Estimating the Effect of Competition on Trait Evolution Using Maximum Likelihood Inference

—Many classical ecological and evolutionary theoretical frameworks posit that competition between species is an important selective force. For example, in adaptive radiations, resource competition between evolving lineages plays a role in driving phenotypic diversification and exploration of novel ecological space. Nevertheless, current models of trait evolution fit to phylogenies and comparative data sets are not designed to incorporate the effect of competition. The most advanced models in this direction are diversity-dependent models where evolutionary rates depend on lineage diversity. However, these models still treat changes in traits in one branch as independent of the value of traits on other branches, thus ignoring the effect of species similarity on trait evolution. Here, we consider a model where the evolutionary dynamics of traits involved in interspecific interactions are influenced by species similarity in trait values and where we can specify which lineages are in sympatry. We develop a maximum likelihood based approach to fit this model to combined phylogenetic and phenotypic data. Using simulations, we demonstrate that the approach accurately estimates the simulated parameter values across a broad range of parameter space. Additionally, we develop tools for specifying the biogeographic context in which trait evolution occurs. In order to compare models, we also apply these biogeographic methods to specify which lineages interact sympatrically for two diversity-dependent models. Finally, we fit these various models to morphological data from a classical adaptive radiation (Greater Antillean Anolis lizards). We show that models that account for competition and geography perform better than other models. The matching competition model is an important new tool for studying the influence of interspecific interactions, in particular competition, on phenotypic evolution. More generally, it constitutes a step toward a better integration of interspecific interactions in many ecological and evolutionary processes. [Adaptive radiation; Anolis; community phylogenetics; interspecific competition; maximum likelihood; phylogenetic comparative methods; trait evolution.] Interactions between species can be strong selective forces. Indeed, many classical evolutionary theories assume that interspecific competition has large impacts on fitness. Character displacement theory (Brown and Wilson 1956; Grant 1972; Pfennig and Pfennig 2009), for example, posits that interactions between species, whether in ecological or social contexts, drive adaptive changes in phenotypes. Similarly, adaptive radiation theory (Schluter 2000) has been a popular focus of investigators interested in explaining the rapid evolution of phenotypic disparity (Grant and Grant 2002; Losos 2009; Mahler et al. 2013; Weir and Mursleen 2013), and competitive interactions between species in a diversifying clade are a fundamental component of adaptive radiations (Schluter 2000; Losos and Ricklefs 2009; Grant and Grant 2011). Additionally, social interactions between species, whether in reproductive (Gröning and Hochkirch 2008; Pfennig and Pfennig 2009) or agonistic (Grether et al. 2009, 2013) contexts, are important drivers of changes in signal traits used in social interactions. Several evolutionary hypotheses predict that geographical overlap with closely related taxa should drive divergence in traits used to distinguish between conspecifics and heterospecifics (e.g., traits involved in mate recognition; Wallace 1889; Fisher 1930; Dobzhansky 1940; Mayr 1963; Gröning and Hochkirch 2008; Ord and Stamps 2009; Ord et al. 2011). Moreover, biologists interested in speciation have often argued that interspecific competitive interactions are important drivers of divergence between lineages that ultimately leads to reproductive isolation. Reinforcement (Dobzhansky 1937, 1940), for example, is often thought to be an important phase of speciation (Grant 1999; Coyne and Orr 2004; Rundle and Nosil 2005; Pfennig and Pfennig 2009) wherein selection against hybridization leads to a reduction in interspecific mate competition as a result of concomitant divergence in traits involved in mate recognition. In addition to the importance of interspecific competition in driving phenotypic divergence between species, competitive interactions are also central to many theories of community assembly, which posit that species with similar ecologies exclude each other from the community (Elton 1946). In spite of the importance of interspecific competition to these key ecological and evolutionary theories, the role of competition in driving adaptive divergence and species exclusion from ecological communities has been historically difficult to measure (Losos 2009), because both trait divergence and species exclusion resulting from competition between lineages during their evolutionary history have the effect of eliminating competition between those lineages at the present (i.e., the contemporary distribution of traits hold a signature of the “ghost of competition past,” Connell 1980). Community phylogeneticists have aimed to solve part of this conundrum by analyzing the phylogenetic structure of local communities: assuming that phylogenetic similarity between two species is a good proxy for their ecological similarity, competitive interactions are considered to have been more important in shaping communities comprising phylogenetically 700 by Jlien C lvel on Jne 0, 2016 http://sysbfordjournals.org/ D ow nladed from 2016 DRURY ET AL.—INTERSPECIFIC COMPETITION AND TRAIT EVOLUTION 701 (and, therefore, ecologically) distant species (Webb et al. 2002; Cavender-Bares et al. 2009). However, there is an intrinsic contradiction in this reasoning, because using phylogenetic similarity as a proxy for ecological similarity implicitly (or explicitly) assumes that traits evolved under a Brownian model of trait evolution, meaning that species interactions had no effect on trait divergence (Kraft et al. 2007; Cavender-Bares et al. 2009; Mouquet et al. 2012; Pennell and Harmon 2013). More generally, and despite the preponderance of classical evolutionary processes that assume that interspecific interactions have important fitness consequences, existing phylogenetic models treat trait evolution within a lineage as independent from traits in other lineages. For example, in the commonly used Brownian motion (BM) and Ornstein–Uhlenbeck models of trait evolution (Cavalli-Sforza and Edwards 1967; Felsenstein 1988; Hansen and Martins 1996), once an ancestor splits into two daughter lineages, the trait values in those daughter lineages do not depend on the trait values of sister taxa. Some investigators have indirectly incorporated the influence of interspecific interactions by fitting models where evolutionary rates at a given time depend on the diversity of lineages at that time (e.g., the “diversity-dependent” models of Mahler et al. 2010; Weir and Mursleen 2013). While these models capture some parts of the interspecific processes of central importance to evolutionary theory, such as the influence of ecological opportunity, they do not explicitly account for trait-driven interactions between lineages, as trait values in one lineage do not vary directly as a function of trait values in other evolving lineages. Recently, Nuismer and Harmon (2015) proposed a model where the evolution of a species’ trait depends on other species’ traits. In particular, they consider a model, which they refer to as the model of phenotype matching, where the probability that an encounter between two individuals has fitness consequences declines as the phenotypes of the individuals become more dissimilar. The consequence of the encounter on fitness can be either negative if the interaction is competitive, resulting in character divergence (matching competition, e.g., resource competition), or positive if the interaction is mutualistic, resulting in character convergence (matching mutualism, e.g., Müllerian mimicry). Applying Lande’s formula (Lande 1976) and given a number of simplifying assumptions— importantly that all lineages evolve in sympatry and that variation in competitors’ phenotypes does not strongly influence the outcome of competition—this model yields a simple prediction for the evolution of a population’s mean phenotype. Here, we develop inference tools for fitting a simple version of the matching competition model (i.e., the phenotype matching model of Nuismer and Harmon incorporating competitive interactions between lineages) to combined phylogenetic and trait data. We begin by showing how to compute likelihoods associated with this model. Next, we use simulations to explore the statistical properties of maximum likelihood (ML) estimation of the matching competition model (parameter estimation as well as model identifiability). While the inclusion of interactions between lineages is an important contribution to quantitative models of trait evolution, applying the matching competition model to an entire clade relies on the assumption that all lineages in the clade are sympatric. However, this assumption will be violated in most empirical cases, so we also developed a method for incorporating data on the biogeographical overlap between species for the matching competition model. We also implemented these biogeographical tools for the linear and exponential diversity-dependent trait models of Weir and Mursleen (2013), wherein the evolutionary rate at a given time in a tree varies as a function of the number of lineages in the reconstructed phylogeny at that time (see also Mahler et al. 2010), so that rates vary only as a function of the number of sympatric lineages. We then fit the model to data from a classical adaptive radiation: Greater Antillean Anolis lizards (Harmon et al. 2003; Losos 2009). Many lines of evidence support the hypothesis that resource competition is responsible for generating divergence 

given a number of simplifying assumptions-importantly that all lineages evolve in sympatry-75 this model yields a simple prediction for the evolution of a population's mean phenotype. 76 Here, we develop inference tools for fitting the matching competition model (i.e., the 77 phenotype matching model of Nuismer and Harmon incorporating competitive interactions 78 between lineages) to combined phylogenetic and trait data. We begin by showing how to 79 compute likelihoods associated with this model. Next, we use simulations to explore the 80 statistical properties of maximum likelihood estimation of the matching competition model 81 (parameter estimation as well as the power and type I error rates). While the inclusion of 82 interactions between lineages is an important contribution to quantitative models of trait 83 evolution, applying the matching competition model to an entire clade relies on the assumption 84 that all lineages in the clade are sympatric. However, this assumption will be violated in most 85 empirical cases, so we also developed a method for incorporating data on the biogeographical 86 overlap between species for this model and for the linear and exponential diversity-dependent 87 trait models of Weir & Mursleen (2013), wherein the evolutionary rate at a given time in a tree 88 varies as a function of the number of lineages in the reconstructed phylogeny at that time (see 89 also Mahler et al. 2010). 90 We then fit the model to data from a classical adaptive radiation: Greater Antillean Anolis 91 lizards (Harmon et al. 2003;Losos 2009). Many lines of evidence support the hypothesis that 92 resource competition is responsible for generating divergence between species in both habitat use 93 (e.g., Pacala and Roughgarden 1982) and morphology (Schoener 1970;Williams 1972;see 94 review in Losos 1994). Thus, we can make an a priori prediction that model comparison will 95 uncover a signature of competition in morphological traits that vary with habitat and resource 96 use. However, in spite of the importance of interspecific competition to adaptive radiation 97 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint theory, the role of competition in driving adaptive divergence has been historically difficult to 98 measure (Losos 2009) because trait divergence resulting from competition between lineages 99 during their evolutionary history should have the effect of eliminating competition between those 100 lineages at the present. Nevertheless, the signature of such competition may be detectable in the 101 contemporary distribution of trait values and their covariance structure (Hansen and Martins 102 1996;Nuismer and Harmon 2015), and as such, the matching competition model should provide 103 a good fit to morphological data collected on anoles. Given the well-resolved molecular 104 phylogeny (Mahler et al. 2010(Mahler et al. , 2013 and the relatively simple geographical relationships 105 between species (i.e., species are restricted to single islands, Rabosky and Glor 2010; Mahler and 106 Ingram 2014), the Greater Antillean Anolis lizards provide a good test system for exploring the 107 effect of competition on trait evolution using the matching competition model. 108

Likelihood Estimation of the Matching Competition Model 111
We consider the evolution of a quantitative trait under the matching competition model of 112 Nuismer & Harmon (2015) wherein trait divergence between lineages will be favored by 113 selection. In our version of the model, we remove stabilizing selection to focus on the effect of 114 competition. The evolution of a population's mean phenotype is thus given by (Eq. S38 in 115 Nuismer & Harmon 2015 with = 0): 116 117 ! ′ = ! + − ! + (Eq. 1) 118 119 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint where drift is incorporated as a Gaussian variable with mean = 0 and variance = ! (the 120 Brownian motion rate parameter) and S measures the strength of interaction (more intense 121 competitive interactions are represented by larger negative values), ! ′ is the mean trait value for 122 lineage i after a infinitesimally small time step, and is the mean trait value for the entire clade 123 at the beginning of that time step. Hence, if a species trait value is greater (respectively smaller) 124 than the trait value average across species in the clade, the species' trait will evolve towards even 125 larger (respectively smaller) trait values. 126 Like several other models of quantitative trait evolution (Harmon et al. 2010, Weir & 127 Mursleen 2013), the expected distribution of trait values on a given phylogeny under the 128 matching competition model follows a multivariate normal distribution with mean vector made 129 of terms each equal to the character value at the root of the tree and variance-covariance matrix 130 determined by the parameter values and phylogeny. Nuismer & Harmon (2015) provide the 131 system of ordinary differential equations describing the evolution of the variance and covariance 132 terms through time (their Eqs.10b and 10c). These differential equations can be integrated 133 numerically from the root to the tips of phylogenies to compute expected variance-covariance 134 matrices and the associated likelihood values given by the multivariate normal distribution. 135 Additionally, to relax the assumption that all of the lineages in a clade coexist 136 sympatrically, we included a term to specify which lineages co-occur at any given time-point in 137 the phylogeny, which can be inferred, e.g., by biogeographical reconstruction. The resulting 138 system of ordinary differential equations describing the evolution of the variance and covariance 139 terms through time are: 140 where ! is the variance for each species i at time t and !,! is the covariance for each species pair 147 i,j at time t. Eq. 2b describes the evolution of the covariance at time t if species i and j are in 148 sympatry at time t . If they are not, = 0 such that the covariance between species i and j 149 remains fixed, The terms ! and S are as specified above, A i,j equals 1 at time t if i and j are 150 sympatric at that time, and 0 otherwise (Fig. 1), and ! = !" ! !!! is the number of lineages 151 interacting with lineage i at time t (equal to the number n of lineages in the reconstructed 152 phylogeny at time t if all species are sympatric). Here, we consider the matrix A to be a block 153 diagonal matrix, such that sets of interacting species are non-overlapping, as in the case of Anolis 154 lizards ( Fig. 1). This implies that if species i and j are sympatric, then ! = ! and !,! = !,! . 155 Note that when S = 0 or n = 1 (i.e., when a species is alone), this model reduces to Brownian 156 motion. This model has three free parameters: ! , S and the ancestral state ! at the root. As with 157 other models of trait evolution, the maximum likelihood estimate for the ancestral state is 158 computed through GLS using the estimated variance-covariance matrix (Grafen 1989;Martins 159 and Hansen 1997). 160 We used the ode function in the R package deSolve (Soetaert et al. 2010) to perform the 161 numerical integration of the differential equations using the "lsoda" solver, and the Nelder-Mead 162 algorithm implemented in the optim function to perform the maximum likelihood optimization. 163 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. Using the same geography matrix A described above for the matching competition model 168 ( Fig. 1), we modified the diversity-dependent linear and exponential models of Weir & Mursleen 169 (2013) to incorporate biological realism into the models, because ecological opportunity is only 170 relevant within rather than between biogeographical regions. The resulting variance-covariance 171 matrices, V, of these models have the elements: 172 for the diversity-dependent linear model, and 176 for the diversity-dependent exponential model, where ! ! is the rate parameter at the root of the 180 tree, b and r are the slopes in the linear and exponential models, respectively, s ij is the shared 181 path length of lineages i and j from the root of the phylogeny to their common ancestor, ! is the 182 number of sympatric lineages (as above) between nodes at times t m-1 and t m (where t 1 is 0, the 183 time at the root, and t M is the total length of the tree) (Weir & Mursleen 2013). When b or r = 0, 184 these models reduce to Brownian motion. 185 186 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Simulation-based Analysis of Statistical Properties of the Matching Competition Model 187
To verify that the matching competition model can be reliably fit to empirical data, we 188 simulated trait datasets to estimate its statistical properties (i.e., parameter estimation, power, and 189 type I error rates). For all simulations, we began by first generating 100 pure-birth trees using 190 TreeSim (Stadler 2014). To determine the influence of the number of tips in a tree, we ran 191 simulations on trees of size n = 20, 50, 100, and 150. We then simulated continuous trait datasets 192 by applying the matching competition model recursively from the root to the tip of each tree 193 (Paradis 2011), following Eq. 1. For these simulations, we set ! = 0.05 and systematically 194 varied S (-1.5, -1, -0.5, -0.1, or 0). Finally, we fit the matching competition model to these 195 datasets using the ML optimization described above. 196 To determine the ability of the approach to accurately estimate simulation parameter 197 values, we first compared estimated parameters to the known parameters used to simulate 198 datasets under the matching competition model (S and ! ). We also quantified the robustness of 199 these estimates in the presence of extinction by estimating parameters for datasets simulated on 200 birth-death trees; in addition, we compared the robustness of the matching competition model to 201 extinction to that of the diversity-dependent models. These two latter sets of analyses are 202 described in detail in the Supplementary Appendix 1. 203 To assess the power of the matching competition model, we compared the fit of this 204 model to other commonly used trait models on the same data (i.e. data simulated under the 205 matching competition model). Specifically, we compared the matching competition model to (1) 206 Brownian motion (BM), (2) Ornstein-Uhlenbeck/single-stationary peak model (OU, Hansen & 207 Martin 1996), (3) exponential time-dependent (TD exp , i.e., the early burst model, or the ACDC 208 model with the rate parameter set to be negative, Blomberg et al. 2003;Harmon et al. 2010), (4) 209 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. slope determined by the model and tree (e.g., for time-dependent models, the slope is a function 223 of the total height of the tree; for the TD exp model, these parameters result in a total of 5.9 half-224 lives elapsing from the root to the tip of the tree, Slater and Pennell 2014). These simulations 225 were performed using our own codes, available in RPANDA (Morlon 2014). As above, we 226 calculated the Akaike weights for all models for each simulated dataset. 227 228

Fitting the Matching Competition Model of Trait Evolution to Caribbean Anolis Lizards 229
To determine whether the matching competition model is favored over models that ignore 230 interspecific interactions in an empirical system where competition is known to have influenced 231 character evolution, we fit the matching competition model to a morphological dataset on adult 232 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint males of 100 species of Greater Antillean Anolis lizards and the time calibrated, maximum clade 233 credibility tree calculated from a Bayesian sample of molecular phylogenies (Mahler et al. 2010(Mahler et al. , 234 2013Mahler and Ingram 2014). We included the first four size-corrected phylogenetic principal 235 components from a set of 11 morphological measurements, collectively accounting for 93% of 236 the cumulative variance explained (see details in Mahler et al. 2013). Together, the shape axes 237 quantified by these principal components describe the morphological variation associated with 238 differences between classical ecomorphs in Caribbean anoles (Williams 1972). In addition to the 239 matching competition model, we fit the six previously mentioned models (BM, OU, TD exp , TD lin , 240 DD exp , and DD lin ) to the Anolis dataset. 241 For the matching competition model and diversity-dependent models, to determine the 242 influence of designating clades as sympatric and allopatric, we fit the model for each trait using 243 two sets of geography matrices (i.e., A in Eq. 1b, 2, & 3, see of parameter values (Fig. 2). As the number of tips increases, so does the reliability of maximum 253 likelihood parameter values (Fig. 2). Parameter estimates remain reliable in the presence of 254 extinction, unless the extinction fraction is very large (Supplementary Appendix 1). When 255 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint datasets are simulated under the matching competition model, model selection generally picks 256 the matching competition model as the best model (Figs. 3, S1); the strength of this 257 discrimination depends on both the S value used to simulate the data and the size of the tree 258 (Figs. 3, S1). 259 We found that the matching competition model behaves similarly to a model of declining 260 evolutionary rates through time, as the rate parameter values of the TD exp model estimated on 261 data generated under the matching competition model became increasingly negative under 262 increasing levels of competition (Fig. S2). The dynamics of this declining rate were better 263 described by an exponential decline in rates than by a linear one (Table S1). 264 Simulating datasets under BM, OU, DD exp , and DD lin generating models, we found that 265 there is a reasonably low type I error rate for the matching competition model (Fig. 4a,b,e,f). 266 When character data were simulated under a TD lin model of evolution, the matching competition 267 and/or the diversity-dependent models tended to have lower AICc values than the TD lin model, 268 For the first four phylogenetic principal components describing variation in Anolis 273 morphology, we found that models that incorporate species interactions fit the data better than 274 models that ignore them (Table 1). PC1 and PC3, which describe variation in hindlimb/hindtoe 275 and forelimb/foretoe length (Mahler et al. 2013), respectively, are fit best by the matching 276 competition model. PC2 and PC4, which describe variation in body size (snout vent length) and 277 lamellae number, respectively, are fit best by the exponential diversity-dependent model. 278 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint Additionally, in every case, models that incorporated the geographic relationships among species 279 in the tree outperformed models that assumed that lineages interact in sympatry at the clade level 280 (Table 1). 281

DISCUSSION 283
The inference methods we present here represent an important new addition to the 284 comparative trait analysis toolkit. Whereas previous models had not accounted for the influence 285 of trait values in other lineages on character evolution, the matching competition model takes 286 these into account. Furthermore, extending both the matching competition model and two 287 diversity-dependent trait evolution models to incorporate geographic networks of sympatry 288 further extends the utility and biological realism of these models. 289 We found that the matching competition property has increasing power and accuracy of 290 parameter estimation with increasing tree sizes and competition strength. We also found that, for 291 most of the generating models we tested (but see below), the matching competition model is not 292 erroneously favored using model selection (i.e., there is a reasonably low type-I error rate). As 293 with all other models, the statistical properties of the matching competition model will depend on 294 the size and shape of a particular phylogeny as well as specific model parameter values. We found that data generated under time-dependent models were often fit better by 299 models that incorporate interspecific interactions (i.e., density-dependent and matching 300 competition models) (Fig. 4c,d). This was especially true for the TD exp model, often referred to 301 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint as the early-burst model-the matching competition model nearly always fit data generated 302 under the TD exp model better than the TD exp model (Fig. 4c). We do not view this as a major 303 limitation of the model for two reasons. First, the TD exp model is known to be statistically 304 difficult to estimate on neontological data alone (Harmon et al. 2010;Slater et al. 2012;Slater 305 and Pennell 2014). Accordingly, since the matching competition model also describes declining 306 evolutionary rates from the root to the tip of the tree (Fig. S2, Table S1), it is perhaps not 307 surprising that the matching competition model fits data generated under the TD exp model well. 308 Secondly, and more importantly, time-dependent models are not process-based models, but 309 rather incorporate time since the root of a tree as a proxy for ecological opportunity or available 310 niche space (Harmon et al. 2010;Mahler et al. 2010;Slater 2015). The matching competition and 311 density-dependent models explicitly account for the interspecific competitive interactions that 312 time-dependent models purport to model, thus we argue that these process-based models are 313 more biologically meaningful than time-dependent models (Moen and Morlon 2014). 314 Because the matching competition model depends on the mean trait values in an evolving 315 clade, maximum likelihood estimation is robust to extinction, whereas the diversity-dependent 316 models are less so (Appendix S1, Figs. S3-S6). Nevertheless, given the failure of maximum 317 likelihood to recover accurate parameter estimations at high levels of extinction ( : ≥ 0.6), we 318 suggest that these models should not be used in clades where the extinction rate is known to be 319 particularly high. In such cases, it would be preferable to adapt the inference framework 320 presented here to include data from fossil lineages (Slater et al. 2012), which could be easily 321 done by integrating the ordinary differential equations described in Eq. 2a and 2b on non-322 ultrametric trees. 323 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint For all of the traits we analyzed, we found that models incorporating both the influence of 324 other lineages and the specific geographical relationships among lineages were the most strongly 325 supported models (though less strikingly for PC3 and PC4). The matching competition model is 326 favored in the PCs describing variation in relative limb size. Previous research demonstrates that 327 limb morphology explains between-ecomorph variation in locomotive capabilities and perch 328 characteristics (Losos 1990(Losos , 2009Irschick et al. 1997 We imagine that the matching competition model and biogeographical implementations 337 of diversity-dependent models will play a substantial role in comparative studies of interspecific 338 competition. There are many possible extensions of the tools developed in this paper. In the 339 current implementation of interspecific geographic overlap, the matrices of sympatry/allopatry 340 are assumed to be block diagonal matrices. In the future, the model can be extended to more 341 complex geography matrices that are more realistic for mainland taxa (e.g., using ancestral 342 biogeographical reconstruction, Ronquist and Sanmartín 2011;Landis et al. 2013), and can also 343 specify degrees of sympatric overlap (i.e., syntopy). Additionally, although the current version of 344 the model is rather computationally expensive with larger trees (with 100 or more tips), future 345 work to speed up the calculation of the variance-covariance matrix under the matching 346 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
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The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint with biologically unrealistic variances. Under both diversity-dependent models, the magnitude of 609 both the slope and ! parameter values are increasingly underestimated with increasing 610 extinction fractions (Fig. S5, S6). 611 612 613 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
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The copyright holder for this preprint (which was not this version posted July 31, 2015. The copyright holder for this preprint (which was not this version posted July 31, 2015. ; https://doi.org/10.1101/023473 doi: bioRxiv preprint