1
2
3
4
Supplemental online material for S.L. Nuismer and L.J. Harmon, “Predicting rates
of interspecific interaction from phylogenetic trees”
7
We study species interactions and evolution within a community of n species, each of
which is characterized by a population abundance, N, and a distribution of phenotypes 𝜙𝑧 for a
focal trait, z. We assume this trait plays a central role in mediating interactions among species
within the community.
8
Section 1: Predicting interaction rates
9
15
We assume ecological interactions occur at random, with each individual encountering
one other individual over the course of its lifetime. When a focal individual with phenotype 𝑧𝑖
encounters another individual with phenotype 𝑧𝑗 , they interact with probability 𝑃(𝑧𝑖 , 𝑧𝑗 ). The
probability that a successful interaction occurs, P, is assumed to be given by a model of
phenotype matching or a model of phenotype differences. The model of phenotype matching
assumes the probability of interaction depends on the distance between the phenotypes of the
individuals:
16
𝑃𝑖,𝑗 = 𝐸𝑥𝑝 [−𝛼(𝑧𝑖 − 𝑧𝑗 ) ]
17
20
where 𝑧𝑖 is the phenotype of the focal individual, 𝑧𝑗 is the phenotype of the individual it
encounters, and 𝛼 determines how sensitive the probability of interaction is to the phenotypes
of the individuals. The model of phenotype differences assumes the probability of interaction
depends on the extent to which the phenotype of one individual exceeds that of the other:
21
𝑃𝑖,𝑗 = 1+𝐸𝑥𝑝[𝛼(𝑧 −𝑧
22
25
where 𝑧𝑖 is the phenotype of the focal individual, 𝑧𝑗 is the phenotype of the individual it
encounters, and 𝛼 determines how sensitive the probability of interaction is to the phenotypes
of the individuals. In the next sections, we use these functions, P, to predict rates of interaction
between species within the community.
26
Phenotype Matching
27
28
If 𝛼 is sufficiently small, and the distributions of 𝑧𝑖 and 𝑧𝑗 not too disparate, equation
(S1) can be accurately approximated by a first-order Taylor Series expansion:
29
𝑃𝑖,𝑗 ≈ 1 − 𝛼(𝑧𝑖 − 𝑧𝑗 )
30
Because we assume random encounters among individuals, we can calculate the rate of
interaction between species i and j by integrating (S3) over their phenotype distributions:
5
6
10
11
12
13
14
18
19
23
24
31
2
1
𝑖
(S1)
(S2)
𝑗 )]
2
Supplemental online material for S.L. Nuismer and L.J. Harmon
(S3)
1
32
2
𝑃̅𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 ∫ ∫ (1 − 𝛼(𝑧𝑖 − 𝑧𝑗 ) ) 𝜙𝑧𝑖 𝜙𝑧𝑗 𝑑𝑧𝑖 𝑑𝑧𝑗
33
Here 𝑓𝑖 is the frequency of species i within the community and 𝜙𝑧𝑖 is the probability density for
34
35
the phenotypic trait value 𝑧𝑖 of species i. We can now solve the integrals in (S4) for a probability
distribution function 𝜙𝑧𝑖 with mean 𝑧̅𝑖 and variance 𝜎𝑧2𝑖 :
36
2
𝑃̅𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 (1 − 𝛼(𝑧̅𝑖 − 𝑧̅𝑗 ) − 𝛼 (𝜎𝑧2𝑖 + 𝜎𝑧2𝑗 ))
(S4)
(S5)
43
Next, we wish to know the expected rate of interaction between two species, say i and j, that
are of known evolutionary relationship (e.g. if we know the time since species i and j have
shared a common ancestor, τ-tij). Because the trait values of species that share an evolutionary
history are not independent, we must now describe the contemporary values of species mean
phenotypes using a bivariate distribution. Integrating the rate of interaction 𝑃̅ between a pair
of species i and j over this bivariate distribution yields the following expression for the expected
rate of interaction between a pair of related species:
44
̂𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 ∫ ∫ (1 − 𝛼(𝑧̅𝑖 − 𝑧̅𝑗 )2 − 𝛼 (𝜎𝑧2 + 𝜎𝑧2 )) 𝜙𝑧̅ ,𝑧̅ 𝑑𝑧̅ 𝑑𝑧̅
P
𝑖
𝑗
𝑖 𝑗
𝑖
𝑗
45
where 𝜙𝑧̅𝑖 ,𝑧̅𝑗 is the bivariate frequency distribution of the trait means 𝑧̅𝑖 and 𝑧̅𝑗 . Carrying out the
46
integrals in (S6) yields:
47
̂𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 (1 − 𝛼(𝜇𝑖 − 𝜇𝑗 )2 − 𝛼 (𝜎𝑧̅2 + 𝜎𝑧̅2 − 2𝜎𝑧̅ ,𝑧̅ ) − 𝛼 (𝜎̅𝑧2 + 𝜎̅𝑧2 ))
P
𝑖 𝑗
𝑖
𝑗
𝑖
𝑗
37
38
39
40
41
42
(S6)
(S7)
49
Here 𝜇𝑖 is the expected population mean phenotype for species i across realizations of the
evolutionary process, 𝜎𝑧̅2𝑖 is the variance in the mean phenotype for species i across realizations
50
of the evolutionary process, 𝜎𝑧̅𝑖 ,𝑧̅𝑗 is the covariance in the mean phenotypes of species i and j
51
across realizations, and 𝜎̅𝑧2𝑖 is the average phenotypic variance within species i across
52
54
realizations. This result reveals that the rate of interaction between any pair of species i and j
depends on their frequencies within the community along with the model of trait evolution and
the phylogenetic relationships between them, which together determine variances 𝜎𝑧̅2𝑖 and 𝜎𝑧̅2𝑖
55
as well as the covariance 𝜎𝑧̅𝑖 ,𝑧̅𝑗 .
56
57
Result (S7) can also be used to predict the average rate of interaction within the entire
community:
58
̿ = ∑𝑛𝑖=1 ∑𝑖𝑗=1 (𝑓𝑖 𝑓𝑗 (1 − 𝛼(𝜇𝑖 − 𝜇𝑗 )2 − 𝛼 (𝜎𝑧̅2 + 𝜎𝑧̅2 − 2𝜎𝑧̅ ,𝑧̅ ) − 𝛼 (𝜎̅𝑧2 + 𝜎̅𝑧2 )))
P
𝑖 𝑗
𝑖
𝑗
𝑖
𝑗
48
53
Supplemental online material for S.L. Nuismer and L.J. Harmon
(S8)
2
60
Result (S8) shows that the overall rate of interaction within the community depends upon
variances 𝜎𝑧̅2𝑖 and 𝜎𝑧̅2𝑖 and covariances 𝜎𝑧̅𝑖 ,𝑧̅𝑗 of interacting species.
61
Phenotype Differences
62
63
As long as α is small and the distributions of 𝑧𝑖 and 𝑧𝑗 not too disparate, (S2) can be
accurately approximated by:
64
𝑃𝑖,𝑗 ≈ 2 + 4 (𝑧𝑗 − 𝑧𝑖 )
65
Because we assume random encounters among individuals, we can calculate the rate of
interaction between species i and j by integrating (S9) over their phenotype distributions:
59
66
67
68
69
70
71
72
1
𝛼
1
𝛼
𝑃̅𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 ∫ ∫ (2 + 4 (𝑧𝑗 − 𝑧𝑖 )) 𝜙𝑧𝑖 𝜙𝑧𝑗 𝑑𝑧𝑖 𝑑𝑧𝑗
1
𝛼
𝑃̅𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 (2 − 4 (𝑧̅𝑖 − 𝑧̅𝑗 ))
74
Evaluating the integrals yields:
75
1
𝛼
̂
P𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 (2 − 4 (𝜇𝑖 − 𝜇𝑗 ))
78
79
80
81
82
83
(S11)
As for phenotype matching, we next calculate the expected rate of interaction between two
species i and j that whose trait value are not independent due to common ancestry:
1
𝛼
̂
P𝑖,𝑗 = 𝑓𝑖 𝑓𝑗 ∫ ∫ (2 − 4 (𝑧̅𝑖 − 𝑧̅𝑗 )) 𝜙𝑧̅𝑖 ,𝑧̅𝑗 𝑑𝑧̅𝑖 𝑑𝑧̅𝑗
77
(S10)
We can now solve the integrals in (S4) for a probability distribution function 𝜙𝑧𝑖 with mean 𝑧̅𝑖
and variance 𝜎𝑧2𝑖 :
73
76
(S9)
(S12)
(S13)
Notice that all variance and covariance terms among species have dropped out. This simple
result demonstrates that phylogenetic relationships have no consequence for the rate of
interaction when species interact through a mechanism of phenotype differences.
Result (S13) can be used to predict the overall rate of interaction within the entire
community:
1
𝛼
1
̿
P = ∑𝑛𝑖=1 ∑𝑖𝑗=1 (𝑓𝑖 𝑓𝑗 (2 − 4 (𝜇𝑖 − 𝜇𝑗 ))) = 2
(S14)
revealing that for interactions mediated by phenotype differences, phylogenetic relationships
have no consequences for expected interaction rates.
Supplemental online material for S.L. Nuismer and L.J. Harmon
3
84
Section 2: Predicting trait evolution
85
92
In this section we study how population mean phenotypes evolve in response to: 1)
genetic drift, 2) genetic drift and stabilizing selection, and 3) genetic drift, stabilizing selection
and species interactions. Our goal is to generate predictions for how the multivariate
distribution describing species mean phenotypes within the community evolves, allowing us to
use the expressions from the previous section to predict rates of interaction for specific
phylogenies. To this end, we will make use of a classical quantitative genetics approach that
assumes weak selection, fixed additive genetic variance, and Gaussian distributions of
phenotypes.
93
Brownian motion caused by random genetic drift
94
96
Here, we assume evolutionary change in each species within the community is driven by
random genetic drift alone. If the additive genetic variance (𝐺𝑖 ) and population size (𝑁𝑖 ) for
species i is constant, the population mean phenotype in the next generation is given by:
97
𝑧̅𝑖 ′ = 𝑧̅𝑖 + 𝛿𝑖
98
where 𝛿𝑖 is a random variable with mean zero and variance equal to 𝐺𝑖 ⁄𝑁𝑖 .
99
100
Equation (S15) can be used to predict the expected value of the trait mean in the next
generation:
101
𝜇𝑖′ = 𝜇𝑖
102
103
verifying, as expected, that genetic drift does not cause directional change in mean phenotype
over time. Next, we can use (S16) to predict the variance in the trait mean for species i:
104
𝜎𝑧̅2𝑖 = 𝜎𝑧̅2𝑖 + 𝑁𝑖
105
and the covariance between trait means in species i and species j in the next generation:
106
𝜎𝑧̅𝑖 ,𝑧̅𝑗 ′ = 𝜎𝑧̅𝑖 ,𝑧̅𝑗 .
107
Because we have assumed weak evolutionary forces such that changes in mean phenotypes are
small across generations, we can equally well describe equations (S16-S18) using the following
system of ordinary differential equations:
86
87
88
89
90
91
95
108
109
110
111
′
(S15)
(S16)
𝐺
𝑖
𝑑𝜇𝑖
=0
𝑑𝑡
𝑑𝜎𝑧̅2
𝑖
𝑑𝑡
=
𝐺𝑖
𝑁
Supplemental online material for S.L. Nuismer and L.J. Harmon
(S17)
(S18)
(S19)
(S20)
4
112
𝑑𝜎𝑧̅𝑖 ,𝑧̅𝑗
𝑑𝑡
=0
(S21)
113
114
Ornstein-Uhlenbeck caused by random genetic drift and stabilizing selection
115
118
Here, we assume evolutionary change in each species within the community is driven by
random genetic drift and stabilizing selection toward some fixed adaptive optimum 𝜃𝑖 . We will
model genetic drift as in the last section, and integrate stabilizing selection using the classical
quantitative genetic expression:
119
∆𝑧̅ = 𝐺 𝑊
̅
120
This expression predicts the value of the population mean trait value in the next generation as a
̅ , and its additive genetic variance (Lande 1976).
function of its population means fitness, 𝑊
Clearly, in order to use equation (S22) to predict evolution, we must formulate expressions for
̅ , of each species within the community.
the population mean fitness, 𝑊
116
117
121
122
123
̅
1 𝜕𝑊
𝜕𝑧̅
(S22)
125
We model stabilizing selection toward some fixed optimum using the standard
expression:
126
𝑊𝑖 = 𝐸𝑥𝑝[−𝛾(𝑧𝑖 − 𝜃𝑖 )2 ]
127
129
where 𝛾 is the intensity of stabilizing selection toward the phenotypic optimum for species i, 𝜃𝑖 .
So long as stabilizing selection is relatively weak (𝛾 of small order 𝜖), individual fitness can be
approximated by:
130
𝑊𝑖 ≈ 1 − 𝛾(𝑧𝑖 − 𝜃𝑖 )2
131
132
Equation (S24) can then be used to calculate the population mean fitness of the focal species, i,
by integrating it over the phenotype frequency distribution of the species:
133
̅𝑖 ≈ 1 − ∫ 𝛾(𝑧𝑖 − 𝜃𝑖 )2 𝜙𝑧 𝑑𝑧 = (1 − 𝛾((𝑧̅𝑖 − 𝜃𝑖 )2 + 𝜎𝑧2 ))
𝑊
𝑖
𝑖
𝑖
134
137
where 𝜎𝑧2𝑖 is the phenotypic variance of trait 𝑧𝑖 within species i. Applying Lande’s formula (S22)
and incorporating random genetic drift as a random variable, 𝛿𝑖 , with mean zero and variance
equal to 𝐺𝑖 ⁄𝑁𝑖 , yields an expression for the population mean phenotype of species i in the next
generation:
138
𝑧̅𝑖 ′ = 𝜓𝑖 (𝜃𝑖 − 𝑧̅𝑖 ) + 𝛿𝑖
139
where 𝜓𝑖 = 2𝛾𝐺𝑖 .
124
128
135
136
Supplemental online material for S.L. Nuismer and L.J. Harmon
(S23)
(S24)
(S25)
(S26)
5
141
Equation (S25) can now be used to predict the expected value of the trait mean in the
next generation:
142
𝜇𝑖′ = 𝜇𝑖 + 𝜓𝑖 (𝜃𝑖 − 𝜇𝑖 )
143
145
showing that stabilizing selection pulls the expected value of the trait mean in toward the
optimal value 𝜃𝑖 over time. Next, we can use (S25) to predict the variance in the trait mean for
species i:
146
𝜎𝑧̅2𝑖 = (1 − 2𝜓)𝜎𝑧̅2𝑖 + 𝑁𝑖
147
and the covariance between trait means in species i and species j in the next generation:
148
𝜎𝑧̅𝑖 ,𝑧̅𝑗 ′ = (1 − 2𝜓)𝜎𝑧̅𝑖 ,𝑧̅𝑗 .
149
Because we have assumed weak evolutionary forces such that changes in mean phenotypes are
small across generations, we can equally well describe equations (S27-S28) using the following
system of ordinary differential equations:
140
144
150
151
152
153
154
155
′
(S27)
𝐺
(S28)
𝑖
𝑑𝜇𝑖
(S29)
= 𝜓𝑖 (𝜃𝑖 − 𝜇𝑖 )
𝑑𝑡
𝑑𝜎𝑧̅2
𝑖
𝑑𝑡
𝐺
= −2𝜓𝜎𝑧̅2𝑖 + 𝑁
𝑑𝜎𝑧̅𝑖 ,𝑧̅𝑗
𝑑𝑡
(S30)
(S31)
= −2𝜓𝜎𝑧̅𝑖 ,𝑧̅𝑗
(S32)
where (S30-S32) assume all species are interchangeable.
156
157
Coevolution mediated by phenotype matching
158
160
We now assume that in addition to random genetic drift and stabilizing selection,
evolution is driven by interactions among species. Specifically, we study coevolution mediated
by a model of phenotype matching where the fitness of each species is given by:
161
𝑊𝑖 = 𝐸𝑥𝑝[−𝛾(𝑧𝑖 − 𝜃𝑖 )2 ] ∑𝑛𝑗=1 𝑓𝑗 (1 + 𝜉𝑖 𝐸𝑥𝑝 [−𝛼(𝑧𝑖 − 𝑧𝑗 ) ])
162
where 𝜉𝑖 measures the fitness consequences of species interactions for the focal species, 𝑓𝑗 is
the frequency of species j within the community; all other parameters are as defined
previously. So long as stabilizing selection is weak weak (𝛾 of small order 𝜖), and the outcome of
encounters not too sensitive to individual phenotypes (𝛼 of small order 𝜖), individual fitness can
be approximated by:
159
163
164
165
166
2
Supplemental online material for S.L. Nuismer and L.J. Harmon
(S33)
6
2
167
𝑊𝑖 ≈ (1 + 𝜉𝑖 )(1 − 𝛾(𝑧𝑖 − 𝜃𝑖 )2 ) − 𝛼𝜉𝑖 ∑𝑛𝑗=1 𝑓𝑗 (𝑧𝑖 − 𝑧𝑗 )
168
169
If individuals of species i encounter individuals of each species j at random, equation (S32) can
be used to calculate the expected fitness of an individual of species i with phenotype 𝑧𝑖 :
170
𝑊𝑖 ≈ (1 + 𝜉𝑖 )(1 − 𝛾(𝑧𝑖 − 𝜃𝑖 )2 ) − 𝛼𝜉𝑖 ∑𝑛𝑗=1 𝑓𝑗 ((𝑧𝑖 − 𝑧̅𝑗 ) + 𝜎𝑧2𝑗 )
171
where 𝜎𝑧2𝑗 is the phenotypic variance of trait 𝑧𝑗 within species j. Equation (S32) can then be used
172
to calculate the population mean fitness of the focal species, i, which equals:
173
̅𝑖 ≈ (1 + 𝜉𝑖 ) (1 − 𝛾((𝑧̅𝑖 − 𝜃𝑖 )2 + 𝜎𝑧2 )) − 𝛼𝜉𝑖 ∑𝑛𝑗=1 𝑓𝑗 ((𝑧̅𝑖 − 𝑧̅𝑗 )2 + 𝜎𝑧2 + 𝜎𝑧2 )
𝑊
𝑖
𝑗
𝑖
(S34)
2
(S35)
(S36)
175
Applying Lande’s formula (S21) yields an expression for the population mean phenotype of
species i in the next generation:
176
z̅𝑖′ = z̅𝑖 + 𝜓𝑖 (𝜃𝑖 − 𝑧̅𝑖 ) + 𝑆𝑖 ∑𝑛𝑗=1 𝑓𝑗 (𝑧̅𝑗 − 𝑧̅𝑖 ) + 𝛿𝑖
177
where 𝜓𝑖 = 2𝛾𝐺𝑖 and 𝑆𝑖 =
178
Assuming all species parameters are interchangeable, (S37) simplifies to:
179
z̅𝑖 ′ = z̅𝑖 + 𝑆(𝜇 − z̅𝑖 ) + 𝜓(𝜃 − z̅𝑖 ) + 𝛿
180
where 𝜇 is the expected value of mean phenotypes within the community.
181
182
We can now use (S38) to predict the expected value of a species’ mean phenotype in
the next generation:
183
𝜇𝑖′ = 𝜇𝑖 + 𝜓(𝜃 − 𝜇𝑖 )
184
Next, we can use (S39) to predict the variance in the trait mean for species i:
185
𝜎𝑧̅2𝑖 = (1 −
174
′
2(𝑛−1)
𝑛
2𝛼𝜉𝑖 𝐺𝑖
(1+𝜉𝑖 )
𝑆 − 2𝜓) 𝜎𝑧̅2𝑖 +
(S37)
.
(S38)
(S39)
2𝑆
𝑛
𝐺
(∑𝑛𝑗=1 𝜎𝑧̅𝑖 ,𝑧̅𝑗 ) + 𝑁
(S40)
(𝑗≠𝑖)
186
and the covariance between trait means in species i and species j in the next generation:
187
𝜎𝑧̅𝑖 ,𝑧̅𝑗 ′ = (1 −
188
189
190
2(𝑛−1)
𝑛
𝑆
𝑆 − 2𝜓) 𝜎𝑧̅𝑖 ,𝑧̅𝑗 + 𝑛 (∑𝑛𝑘=1 𝜎𝑧̅𝑖 ,𝑧̅𝑘 + ∑𝑛𝑙=1 𝜎𝑧̅𝑗,𝑧̅𝑙 )
𝑘≠𝑗
(S41)
𝑙≠𝑖
Because we have assumed weak evolutionary forces such that changes in mean
phenotypes are small across generations, we can equally well describe equations (S39-S41)
using the following system of ordinary differential equations:
Supplemental online material for S.L. Nuismer and L.J. Harmon
7
191
192
193
194
195
𝑑𝜇𝑖
= 𝜓(𝜃 − 𝜇𝑖 )
𝑑𝑡
𝑑𝜎𝑧̅2
𝑖
𝑑𝑡
𝑑𝜎𝑧̅𝑖 ,𝑧̅𝑗
𝑑𝑡
2(𝑛−1)
= −(
𝑛
(S42)
𝑆 + 2𝜓) 𝜎𝑧̅2𝑖 +
2(𝑛−1)
= −(
𝑛
2𝑆
𝑛
𝐺
(∑𝑛𝑗=1 𝜎𝑧̅𝑖 ,𝑧̅𝑗 ) + 𝑁
(S43)
(𝑗≠𝑖)
𝑆
𝑆 + 2𝜓) 𝜎𝑧̅𝑖 ,𝑧̅𝑗 + 𝑛 (∑𝑛𝑘=1 𝜎𝑧̅𝑖 ,𝑧̅𝑘 + ∑𝑛𝑙=1 𝜎𝑧̅𝑗,𝑧̅𝑙 )
𝑘≠𝑗
(S44)
𝑙≠𝑖
Expressions (S42-S44) appear as equations (10) in the main text.
196
197
Coevolution mediated by phenotype differences
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As was the case for coevolution mediated by phenotype matching, the fitness of each
species depends on both coevolutionary selection and stabilizing selection toward some fixed
optimum:
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𝑊𝑖 = 𝐸𝑥𝑝[−𝛾(𝑧𝑖 − 𝜃𝑖 )2 ] ∑𝑛𝑗=1 𝑓𝑗 (1 + 𝜉𝑖 1+𝐸𝑥𝑝[𝛼(𝑧 −𝑧 )])
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where all parameters are as previously described. So long as stabilizing selection is weak (𝛾 of
small order 𝜖), and the outcome of encounters not too sensitive to individual phenotypes (𝛼 of
small order 𝜖), individual fitness can be approximated by:
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𝑊𝑖 ≈ (1 + 2𝑖 ) (1 − 𝛾(𝑧𝑖 − 𝜃𝑖 )2 ) +
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If individuals of species i encounter individuals of each species j at random, equation (S46) can
be used to calculate the expected fitness of an individual of species i with phenotype 𝑧𝑖 :
208
𝑊𝑖 ≈ (1 + 2𝑖 ) (1 − 𝛾(𝑧𝑖 − 𝜃𝑖 )2 ) + 𝛼𝜉𝑖 ∑𝑛𝑗=1 𝑓𝑗 (𝑧̅𝑗 − 𝑧𝑖 )
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Equation (S47) can then be used to calculate the population mean fitness of the focal species, i,
which equals:
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̅𝑖 ≈ (1 + 𝜉𝑖 ) (1 − 𝛾((𝑧̅𝑖 − 𝜃𝑖 )2 + 𝑉[𝑧𝑖 ])) + 𝛼𝜉𝑖 ∑𝑛𝑗=1 𝑓𝑗 (𝑧̅𝑗 − 𝑧̅𝑖 )
𝑊
2
4
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Applying Lande’s formula (S22) yields an expression for the population mean phenotype of
species i in the next generation:
214
z̅𝑖′ = z̅𝑖 + 𝜓𝑖 (𝜃𝑖 − 𝑧̅𝑖 ) + 𝑆𝑖 + 𝛿𝑖
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1
𝑖
𝜉
𝛼𝜉𝑖
4
𝑗
∑𝑛𝑗=1 𝑓𝑗 (𝑧𝑗 − 𝑧𝑖 )
𝜉
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(S45)
(S46)
(S47)
(S48)
(S49)
8
𝛼𝜉 𝐺
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𝑖 𝑖
where 𝜓𝑖 = 2𝛾𝐺𝑖 and 𝑆𝑖 = 2(2+𝜉
.
)
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Assuming all species parameters are interchangeable (S49) simplifies to:
217
z̅𝑖 ′ = z̅𝑖 + 𝑆 + 𝜓(𝜃 − z̅𝑖 ) + 𝛿
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219
We can now use (S50) to predict the expected value of a species’ mean phenotype in
the next generation:
220
𝜇𝑖 ′ = 𝜇𝑖 + 𝜓(𝜃 − 𝜇𝑖 ) + 𝑆
221
the variance of a species’ mean phenotype in the next generation:
222
𝜎𝑧̅2𝑖 = (1 − 2𝜓)𝜎𝑧̅2𝑖 + 𝑁
223
and the covariance between mean phenotypes in species i and j:
224
𝜎𝑧̅𝑖 ,𝑧̅𝑗 ′ = (1 − 2𝜓)𝜎𝑧̅𝑖 ,𝑧̅𝑗
225
Because we have assumed weak evolutionary forces such that changes in mean
phenotypes are small across generations, we can equally well describe equations (S51-S53)
using the following system of ordinary differential equations:
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𝑖
′
𝑑𝜇𝑖
𝐺
= 𝜓(𝜃 − Z̅) + 𝑆
𝑑𝑡
𝑑𝜎𝑧̅2
𝑖
𝑑𝑡
𝑑𝜎𝑧̅𝑖 ,𝑧̅𝑗
𝑑𝑡
𝐺
= −2𝜓𝜎𝑧̅2𝑖 + 𝑁
= −2𝜓𝜎𝑧̅𝑖 ,𝑧̅𝑗
(S50)
(S51)
(S52)
(S53)
(S54)
(S55)
(S56)
231
Expressions (S54-S56) appear as equations (11) in the main text.
232
Section 3. Merging models of trait evolution with phylogenetic trees
233
In the section above, we derived systems of ordinary differential equations describing
how the multivariate distribution describing species mean trait values within a community of
fixed size evolve. Thus, these systems of equations describe evolution during time intervals
where no species form or go extinct. To consider these models along a phylogenetic tree, we
divide the phylogenetic tree into time intervals with speciation events and time intervals
without speciation events. We assume that speciation is instantaneous and that only one
speciation event can occur in each such interval; e.g. intervals that include speciation involve
only a transition of one lineage to two, and nothing else. Furthermore, we assume that both
daughter lineages inherit the traits of the ancestral lineage. Any phylogenetic tree can be thus
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divided into (n-1) speciation intervals and (n-1) between-speciation intervals. We ignore the
potential for extinction. By considering evolution in each of these intervals, and progressing
from the root of the tree to the tips, we can predict contemporary values for the variances and
covariances that define the multivariate distribution of species’ mean trait values. Specifically,
within each time interval where speciation does not occur, we solve the system of differential
equations for the current number of species in the community. When a time interval in which
speciation occurs is reached, we increase the number of differential equations and use the
solutions from the previous time interval and our assumption that new species initially share
their parental species trait value to provide initial conditions. We then solve this new system of
differential equations. This process continues until the present time is reached, yielding
expressions for the variance covariance matrix describing contemporary species mean trait
values within the community. With this information in hand, we can use the expressions
derived in the first section to predict interaction rates and ecosystem function.
263
When trait evolution does not involve coevolution, instead resulting from genetic drift
in isolation (Brownian motion) or genetic drift and stabilizing selection (Ornstein-Uhlenbeck),
deriving the solution for the variance covariance matrix is straightforward and has been
previously solved (Hansen and Martins 1996; Hansen 1997). The reason the solution is so
straightforward in these cases is that all lineages evolve independently of one another. Solving
equations (S19-S21) for a phylogeny described by a total time since clade origin τ and set of
times from clade origin to speciation, tij, for all species pairs i and j yields the following
equations describing the distribution of species mean trait values expected when evolution is
driven by random genetic drift (Brownian motion):
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𝜇𝑖 = 𝜇𝑗 = 𝜇0
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𝜎𝑧̅2𝑖 = 𝜎𝑧̅2𝑗 = 𝑁 𝜏
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𝜎𝑧̅𝑖 ,𝑧̅𝑗 = 𝑁 𝑡𝑖,𝑗
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Equations (S57-59) can be substituted into expressions (S7 or S13) to predict pairwise rates of
interaction or into expressions (S8 or S14) to predict overall rates of interaction within the
community. Solving equations (S30-S32) for a phylogeny described by a total time since clade
origin, τ, and set of times from clade origin to speciation, tij, for all species pairs i and j yields the
following equations describing the distribution of species mean trait values expected when
evolution is driven by random genetic drift and stabilizing selection (Ornstein-Uhlenbeck).
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𝜇𝑖 = 𝜇𝑗 = θ
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𝐺
𝐺
(S57)
(S58)
(S59)
(S60)
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1−𝐸𝑥𝑝[−2𝜓𝜏]
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𝜎𝑧̅2𝑖 = 𝜎𝑧̅2𝑗 =
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𝜎𝑧̅𝑖 ,𝑧̅𝑗 =
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Equations (S60-62) can be substituted into expressions (S7 or S13) to predict pairwise rates of
interaction or into expressions (S8 or S14) to predict overall rates of interaction within the
community.
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𝜓𝑁
𝐺
(S61)
𝐸𝑥𝑝[−2𝜓𝜏](𝐸𝑥𝑝[2𝜓𝑡𝑖,𝑗 ]−1)
2𝑁𝜓
𝐺
(S62)
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When coevolution plays a role in trait evolution, solutions become more elusive and
quite complex. In fact, the solutions for even very small trees are much too complex to include
in this appendix. For this reason, our analytical solutions for small phylogenetic trees in which
coevolution plays a role are provided in the form of a Mathematica notebook. For larger trees,
solutions require numerical or simulation based approaches.
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Section 4: Individual based simulations
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To test our analytic predictions, we simulated communities of n interacting species where each
species is composed of Ni individuals, each with a particular value of the continuous phenotype
zik. Each simulation begins with a simple community containing only a single ancestral species.
Over evolutionary time, the size of this community grows as speciation occurs. Within each
generation, individuals follow a life cycle consisting of stabilizing selection, species interactions,
selective mortality, and finally, reproduction. Each of these events in the simulated life cycle is
described in greater detail in the following sections.
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Stabilizing Selection
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At the beginning of each generation, all individuals are assumed to have fitness equal to W0.
Next, all individuals experience stabilizing selection, which reduces the fitness of individuals
deviating from the phenotypic optimum:
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𝑊 ′ = 𝑊0 𝐸𝑥𝑝[−𝛾(𝑧𝑖,𝑘 − 𝜃) ]
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Where 𝛾 is the strength of stabilizing selection, 𝜃 is the phenotypic optimum, and 𝑧𝑖𝑘 is the
phenotype of the individual k of species i.
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Species Interactions
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Next, individuals encounter a randomly selected individual from a randomly selected species
within the community. We can consider a focal individual with phenotype 𝑧𝑖 and another
randomly selected individual with phenotype 𝑧𝑗 . The probability that this encounter causes
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2
(S63)
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fitness consequences for the focal individual depends on our model of species interactions. If
the interaction is mediated by a mechanism of phenotype matching:
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𝑃𝑖,𝑗 = 𝐸𝑥𝑝 [−𝛼(𝑧𝑖 − 𝑧𝑗 ) ]
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If the interaction is mediated by a mechanism of phenotype differences:
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𝑃𝑖,𝑗 = 1+𝐸𝑥𝑝[𝛼(𝑧 −𝑧
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In both cases, 𝛼 measures the sensitivity of interaction outcome to the phenotypes of the
individuals. After calculating the probability the interaction has fitness consequences, 𝑃𝑖,𝑗 , a
random number is drawn from a uniform distribution on {0,1}. If this number is less than 𝑃𝑖,𝑗 , an
interaction occurs, such that the fitness of the focal individual is now:
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𝑊 ′′ = 𝑊 ′ (1 + 𝜉𝑖 )
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309
310
2
1
𝑖
𝑗 )]
(S64)
(S65)
(S66)
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Where 𝜉𝑖 measures the fitness consequences of interacting for species i and is positive if the
interaction is beneficial but negative if the interaction is harmful. If, instead, the random
number is greater than 𝑃𝑖,𝑗 , no interaction occurs and the fitness of the focal individual remains
unchanged:
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𝑊 ′′ = 𝑊 ′
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Interacting individuals are drawn at random until every individual within every species has
encountered exactly one other individual.
321
Selective Mortality
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Following species interactions, selective mortality occurs based on individual fitness.
Specifically, for each individual a random number is drawn from a uniform distribution on {0,1}.
If this number falls below the individual’s fitness, 𝑊 ′′ , the individual survives. If, instead, the
random number exceeds the individual’s fitness they are eliminated.
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Reproduction
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Following selective mortality surviving individuals mate and reproduce. This occurs by drawing
a pair of random individuals within a species and using them to produce a single offspring with
a phenotype drawn from a Gaussian distribution with mean equal to the phenotypic average of
the parents and variance equal to the segregational variance 𝜎𝑆2 . The segregational variance
captures phenotypic variation which arises afresh every generation as a consequence of
segregation, recombination, and mutation. Individuals continue to mate and reproduce until N
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(S67)
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offspring are formed. These N offspring form the next generation for the species with all
parental individuals dying post-reproduction.
335
Speciation
336
Speciation is integrated into simulations according to a pre-determined phylogeny. When the
simulation reaches a bifurcation within the phylogeny, the appropriate existing species gives
rise to a daughter species. This new species is created by sampling a number, N, of individuals
at random from the ancestral species. Thus, the population mean phenotype of the new species
is, on average, equal to that of its progenitor. However, because the individuals of the new
species are selected at random, the population mean phenotype of the new species is
stochastic, with some amount of variance caused by random genetic drift. Consequently, the
change in mean phenotype which occurs at speciation will be greatest when the population size
of the new species is small.
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References
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Hansen, T. F. 1997. Stabilizing selection and the comparative analysis of adaptation. EVOLUTION
51:1341-1351.
Hansen, T. F., and E. P. Martins. 1996. Translating between microevolutionary process and
macroevolutionary patterns: The correlation structure of interspecific data. EVOLUTION
50:1404-1417.
Lande, R. 1976. Natural selection and random genetic drift in phenotypic evolution. Evolution 30:314334.
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