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 (REFS). 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 (REFS). Moreover, recent
work demonstrates that population fluctuations in this system are tied to changes in gene frequencies
(REFS), 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
of the interaction between Flax and Flax Rust is adequately described by the (slightly modified) LotkaVolterra model of predator-prey interactions:
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
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:
𝑁𝑋 = 0 𝑎𝑛𝑑 𝑁𝑌 = 0
(3a)
2
𝑁𝑋 = 𝑘 𝑎𝑛𝑑 𝑁𝑌 = 0
𝑑
𝑁𝑋 = 𝛼𝜉 𝑎𝑛𝑑 𝑁𝑌 =
(3b)
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
(6c)
respectively.
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.
3
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
𝑑𝑁𝑌,𝑉
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)
= 𝛼𝑅,𝑉 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑉 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝑑𝑁𝑌,𝑉
(7c)
4
𝑑𝑁𝑌,𝑣
𝑑𝑡
= 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 =
𝑁𝑋,𝑅
(8c)
𝑁𝑋,𝑅 +𝑁𝑋,𝑟
𝑝𝑌 = 𝑁
𝑁𝑌,𝑉
(8d)
𝑌,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
(XXXX) are virtually identical to those describing coevolution here (11c,d). How cool is that? The only
thing demography and population size does in this case is 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
x). 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 1).
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 (REFS).
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 (REFS). 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.
One particularly interesting question we could now ask is whether or not coevolution can
potentially prevent the parasite from ever being able to invade the host population. In other words, how
great would the cost of carrying a virulent allele need to be for the Flax population to win the
coevolutionary race and drive the pathogen to extinction? To answer this question we need to evaluate
the local stability of the equilibrium where the Flax population is at its carrying capacity (k) and fixed for
the resistant allele (R) while the Rust population is extinct and fixed for the avirulent allele (v). This
equilibrium represents a case where the Flax population has won the coevolutionary race (the average
rate of infection is zero), and as a result, the Rust population has been driven to extinction. If this
equilibrium is locally stable, it means that even if we introduce a handful of Rust individuals fixed for the
virulent allele (V), they will be unable to increase in numbers even though they can infect all of the
available hosts. If, on the other hand this equilibrium is unstable, it means that introducing this same
handful of virulent Rust individuals causes the community to move away from this equilibrium. We will
see later, using numerical solutions to our equations, that when this equilibrium is unstable, the Rust is
able to hold its own in the coevolutionary race and maintain a viable population.
To analyze the local stability of this equilibrium, 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. Since we have now been through this exercise a few times, I will just skip ahead
and present the eigenvalues that emerge from this analysis:
𝜆1 = 0
(14a)
𝜆2 = −𝑑𝑣
(14b)
𝜆3 = 𝑘𝜉 − 𝑑𝑉 + 𝑑𝑣
(14c)
𝜆4 = −𝑟𝑅
(14d)
One thing you might notice is that we now have four eigenvalues whereas in past analyses we have had
only two. The reason for this is that the number of eigenvalues always matches the dimensions of the
Jacobian Matrix, which in turn always matches the number of equations in your dynamical system. Since
we now have four equations, we now have a 4 × 4 Jacobian matrix and four eigenvalues.
Otherwise, everything is as before and we need only identify the conditions under which all four
eigenvalues are negative in order to establish stability. Clearly, the critical eigenvalue is the third (14c),
since all the other eigenvalues are always zero or negative. For this critical eigenvalue to be negative,
the equilibrium locally stable, and the Rust doomed to lose the coevolutionary race and confront
extinction, the following condition must hold:
𝑘𝜉 < 𝜏𝑉
(15)
10
where 𝜏𝑉 = 𝑑𝑉 − 𝑑𝑣 measures the cost of carrying the virulent allele. Why does this condition ensure
the demise of the Rust? Simply because when this condition holds, the cost of carrying the virulent allele
is too great for it to spread through the population??? CHECK ON THIS…
Generalization 2: Matching alleles interaction
A return to Daphnia, pasteuria, and Matching alleles
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
(14b)
𝑑𝑝𝑋
𝑑𝑡
= 𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 (1 − 2𝑝𝑌 )
(14c)
𝑑𝑝𝑌
𝑑𝑡
= −𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 (1 − 2𝑝𝑋 )
(14d)
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(14a)
where 𝛼̅ = 𝑝𝑋 𝑝𝑌 + 𝑞𝑋 𝑞𝑌
HERE WE SEE THAT COEVOLUTION INFLUENCES ECOLOGY, BUT NOT VICE VERSA
Generalization 3: Quantitative traits
A return to cuckoos and their hosts and an introduction to adaptive dynamics
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝛼(𝑥, 𝑦)𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌
𝛽𝛼(𝑥, 𝑦)𝑁𝑋 𝑁𝑌
(15a)
(15b)
Specify alpha:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝐸𝑥𝑝[−𝛼(𝑥 − 𝑦)2 ]𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌
𝛽𝐸𝑥𝑝[−𝛼(𝑥 − 𝑦)2 ]𝑁𝑋 𝑁𝑌
(16a)
(16b)
Find ecological equilibria:
11
̂𝑋 = 0 𝑎𝑛𝑑 𝑁
̂𝑌 = 0
𝑁
(17a)
̂𝑋 = 𝑘 𝑎𝑛𝑑 𝑁
̂𝑌 = 0
𝑁
(17b)
̂𝑋 = 𝑑ⅇ
𝑁
(𝑥−𝑦)2 𝜔
𝜉
̂𝑌 = ⅇ
𝑎𝑛𝑑 𝑁
(𝑥−𝑦)2 𝜔 𝑟(−𝑑ⅇ (𝑥−𝑦)2 𝜔 +𝑘𝜉)
(17c)
𝑘𝛽𝜉
The interesting equilibrium, (17c), where both species are present, can exist only if d < k and even
then, only when x=y.
Define per capita fitness:
𝑊𝑋 =
1 𝑑𝑁𝑋
𝑁𝑋 𝑑𝑡
1 𝑑𝑁𝑌
𝑌 𝑑𝑡
𝑊𝑌 = 𝑁
= 𝑟 (1 −
𝑁𝑋
)
𝑘
− 𝛽𝐸𝑥𝑝[−𝛼(𝑥 − 𝑦)2 ]𝑁𝑌
= 𝜉𝐸𝑥𝑝[−𝛼(𝑥 − 𝑦)2 ]𝑁𝑋 − 𝑑
(18a)
(18b)
Use these to calculate a “selection gradient” or is this invasion fitness? I don’t like calling it a selection
gradient because it isn’t
𝜕𝑊𝑋
𝜕𝑥
= 2𝛼𝛽ⅇ −(𝑥−𝑦) 𝛼 (𝑥 − 𝑦)𝑁𝑌
𝜕𝑊𝑌
𝜕𝑦
= 2𝛼𝜉ⅇ −(𝑥−𝑦) 𝛼 (𝑥 − 𝑦)𝑁𝑋
2
(19a)
2
(19b)
And evaluate these at the ecological equilibrium of interest… Let’s focus on 17c where both species
coexist…
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
(20a)
(20b)
OK. So what can we learn from these invasion thingers? Maybe we could identify coevolutionary
equilibra?
𝑦=𝑥
(21)
This equilibrium should look familiar, being identical to the equilibrium we identified for cuckoo and
warbler in Chapter 3.
WHAT ARE THE POPULATION DENSTIES AT THIS EQUILIBRIUM???
𝑑
{NX → 𝜉 , NY →
𝑟(−𝑑+𝑘𝜉)
}
𝑘𝛽𝜉
(22)
12
Are they stable?
{−
𝑑𝑟 + √𝑑√𝑟√𝑑𝑟 + 4𝑑𝑘𝜉 − 4𝑘 2 𝜉 2 −𝑑𝑟 + √𝑑√𝑟√−4𝑘 2 𝜉 2 + 𝑑(𝑟 + 4𝑘𝜉)
,
}
2𝑘𝜉
2𝑘𝜉
It is VERY important that you make certain your equilibrium exists (and is stable) and continues to exist
and be stable as you move through your adaptive dynamics analysis. All too often this part is ignored
yielding dubious conclusions…
Let’s focus on the first and ask whether it is CONVERGENT, or locally stable to normal people. To do this,
we must perform a standard local stability analysis. Doing so reveals that the eigenvalues are:
{0,2(𝑟 −
𝑑(𝑟+𝑘𝜉)
)𝜔}
𝑘𝜉
(23)
Stable if
𝜔𝑑 > 𝜔𝑟
(𝑘𝜉−𝑑)
𝑘𝜉
(24)
This shows that the matching equilibrium is stable anytime:
>
where  =  d and  = (k -d)/(k )  r and the terms  and  measure the strength of selection
acting on the parasite and host, respectively.
Because the Ecological equilibrium upon which this solution is based exists iff d < k the right hand
side is bounded on {0,r} guaranteeing stability anytime d>r??? The reason increasing death rates
stabilize the equilibrium is because they correspond to this evolutionary equilibrium is stable iff 𝛽𝜇𝑋 >
𝜉𝜇𝑌 . Here too, you might notice some similarity to our results from Chapter 3 where we studied
coevolution between cuckoo and warbler using a non-demographic quantitative genetic approach.
Specifically, this result suggests that anytime the response to selection is greater for the cuckoo than the
warbler, this matching egg coloration in the two species will be evolutionary stable. Under such
conditions, the cuckoo has effectively won the evolutionary race. If in contrast, the warbler has the
greater response to selection, this matching equilibrium is unstable.
Finally, because it is rather an obsession of practicioners of adaptive dynamics, we will evaluate whether
or not the matching equilibrium represents a “branching point”. Although almost invariably portrayed as
an indication of incipient speciation, branching points are actually nothing more than locally stable
evolutionary equilibria characterized by a pattern of disruptive selection. Not surprisingly then, the way
in which they can be identified is by taking the second derivative of Equation (X) with respect to the trait
and evaluating at the eco-evo equilibrium.
̂𝑋 ,𝑁
̂𝑌 )
𝜕2 𝑊𝑋 (𝑁
𝜕𝑥 2
=−
̂𝑋 ,𝑁
̂𝑌 )
𝜕2 𝑊𝑌 (𝑁
2
𝜕𝑦
=
2𝑑𝑟(𝑑−𝑘𝜉)𝜔
𝑘𝜉 2
(25a)
2𝑑𝑟(𝑑−𝑘𝜉)𝜔
𝑘𝛽𝜉
(25b)
13
Since we know the ecological equilibrium where both species coexist exists only when d < k we can
immediately infer that the sign of (Xa) is positive and the sign of (Xb) negative implying that the host
species experiences disruptive selection whereas the parasite species experiences stabilizing selection.
Here too, these results should seem familiar. The reason for this familiarity is that our previous analysis
of cuckoo-warbler coevolution also demonstrated disruptive selection acting on warblers but stabilizing
selection acting on cuckoos. In light of this result indicating the matching equilibrium is a “branching
point” for the warbler, do we expect speciation to occur? The answer here is almost surely, NO.
Although for a clonal species divergence into two lineages would be expected, for a sexual species like
our warbler many additional hurdles must be overcome for speciation to occur. The primary reason for
this is recombination. Unless assortative mating by phenotype is present within the population and
already quite strong, mating between egg color morphs in each generation will allow recombination to
erode linkage disequilbirum and continually homogenize egg-coloration. A more realistic expectation at
this equilibrium is that, all else being equal, the warbler population should have somewhat greater
phenotypic variation than should the cuckoo population (REFS).
Conclusions and Synthesis
14
References
Figure Legends
15
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
16
Biological consequences
Extinction of Rust. Flax at
carrying capacity
Coexistence of Flax and Rust
Coexistence of Flax and Rust.
Transient cycles likely.