Chapter 4. Coevolution and Population Dynamics
Biological Motivation
The previous chapters considered the process of coevolution without touching on changes in
population sizes. Although there are many cases where this may be a good approximation of reality,
there are clearly other systems where it is unlikely to be true. One obvious example occurs in
interactions between infectious diseases and their hosts. In these types of interactions, pathogen
populations often go through boom and bust cycles or at least show significant fluctuations in
population size over time (REFS). RETURN TO FLAX AND GFG!!! FIRST DO POP GEN THEN DO AD. USE
BURDON/THRALL AS MOTIVATION FOR CYCLES. These population fluctuations could have important
consequences for coevolution if they cause the intensity or form of reciprocal selection to change over
time.
COULD ALSO PROVIDE A GENERAL INTRODUCTION POINTING TO EXAMPLES OF PUTATIVE
COEVOLVING INTERACTIONS IN WHICH POP SIZES CHANGE. THEN INTRODUCE LV MODEL AND ASK:
“WHAT HAPPENS WHEN THE PAREMETERS EVOLVE?COEVOLVE?)
WOULD PLANT/PATHOGEN INTERACTION BE SI?
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:



Does coevolution influence population dynamics?
When should coevolution cause population cycles?
Can coevolution drive a species to extinction?
Building a Model of X-X Coevolution
Our general approach 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, creating the possibility of coevolution. We will then use this ecocoevolutionary model to answer our key questions.
Preliminaries — developing the ecological context for coevolution
A logical place to begin is by formulating a model that describes the ecology of the interaction between
these species. How can we go about defining such a model? The simplest possible approach is to grab a
well-studied model from ecology and bend it to our purposes. An obvious candidate is the classical
Lotka-Volterra model of predator-prey interactions:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝛼𝛽𝑁𝑋 𝑁𝑌
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
(1a)
𝑑𝑁𝑌
𝑑𝑡
= 𝛼𝜉𝑁𝑋 𝑁𝑌 − 𝑑𝑁𝑌
(1b)
DIFFERENTIAL EQUATIONS where 𝑁𝑋 is the population abundance of flax, 𝑁𝑌 is the population
abundance of rust, k is the carrying capacity of the flax population r is the growth rate of the flax
population, d is the death rate of rust, 𝛽 is the probability that an encounter between rust and flax
leads to death of flax, and 𝜉 is the rate at which encounters between flax and rust cause infection and
production of new rust individuals (REFS). No doubt, you are now wondering why we would use a
classical model of a predator prey interaction for the host-pathogen interaction between flax and flaxrust.
What can we learn about the ecological dynamics of this interaction from this simple model?
Hopefully, we will be able to answer some simple questions such as whether or not cycles are possible, if
there are stable equilibria, and when both of the species are likely to coexist. The simplest way to tackle
these questions would be to solve the system of differential equations (1) directly. This would give us a
complete solution telling us the abundance of flax and rust at any possible point in time as a function of
the model parameters. Unfortunately, however, such a general solution is really not practical. Instead,
we will have to satisfy our curiosity by identifying equilibria and assessing their stability.
Even though we are now working with differential equations rather than the discrete 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
𝛽𝑁𝑋 𝑁𝑌
(2a)
0 = 𝜉𝑁𝑋 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)
𝑁𝑋 = 𝑘 𝑎𝑛𝑑 𝑁𝑌 = 0
(3b)
𝑑
𝑁𝑋 = 𝛼𝜉 𝑎𝑛𝑑 𝑁𝑌 =
𝑟(𝛼𝑘𝜉−𝑑)
𝛼 2 𝑘𝜉𝛽
(3c)
where the first represents extinction of both species, the second the extinction of the rust only, and the
third coexistence of both species. But when, if ever, DOES THE COEXISTENCE EQ EXIST?
Only if:
will each of these equilibria occur? To answer this question, we must analyze the local stability of these
equilibria. 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:
2
𝑟−
𝐽=[
2𝑟𝑁𝑋
𝑘
− 𝛼𝛽𝑁𝑌
𝛼𝜉𝑁𝑌
−𝛼𝛽𝑁𝑋
𝛼𝜉𝑁𝑋 − 𝑑
].
(4)
For the three ecological equilibria (3), the eigenvalues of this Jacobian are:
𝜆1 = 𝑟 𝑎𝑛𝑑 𝜆2 = −𝑑
(5a)
𝜆1 = −𝑟 𝑎𝑛𝑑 𝜆2 = 𝛼𝜉𝑘 − 𝑑
(5b)
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
(5c)
respectively. Thankfully, we are done with the math for a bit and can now sit back and think about what
these results tell us about the ecological dynamics of our interaction. 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 only Flax is presence
can be locally stable, but only if the rate at which the Flax rust dies, d, exceeds the rate at which it can
be produced by infections of a Flax population at its carrying capacity k. So far so good, right?
Unfortunately, now we need to spend a bit of time understanding when the third equilibrium is stable,
allowing us to identify those conditions that allow Flax and Flax rust to potentially coexist.
With those simple results in hand, we can move on to trying to figure out when the third
equilibrium is likely to evolve, and what ecological dynamics should 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 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. If the quantity under the more
complicated root is negative, the real part of both eigenvalues must be negative because both r and d
are positive (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 then depends on magnitude of the positive root relative to the
product r×d. Together, these considerations and some simple algebra lead to biological predictions for
the behavior of the interacting populations (Table 1).
Table 1. Summary of stability conditions and ecological dynamics
Condition
Implications for stability
Equilibrium (3c) unstable;
𝑘 < 𝑑⁄𝛼𝜉
Equilibrium (3b) stable
Equilibrium (3c) stable;
𝑘 > 𝑑⁄𝛼𝜉
Equilibrium (3b) unstable
Equilibrium (3c) stable and cyclical;
𝑘 > (𝑑 + √𝑑(𝑑 + 𝑟))⁄2𝛼𝜉
Equilibrium (3b) unstable
3
Biological consequences
Extinction of Rust. Flax at
carrying capacity
Coexistence of Flax and Rust
Coexistence of Flax and Rust.
Transient cycles likely.
We are now to a point where we can summarize the ecological dynamics of the interaction
between Flax and Flax rust in the absence of any coevolution. 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.
Preliminaries — Integrating coevolution
𝑑𝑁𝑋,𝑅
𝑑𝑡
𝑑𝑁𝑋,𝑟
𝑑𝑡
𝑑𝑁𝑌,𝑉
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
(6a)
) − 𝛼𝑟,𝑉 𝛽𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝛼𝑟,𝑣 𝛽𝑁𝑋,𝑟 𝑁𝑌,𝑣
(6b)
= 𝛼𝑅,𝑉 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑉 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑉 − 𝑑𝑁𝑌,𝑉
(6c)
= 𝛼𝑅,𝑣 𝜉𝑁𝑋,𝑅 𝑁𝑌,𝑣 + 𝛼𝑟,𝑉 𝜉𝑁𝑋,𝑟 𝑁𝑌,𝑣 − 𝑑𝑁𝑌,𝑣
(6d)
Analyzing the Model
CHANGE OF VARIABLES
1) Define new variables:
𝑁𝑋 = 𝑁𝑋,𝑅 + 𝑁𝑋,𝑟
(7a)
𝑁𝑌 = 𝑁𝑌,𝑉 + 𝑁𝑌,𝑣
(7b)
𝑝𝑋 = 𝑁
𝑁𝑋,𝑅
(7c)
𝑋,𝑅 +𝑁𝑋,𝑟
𝑝𝑌 = 𝑁
𝑁𝑌,𝑉
(7d)
𝑌,𝑉 +𝑁𝑌,𝑣
2) Apply chain rule:
𝑑𝑁𝑋
𝑑𝑡
𝜕𝑁
= 𝜕𝑁 𝑋
𝑋,𝑅
𝑑𝑁𝑋,𝑅
𝑑𝑡
𝜕𝑁
+ 𝜕𝑁 𝑋
𝑋,𝑟
𝑑𝑁𝑋,𝑟
(8a)
𝑑𝑡
4
𝜕𝑁
𝑑𝑁𝑌,𝑉
= 𝜕𝑁 𝑌
𝑑𝑝𝑋
𝑑𝑡
=
𝑑𝑝𝑌
𝑑𝑡
= 𝜕𝑁 𝑌
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
(9b)
𝑑𝑝𝑋
𝑑𝑡
= −𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 (𝑝𝑌 (𝛼𝑅,𝑉 − 𝛼𝑟,𝑉 ) + 𝑞𝑌 (𝛼𝑅,𝑣 − 𝛼𝑟,𝑣 ))
(9c)
𝑑𝑝𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 (𝑝𝑋 (𝛼𝑅,𝑉 − 𝛼𝑅,𝑣 ) + 𝑞𝑋 (𝛼𝑟,𝑉 − 𝛼𝑟,𝑣 ))
(9d)
𝑌,𝑉
𝑑𝑡
𝑌,𝑉
𝑑𝑁𝑌,𝑉
𝑑𝑡
+ 𝜕𝑁 𝑌
𝑌,𝑣
𝜕𝑝𝑋 𝑑𝑁𝑋,𝑅
𝜕𝑁𝑋,𝑅 𝑑𝑡
𝜕𝑝
𝜕𝑁
𝑑𝑁𝑌,𝑣
𝑑𝑁𝑌
𝑑𝑡
+
(8b)
𝑑𝑡
𝜕𝑝𝑋 𝑑𝑁𝑋,𝑟
𝜕𝑁𝑋,𝑟 𝑑𝑡
𝜕𝑝
+ 𝜕𝑁 𝑌
𝑌,𝑣
𝑁𝑋
)−
𝑘
(8c)
𝑑𝑁𝑌,𝑣
(8d)
𝑑𝑡
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(9a)
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.
First and foremost, it appears that at least for this particular model of ecology, it is coevolution that
drives ecology and not vice versa. Specifically, the only impact of ecology on coevolution is to modulate
the rate of coevolutionary change. RELATE COEVOLUTION OF ALPHA BAR TO TABLE 1 CONDITIONS
FOR THE SPECIFIC CASE OF THE GFG MODEL:
𝑑𝑁𝑋
𝑑𝑡
= 𝑟𝑁𝑋 (1 −
𝑁𝑋
)−
𝑘
𝑑𝑁𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑁𝑌 (𝛼̅) − 𝑑𝑁𝑌
(10b)
𝑑𝑝𝑋
𝑑𝑡
= 𝛽𝑁𝑌 𝑝𝑋 𝑞𝑋 𝑞𝑌
(10c)
𝑑𝑝𝑌
𝑑𝑡
= 𝜉𝑁𝑋 𝑝𝑌 𝑞𝑌 𝑞𝑌
(10d)
𝛽𝑁𝑋 𝑁𝑌 (𝛼̅)
(10a)
OK, so the cool thing is this: look back at chapter two where we modeled coevolution without including
any explicit ecology. How cool is that? Exact same equations but modified by the population size of the
opposing species!
5
Answers to 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?
New Questions Arising:
. At the same time, however, our model analyses raise several important questions:



What keeps arms races from escalating in perpetuity?
Do our results hold for antagonistic interactions with a different mechanistic basis?
How would our results differ for other types of ecological interaction?
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 arms races from escalating in perpetuity?
Generalization 2: Interactions with a different mechanistic basis
Generalization #3: Other forms of ecological interaction
Conclusions and Synthesis
6
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 oviposition by the weevil to fall off more rapidly as the thickness of
the camellia pericarp increases.
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 pericarp thickness in the Camellia
and rostrum length in the weevil, 𝛿 = 𝑦̅ − 𝑥̅ , over 100 generations. The solid line shows a case where
𝐾 > 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 𝐾 < 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 4. Gaussian stabilizing selection for three different values of the parameter γ. In all cases, the
optimal phenotype is θ = 0.
Figure 5. The exact fitness function 𝑊(𝑧) (solid line), and its quadratic approximation (dashed line), for
three different values of the parameter γ. As the value of γ increases, the range of phenotypes over
which the quadratic approximation remains accurate decreases.
Figure 6. 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 7. 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.
7
Figure 8. 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 9. 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 𝐾 > 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 𝐾 < 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 10. 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 11. 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.
8