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 ππ§ 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 π§π encounters another individual with phenotype π§π , they interact with probability π(π§π , π§π ). 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 ππ,π = πΈπ₯π [−πΌ(π§π − π§π ) ] 17 20 where π§π is the phenotype of the focal individual, π§π is the phenotype of the individual it encounters, and πΌ 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 ππ,π = 1+πΈπ₯π[πΌ(π§ −π§ 22 25 where π§π is the phenotype of the focal individual, π§π is the phenotype of the individual it encounters, and πΌ 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 πΌ is sufficiently small, and the distributions of π§π and π§π not too disparate, equation (S1) can be accurately approximated by a first-order Taylor Series expansion: 29 ππ,π ≈ 1 − πΌ(π§π − π§π ) 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 π (S1) (S2) π )] 2 Supplemental online material for S.L. Nuismer and L.J. Harmon (S3) 1 32 2 πΜ π,π = ππ ππ ∫ ∫ (1 − πΌ(π§π − π§π ) ) ππ§π ππ§π ππ§π ππ§π 33 Here ππ is the frequency of species i within the community and ππ§π is the probability density for 34 35 the phenotypic trait value π§π of species i. We can now solve the integrals in (S4) for a probability distribution function ππ§π with mean π§Μ π and variance ππ§2π : 36 2 πΜ π,π = ππ ππ (1 − πΌ(π§Μ π − π§Μ π ) − πΌ (ππ§2π + ππ§2π )) (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 πΜ 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 Μπ,π = ππ ππ ∫ ∫ (1 − πΌ(π§Μ π − π§Μ π )2 − πΌ (ππ§2 + ππ§2 )) ππ§Μ ,π§Μ ππ§Μ ππ§Μ P π π π π π π 45 where ππ§Μ π ,π§Μ π is the bivariate frequency distribution of the trait means π§Μ π and π§Μ π . Carrying out the 46 integrals in (S6) yields: 47 Μπ,π = ππ ππ (1 − πΌ(ππ − ππ )2 − πΌ (ππ§Μ 2 + ππ§Μ 2 − 2ππ§Μ ,π§Μ ) − πΌ (πΜ π§2 + πΜ π§2 )) P π π π π π π 37 38 39 40 41 42 (S6) (S7) 49 Here ππ is the expected population mean phenotype for species i across realizations of the evolutionary process, ππ§Μ 2π is the variance in the mean phenotype for species i across realizations 50 of the evolutionary process, ππ§Μ π ,π§Μ π is the covariance in the mean phenotypes of species i and j 51 across realizations, and πΜ π§2π 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 ππ§Μ 2π and ππ§Μ 2π 55 as well as the covariance ππ§Μ π ,π§Μ π . 56 57 Result (S7) can also be used to predict the average rate of interaction within the entire community: 58 ΜΏ = ∑ππ=1 ∑ππ=1 (ππ ππ (1 − πΌ(ππ − ππ )2 − πΌ (ππ§Μ 2 + ππ§Μ 2 − 2ππ§Μ ,π§Μ ) − πΌ (πΜ π§2 + πΜ π§2 ))) P π π π π π π 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 ππ§Μ 2π and ππ§Μ 2π and covariances ππ§Μ π ,π§Μ π of interacting species. 61 Phenotype Differences 62 63 As long as α is small and the distributions of π§π and π§π not too disparate, (S2) can be accurately approximated by: 64 ππ,π ≈ 2 + 4 (π§π − π§π ) 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 πΌ 1 πΌ πΜ π,π = ππ ππ ∫ ∫ (2 + 4 (π§π − π§π )) ππ§π ππ§π ππ§π ππ§π 1 πΌ πΜ π,π = ππ ππ (2 − 4 (π§Μ π − π§Μ π )) 74 Evaluating the integrals yields: 75 1 πΌ Μ Pπ,π = ππ ππ (2 − 4 (ππ − ππ )) 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 πΌ Μ Pπ,π = ππ ππ ∫ ∫ (2 − 4 (π§Μ π − π§Μ π )) ππ§Μ π ,π§Μ π ππ§Μ π ππ§Μ π 77 (S10) We can now solve the integrals in (S4) for a probability distribution function ππ§π with mean π§Μ π and variance ππ§2π : 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 πΌ 1 ΜΏ P = ∑ππ=1 ∑ππ=1 (ππ ππ (2 − 4 (ππ − ππ ))) = 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 (πΊπ ) and population size (ππ ) for species i is constant, the population mean phenotype in the next generation is given by: 97 π§Μ π ′ = π§Μ π + πΏπ 98 where πΏπ is a random variable with mean zero and variance equal to πΊπ ⁄ππ . 99 100 Equation (S15) can be used to predict the expected value of the trait mean in the next generation: 101 ππ′ = ππ 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 ππ§Μ 2π = ππ§Μ 2π + ππ 105 and the covariance between trait means in species i and species j in the next generation: 106 ππ§Μ π ,π§Μ π ′ = ππ§Μ π ,π§Μ π . 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) πΊ π πππ =0 ππ‘ πππ§Μ 2 π ππ‘ = πΊπ π Supplemental online material for S.L. Nuismer and L.J. Harmon (S17) (S18) (S19) (S20) 4 112 πππ§Μ π ,π§Μ π ππ‘ =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 ππ . We will model genetic drift as in the last section, and integrate stabilizing selection using the classical quantitative genetic expression: 119 βπ§Μ = πΊ π Μ 120 This expression predicts the value of the population mean trait value in the next generation as a Μ , and its additive genetic variance (Lande 1976). function of its population means fitness, π Clearly, in order to use equation (S22) to predict evolution, we must formulate expressions for Μ , of each species within the community. the population mean fitness, π 116 117 121 122 123 Μ 1 ππ ππ§Μ (S22) 125 We model stabilizing selection toward some fixed optimum using the standard expression: 126 ππ = πΈπ₯π[−πΎ(π§π − ππ )2 ] 127 129 where πΎ is the intensity of stabilizing selection toward the phenotypic optimum for species i, ππ . So long as stabilizing selection is relatively weak (πΎ of small order π), individual fitness can be approximated by: 130 ππ ≈ 1 − πΎ(π§π − ππ )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 Μ π ≈ 1 − ∫ πΎ(π§π − ππ )2 ππ§ ππ§ = (1 − πΎ((π§Μ π − ππ )2 + ππ§2 )) π π π π 134 137 where ππ§2π is the phenotypic variance of trait π§π within species i. Applying Lande’s formula (S22) and incorporating random genetic drift as a random variable, πΏπ , with mean zero and variance equal to πΊπ ⁄ππ , yields an expression for the population mean phenotype of species i in the next generation: 138 π§Μ π ′ = ππ (ππ − π§Μ π ) + πΏπ 139 where ππ = 2πΎπΊπ . 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 ππ′ = ππ + ππ (ππ − ππ ) 143 145 showing that stabilizing selection pulls the expected value of the trait mean in toward the optimal value ππ over time. Next, we can use (S25) to predict the variance in the trait mean for species i: 146 ππ§Μ 2π = (1 − 2π)ππ§Μ 2π + ππ 147 and the covariance between trait means in species i and species j in the next generation: 148 ππ§Μ π ,π§Μ π ′ = (1 − 2π)ππ§Μ π ,π§Μ π . 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) πΊ (S28) π πππ (S29) = ππ (ππ − ππ ) ππ‘ πππ§Μ 2 π ππ‘ πΊ = −2πππ§Μ 2π + π πππ§Μ π ,π§Μ π ππ‘ (S30) (S31) = −2πππ§Μ π ,π§Μ π (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 ππ = πΈπ₯π[−πΎ(π§π − ππ )2 ] ∑ππ=1 ππ (1 + ππ πΈπ₯π [−πΌ(π§π − π§π ) ]) 162 where ππ measures the fitness consequences of species interactions for the focal species, ππ is the frequency of species j within the community; all other parameters are as defined previously. So long as stabilizing selection is weak weak (πΎ of small order π), and the outcome of encounters not too sensitive to individual phenotypes (πΌ of small order π), 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 ππ ≈ (1 + ππ )(1 − πΎ(π§π − ππ )2 ) − πΌππ ∑ππ=1 ππ (π§π − π§π ) 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 π§π : 170 ππ ≈ (1 + ππ )(1 − πΎ(π§π − ππ )2 ) − πΌππ ∑ππ=1 ππ ((π§π − π§Μ π ) + ππ§2π ) 171 where ππ§2π is the phenotypic variance of trait π§π within species j. Equation (S32) can then be used 172 to calculate the population mean fitness of the focal species, i, which equals: 173 Μ π ≈ (1 + ππ ) (1 − πΎ((π§Μ π − ππ )2 + ππ§2 )) − πΌππ ∑ππ=1 ππ ((π§Μ π − π§Μ π )2 + ππ§2 + ππ§2 ) π π π π (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 zΜ π′ = zΜ π + ππ (ππ − π§Μ π ) + ππ ∑ππ=1 ππ (π§Μ π − π§Μ π ) + πΏπ 177 where ππ = 2πΎπΊπ and ππ = 178 Assuming all species parameters are interchangeable, (S37) simplifies to: 179 zΜ π ′ = zΜ π + π(π − zΜ π ) + π(π − zΜ π ) + πΏ 180 where π 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 ππ′ = ππ + π(π − ππ ) 184 Next, we can use (S39) to predict the variance in the trait mean for species i: 185 ππ§Μ 2π = (1 − 174 ′ 2(π−1) π 2πΌππ πΊπ (1+ππ ) π − 2π) ππ§Μ 2π + (S37) . (S38) (S39) 2π π πΊ (∑ππ=1 ππ§Μ π ,π§Μ π ) + π (S40) (π≠π) 186 and the covariance between trait means in species i and species j in the next generation: 187 ππ§Μ π ,π§Μ π ′ = (1 − 188 189 190 2(π−1) π π π − 2π) ππ§Μ π ,π§Μ π + π (∑ππ=1 ππ§Μ π ,π§Μ π + ∑ππ=1 ππ§Μ π,π§Μ π ) π≠π (S41) π≠π 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 πππ = π(π − ππ ) ππ‘ πππ§Μ 2 π ππ‘ πππ§Μ π ,π§Μ π ππ‘ 2(π−1) = −( π (S42) π + 2π) ππ§Μ 2π + 2(π−1) = −( π 2π π πΊ (∑ππ=1 ππ§Μ π ,π§Μ π ) + π (S43) (π≠π) π π + 2π) ππ§Μ π ,π§Μ π + π (∑ππ=1 ππ§Μ π ,π§Μ π + ∑ππ=1 ππ§Μ π,π§Μ π ) π≠π (S44) π≠π Expressions (S42-S44) appear as equations (10) in the main text. 196 197 Coevolution mediated by phenotype differences 198 200 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: 201 ππ = πΈπ₯π[−πΎ(π§π − ππ )2 ] ∑ππ=1 ππ (1 + ππ 1+πΈπ₯π[πΌ(π§ −π§ )]) 202 204 where all parameters are as previously described. So long as stabilizing selection is weak (πΎ of small order π), and the outcome of encounters not too sensitive to individual phenotypes (πΌ of small order π), individual fitness can be approximated by: 205 ππ ≈ (1 + 2π ) (1 − πΎ(π§π − ππ )2 ) + 206 207 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 π§π : 208 ππ ≈ (1 + 2π ) (1 − πΎ(π§π − ππ )2 ) + πΌππ ∑ππ=1 ππ (π§Μ π − π§π ) 209 210 Equation (S47) can then be used to calculate the population mean fitness of the focal species, i, which equals: 211 Μ π ≈ (1 + ππ ) (1 − πΎ((π§Μ π − ππ )2 + π[π§π ])) + πΌππ ∑ππ=1 ππ (π§Μ π − π§Μ π ) π 2 4 212 213 Applying Lande’s formula (S22) yields an expression for the population mean phenotype of species i in the next generation: 214 zΜ π′ = zΜ π + ππ (ππ − π§Μ π ) + ππ + πΏπ 199 203 1 π π πΌππ 4 π ∑ππ=1 ππ (π§π − π§π ) π Supplemental online material for S.L. Nuismer and L.J. Harmon (S45) (S46) (S47) (S48) (S49) 8 πΌπ πΊ 215 π π where ππ = 2πΎπΊπ and ππ = 2(2+π . ) 216 Assuming all species parameters are interchangeable (S49) simplifies to: 217 zΜ π ′ = zΜ π + π + π(π − zΜ π ) + πΏ 218 219 We can now use (S50) to predict the expected value of a species’ mean phenotype in the next generation: 220 ππ ′ = ππ + π(π − ππ ) + π 221 the variance of a species’ mean phenotype in the next generation: 222 ππ§Μ 2π = (1 − 2π)ππ§Μ 2π + π 223 and the covariance between mean phenotypes in species i and j: 224 ππ§Μ π ,π§Μ π ′ = (1 − 2π)ππ§Μ π ,π§Μ π 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: 226 227 228 229 230 π ′ πππ πΊ = π(π − ZΜ ) + π ππ‘ πππ§Μ 2 π ππ‘ πππ§Μ π ,π§Μ π ππ‘ πΊ = −2πππ§Μ 2π + π = −2πππ§Μ π ,π§Μ π (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 234 235 236 237 238 239 240 241 Supplemental online material for S.L. Nuismer and L.J. Harmon 9 242 243 244 245 246 247 248 249 250 251 252 253 254 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): 264 ππ = ππ = π0 265 ππ§Μ 2π = ππ§Μ 2π = π π 266 ππ§Μ π ,π§Μ π = π π‘π,π 267 272 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). 273 ππ = ππ = θ 255 256 257 258 259 260 261 262 268 269 270 271 πΊ πΊ (S57) (S58) (S59) (S60) Supplemental online material for S.L. Nuismer and L.J. Harmon 10 1−πΈπ₯π[−2ππ] 274 ππ§Μ 2π = ππ§Μ 2π = 275 ππ§Μ π ,π§Μ π = 276 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. 277 278 ππ πΊ (S61) πΈπ₯π[−2ππ](πΈπ₯π[2ππ‘π,π ]−1) 2ππ πΊ (S62) 283 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. 284 Section 4: Individual based simulations 285 291 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. 292 Stabilizing Selection 293 295 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: 296 π ′ = π0 πΈπ₯π[−πΎ(π§π,π − π) ] 297 298 Where πΎ is the strength of stabilizing selection, π is the phenotypic optimum, and π§ππ is the phenotype of the individual k of species i. 299 Species Interactions 300 Next, individuals encounter a randomly selected individual from a randomly selected species within the community. We can consider a focal individual with phenotype π§π and another randomly selected individual with phenotype π§π . The probability that this encounter causes 279 280 281 282 286 287 288 289 290 294 301 302 2 (S63) Supplemental online material for S.L. Nuismer and L.J. Harmon 11 304 fitness consequences for the focal individual depends on our model of species interactions. If the interaction is mediated by a mechanism of phenotype matching: 305 ππ,π = πΈπ₯π [−πΌ(π§π − π§π ) ] 306 If the interaction is mediated by a mechanism of phenotype differences: 307 ππ,π = 1+πΈπ₯π[πΌ(π§ −π§ 308 311 In both cases, πΌ measures the sensitivity of interaction outcome to the phenotypes of the individuals. After calculating the probability the interaction has fitness consequences, ππ,π , a random number is drawn from a uniform distribution on {0,1}. If this number is less than ππ,π , an interaction occurs, such that the fitness of the focal individual is now: 313 312 π ′′ = π ′ (1 + ππ ) 303 309 310 2 1 π π )] (S64) (S65) (S66) 317 Where ππ 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 ππ,π , no interaction occurs and the fitness of the focal individual remains unchanged: 318 π ′′ = π ′ 319 320 Interacting individuals are drawn at random until every individual within every species has encountered exactly one other individual. 321 Selective Mortality 322 325 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, π ′′ , the individual survives. If, instead, the random number exceeds the individual’s fitness they are eliminated. 326 Reproduction 327 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 ππ2 . 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 314 315 316 323 324 328 329 330 331 332 (S67) Supplemental online material for S.L. Nuismer and L.J. Harmon 12 334 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. 333 337 338 339 340 341 342 343 344 345 346 347 348 349 Supplemental online material for S.L. Nuismer and L.J. Harmon 13 350 References 351 352 353 354 355 356 357 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. 358 359 Supplemental online material for S.L. Nuismer and L.J. Harmon 14