Chapter 8. Coevolution, major genes, and spatial patterns of genetic variation
Biological Motivation
IN THIS CHAPTER WE EXPLORE WHEN COEVOLUTION GENERATES SPATIAL PATTERNS. Focused on very
simple models that illustrate how we can include spatial structure
START WITH MUTUALISM AND POINT OUT HOW WEIRD IT IS THAT THERE IS VARIATION WITHIN POPS
Until now we have focused on interactions in single pops. We know however, that spatial structure
matters… Let’s look at the snails and trematodes again… One way the importance of space has been
discerened is by conducting a reciprocal cross infection experiment… These experiments have revealed
very large levels of local adaptation. What is the underlying cause of these patterns?
Key Questions:



Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations?
Can coevolution generate genetic differentiation among populations?
How do coevolutionary selection and gene flow interact to shape rates of mutualistic interaction
within and among populations?
Building a model of spatially structured coevolution
When we first developed a model of coevolution between M. truncatula and S. meliloti in
Chapter 2, we assumed that the two species interacted within a single, isolated population. In order to
study how spatial structure influences the maintenance of genetic variation within and between
populations, we will obviously need to relax this assumption. Although it might be tempting to relax this
assumption by expanding our model to include a large number of populations and complex patterns of
gene flow, this would preclude the possibility of mathematical analysis. Instead, we will focus on
developing the simplest possible model that allows us to understand spatially structured coevolution: a
pair of coevolving populations connected by gene flow (Figure 1). Within each population, we will
assume that M. truncatula and S. meliloti encounter one another at random and that their populations
are sufficiently large that the impacts of random genetic drift can be ignored. We will also restrict
ourselves to the case we studied previously in Chapter 2 where both of the interacting species are
haploid and the interaction is mediated by a single diallelic locus in each species. With these
assumptions, results we developed in Chapter 2 show that the frequency of the A allele within the M.
truncatula population after a single round of coevolutionary selection is given by:
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
′
𝑝𝑋,𝑖
=
𝑝𝑋,𝑖 𝑊𝑋,𝐴,𝑖
̅ 𝑋,𝑖
𝑊
(1a)
and the frequency of the B allele within the S. meliloti population is given by:
′
𝑝𝑌,𝑖
=
𝑝𝑌,𝑖 𝑊𝑌,𝐵,𝑖
̅ 𝑌,𝑖
𝑊
(1b)
where 𝑝𝑋,𝑖 is the frequency of the A allele in M. truncatula within population i, 𝑝𝑌,𝑖 is the frequency of
the B allele in S. meliloti within population i, 𝑊𝑋,𝐴,𝑖 is the fitness of the A allele in population i, 𝑊𝑌,𝐵,𝑖 is
̅𝑋,𝑖 is the mean fitness of M. truncatula in population i, and
the fitness of the B allele in population i, 𝑊
̅𝑌,𝑖 is the mean fitness of S. meliloti in population i. Our next challenge in building our model of
𝑊
spatially structured coevolution is to figure out how gene flow between populations changes the
frequencies of the A and B alleles.
If we assume that individuals move between the two populations at random with a rate of mX in
M. truncatula and mY in S. meliloti we can write down expressions for the allele frequencies within each
population after gene flow occurs. The key to making progress is to focus on the proportion of each
population that is made up of resident individuals (1-m) and the proportion of each population made up
of immigrant individuals (m). Keeping this in mind, we can now calculate the allele frequency of each
species within each population after gene flow occurs by calculating the weighted average allele
frequency of residents and immigrants:
′′
′
′
(1 − 𝑚𝑋 ) + 𝑝𝑋,𝑗
(𝑚𝑋 )
𝑝𝑋,𝑖
= 𝑝𝑋,𝑖
(2a)
′′
′
′
(1 − 𝑚𝑌 ) + 𝑝𝑌,𝑗
(𝑚𝑌 )
𝑝𝑌,𝑖
= 𝑝𝑌,𝑖
(2b)
where the double prime notation indicates the frequency of the allele after both selection and
migration. In equations (2) the first term indicates the contribution to allele frequency within population
i made by individuals that did not leave and the second term indicates the contribution made by
individuals that arrived as immigrants from population j. Substituting (1) into (2) yields expressions for
the allele frequencies of each species within each population in the next generation:
′′
𝑝𝑋,𝑖
=
𝑝𝑋,𝑖 𝑊𝑋,𝐴,𝑖
(1 −
̅ 𝑋,𝑖
𝑊
𝑚𝑋 ) +
𝑝𝑋,𝑗 𝑊𝑋,𝐴,𝑗
(𝑚𝑋 )
̅ 𝑋,𝑗
𝑊
(3a)
′′
𝑝𝑌,𝑖
=
𝑝𝑌,𝑖 𝑊𝑌,𝐵,𝑖
(1 −
̅ 𝑌,𝑖
𝑊
𝑚𝑌 ) +
𝑝𝑌,𝑗 𝑊𝑌,𝐵,𝑗
(𝑚𝑌 )
̅ 𝑌,𝑗
𝑊
(3b)
Together, equations (3) provide a very general description of coevolution within two coupled habitat
patches. To make further progress in understanding spatially structured coevolution, however, we will
need to get a bit more specific about the relationship between species interactions and fitness.
As we have seen in previous chapters, there are very many different assumptions we can make
about the relationship between genotypes and fitness in coevolutionary models. Of central importance
is what we assume about the relationship between genotypes and the outcome of encounters between
individuals captured in the interaction matrix, α. As we discussed in Chapter 2, the genetic details
surrounding the interactions between M. truncatula and S. meliloti are not entirely clear. Some evidence
points to a GFG type model? While other points to a matching alleles type (REFS). To keep things
consistent with Chapter 2, we will assume the interaction is mediated by a matching alleles type
mechanism such that the interaction matrix takes the following form:
1
𝛼=(
0
0
)
1
(4)
where M. truncatula genotypes are in ROWS? and S. meliloti genotypes are in columns? and 1’s indicate
an encounter leads to successful interaction and nodulation and 0’s indicate a failed interaction and an
absence of nodulation. Also in keeping with Chapter 2, we will assume the interaction is a strict
mutualism such that both species receive fitness benefits 𝑠𝑋 and 𝑠𝑌 when an encounter leads to a
successful interaction (e.g., nodulation). With these assumptions, the expected fitness of M. truncatula
A and a genotypes is:
𝑊𝑋,𝐴,𝑖 = 1 + 𝑠𝑋 (𝛼𝐴,𝐵 𝑝𝑌,𝑖 + 𝛼𝐴,𝑏 (1 − 𝑝𝑌,𝑖 )) = 1 + 𝑠𝑋 (𝑝𝑌,𝑖 )
(5a)
𝑊𝑋,𝑎,𝑖 = 1 + 𝑠𝑋 (𝛼𝑎,𝐵 𝑝𝑌,𝑖 + 𝛼𝑎,𝑏 (1 − 𝑝𝑌,𝑖 )) = 1 + 𝑠𝑋 (1 − 𝑝𝑌,𝑖 )
(5b)
and the expected fitness of S. meliloti B and b genotypes is:
𝑊𝑌,𝐵,𝑖 = 1 + 𝑠𝑌 (𝛼𝐴,𝐵 𝑝𝑋,𝑖 + 𝛼𝑎,𝐵 (1 − 𝑝𝑋,𝑖 )) = 1 + 𝑠𝑌 (𝑝𝑋,𝑖 )
(5c)
𝑊𝑌,𝑏,𝑖 = 1 + 𝑠𝑌 (𝛼𝐴,𝑏 𝑝𝑋,𝑖 + 𝛼𝑎,𝑏 (1 − 𝑝𝑋,𝑖 )) = 1 + 𝑠𝑌 (1 − 𝑝𝑋,𝑖 )
(5b)
where the quantity in parentheses is the probability of a successful interaction. Substituting (5) into (3)
yields the following set of recursion equations that completely describes the process of coevolution in
our two population system:
′′
𝑝𝑋,𝑖
=
(1−𝑚𝑋 )𝑝𝑋,𝑖 (1+𝑠𝑋 𝑝𝑌,𝑖 )
1+𝑠𝑋 (𝑞𝑋,𝑖 −(1−2𝑝𝑋,𝑖 )𝑝𝑌,𝑖 )
(1−𝑚𝑌 )𝑝𝑌,𝑖 (1+𝑠𝑌 𝑝𝑋,𝑖 )
′′
𝑝𝑌,𝑖
= 1+𝑠
𝑌 (𝑞𝑋,𝑖 −𝑝𝑌,𝑖 (1−2𝑝𝑋,𝑖 ))
+
𝑚𝑋 𝑝𝑋,𝑗 (1+𝑝𝑌,𝑗 𝑠𝑋 )
1+𝑠𝑋 (𝑞𝑋,𝑗 −(1−2𝑝𝑋,𝑗 )𝑝𝑌,𝑗 )
𝑚𝑌 𝑝𝑌,𝑗 (1+𝑝𝑋,𝑗 𝑠𝑌 )
+ 1+(𝑞
𝑋,𝑗 −(1−2𝑝𝑋,𝑗 )𝑝𝑌,𝑗 )𝑠𝑌
(6a)
(6b)
where the first term in each expression represents the post-selection frequency of resident individuals
and the second term the post-selection frequency of immigrant individuals.
Although coevolutionary recursions (6) completely describe the process of coevolution in our
two population system, it probably comes as little surprise that without further simplification they are
largely intractable. A first step in simplifying the problem is to employ a change of variables that allows
us to more easily identify the conditions under which coevolution generates and maintains spatial
genetic differentiation. Specifically, we will define the following new variables:
𝑝̅𝑋 =
𝑝̅𝑌 =
𝑝𝑋,1 +𝑝𝑋,2
2
𝑝𝑌,1 +𝑝𝑌,2
2
(7a)
(7b)
𝛿𝑋 = 𝑝𝑋,1 − 𝑝𝑋,2
(7c)
𝛿𝑌 = 𝑝𝑌,1 − 𝑝𝑌,2
(7d)
where 𝑝̅𝑋 and 𝑝̅𝑌 are the average frequency of the A and B alleles across both populations for M.
truncatula and S. meliloti respectively and 𝛿𝑋 and 𝛿𝑌 are the difference in the frequency of the A and B
alleles across both populations for M. truncatula and S. meliloti respectively. Thus, the new variables 𝛿𝑋
and 𝛿𝑌 explicitly quantify the amount of spatial genetic differentiation that exists for each of the
coevolving species. Using definitions (7), we can write down recursions for these new variables:
𝑝̅𝑋′′ =
′′
′′
𝑝𝑋,1
+𝑝𝑋,2
2
′′
′′
𝛿𝑋′′ = 𝑝𝑋,1
− 𝑝𝑋,2
𝑝̅𝑌′′ =
′′
′′
𝑝𝑌,1
+𝑝𝑌,2
2
′′
′′
𝛿𝑌′′ = 𝑝𝑌,1
− 𝑝𝑌,2
(8a)
(8b)
(8c)
(8d)
and complete are change of variables by making the following substitutions on the right hand side of the
expressions: 𝑝𝑋,1 = 𝑝̅𝑋 + 𝛿𝑋 /2, 𝑝𝑋,2 = 𝑝̅𝑋 − 𝛿𝑋 /2, 𝑝𝑌,1 = 𝑝̅𝑌 + 𝛿𝑌 /2, and 𝑝𝑌,2 = 𝑝̅𝑌 − 𝛿𝑌 /2. We now
have a set of recursions for our new variables written entirely in terms of the new variables. Our next
challenge is to learn something from these new expressions we have created.
Even with the change of variables, the equations (X) describing spatially structured coevolution
remain challenging to analyze. It can be done, but it isn’t easy and it isn’t terribly transparent. Instead,
we will make one final set of assumptions that greatly simplifies our exploration of coevolution.
Specifically, we will assume the fitness consequences of species interactions are relatively small such
that 𝑠𝑋 and 𝑠𝑌 are both of small order 𝜀, and that rates of gene flow are also small such that 𝑚𝑋 and 𝑚𝑌
are also of small order 𝜀. With these assumptions, we can use Taylor Series to approximate equations
(8), ignoring all terms of 𝒪(𝜀 2 ) and smaller. Implementing this approximation and changing from
recursion equations to difference equations yields the following set of more manageable equations:
1
∆𝑝̅𝑋 ≈ 4 𝑠𝑋 ((1 − 2𝑝̅𝑌 )(𝛿𝑋2 − 4𝑝̅𝑋 𝑞̅𝑋 ) + 2(1 − 2𝑝̅𝑋 )𝛿𝑋 𝛿𝑌 ) + 𝒪(𝜀 2 )
1
2
∆𝛿𝑋 ≈ (−𝑠𝑋 (2(1 − 2𝑝̅𝑋 )(1 − 2𝑝̅𝑌 )𝛿𝑋 + (𝛿𝑋2 − 4𝑝̅𝑋 𝑞̅𝑋 )𝛿𝑌 )) − 2𝑚𝑋 𝛿𝑋 + 𝒪(𝜀 2 )
1
∆𝑝̅𝑌 ≈ 4 𝑠𝑌 ((1 − 2𝑝̅𝑋 )(𝛿𝑌2 − 4𝑝̅𝑌 𝑞̅𝑌 ) + 2(1 − 2𝑝̅𝑌 )𝛿𝑋 𝛿𝑌 ) + 𝒪(𝜀 2 )
1
∆𝛿𝑌 ≈ 2 (−𝑠𝑌 (2(1 − 2𝑝̅𝑌 )(1 − 2𝑝̅𝑋 )𝛿𝑌 + (𝛿𝑌2 − 4𝑝̅𝑌 𝑞̅𝑌 )𝛿𝑋 )) − 2𝑚𝑌 𝛿𝑌 + 𝒪(𝜀 2 )
(9a)
(9b)
(9c)
(9d)
With our change of variables complete and our key assumptions implemented, we can finally move on
to trying to learn something about coevolution in spatially structured environments.
Analyzing the Model
A logical first step in analyzing our model is to identify equilibria. Unfortunately, even after all of
our clever changes and approximations it is not possible (for me anyway!) to find equilibrium solutions
using the systematic approach we used previously. Instead, we need to approach the problem in a
different way that relies on making educated guesses using a combination of inspection, simple algebra,
and biological insight. The most obvious and (boring) equilibria we can identify using simple inspection
are the four cases where both M. truncatula and S. meliloti are fixed for a particular allele in both
populations such that spatial genetic structure is absent for both species (Table 1; Equilibria 1-4). We
know these must be equilibria because absent genetic variation within a population there can be no
response to selection and since both populations are fixed for the same allele, gene flow too, has no
consequences for evolution. The fact that these are, indeed, equilibria can be verified by substituting
our guesses for the equilibrium values of the variables into (7) and verifying that the right hand sides all
equal zero. So far, the equilibria we have been able to identify have been fairly obvious and not very
interesting because they preclude genetic variation within and among populations and thus cannot
possibly explain the patterns of variability we observe in the mutualistic interaction between M.
truncatula and S. meliloti. How could we go about identifying more interesting equilibria that allow for
the maintenance of genetic variation? The first step we will take is to simply propose equilibria where
the spatial average allele frequencies 𝑝̅𝑋 and 𝑝̅𝑌 are equal to ½. The next step is to solve for the values of
𝛿𝑋 and 𝛿𝑌 given our proposed values of 𝑝̅𝑋 = 1/2 and 𝑝̅𝑌 = 1/2, yielding five candidate equilibria (Table
1; Equilibria 5-9). Finally, we substitute these candidate equilibria back into the difference equations (7)
and verify that they are indeed equilibria (they are!). We now have an interesting set of equilibria that
may help us to shed light on spatial patterns of genetic variation in mutualistic interactions. iSAY MORE
ABOUT THESE EQUILIBRIA. POINT OUT THAT ONLY THE FINAL FOUR ARE SPATIALLY VARIABLE
Identifying equilibria that correspond to biologically interesting scenarios of spatial variation is
an important step, but in order to proceed we need to evaluate when (or even if) these equilibria exist.
Although finding existence conditions for equilibria sounds rather technical and dull, don’t despair, the
results are almost certain to provide us with interesting insights into the biology of mutualism! Our
general approach to identifying the conditions under which each of the spatially variable equilibria
(Table 1; 6-9) exist is to find conditions on the parameters guaranteeing that 𝛿𝑋 and 𝛿𝑌 lie between -1
and 1. If these variables do not lie within this interval, the equilibrium cannot possibly exist because it
predicts allele frequencies that are either less than zero or greater than 1. Solving these inequalities
shows that the first two spatially variable equilibria (Table 1; 6-7) never exist for mutualistic interactions
because 𝛿𝑋 and 𝛿𝑌 cannot both lie within the required interval {-1, 1} if 𝑠𝑋 and 𝑠𝑌 are both positive as
required for a strictly mutualistic interaction. In contrast, solving the inequalities for the final two
spatially variable equilibria shows that the equilibria can exist as long as the following condition holds:
𝑠𝑋 𝑠𝑌 > 16𝑚𝑋 𝑚𝑌
(10)
This existence condition provides us valuable insight into the conditions that allow spatial variation in
genotype frequencies to be maintained within mutualistic interactions. Specifically, (8) reveals that the
product of the fitness benefits accruing through mutualistic interaction (𝑠𝑋 𝑠𝑌 ) must exceed sixteen
times the product of the rate of gene flow in the interacting species. Thus, even if the fitness benefits of
mutualism to one species are weak, spatial variation can still be maintained if the fitness benefits to the
interacting species are quite strong. Similarly, even if rates of gene flow in one of the species are quite
high, spatial variation can still be maintained in both species if the rate of gene flow in the interacting
species is quite low. This result is an interesting coevolutionary twist on classical single species results
for the balance between spatially variable selection and gene flow (REFS).
We now know that two of the spatially variable equilibria we have identified can exist and thus
are worthy of further investigation. The obvious next step is to take a look at what these equilibria look
like for various combinations of model parameters. Perhaps the most biologically intuitive way to
explore these equilibria is to plot allele frequencies in both species and populations for a handful of
representative parameter combinations (Figure 1). Taking this numerical approach to exploring the
equilibria reveals several important points. First, there appear to be two possible spatial configurations
for any particular combination parameters. Specifically, the frequencies of the A and B alleles can be
larger in population 1 than in population 2 (Equilibrium #8; Figure 1, left hand panels) or they can both
be smaller in population 1 than in population 2 (Equilibrium #8; Figure 1, right hand panels). These two
possible equilibrium states are simply mirror images of one another. The second thing we can learn by
plotting the spatially variable equilibrium frequencies is that, not surprisingly, the greater the overall
fitness impacts of the mutualistic interaction, measured as the product 𝑠𝑋 𝑠𝑌 , the larger the degree of
spatial variation in allele frequencies (Figure 1). Similarly, the lower the overall rate of gene flow,
measured as the product 𝑚𝑋 𝑚𝑌 , the greater the spatial differentiation in allele frequencies (Figure 1).
Finally, our equilibrium results give us a tool that allows us to predict which of the interacting species
will show a greater degree of spatial genetic variation. Specifically, studying plots of equilibrium allele
frequencies suggests that spatial differentiation will be greater in M. truncatula (Species X) anytime:
𝑠𝑋
𝑠𝑌
>
𝑚𝑋 𝑚𝑌
but greater in S. meliloti (Species Y) anytime:
𝑠𝑋
𝑠𝑌
<
𝑚𝑋 𝑚𝑌
This general prediction can be verified by solving for the conditions under which |𝛿𝑋 | > |𝛿𝑌 | and vice
versa. Together, these analyses of the spatially variable equilibria give us a pretty good idea of how
much spatial variation we should expect to see in coevolving mutualists if they have reached the
spatially variable equilibria.
The final step in our mathematical analysis is to evaluate the local stability of the various
equilibria shown in Table 1. These analyses will help us to identify biological scenarios under which we
expect coevolution between mutualists to lead to spatial variation in gene frequencies and genetic
polymorphism within populations as observed for the interaction between M. truncatula and S. meliloti.
As we learned previously in Chapter 2, the first step in performing a local stability analysis is to create a
Jacobian Matrix. The only difference is that we now have a system of four recursion equations, and as a
consequence, our Jacobian is given by the following 4x4 matrix of partial derivatives:
𝜕𝑝̅𝑋∗
𝜕𝑝̅𝑋
𝜕𝑝̅𝑌∗
𝜕𝑝̅𝑋
𝐽=
𝜕𝛿𝑋∗
𝜕𝑝̅𝑋
𝜕𝛿𝑌∗
[𝜕𝑝̅𝑋
𝜕𝑝̅𝑋∗
𝜕𝑝̅𝑌
𝜕𝑝̅𝑌∗
𝜕𝑝̅𝑌
𝜕𝛿𝑋∗
𝜕𝑝̅𝑌
𝜕𝛿𝑌∗
𝜕𝑝̅𝑌
𝜕𝑝̅𝑋∗
𝜕 𝛿𝑋
𝜕𝑝̅𝑌∗
𝜕 𝛿𝑋
𝜕𝛿𝑋∗
𝜕 𝛿𝑋
𝜕𝛿𝑌∗
𝜕 𝛿𝑋
𝜕𝑝̅𝑋∗
𝜕 𝛿𝑌
𝜕𝑝̅𝑌∗
𝜕 𝛿𝑌
𝜕𝛿𝑋∗
𝜕 𝛿𝑌
𝜕𝛿𝑌∗
𝜕 𝛿𝑌 ]
where 𝑝
̅ ∗𝑋 = 𝑝̅ 𝑋 + ∆𝑝̅ 𝑋 , 𝛿𝑋∗ = 𝛿𝑋 + ∆𝛿𝑋 , 𝑝̅ ∗𝑌 = 𝑝̅ 𝑌 + ∆𝑝̅ 𝑌 , and 𝛿𝑌∗ = 𝛿𝑌 + ∆𝛿𝑌 . Next, we substitute in
the equilibrium of interest and calculate the eigenvalues. As before, if the magnitude of the largest
eigenvalues is greater than 1, the equilibrium is unstable; if it is less than one, it is stable. Taking this
approach yields easily interpretable eigenvalues for the first five spatially homogenous equilibria (Table
2).
What our local stability analyses reveal is that only those equilibria corresponding to the fixation
of matching alleles in both populations (1&4) is locally stable. This is not surprising, really, because it is
these scenarios of fixed matching that yield the highest reciprocal benefits of the mutualistic interaction,
and maximize the population mean fitness of both interacting species. In contrast, those equilibria
corresponding to fixed mismatching (2&3) are never stable. This, too, is not surprising since these
equilibria minimize reciprocal benefits of interacting and also population mean fitness of both
interacting species. Finally, the equilibrium characterized by equal levels of polymorphism in both
populations and species (5), is never stable for mutualistic interactions. This result also makes good
sense, intuitively, as one would expect small perturbations of this equilibrium to lead rapidly to the
increase in frequency of matching alleles in both interacting species.
Evaluating the stability of the spatially homogenous equilibria was relatively simple and yielded
very clear insights. In contrast, determining when the spatially variable equilibria (8&9) are stable is not
at all straightforward, and requires interpretation of complicated eigenvalues with expressions much
too long to easily summarize in a table. Because I was unable to simplify these in any elegant way, we
will need to proceed by evaluating them numerically/graphically. Although not elegant, these numerical
analyses of the eigenvalues associated with the spatially variable equilibria (8&9) suggests that as long
as these equilibria exist (defined by (X)) they are guaranteed to be locally stable as well (Figure 2). Thus,
as long as 𝑠𝑋 𝑠𝑌 > 16𝑚𝑋 𝑚𝑌 , the spatially variable equilibria exist, and are locally stable.
The observation that four possible equilibria may all be simultaneously stable is a phenomenon
we have not previously encountered in our previous investigations of coevolutionary interactions
(TRUE?). What this means, is that if — for whatever reason — allele frequencies of M. truncatula and S.
meliloti begin near one of these equilibria, they will remain there in perpetuity. Thus, if allele
frequencies are initially spatially structured, conforming to either equilibrium 8 or 9, we would observe
spatial genetic variation in this mutualistic interaction and also genetic polymorphism and variability in
compatibility within populations. In contrast, if allele frequencies were initially spatially homogenous
and matched the equilibria described by 1 or 4, we would observe no genetic variation across space and
genetic polymorphism and variation in compatibility would be absent from individual populations.
Obviously, only the former scenario is consistent with empirical observation. This raises an important
question: just how differentiated do populations need to be initially for mutualistic coevolution to
maintain spatial genetic variation? Unfortunately, answering this question mathematically is not
straightforward. We can, however, simulate a handful of scenarios to gain some crude insight into the
conditions that allow spatial variation to coevolve within mutualistic interactions.
In order to simulate mutualistic coevolution, we can iterate the exact recursion equations (8) for
particular combinations of parameters and initial conditions. Performing simulations across a range of
initial conditions and strengths of mutualistic interaction reveal several important points. First, the more
symmetrically differentiated the populations of the two mutualists are initially, the more likely it is that
coevolution can drive further spatial genetic differentiation (Figures 3, 4). The reason for this is that,
because of positive frequency dependence, the strength of spatially divergent natural selection
increases with the degree of initial genetic differentiation. Thus, if the two mutualist species within one
of the populations initially have high frequencies of the A and B alleles whereas the two mutualists
within the other population have initially low frequencies of these alleles, spatially divergent selection is
quite strong and maintains genetic differentiation even in the face of gene flow. Second, as the fitness
consequences of the mutualistic interaction increase, the smaller the amount of initial spatial genetic
differentiation required for coevolutionary differentiation to proceed (Figures 3, 4). Together, these
simulations reveal that although mutualistic coevolution can drive the evolution of spatial genetic
differentiation, it cannot do so de novo. Only if spatial genetic differentiation already exists does
mutualistic coevolution lead to anything other than spatially homogenous allele frequencies (e.g., Figure
3c-f and Figure 4e-f). Later in this chapter we will grapple with the problem of how initial spatial genetic
differentiation of this sort might, in principle, be generated.
Answers to Key Questions:
Can mutualistic coevolution maintain or generate genetic differentiation among populations?
Our results suggest that mutualistic coevolution can maintain spatial genetic differentiation but
cannot generate such differentiation de novo. Thus, if genotype frequencies of the mutualists M.
truncatula and S. meliloti, are initially very similar across populations, mutualistic coevolution will simply
reinforce this uniformity. If, in contrast, genotype frequencies are initially somewhat divergent among
populations and the fitness consequences of mutualism are large relative to rates of gene flow,
coevolution can enhance and maintain spatial genetic differentiation.
Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations?
Sometimes. In those cases where coevolution maintains genetic differentiation among
populations, gene flow will continually move alleles between populations and thus maintain genetic
polymorphism. The amount of genetic polymorphism depends on the fitness consequences of
mutualistic interactions relative to the rate of gene flow between populations.
How do coevolutionary selection and gene flow interact to shape rates of mutualistic interaction within
and among populations?
Gene flow reduces rates of mutualistic interaction within populations when spatial
heterogeneity is present. This is because individuals with incompatible alleles are continually
introduced, thus reducing the average compatibility between species within populations. For the
interaction between M. truncatula and S. meliloti this might lead to populations with variation in
compatibility between genotypes. If, however, spatial heterogeneity is absent, gene flow has no
consequence on rates of interaction because immigrants are genetically identical to residents.
New Questions Arising:
Our simple model of spatially structured mutualism between M. truncatula and S. meliloti
suggests that coevolution can reinforce pre-existing spatial genetic differentiation but cannot cause
initially homogenous populations to diverge from one another. Given that we observe spatial genetic
variation within many coevolving interactions, this simple theoretical result raises several important
questions:

Do similar results hold for other forms of ecological interaction?

How would our results change if the environment were heterogeneous?

What is the role of random genetic drift?
In the next three sections, we will generalize our simple model in ways that allow us to answer these
questions.
Generalizations
Generalization 1: Alternative forms of ecological interaction
Our investigation of spatially structured mutualism suggested that coevolution can reinforce
pre-existing spatial genetic structure but cannot cause initially genetically homogenous populations to
diverge. To some extent, this result is intuitive and follows from the positive frequency dependence built
into the model of mutualism we explored where individuals whose genotype matches the predominant
genotype of the interacting species are most likely to interact successfully and accrue the benefits of the
mutualism. But how might things differ for antagonistic interactions such as those between hosts and
parasites or predators and prey that can generate negative frequency dependence? To investigate this
scenario, we now turn our attention back to the interaction between the castrating trematode,
Microphallus, and its snail host, P. antipodarum.
As you might recall from the previous chapter, the interaction between Microphallus and P.
antipodarum has been very well-studied within the context of the Red Queen Hypothesis for the
evolution of sex. These studies have revealed significant genetic differentiation among lakes as well as
substantial genetic polymorphism within lakes. In addition, studies of clone frequencies over time
suggest cyclical dynamics (REFS), and are thus compatible with coevolutionary dynamics we expect from
a matching alleles model. If we are willing to take some liberties with the genetics of the interacting
species, and assume that both are haploid and that interactions involve only a single diallelic locus in
each species, we can apply the same mathematical framework we used to study the spatially structured
mutualism between M. truncatula and S. meliloti. In fact, as long as we remain comfortable with our key
assumptions of weak selection and infrequent gene flow, we can even use the exact same equations!
The only change we need to make is to recognize that the fitness consequences of the interaction are
now negative for P. antipodarum. Thus, if we label P. antipodarum as species X and Microphallus as
species Y, the fitness consequences of interacting for P. antipodarum must be negative such that 𝑠𝑋 < 0
and the fitness consequences of interacting for Microphallus must be positive such that 𝑠𝑌 > 0. With
this small change in the ranges of these key parameters, we can repurpose our previous analyses of
mutualistic interactions in a way that allows us to explore coevolution between P. antipodarum and
Microphallus. Our first step will be to take a fresh look at the equilibria we identified previously as well
as their local stability.
A logical first step in our analysis of coevolution between P. antipodarum and Microphallus is to
identify equilibria. Because the equations describing coevolution between these species are identical to
those we used to study coevolution between M. truncatula and S. meliloti, the equilibria must also be
the same (e.g., Table 1). Although we can be certain that the equilibria themselves remain identical, this
need not be the case for the conditions under which the equilibria exist. The reason the existence
conditions may no longer be the same as they were for the mutualistic interaction between M.
truncatula and S. meliloti is that the parameter 𝑠𝑋 is now negative, capturing the reduction in fitness of
P. antipodarum individuals unfortunate enough to become infected by the castrating trematode
Microphallus. Thus, we must again identify the conditions under which each equilibrium exists by solving
for parameter combinations that yield equilibrium values of host and parasite allele frequencies that lie
between 0 and 1, inclusive. Taking this approach for each of the equilibria we identified previously
shows that fewer equilibria exist when the interaction is between host and parasite rather than
between mutualists (Table 3). In fact, the only equilibria that can now possibly exist are those where
allele frequencies are spatially homogenous. This result suggests that if our model is to explain the
spatial variation in genotype frequencies observed in the interaction between P. antipodarum and
Microphallus, it must be through temporally fluctuating patterns rather than static equilibria. In order to
investigate this possibility, we need to turn our attention from finding equilibria to studying their local
stability.
By now, the steps involved in analyzing local stability should be old hat: 1) generate a Jacobian
matrix, 2) substitute the equilibrium of interest, and 3) calculate the eigenvalues of the Jacobian. In this
case, our life is made easier by recognizing that, because the equations remain unchanged from our
previous study of mutualism, the eigenvalues associated with each equilibrium must remain the same.
Thus, the only challenge we face in our transition from mutualism to antagonism is to re-interpret the
local stability of each equilibrium while recognizing that one of the species is now harmed by the
interaction such that 𝑠𝑋 < 0 (Table 4). Taking this approach reveals that changing the sign of this single
parameter has significant consequences for coevolution, destabilizing the matching equilibria (1&4), and
making oscillatory dynamics near the polymorphic equilibrium (5) inevitable. These results strongly
suggest the outcome of spatially structured coevolution between P. antipodarum and Microphallus will
be oscillations in allele frequencies. The crucial question that remains unanswered by these analyses,
however, is whether allele frequency oscillations will be out of phase in the two populations such that
allele frequencies would vary across space as observed in the real world interaction between P.
antipodarum and Microphallus? To answer these questions, we will need to carry our local stability
analysis of the polymorphic equilibrium (5) a bit further.
Up until now, we have interpreted the results of our local stability analyses by focusing on only
the leading (largest) eigenvalue. If this leading eigenvalue had a magnitude greater than one, we
deemed the system unstable; if it had a leading eigenvalue less than one, we deemed the system stable.
Using this (entirely correct) interpretation, the results in table 4 show that the polymorphic equilibrium
is unstable, and — because it has an imaginary component — also oscillatory. But does this imply that
allele frequencies will oscillate over time but remain spatially homogenous? Or does it imply that allele
frequencies will oscillate in a way that generates spatial heterogeneity? How can we tell? The key to
answering these questions is to recognize that the Jacobian Matrix has a block form when we substitute
in the polymorphic equilibrium:
1
𝑠
2 𝑋
0
0
1
0
0
0
0
(1 − 2𝑚𝑋 )
[ 0
0
1
𝑠
2 𝑌
1
𝑠
2 𝑋
1
1
𝑠𝑌
𝐽= 2
(1 − 2𝑚𝑌 )]
What this means is that the fate of perturbations to the average allele frequencies (𝑝̅𝑋 and 𝑝̅𝑌 ) can be
studied independently of perturbations to the spatial differences in allele frequencies (𝛿𝑋 and 𝛿𝑌 ).
Specifically, we can analyze the local stability of average allele frequencies by calculating the eigenvalues
of the 2×2 matrix defined by the upper left block, and the local stability of the spatial difference in allele
frequencies by calculating the eigenvalues of the 2×2 matrix defined by the lower right hand block.
Taking this approach reveals the following pair of eigenvalues for average allele frequencies:
1
1 ± 2 √𝑠𝑋 √𝑠𝑌
(12a)
and the following pair of eigenvalues for the spatial differences in allele frequencies:
1
1 − (𝑚𝑋 + 𝑚𝑌 ) ± 2 √4(𝑚𝑋 − 𝑚𝑌 )2 + 𝑠𝑋 𝑠𝑌
(12b)
By decomposing our Jacobian matrix in this way and analyzing the dynamics of each block separately,
we gain two important insights. First, since the leading eigenvalue associated with the average allele
frequencies (12a) is always imaginary and greater than one in magnitude, we know that average allele
frequencies will oscillate around the polymorphic equilibrium with ever increasing amplitude. Second,
since the leading eigenvalue associated with the spatial difference in allele frequency (12b) is
guaranteed to have a magnitude less than one, we know that any small difference in allele frequencies
across populations will ultimately decay.
We are now at a point where we can bring our mathematical analyses together and use them to
sketch a rough picture of the coevolutionary dynamics we expect for the spatially structured interaction
between P. antipodarum and Microphallus. What this picture depicts is the inevitable emergence of
spatially synchronized cycles that grow in amplitude over time. Even if allele frequencies initially differ
among populations by some small amount, our mathematical results suggest this initial spatial
heterogeneity will inevitably decay over time. Somewhat surprisingly, these results suggest that
antagonistic interactions may be even less capable of generating and maintaining spatially
heterogeneity in allele frequencies than are mutualistic interactions, at least under the conditions of
weak selection we have considered here. This result is uncomfortable because it is not at all consistent
with what is actually observed in empirical studies of the interaction between P. antipodarum and
Microphallus, as well as in many other well-studied antagonistic interactions (REFS). Where could our
model have gone wrong? One obvious possibility is our assumption that the fitness consequences of
interactions are weak.
Although tackling strong selection mathematically is challenging, simulating coevolution under
conditions of strong selection is really pretty easy. All we need to do is write a simple loop that iterates
the exact recursion equations (6), over and over again for specific parameter combinations and initial
conditions. Taking this approach for a smattering of parameter values and initial conditions suggests
that our mathematical results are pretty darn robust: in virtually every case simulated, spatial
heterogeneity in allele frequencies ultimately decays over time (Figure 5). At the same time, however,
simulations can only ever investigate specific parameter combinations, so the generality of this result is
unclear. It may well be that specific combinations of parameters exist that do maintain spatial
heterogeneity in allele frequencies (E.G., see Gav and Mik). What seems clear, however, is that our
model lends little support to the idea that coevolution between hosts and parasites can generate
significant levels of spatial genetic differentiation, even with strong selection. What we will see in the
next two sections, however, is that by relaxing our assumptions that the fitness consequences of
interactions are constant across space and that population sizes are infinite, creates substantially more
scope for coevolution to drive genetic differentiation among populations.
Generalization 2: Heterogeneous environments and selection mosaics
When we first started writing down expressions describing the fitness of M. truncatula and S.
meliloti at the beginning of this chapter (i.e., equations 5), we made a critical assumptions that might
not have been entirely obvious at the time. Specifically, we assumed that the fitness consequences of
the interaction, 𝑠𝑋 and 𝑠𝑌 , do not vary across space. Biologically, this means that a successful interaction
between M. truncatula and S. meliloti has exactly the same fitness consequences in different geographic
regions. Although that might sound reasonable at first glance, imagine what would happen if the
amount of available nitrogen in the soil differed between geographic locations, perhaps due to more
severe run-off from agricultural fields in some populations than in others. Under such conditions the
benefits of interacting with the nitrogen fixing rhizobium, S. meliloti, may be significantly reduced for
populations of M. truncatula living in nitrogen rich populations such that the value of the parameter 𝑠𝑋
is reduced. We now know that such selection mosaics are quite common (REFS), and can have important
consequences for coevolution (REFS). Our goal here is to integrate selection mosaics into our model of
spatially structured coevolution, and ask if they can facilitate the emergence of spatial differentiation
among coevolving populations.
The first step in our exploration of selection mosaics is to decide how we can integrate spatial
variation in the fitness consequences of an interaction into our fitness expressions. The simplest way to
do this is to modify our original expressions for fitness (5) to allow the values of the parameters 𝑠𝑋 and
𝑠𝑌 to vary in the two populations, yielding the following equations for fitness in the coevolving species:
𝑊𝑋,𝐴,𝑖 = 1 + 𝑠𝑋,𝑖 (𝑝𝑌,𝑖 )
(13a)
𝑊𝑋,𝑎,𝑖 = 1 + 𝑠𝑋,𝑖 (1 − 𝑝𝑌,𝑖 )
(13b)
𝑊𝑌,𝐵,𝑖 = 1 + 𝑠𝑌,𝑖 (𝑝𝑋,𝑖 )
(13c)
𝑊𝑌,𝑏,𝑖 = 1 + 𝑠𝑌,𝑖 (1 − 𝑝𝑋,𝑖 )
(13d)
where the subscript i indicates population. Next, we can follow the same sequence of mathematical
steps we used earlier in this chapter to explore coevolution between S. meliloti and M. truncatula until
we arrive at expressions like (8) describing the allele frequency of both species in each of the
populations. At this point, however, we need to diverge from the path we took previously if we are
going to develop useful mathematical results. The reason is that in the presence of selection mosaics,
even our previous assumptions of weak fitness consequences and weak gene flow fail to simplify the
problem sufficiently for any substantive mathematical progress to be made.
Because our previous approximation based on assumptions of weak selection and weak gene
flow failed to sufficiently simplify the problem, we need to identify a fresh approach. One idea might be
to flip our worldview and assume that rather than being infrequent, gene flow is actually quite common.
Superficially, this seems like a dumb idea; how could assuming a parameter is large possibly simplify the
problem more than assuming it is small? To see why this superficially counter-intuitive approximation
might work, we need to focus our attention on the amount of genetic differentiation among populations
we expect to see. Specifically, as the rate of gene flow increases, we expect the level of genetic
differentiation among populations to be reduced. In fact, if selection is relatively weak (of small order 𝜀),
and gene flow is substantial, we expect the difference in allele frequencies between populations (𝛿𝑋
and 𝛿𝑌 terms) to also be small and of order 𝜀 (REFS). What this means, is that if we are willing to assume
gene flow is relatively strong and selection is relatively weak, we can approximate the dynamics of
spatially structured coevolution by using a Taylor Series to approximate equations (8), ignoring all terms
of 𝒪(𝜀 2 ). Taking this approach yields the following expressions for the change in average allele
frequencies and the difference in allele frequencies:
∆𝑝̅𝑋 ≈ 𝑠̅𝑋 𝑝̅𝑋 𝑞̅𝑋 (1 − 2𝑝̅𝑌 ) + 𝒪(𝜀 2 )
(14a)
∆𝛿𝑋 ≈ 𝛿𝑠𝑋 𝑝̅𝑋 𝑞̅𝑋 (1 − 2𝑝̅𝑌 ) − 2𝑚𝑋 𝛿𝑋 + 𝒪(𝜀 2 )
(14b)
∆𝑝̅𝑌 ≈ 𝑠𝑌 𝑝̅𝑌 𝑞̅𝑌 (1 − 2𝑝̅𝑋 ) + 𝒪(𝜀 2 )
(14c)
∆𝛿𝑌 ≈ 𝛿𝑠𝑌 𝑝̅𝑌 𝑞̅𝑌 (1 − 2𝑝̅𝑋 ) − 2𝑚𝑌 𝛿𝑌 + 𝒪(𝜀 2 )
(14d)
where the terms 𝛿𝑠𝑋 = 𝑠𝑋,1 − 𝑠𝑋,2 and 𝛿𝑠𝑌 = 𝑠𝑌,1 − 𝑠𝑌,2 measure the intensity of the selection mosaic
acting on each species.
A remarkable fact about our new approximation (14) is that the spatial genetic differentiation
and average allele frequencies are now uncoupled! Just as we saw in Chapter 6 when we explored the
QLE approximation, we have a situation where we expect one of the variables, in this case the degree of
spatial genetic differentiation (𝛿𝑋 and 𝛿𝑌 ), to approach a quasi-equilibrium state where its value
depends only on that of the other variable. To see this more clearly, we can solve for the conditions
under which (14b) and (14d) are equal to zero, yielding the following Quasi Panmixia approximation for
levels of genetic differentiation between populations:
(1−2𝑝̅𝑌 )
𝛿𝑠 𝑝̅𝑋 𝑞̅𝑋
𝛿̃𝑋 ≈ − 𝑋 2𝑚
𝑋
(1−2𝑝̅𝑋 )
𝛿𝑠 𝑝̅𝑌 𝑞̅𝑌
𝛿̃𝑌 ≈ − 𝑌 2𝑚
𝑌
(15a)
(15b)
A quick inspection of equations (15) reveals a very important result: as long as average allele frequencies
do not become stuck at zero, one, or one half, the presence of a selection mosaic guarantees that allele
frequencies will differ across populations. Thus, we need only identify those conditions where average
allele frequencies do not approach and remain at these values in order to nail down conditions
guaranteeing we see the type of genetic variation among populations observed in many coevolving
systems. The logical way to identify these conditions is to turn again to an analysis of equilibria and their
local stability.
Because the dynamics of average allele frequencies are decoupled from differences in allele
frequencies, we can identify these conditions by analyzing the equilibria and stability defined by the pair
of equations (14a and 14c) in isolation. Following the usual steps reveals that there are five possible
equilibria, none of which are ever stable for antagonistic interactions like those between P. antipodarum
and Microphallus (Table 5). However, one of the equilibria (where average allele frequencies are near ½
in both species), is associated with imaginary eigenvalues, suggesting that average allele frequencies
should oscillate around ½ in both species. GIVEN THAT, WHAT SHOULD WE SEE? IT APPEARS WE
SHOULD SEE FLUCTUATING LEVELS OF SPATIAL DIFFERENTIATION. WHEN AVERAGE ALLELE
FREQUENCIES ARE FAR FROM ½ DIFFERENTIATION IS LARGE, WHEN IT IS NEAR ½ small, on the other side
of ½ the sign fluctuates. Thus, our results suggest that adding a selection mosaic can greatly increase the
scope for anatogianitic coevolution to generate temporally dynamics spatial genetic diffrerentiation
(FIGURE 6)
Generalization 3: Random genetic drift
Conclusions and Synthesis
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Figure Legends
Table 1. A subset of equilibria for spatially structured mutualism
Equilibrium
𝑝̅𝑋
𝑝̅𝑌
𝛿𝑋
𝛿𝑌
Existence
1
0
0
0
0
Always
2
0
1
0
0
Always
3
1
0
0
0
Always
4
1
1
0
0
Always
5
½
½
0
0
Always
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 + 4√𝐾𝑋
√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 + 4√𝐾𝑌
√
𝑠𝑌2 𝑠𝑋
Never
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 + 4√𝐾𝑋
√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 + 4√𝐾𝑌
−√
𝑠𝑌2 𝑠𝑋
Never
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 − 4√𝐾𝑋
−√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 − 4√𝐾𝑌
√
𝑠𝑌2 𝑠𝑋
𝑠𝑋 𝑠𝑌 > 16𝑚𝑋 𝑚𝑌
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 − 4√𝐾𝑋
−√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 − 4√𝐾𝑌
−√
𝑠𝑌2 𝑠𝑋
𝑠𝑋 𝑠𝑌 > 16𝑚𝑋 𝑚𝑌
6
7
8
9
½
½
𝐾𝑋 = 𝑚𝑋2 (4𝑚𝑌2 𝑠𝑋2 − 8𝑚𝑋 𝑚𝑌 𝑠𝑋 𝑠𝑌 + (4𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌2 ) and 𝐾𝑌 = 𝑚𝑌2 (4𝑚𝑋2 𝑠𝑌2 − 8𝑚𝑌 𝑚𝑋 𝑠𝑌 𝑠𝑋 + (4𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋2 )
Table 2. Spatially homogenous equilibria and associated eigenvalues
Equilibrium
𝑝̅𝑋
𝑝̅𝑌
𝛿𝑋
𝛿𝑌
Eigenvalues
Stability
1
0
0
0
0
1 − 2𝑚𝑋 − 𝑠𝑋 , 1 − 𝑠𝑋 , 1 − 2𝑚𝑌 − 𝑠𝑌 , 1 − 𝑠𝑌
Stable
2
0
1
0
0
1 + 𝑠𝑋 , 1 − 2𝑚𝑋 + 𝑠𝑋 , 1 + 𝑠𝑌 , 1 − 2𝑚𝑌 + 𝑠𝑌
Unstable
3
1
0
0
0
1 + 𝑠𝑋 , 1 − 2𝑚𝑋 + 𝑠𝑋 , 1 + 𝑠𝑌 , 1 − 2𝑚𝑌 + 𝑠𝑌
Unstable
4
1
1
0
0
1 − 2𝑚𝑋 − 𝑠𝑋 , 1 − 𝑠𝑋 , 1 − 2𝑚𝑌 − 𝑠𝑌 , 1 − 𝑠𝑌
Stable
0
1
1
1 − √𝑠𝑋 √𝑠𝑌 , 1 + √𝑠𝑋 √𝑠𝑌 ,
2
2
1
1
1 − 𝑚𝑋 − 𝑚𝑌 − √𝐶, 1 − 𝑚𝑋 − 𝑚𝑌 + √𝐶
2
2
Unstable
5
½
½
𝐶 = 4(𝑚𝑋 − 𝑚𝑌 )2 + 𝑠𝑋 𝑠𝑌
0
Table 3. A subset of equilibria for spatially structured antagonism
Equilibrium
𝑝̅𝑋
𝑝̅𝑌
𝛿𝑋
𝛿𝑌
Existence
1
0
0
0
0
Always
2
0
1
0
0
Always
3
1
0
0
0
Always
4
1
1
0
0
Always
5
½
½
0
0
Always
6
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 + 4√𝐾𝑋
√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 + 4√𝐾𝑌
√
𝑠𝑌2 𝑠𝑋
Never
7
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 + 4√𝐾𝑋
√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 + 4√𝐾𝑌
−√
𝑠𝑌2 𝑠𝑋
Never
8
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 − 4√𝐾𝑋
−√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 − 4√𝐾𝑌
√
𝑠𝑌2 𝑠𝑋
Never
9
½
½
−8𝑚𝑋 𝑚𝑌 𝑠𝑋 + (8𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌 − 4√𝐾𝑋
−√
𝑠𝑋2 𝑠𝑌
−8𝑚𝑌 𝑚𝑋 𝑠𝑌 + (8𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋 − 4√𝐾𝑌
−√
𝑠𝑌2 𝑠𝑋
Never
𝐾𝑋 = 𝑚𝑋2 (4𝑚𝑌2 𝑠𝑋2 − 8𝑚𝑋 𝑚𝑌 𝑠𝑋 𝑠𝑌 + (4𝑚𝑋2 + 𝑠𝑋2 )𝑠𝑌2 ) and 𝐾𝑌 = 𝑚𝑌2 (4𝑚𝑋2 𝑠𝑌2 − 8𝑚𝑌 𝑚𝑋 𝑠𝑌 𝑠𝑋 + (4𝑚𝑌2 + 𝑠𝑌2 )𝑠𝑋2 )
Table 4. Spatially homogenous equilibria and associated eigenvalues
Equilibrium
𝑝̅𝑋
𝑝̅𝑌
𝛿𝑋
𝛿𝑌
Eigenvalues
Stability
1
0
0
0
0
1 − 2𝑚𝑋 − 𝑠𝑋 , 1 − 𝑠𝑋 , 1 − 2𝑚𝑌 − 𝑠𝑌 , 1 − 𝑠𝑌
Unstable
2
0
1
0
0
1 + 𝑠𝑋 , 1 − 2𝑚𝑋 + 𝑠𝑋 , 1 + 𝑠𝑌 , 1 − 2𝑚𝑌 + 𝑠𝑌
Unstable
3
1
0
0
0
1 + 𝑠𝑋 , 1 − 2𝑚𝑋 + 𝑠𝑋 , 1 + 𝑠𝑌 , 1 − 2𝑚𝑌 + 𝑠𝑌
Unstable
4
1
1
0
0
1 − 2𝑚𝑋 − 𝑠𝑋 , 1 − 𝑠𝑋 , 1 − 2𝑚𝑌 − 𝑠𝑌 , 1 − 𝑠𝑌
Unstable
0
1
1
1 − √𝑠𝑋 √𝑠𝑌 , 1 + √𝑠𝑋 √𝑠𝑌 ,
2
2
1
1
1 − 𝑚𝑋 − 𝑚𝑌 − √𝐶, 1 − 𝑚𝑋 − 𝑚𝑌 + √𝐶
2
2
Unstable and Oscillatory
5
½
½
𝐶 = 4(𝑚𝑋 − 𝑚𝑌 )2 + 𝑠𝑋 𝑠𝑌
0
Table 5. Spatially homogenous equilibria and associated eigenvalues
Equilibrium
𝑝̅𝑋
𝑝̅𝑌
𝛿𝑋
𝛿𝑌
Eigenvalues
Stability for host-parasite
1
0
0
0
0
1 − 𝑠̅𝑋 , 1 − 𝑠̅𝑌
Unstable
2
0
1
0
0
1 + 𝑠̅𝑋 , 1 + 𝑠̅𝑌
Unstable
3
1
0
0
0
1 + 𝑠̅𝑋 , 1 + 𝑠̅𝑌
Unstable
4
1
1
0
0
1 − 𝑠̅𝑋 , 1 − 𝑠̅𝑌
Unstable
5
½
½
0
0
1
1
1 − √𝑠̅𝑋 √𝑠̅𝑌 , 1 + √𝑠̅𝑋 √𝑠̅𝑌 ,
2
2
Unstable and Oscillatory