suggest that — at least in the neighborhood of this equilibrium — coevolution
and demography are essentially independent. This general prediction is borne out by numerical solution
(Figure 4). Of course, it is important to realize that these conclusions apply only near the equilibrium
(16). If population sizes or allele frequencies begin far away from this equilibrium there may be
significantly more scope for interesting interactions between coevolution and demography to arise.
Generalization 3: Quantitative traits
So far we have studied the interaction between coevolution and demography only for scenarios
where coevolution depends on a single genetic locus in each species. Of course we know that this is not
the case for many other interacting species such as the interaction between the cuckoo Cuculus canorus
and its host bird Acrocephalus scirpaceus (Aviles et al. 2012) we studied in Chapter 3. How could we
study an interaction such as this in a framework that merges explicit ecology with the coevolution of
quantitative traits? One straightforward way to do this would be to simply marry the quantitative
genetic approach developed in Chapter 3 with the classical Lotka-Volterra predator-prey model
introduced at the beginning of this chapter. Such an approach has been used profitably to address a
number of questions in evolutionary and coevolutionary biology (Frank 1991; Abrams 2001; Nuismer
and Kirkpatrick 2003). Alternatively, we could use the framework of “Adaptive Dynamics” which makes a
somewhat different set of assumptions. Because adaptive dynamics is currently quite popular and
heavily used in the development of coevolutionary theory (Dieckmann and Law 1996; Best et al. 2009;
Jones et al. 2009; Carval and Ferriere 2010), we will take this latter approach here.
The first step in our adaptive dynamics approach is to reformulate the basic Lotka-Volterra
model with key parameters replaced by functions of quantitative traits in the interacting species. To
keep things as parallel as possible with our studies of coevolution between these species in Chapter 3,
we will assume that the probability that a warbler with egg coloration x accepts and rears a cuckoo egg
(𝛼) depends on the coloration of the cuckoo egg y. Rearing a cuckoo egg has a negative impact on the
warbler measured by the parameter 𝛽 and a positive impact on the cuckoo measured by the parameter
13
𝜉. If we are now willing to assume that the population of warblers and the population of cuckoos are
monomorphic (all birds of each species have the same egg color) we can write down expressions for the
change in the abundance of warblers and cuckoos:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝛼(𝑥, 𝑦)𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌
𝛽𝛼(𝑥, 𝑦)𝑁𝑋 𝑁𝑌
(18a)
(18b)
where the population dynamics now depend on the phenotype of each species. Because we know from
our discussion of the cuckoo-warbler interaction in Chapter 3 that warblers are less likely to recognize
cuckoo eggs with coloration similar to their own, we can further specify the population dynamics by
replacing the function 𝛼(𝑥, 𝑦) with a function appropriate for a model of phenotype matching:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝐸𝑥𝑝[−𝜔(𝑥 − 𝑦)2 ]𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌
𝛽𝐸𝑥𝑝[−𝜔(𝑥 − 𝑦)2 ]𝑁𝑋 𝑁𝑌
(19a)
(19b)
where ω measures the ability of a warbler to discriminate between eggs of differing coloration. We now
have a model that predicts how the population densities of two interacting species change if each is
entirely monomorphic.
The next step in using adaptive dynamics to predict coevolution is to define the per capita
fitness of individuals within each species. This is easily accomplished by dividing (19) by the population
size of the relevant species such that:
1 𝑑𝑁𝑋
𝑋 𝑑𝑡
= 𝑟 (1 −
1 𝑑𝑁𝑌
𝑌 𝑑𝑡
= 𝜉𝐸𝑥𝑝[−𝜔(𝑥 − 𝑦)2 ]𝑁𝑋 − 𝑑
𝑊𝑋 = 𝑁
𝑊𝑌 = 𝑁
𝑁𝑋
)
𝑘
− 𝛽𝐸𝑥𝑝[−𝜔(𝑥 − 𝑦)2 ]𝑁𝑌
(20a)
(20b)
The problem we now face, however, is that equations (20) accurately describe fitness only if both
species are monomorphic. How, then, can we study coevolution when our fitness equations are
predicated upon our populations being monomorphic? Adaptive dynamics overcomes this problem by
assuming evolution operates on genetic variation introduced by new mutations of very small effect that
occur very rarely. More specifically, new mutations must occur sufficiently rarely that only the resident
phenotype and a single mutant segregate at any time and must be of sufficiently small phenotypic effect
that the fitness functions (20) can be accurately approximated by their slope. With these assumptions,
we can write down expressions for the “invasion fitness” of mutant individuals:
𝜕𝑊𝑋
𝜕𝑥
= 2𝜔𝛽ⅇ −(𝑥−𝑦)
𝜕𝑊𝑌
𝜕𝑦
= 2𝜔𝜉ⅇ −(𝑥−𝑦)
2𝜔
2𝜔
(𝑥 − 𝑦)𝑁𝑌
(21a)
(𝑥 − 𝑦)𝑁𝑋
(21b)
14
We now know the growth rate of rare mutants that do not deviate much from the phenotype of
resident individuals. What we do not know, however, is the total population size of the two species
(𝑁𝑋 , 𝑁𝑌 ) upon which fitness depends. Although we could actually follow population dynamics and
evolutionary change simultaneously as we did in the previous sections, adaptive dynamics makes an
additional assumption to simplify mathematical analysis. Specifically, adaptive dynamics assumes
evolutionary change is slow relative to ecological dynamics, allowing for a separation of time scales
approximation.
In order to implement the separation of time scales approximation, we first need to solve for
the equilibrium population sizes of the interacting species using equations (19). To do this, set the left
hand sides of these equations equal to zero, and solve for the variables 𝑁𝑋 and 𝑁𝑌 . The result is
identification of three possible equilibrium states:
̂𝑋 = 0 𝑎𝑛𝑑 𝑁
̂𝑌 = 0
𝑁
(22a)
̂𝑋 = 𝑘 𝑎𝑛𝑑 𝑁
̂𝑌 = 0
𝑁
(22b)
̂𝑋 = 𝑑ⅇ
𝑁
(𝑥−𝑦)2 𝜔
𝜉
̂𝑌 =
𝑎𝑛𝑑 𝑁
2
2
ⅇ (𝑥−𝑦) 𝜔 𝑟(−𝑑ⅇ (𝑥−𝑦) 𝜔 +𝑘𝜉)
𝑘𝛽𝜉
(22c)
where the “hats” over the population sizes indicate equilibria. The first of these equilibria is rather dull
and corresponds to a case where both warbler and cuckoo have gone extinct. The second equilibrium is
also not very interesting, corresponding to a case where the cuckoo has been driven to extinction. Only
the third equilibrium offers the opportunity to study coevolution between warbler and cuckoo when
both species coexist within a single community. Even then, it is important to realize that coexistence
occurs only when equilibrium (22c) exists, such that the population densities of both species are greater
2
than zero. This requires that 𝑑ⅇ (𝑥−𝑦) 𝜔 < k suggesting that the further apart the two species are
phenotypically, the less likely it is that the cuckoo will be likely to persist.
We can now use our equilibrium solution to study coevolutionary dynamics by substituting the
equilibrium expressions for population sizes (22c) into our expressions for invasion fitness (21). The
result is a pair of equations that predict the way in which the two species should coevolve:
2
2𝑟(𝑥−𝑦)(−𝑑ⅇ (𝑥−𝑦) 𝜔 +𝑘𝜉)𝜔
𝑘𝜉
̂𝑋 ,𝑁
̂𝑌 )
𝜕𝑊𝑋 (𝑁
𝜕𝑥
=
̂𝑋 ,𝑁
̂𝑌 )
𝜕𝑊𝑌 (𝑁
𝜕𝑦
= 2𝑑(𝑥 − 𝑦)𝜔
(23a)
(23b)
Usually, within the framework of adaptive dynamics, the next step is to identify evolutionary equilibria.
For equations (23) this is straightforward and reveals a single equilibrium where both species have
identical egg coloration:
𝑦=𝑥
(24)
15
Perhaps not surprisingly, this coevolutionary equilibrium is identical to that we identified in Chapter 3
for the interaction between these species, even though we there utilized the framework of quantitative
genetics and ignored demography. At this evolutionary equilibrium the population sizes of warbler and
cuckoo are:
̂𝑋 = 𝑑 𝑎𝑛𝑑 𝑁
̂𝑌 = 𝑟(−𝑑+𝑘𝜉)
𝑁
𝜉
𝑘𝛽𝜉
(25)
such that the equilibrium exists and is biologically relevant as long as 𝑑 < k 

Much as we would do with any standard mathematical analysis, we can now evaluate the local
stability (or convergence stability in the jargon of adaptive dynamics) of this equilibrium. To do this, we
employ the same old routine: 1) generate a Jacobian matrix, 2) solve for the eigenvalues of this matrix,
and 3) evaluate these eigenvalues at the equilibrium of interest. The result, in this case, is the following
pair of eigenvalues:
𝜆1 = 0
𝜆2 = 2𝜔(𝑟 −
(26a)
𝑑(𝑟+𝑘𝜉)
)
𝑘𝜉
(26b)
demonstrating that the matching equilibrium (24) is locally stable anytime:
𝑑>𝑟
(𝑘𝜉−𝑑)
𝑘𝜉
(27)
Although it is not immediately obvious, this result effectively reveals that the matching equilibrium is
locally/convergent stable only when the maximal rate of evolution in the cuckoo exceeds that in the
warbler, just as we found in Chapter 3 (Figure 5). If, instead, the warbler has the greater maximum rate
of evolution, the matching equilibrium is unstable and egg coloration of cuckoo and warbler will diverge.
Unlike our predication from classical quantitative genetic approaches used in Chapter 3, however,
numerical solutions indicate that trait divergence does not proceed in perpetuity. Instead, it appears
that once the warbler has diverged sufficiently in egg coloration, the two species continue to evolve, but
the difference in egg coloration remains approximately constant (Figure 5). Based on our expressions for
equilibrium population size (25), we can see this latter scenario might drive the cuckoo to extinction as
we suggested in Chapter 3, but that this might not be a general expectation. Instead, numerical
solutions suggest that even when the warbler is “winning” the coevolutionary race, the cuckoo
population may persist even though its eggs poorly match the eggs of its host (Figure 5). The reason for
this appears to be an interesting feedback between demography and coevolution where warbler escape
results in a larger number of warblers and thus a greater number of potential hosts for the cuckoo. Thus,
we end up in a situation where the cuckoos don’t match their hosts too terribly well, but they have
enough potential hosts to compensate for their coevolutionary inadequacy and maintain a viable
population!
Finally, we can carry our analysis one step further than we have previously and investigate the
form of selection experienced by warbler and cuckoo near the matching equilibrium (24). Specifically,
16
we will evaluate whether or not this equilibrium represents a “branching point” for either species,
indicating that selection is disruptive in form at the same time the equilibrium is locally stable.
Evaluating whether the matching equilibrium is a branching point requires calculating the second
derivative of the per capita fitness functions given by Equation (20), and evaluating this second
derivative at the ecological and evolutionary equilibrium of interest:
̂𝑋 ,𝑁
̂𝑌 )
𝜕2 𝑊𝑋 (𝑁
𝜕𝑥 2
=−
̂𝑋 ,𝑁
̂𝑌 )
𝜕2 𝑊𝑌 (𝑁
𝜕𝑦 2
=
2𝑑𝑟(𝑑−𝑘𝜉)𝜔
𝑘𝜉 2
(28a)
2𝑑𝑟(𝑑−𝑘𝜉)𝜔
𝑘𝛽𝜉
(28b)
Because we know the ecological equilibrium upon which our analysis rests exists only when d < k we
can immediately see that the sign of (28a) is always positive and the sign of (28b) always negative. Thus,
the warbler experiences selection that is disruptive in form whereas the cuckoo experiences selection
that is stabilizing in form. These results are identical to those we found in Chapter 3 using classical
quantitative genetic approaches. Together, these results show that anytime the matching equilibrium is
locally stable, it is a branching point for the warbler population.
In light of this result indicating the matching equilibrium is a “branching point” for the warbler,
should we expect speciation to occur in this species? Although it would be exciting to imagine that
coevolution between these species might readily lead to speciation, the reality is that this would be a
very unlikely outcome. The primary reason speciation is unlikely in a sexual species like the warbler A.
scirpaceus, is that recombination will continually break down linkage disequilibrium that disruptive
selection builds up (Kirkpatrick and Ravigne 2002). Without an effective mechanism of assortative
mating based on egg coloration, recombination will prevent linkage disequilibrium from building up to
levels sufficient for the emergence of a bimodal distribution of egg coloration (Dieckmann and Doebeli
1999; Kondrashov and Kondrashov 1999; Gavrilets 2004). As a consequence, the impact of disruptive
selection at this “branching point” will likely be limited to an increase in the phenotypic variation within
the warbler population, a prediction that has been made previously for quantitative genetic models of
this type of coevolutionary interaction (Nuismer et al. 2005; Kopp and Gavrilets 2006).
Conclusions and Synthesis
The models we have developed and analyzed in this chapter reveal that, in some cases,
coevolution and demography interact in an interesting way. In other biologically plausible scenarios,
however, coevolutionary and demographic dynamics can be effectively decoupled. The latter set of
results lends support to coevolutionary modeling approaches that ignore explicit demography and
instead rely on well-established population and quantitative genetic approaches (Mode 1958; Kiester et
al. 1984; Seger 1988). At the same time, however, our results do show that sometimes models that
ignore demography will be insufficient and potentially produce inaccurate or incomplete predictions
about the coevolutionary process. Is there a way to predict those cases where including explicit ecology
and demography will influence coevolutionary dynamics and outcomes?
17
Based on the simple set of models we have studied in this chapter, there does seem to be a
general principle at work. Specifically, of the models we considered, only the gene-for-gene model with
costs of resistance and the phenotype matching model lead to coevolutionary dynamics that differ
qualitatively and substantially from those we predicted using the non-demographic approaches in
earlier chapters. The reason these models are strongly influenced by demography is that the underlying
coevolutionary model has the potential to produce qualitatively different types of coevolutionary
dynamics depending on parameter values. For instance, if costs of resistance and virulence are weak, we
expect fixation of the resistant and virulent alleles. If, on the other hand, costs of resistance and
virulence are asymmetric and moderate in strength, the result is often coevolutionary cycles. Similarly,
for the phenotype matching model, when selection is stronger on the parasite, matched phenotypes
represents a stable state. In contrast, when selection on the host is stronger, matching phenotypes
becomes unstable. When we include ecology and explicit demography in these models, the effective
strengths of selection fluctuate with the population sizes of the interacting species, allowing the model
to move between different coevolutionary regimes. In contrast, the gene-for-gene model without costs
and the matching alleles model predict only one type of coevolutionary dynamic such that including
demography and ecology generally does little more than adjust the rate of coevolution.
In summary, the models we have studied in this chapter suggest that interesting feedbacks
between ecology and coevolution can, but do not always, occur. Whether such feedbacks occur depends
on the mechanistic basis of coevolution and may also depend on the approaches we use to analyze the
models. Developing a robust understanding of when coevolution and demography interact, and by how
much, provide interesting avenues for future modeling and empirical investigation.
18
References
Abrams, P. A. 2001. Modelling the adaptive dynamics of traits involved in inter- and intraspecific
interactions: An assessment of three methods. Ecology Letters 4:166-175.
Aviles, J. M., J. R. Vikan, F. Fossoy, A. Antonov, A. Moksnes, E. Roskaft, J. A. Shykoff, A. P. Moller, and B.
G. Stokke. 2012. Egg phenotype matching by cuckoos in relation to discrimination by hosts and
climatic conditions. Proceedings of the Royal Society B-Biological Sciences 279:1967-1976.
Best, A., A. White, and M. Boots. 2009. The Implications of Coevolutionary Dynamics to Host-Parasite
Interactions. American Naturalist 173:779-791.
Burdon, J. J., and P. H. Thrall. 1999. Spatial and temporal patterns in coevolving plant and pathogen
associations. American Naturalist 153:S15-S33.
Burdon, J. J., and P. H. Thrall. 2000. Coevolution at multiple spatial scales: Linum marginale- Melampsora
lini - from the individual to the species. Evolutionary Ecology 14:261-281.
Carval, D., and R. Ferriere. 2010. A UNIFIED MODEL FOR THE COEVOLUTION OF RESISTANCE,
TOLERANCE, AND VIRULENCE. EVOLUTION 64:2988-3009.
Dieckmann, U., and M. Doebeli. 1999. On the origin of species by sympatric speciation. Nature 400:354357.
Dieckmann, U., and R. Law. 1996. The dynamical theory of coevolution: A derivation from stochastic
ecological processes. Journal of Mathematical Biology 34:579-612.
Frank, S. A. 1991. ECOLOGICAL AND GENETIC MODELS OF HOST PATHOGEN COEVOLUTION. Heredity
67:73-83.
Gavrilets, S. 2004. Fitness landscapes and the origin of species. Princeton University Press, Princeton.
Jones, E. I., R. Ferriere, and J. L. Bronstein. 2009. Eco-Evolutionary Dynamics of Mutualists and
Exploiters. AMERICAN NATURALIST 174:780-794.
Kiester, A. R., R. Lande, and D. W. Schemske. 1984. MODELS OF COEVOLUTION AND SPECIATION IN
PLANTS AND THEIR POLLINATORS. AMERICAN NATURALIST 124:220-243.
Kirkpatrick, M., and V. Ravigne. 2002. Speciation by natural and sexual selection: Models and
experiments. American Naturalist 159:S22-S35.
Kondrashov, A. S., and F. A. Kondrashov. 1999. Interactions among quantitative traits in the course of
sympatric speciation. Nature 400:351-354.
Kopp, M., and S. Gavrilets. 2006. Multilocus genetics and the coevolution of quantitative traits.
Evolution 60:1321-1336.
Luijckx, P., H. Fienberg, D. Duneau, and D. Ebert. 2013. A Matching-Allele Model Explains Host
Resistance to Parasites. Current Biology 23:1085-1088.
Mode, C. J. 1958. A MATHEMATICAL-MODEL FOR THE CO-EVOLUTION OF OBLIGATE PARASITES AND
THEIR HOSTS. EVOLUTION 12:158-165.
Nuismer, S. L., M. Doebeli, and D. Browning. 2005. The coevolutionary dynamics of antagonistic
interactions mediated by quantitative traits with evolving variances. Evolution 59:2073-2082.
Nuismer, S. L., and M. Kirkpatrick. 2003. Gene flow and the coevolution of parasite range. Evolution
57:746-754.
Seger, J. 1988. Dynamics of some simple host-parasite models with more than two genotypes in each
species. Phil. Trans. R. Soc. Lond. B. 319:541-555.
Thrall, P. H., A.-L. Laine, M. Ravensdale, A. Nemri, P. N. Dodds, L. G. Barrett, and J. J. Burdon. 2012. Rapid
genetic change underpins antagonistic coevolution in a natural host-pathogen metapopulation.
Ecology Letters 15:425-435.
Tian, D., M. B. Traw, J. Q. Chen, M. Kreitman, and J. Bergelson. 2003. Fitness costs of R-gene-mediated
resistance in Arabidopsis thaliana. Nature 423:74-77.
19
Yoshida, T., L. E. Jones, S. P. Ellner, G. F. Fussmann, and N. G. Hairston, Jr. 2003. Rapid evolution drives
ecological dynamics in a predator-prey system. Nature 424:303-306.
Figure Legends
Figure 1. Population dynamics predicted by the Lotka-Volterra model for three different rust death
rates, d. In all cases, the dashed line indicates rust population size and the solid line indicates flax
population size. Parameters held constant across panels were 𝛼 = .1, 𝛽 = .03, 𝜉 = .01, 𝑟 = .1,
and 𝑘 = 100. Rust death rates were: A) 𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 2. Population dynamics and allele frequencies for the gene-for-gene model over three different
rust death rates, d. In all cases, dashed lines indicate rust population size or frequency of the virulent
allele and solid lines indicate flax population size or frequency of the resistant allele. Parameters held
constant across panels were 𝛽 = .003, 𝜉 = .001, 𝑟 = .1, and 𝑘 = 100. Rust death rates were: A) 𝑑 =
.005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 3.
Figure 4.
Figure 5.
20
Table 1. Summary of stability conditions and ecological dynamics
Condition
Implications for stability
Equilibrium (3c) unstable;
𝑑 > 𝑘𝛼𝜉
Equilibrium (3b) stable
Equilibrium (3c) stable;
𝑑 < 𝑘𝛼𝜉
Equilibrium (3b) unstable
Equilibrium
(3c) stable and
4𝑘𝛼𝜉
oscillatory;
𝑑 < 𝑘𝛼𝜉 (
)
𝑟 + 4𝑘𝛼𝜉
Equilibrium (3b) unstable
21
Biological consequences
Extinction of Rust. Flax at
carrying capacity
Coexistence of Flax and Rust
Coexistence of Flax and Rust.
Transient cycles likely.
Table 2. Equilibria corresponding to parasite extinction. Because we assume carrying the virulent allele is
costly, 𝑑𝐵 > 𝑑𝑏 .
Equilibrium
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 0, 𝑝𝑉 = 0
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 0, 𝑝𝑉 = 1
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1, 𝑝𝑉 = 0
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1, 𝑝𝑉 = 1
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1 − (𝑑𝑏 ⁄𝑘𝜉 ), 𝑝𝑉 = 0
Conditions for instability
𝑑𝑏 < 𝑘𝜉
Always unstable
𝑑𝐵 − 𝑑𝑏 < 𝑘𝜉
𝑘𝜉 + 𝑑𝑏 < 𝑑𝐵 𝑜𝑟 𝑑𝐵 < 𝑘𝜉
𝑑𝐵 < 𝑘𝜉
22