Evolution, 58(8), 2004, pp. 1664-1673 APPLICATION OF JOHNSON ET AL.'S SPECIATION THRESHOLD MODEL TO APPARENT COLONIZATION TIMES OF ISLAND BIOTAS ROBERT E. RICKLEFS1 AND ELDREDGE BERMINGHAM2 1Department of Biology, University of Missouri-St. Louis, 8001 Natural Bridge Road, St. Louis, Missouri 63121-4499 E-mail: ricklefs@umsl.edu ^Smithsonian Tropical Research Institute, Unit 0948, APO AA 34002-0948, Panama E-mail: eb@stri.org, eb@naos.si.edu Abstract.?Understanding patterns of diversity can be furthered by analysis of the dynamics of colonization, speciation, and extinction on islands using historical information provided by molecular phylogeography. The land birds of the Lesser Antilles are one of the most thoroughly described regional faunas in this context. In an analysis of colonization times, Ricklefs and Bermingham (2001) found that the cumulative distribution of lineages with respect to increasing time since colonization exhibits a striking change in slope at a genetic distance of about 2% mitochondrial DNA sequence divergence (about one million years). They further showed how this heterogeneity could be explained by either an abrupt increase in colonization rates or a mass extinction event. Cherry et al. (2002), referring to a model developed by Johnson et al. (2000), argued instead that the pattern resulted from a speciation threshold for reproductive isolation of island populations from their continental source populations. Prior to this threshold, genetic divergence is slowed by migration from the source, and species of varying age accumulate at a low genetic distance. After the threshold is reached, source and island populations diverge more rapidly, creating heterogeneity in the distribution of apparent ages of island taxa. We simulated of Johnson et al.'s speciation-threshold model, incorporating genetic divergence at rate k and fixation at rate M of genes that have migrated between the source and the island population. Fixation resets the divergence clock to zero. The speciation-threshold model fits the distribution of divergence times of Lesser Antillean birds well with biologically plausible parameter estimates. Application of the model to the Hawaiian avifauna, which does not exhibit marked heterogeneity of genetic divergence, and the West Indian herpetofauna, which does, required unreasonably high migration-fixation rates, several orders of magnitude greater than the colo- nization rate. However, the plausibility of the speciation-divergence model for Lesser Antillean birds emphasizes the importance of further investigation of historical biogeography on a regional scale for whole biotas, as well as the migration of genes between populations on long time scales and the achievement of reproductive isolation. Key words.?Colonization, genetic divergence, Hawaiian Islands, migration, speciation, speciation-threshold model, West Indies. Received July 30, 2003. Accepted May 4, 2004. Species richness in island biotas represents a history of colonization, radiation, and extinction of taxa. Understanding the dynamics of community assembly depends on estimating the rates of each of these processes and their variation over time. Molecular assessments of genetic distance between populations provide estimates of colonization times of island populations from their continental source populations. Rick- lefs and Bermingham (2001) showed that the dynamics of colonization and extinction could be explored by analyzing the distribution of colonization times of extant island line- ages. Cherry et al. (2002) have cautioned, however, that ge- netic distance does not reveal the history of a particular pop- ulation on an island when continuing gene flow from the source prevents genetic divergence. Only after a speciation threshold of genetic divergence has been reached do source and island populations diverge at a rate determined by in- dependent evolutionary change in both (Johnson et al. 2000). The degree to which the speciation threshold masks the his- tory of colonization is a matter of concern for understanding both the history of island biotas and the dynamics of colo- nization and gene flow between source and island popula- tions. In this study, we fit a simple speciation-threshold model to the distribution of genetic distances between source and island populations in three systems for which data are avail- able for most of the island lineages: land birds of the Lesser Antilles, reptiles and amphibians of the West Indies, pri- marily the Greater Antilles, and birds of the Hawaiian ar- chipelago. We then discuss whether the fitted parameters de- scribing rates of colonization, extinction, migration, and the speciation threshold are biologically realistic, focusing pri- marily on the land birds of the Lesser Antilles. Ricklefs and Bermingham (2001) examined the accumu- lation of lineages of land birds in the Lesser Antilles as a function of increasing relative age of colonization, which they inferred from genetic divergence between Lesser Antillean lineages and South American or Greater Antillean sources. When probabilities of colonization and extinction are con- stant over time and homogeneous over lineages, the accu- mulation curve increases exponentially toward an equilibri- um value. Ricklefs and Bermingham (2001) found instead that the accumulation curve did not reach equilibrium and that its slope changed abruptly at a mitochondrial DNA (mtDNA) genetic distance (d) of about 0.02 (2% sequence divergence). As Cherry et al. (2002) pointed out, however, the non- homogeneity in the lineage accumulation curve could also be explained by the speciation-threshold model developed by Johnson et al. (2000). Accordingly, following initial colo- nization of an island, genes carried by individuals continuing to arrive from the source would occasionally become fixed in the gene pool of the island population by drift or selection. Such a migration-fixation event would eliminate any diver- gence between the mainland and island populations and set the divergence clock back to zero. Thus, the apparent rate of divergence among recent colonists would be low due to mi- gration, and seemingly new colonist populations would ac- 1664 ? 2004 The Society for the Study of Evolution. All rights reserved. SPECIATION AND DIVERGENCE 1665 cumulate at low genetic distances from source populations. Once a speciation threshold of genetic distance had been achieved, however, migrant alleles could no longer become fixed in the island population and divergence would proceed at a more rapid pace determined by independent evolutionary change in each population. Thus, the distribution of genetic divergence values in an island biota with continuous colo- nization and migration would exhibit a high density prior to the speciation threshold and a sparser distribution over a wider range of divergence values above that threshold. The slope of the curve relating the cumulative number of species as a function of increasing genetic divergence, which was the basis of analyses by Ricklefs and Bermingham (2001), would appear to change abruptly at the speciation threshold. Assuming that the model of Johnson et al. (2000) could replicate the observed lineage accumulation curve, testing the model as a viable alternative to heterogeneous coloni- zation and extinction would depend on (1) the plausibility of the particular parameters in the successful model, (2) whether the threshold genetic divergence for speciation corresponds to a level that is reasonable for species-level distinction, and (3) whether geographic patterns of genetic variation are con- sistent with predictions of the model. This last point arises because Johnson et al.'s speciation-divergence model should apply to relations between island populations within archi- pelagoes as well as to relations between the archipelago and the source areas. Specifically, and depending on the distances involved, genetic divergence between island populations should sometimes exceed the genetic divergence between any island and the source (Ricklefs and Bermingham 2002). Regardless of whether Johnson et al.'s speciation-diver- gence model applies in the end, we have found it useful to evaluate parameters of the model that provide a close fit to data. We have tried to determine whether these parameters are consistent with other information on colonization, di- vergence between populations in allopatry, and geographic structure in the distribution of genetic variation within lin- eages distributed over archipelagoes, particularly the Lesser Antilles. In this analysis, we construct a simple simulation of the Johnson et al. model and examine the sensitivity of the model's output to variation in migration rates and the genetic divergence required for speciation. MODELS OF COLONIZATION TIME DISTRIBUTIONS The Speciation-Threshold Model We consider lineages present in a source area that colonize a single island. Our simulation follows the fate of 1000 lin- eages (biological species) over 10,000 time steps. The fate of each lineage is independent of all others in the simulation; that is, there are no competitive or precedence effects. Prob- abilities of genetic divergence and migration are constant over the course of the simulation. We found that it was un- necessary to include background extinction to provide ade- quate fits of the model to the data examined in this analysis. For each lineage, the time of initial colonization of the island was drawn from a random uniform distribution over the 10,000 time steps in the simulation in one set of simu- lations, and it was the initial time step in another set of simulations. For convenience, we considered nucleotide sub- stitutions in a 1000 base-pair sequence of mtDNA. Substi- tutions were additive, as in an infinite alleles model. At each time step, the model determined whether initial colonization occurred by comparing a uniform random variate to the prob- ability of initial colonization during a single time step (0.0001). Following colonization of the island by each lin- eage, at each subsequent time step we determined whether either a nucleotide substitution or a migration-fixation event occurred. Substitutions were assumed to be neutral. Mutation events (probability \L) were based on the probability of a single mutation over the entire mtDNA sequence; the mu- tation rate was low enough that the probability of multiple mutations was close to zero. Nucleotide substitutions were accumulated over time to determine sequence divergence (d), which occurred at an average rate of k = 2\L. A migration- fixation event (probability M) resulted in the fixation of the source haplotype in the island population, which reset genetic divergence between the source and island population to zero. Speciation occurred when the number of nucleotide sub- stitutions exceeded a threshold. The probability of speciation (ps) remained at zero until the number of substitutions (n) had accumulated to a minimum required for reproductive isolation (the speciation threshold, ns), after which it in- creased, approaching one exponentially according to Ps Ps 0 [1 -bs(n-ns)i when n < ns when n > nr and (1) where bs is a scaling parameter that determines the rate at which ps approaches one. When bs = 0.25, which was the typical value in our simulations, ps exceeded 0.5 after ns + 3 nucleotide substitutions and it exceeded 0.9 after ns + 10 nucleotide substitutions. The model of speciation employed by Johnson et al. (2000) is more complex than this, but pro- duces a similar exponential approach to a speciation proba- bility of one. For each simulation, we recorded the proportion of lineages that had achieved species status by the 10,000 time steps, the colonization time of lineages that had become new species (differentiated) and those that had not (undifferentiated), the average time to speciation among those lineages that had become new species, and the nucleotide divergence (n) be- tween source and island populations for both sets of lineages. Because we simulated a 1000-nucleotide sequence, d = nl 1000. We also produced a plot of the cumulative number of lineages as a function of genetic divergence to compare with the observed data. The Homogeneous Colonization-Extinction Model When the arrival of new species on an island is constant at rate C per unit time (f) and resident island species go extinct at rate E per species per unit time, that is, species survival S = exp(-Et), the cumulative number of species (S) with colonization times up to t is S = (C/E)(l ? exp[?Et]) (Rick- lefs and Bermingham 2001). Thus, S exponentially approach- es an asymptotic equilibrium number of species (C/E) with increasingly older colonists included in the sample. We fitted this model to the cumulative species curves by nonlinear regression (SAS Proc NLIN). 1666 R. E. RICKLEFS AND E. BERMINGHAM The Mass Extinction Model The mass extinction model is identical to the homogeneous colonization-extinction model except that a mass extinction event survived by proportion S of the resident biota occurs at time ts in the past. The consequence of this event is that the rate of accumulation of extant lineages colonizing the island prior to ts is proportion S of the rate determined by the balance of C and E. Fits of this model to data were obtained by nonlinear regression. The mass extinction model is identical to a model with an increase in colonization at tM by a factor of \IS or a decrease in extinction rate by proportion (1-S). All simulations and statistical analyses were carried out with the Statistical Analysis System, version 8.12 (SAS In- stitute, Gary, NC). EMPIRICAL DATA Land Birds of the Lesser Antilles The observed data were the mtDNA genetic distances be- tween source and Lesser Antillean populations of 38 lineages of small land birds. A 39th lineage, representing the endemic Antillean crested hummingbird Orthorhynchus cristatus was not included because we were unsure of its source taxon, hence the genetic distance to the source population repre- senting its initial entry into the Lesser Antilles. Observed genetic distances were based on 842 bp of the overlapping ATPase 6 and ATPase 8 protein-coding mitochondrial gene regions. We used the Tamura-Nei model of nucleotide sub- stitution (Tamura and Nei 1993) to calculate genetic dis- tances. These distances are corrected for multiple substitu- tions and are therefore comparable to the additive distances produced by the simulation. The observed lineage accumu- lation curve is linear over the range of genetic divergence (d) between approximately 0.025 (2.5%) and the maximum value of 0.174 (17.4%) in our sample (Ricklefs and Ber- mingham 2001). Accordingly, we chose 0.025 X 1000 = 25 as the minimum number of nucleotide substitutions (ns) for speciation to occur with probability ps > 0 (eq. 1). Among the observed data, 21 of 38 Lesser Antillean lin- eages (55%) had mtDNA nucleotide divergence values under d = 0.025. Thus, one benchmark for judging the fit of a simulation to the observed data was the approximately 55% of lineages that had not achieved species-level differentiation. In comparing the overall fit graphically, the cumulative num- ber of lineages in the simulation was multiplied by 38/1000 = 0.038 to make the output comparable to the observed data. The Hawaiian Avifauna Fleischer and Mclntosh (2001) summarized what is known about genetic divergence between Hawaiian taxa and their continental sister taxa. Of 22 avian lineages that colonized the Hawaiian Islands, genetic divergences based on varying amounts of mtDNA sequence have been estimated for 13 lineages. Fleischer and Mclntosh assumed that three addi- tional lineages (Fulica alai, Gallinula chloropus sandvicensis, Nycticorax nycticorax hoactii) were recent colonists based on their lack of differentiation in the Hawaiian Islands, and we arbitrarily set their sequence divergence at 1%. Of the re- 40 E 3 u 10 Observed data Simulated data fc = 0.019 m = 0.0021 d, = 25 b =0.25 0.00 0.05 0.10 0.15 0.20 Genetic distance (d) FIG. 1. The speciation-divergence model fits the observed cu- mulative distribution of genetic divergence values of land birds in the Lesser Antilles closely when the divergence rate k = 0.019 and the migration-fixation rate M = 0.0021. Simulated data are based on 1000 lineages over 10,000 time steps. maining six lineages, one (Asio flammeus sandwichensis) is almost certainly a recent colonist and the other five {Circus dossenus^, Grallistrix sp.\ Acrocephalus familiaris, Chas- iempsis sandwichensis, Moho [Chaetoptila]^) are well differ- entiated but of unknown age (t indicates extinct). We ana- lyzed the relationship between colonization time and the spe- cies accumulation curve both without these species in = 16), and with estimates of genetic differentiation of 1 % for sub- specific distinction (Asio), 5% for specific distinction (Circus, Acrocephalus), and 10% for generic distinction (Grallistrix, Chasiempsis, Moho) (n = 22). The West Indian Herpetofauna Hedges (1996) estimated colonization times in millions of years (my) for 38 of 42 endemic lineages of reptiles and amphibians of the West Indies, primarily the Greater Antilles, based on immunological distance. In many cases these es- timates were presented as ranges, often very broad, from which we chose the midpoint. We arbitrarily set the colo- nization times of 35 nonendemic lineages at 1 my, but spread- ing these colonization times over 5 my did not substantially affect the fitted constants in the model. Although immuno- logical distance provides a scale of relative age (Prager 1993), the accuracy and applicability of this approach to the West Indian herpetofauna are not uniformly agreed upon (Crother and Guyer 1996; Hedges 1996). RESULTS Lesser Antillean Land Birds For our initial simulations, we used bs = 0.25 as the ex- ponential rate of approach to complete speciation and a spe- ciation threshold (ds) of 0.025 (ns = 25 nucleotide substi- tutions). Using a systematic search over the potential param- eter space, we determined that a divergence rate of k = 0.019 and a migration-fixation rate of M = 0.0021 produced an output that described the observed data well (Fig. 1). SPECIATION AND DIVERGENCE 1667 250 en c o X! 3 (/) m o o\ r- r- (N (N in (N ?i oo TJ- ? In _ tg 't O in vi r~> H (S ^O Tf -H m o (N m oo TJ- 1- ?* ?* ?* ^ K ID ON In \o 3 33*^ 62 86 23 40 23 14 13 19 10 1 45 B & O %o go- ?i 00 TJ- ^t (N *?> 00 xl" CO 00 ?I m ? o5 \D O^M h (Noom\c\0\C' O O oo oo O ^ "S o ?* 3K-^ O t? O OS ON oo O ON NO (N O ^J- '3 a a '5 a __ Q TABLE 2. The effect of varying the exponential rate of approach (bs) to a speciation probability (ps) of 1 on the time to ps = 0.90 and the proportion of undifferentiated lineages in simulations with k = 0.019, ns = 25, and bs = 0.25. The simulations included 1000 individuals and 10,000 time steps. n90 is the genetic divergence at which the speciation probability reaches 90%. #(MS) ?,in 0.0625 0.125 0.25 0.5 1 558 556 560 583 546 62 44 35 30 28 bution of colonization times well (Fig. 5). We do not report the results of fitting a mass-extinction model because the data did not indicate such an event. With 16 species, nonlinear least squares regression fitting of the homogeneous coloni- zation-extinction model provides parameter estimates of col- onization rate (C) of 386 ? 38 SE (95% CI, 305-468) and extinction rate (E) of 22.6 ? 4.0 (14.1-31.1). The units are the probability of a colonization event in the archipelago, and the probability that a resident lineage in the archipelago will become extinct, per unit (100%) of mtDNA sequence divergence. Assuming approximately 2% sequence diver- gence per 1 million years, these estimates divided by 50 are equivalent to rates per million years, that is, C = 7.7 and E = 0.45 per million years. With all 22 species included in the analysis, estimates were C = 388 ? 29 (327-449) and E = 15.1 ? 2.4 (10.1-20.1), or 7.8 and 0.3 per million years, respectively. The ratio CIE defines the number of lineages at the colonization-extinction equilibrium. These ratios were 17.1 and 25.7, respectively, which indicates that the lineage diversity of the islands is close to the colonization-extinction equilibrium. The colonization rate is much lower than the value of 1600-1700 per unit mtDNA sequence divergence estimated for the subset of land birds in the Lesser Antilles (Ricklefs and Bermingham 2001). The observed colonization times for Hawaiian island birds also conform closely to a speciation-threshold model (Fig. 5). With all 22 lineages included, the data are modeled over 10,000 time steps with rates per time step for divergence of k = 0.011 and migration-fixation of M = 0.00045, setting the speciation threshold at 25 nucleotide substitutions with bs = 0.25. The statistics for lineages are shown in Table 5. The West Indian Herpetofauna The cumulative distribution of colonization times for rep- tiles and amphibians within the West Indies as a whole re- TABLE 3. Fitted values of M that produce about 55% undiffer- entiated lineages when the speciation threshold (ns) was varied be- tween 15 and 30 (1.5 and 3.0% sequence divergence). For these simulations, k = 0.019 and bs = 0.25. 15 20 25 30 M 0.0046 0.0030 0.0021 0.0015 SPECIATION AND DIVERGENCE 1669 TABLE 4. Statistics for the case in which all lineages colonize the island from the source at the beginning of the simulation period. th age of lineages (time steps since colonization); ts, time steps to speciation; n, genetic divergence between island and source populations (number of nucleotide substitutions). Lineages SD SD Undifferentiated Differentiated 530 470 10,000 10,000 5216.5 2512.5 629 S8.24 5 95 33.16 sembles the distribution for birds of the Lesser Antilles (Fig. 6). Accordingly, a homogeneous colonization-extinction model would not provide a good fit to the data. We fitted a mass-extinction model with homogenous rates of coloniza- tion (C) and background extinction (E) overlaid by a mass extinction event at 2 my survived by proportion S of lineages. The close fit (Fig. 6) provides parameter estimates of C = 19.6 ? 0.9 (95% CI, 17.7-21.5), E = 0.027 ? 0.014 (-0.001- 0.054), and S = 0.062 ? 0.020 (0.021-0.103). Background extinction for reptiles and amphibians is close to zero, with an estimated average persistence time of lineages of almost 40 my, and the contemporary (post mass extinction) rate of colonization in the herpetofauna as a whole is about 20 lin- eages per my. This colonization rate estimated under a mass- extinction model is not substantially less than that estimated for birds in the Lesser Antilles (about 32-34 my-1). Hedges (1996) estimated the oldest lineages of West Indian reptiles and amphibians to be 70 my, which corresponds ap- proximately to the Cretaceous-Tertiary boundary. For sim- ulations of the speciation-threshold model, we set the time scale to 0.0065 my per time step, which gives an estimated time over 10,000 time steps of 65 my. The model provides a close fit to the data with the following variables: divergence rate k = 0.0065 my per time step, migration-fixation rate M = 0.03 per time step, speciation threshold ds = 2.4 my, and bs = 0.25 (Fig. 6). The estimated frequency of migration- fixation is equivalent to M = 4.6 per my, or a migration- fixation event within each island resident population every 0.22 my. Statistics for the lineages that had and had not achieved species status are presented in Table 6. Divergence (d) is expressed in my and the times are expressed in number of time steps, as in related tables. When the colonization times of the nonendemic lineages were spread uniformly over 5 my, a reasonable fit to the data was obtained with fitted parameters ds = 10, bs = 0.25, k = 0.0065, and M = 0.0022 (results not shown). In this case, the predicted curve initially rises more steeply than the data. The fit is improved by increasing ds to 20 (k = 0.0065, M = 0.00085) but, as a result, many endemics are classified as undifferentiated (73% of all lineages). DISCUSSION For the land birds of the Lesser Antilles, the models pro- posed by Ricklefs and Bermingham (2001) employed change in colonization or extinction rates, or a mass extinction event, to explain the abrupt change in the lineage accumulation rate prior to a genetic distance of 2.5% mtDNA sequence diver- gence. Thus, the heterogeneity in the lineage accumulation curve was caused by changes in external environmental fac- tors. The model of Johnson et al. (2000) differs in that neither changes in rates nor a transient event is required to account for the apparent heterogeneity in the lineage accumulation rate. This is produced in their model by an internal factor: a speciation threshold that separates lineages in two groups, one of which diverges slowly (before speciation) and the other rapidly (after speciation) from source populations. Be- cause the different models fit the observed data equally well, evaluating the applicability of the models will depend on how well the fitted parameters conform to other information about rates of migration and divergence and the genetic dis- 40 (/) CD CO CO CD c 30 - CD .o E 3 C CD > 20 J2 10 3 o FIG. 4. observed colonize as in Fig 0.05 0.10 0.15 Genetic distance (d) 0.20 The fit of the simulated lineage accumulation curve to the data for land birds of the Lesser Antilles when all lineages at time 0 and k = 0.013 and M = 0.0018. Other parameters ure 1 (n, = 25, bs = 0.25). 25 CD O) TO CD c 0) E 3 C CD > E 3 o 20 15 10 Hawaiian land birds observed data speciation-divergence model colonization-extinction model 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Genetic distance (d) FIG. 5. Three models fit to the lineage accumulation with genetic distance in Hawaiian birds. Data from Fleischer and Mclntosh (2001); see text for explanation and fitted constants. 1670 R. E. RICKLEFS AND E. BERMINGHAM TABLE 5. Simulation statistics for a speciation-divergence model of colonization times of birds of the Hawaiian Islands. Colonization times are drawn from a uniform random distribution over the 10,000 time steps of the simulation. th age of lineages (time steps since colonization); ts, time steps to speciation; n, genetic divergence between island and source populations (number of nucleotide substitutions). Lineages t, SD t, SD n SD Undifferentiated Differentiated 399 601 2730 6785 2210 2131 3295 1536 9.7 63.9 7.4 243 tance associated with species formation. The land birds of the Lesser Antilles are particularly well suited for comparing these models in that they offer additional predictions con- cerning the geographic structure of genetic variation among islands within the archipelago. We shall return to this prop- erty below. The Migration-Fixation Rate For the land birds of the Lesser Antilles, we ask whether the fitted migration-fixation rate of M = 0.0021 per time step is reasonable in the context of what we know about the col- onization of islands and the genetic relationships among is- land populations in the Lesser Antilles. When we equate the time step in our simulations to 1000 years, the rate of genetic divergence over 1000 nucleotides of mtDNA sequence is equivalent to genetic divergence at a rate of 1.9% per my. This value is similar to estimates based on calibrations for birds (Shields and Wilson 1987; Klicka and Zink 1997; Fleischer et al. 1998; Lovette 2004) and suggests that a 10 million-year simulation period reasonably approximates the history of the contemporary Lesser Antillean avifauna. Ac- cordingly, a rate of M = 0.0021 per time step (1000 years) would be equivalent to the fixation of continental nucleotide substitutions in an island population at a rate of 2.1 events per my and an average waiting time between successful mi- gration-fixation events of about 0.48 my. The fitted migration-fixation interval of 0.48 my implies that the accumulation of nucleotide differences over the av- erage interval between migration events would be 0.48 my X 0.019 substitutions per nucleotide per my X 1000 nucle- eu 80 m re 3 20 E o 0 West Indian reptiles and amphibians A^"^ ? observed data mass extinction model 20 40 60 80 Age of lineage (10 years) FIG. 6. Speciation-divergence and mass-extinction models fitted to the cumulative distribution of lineages with respect to age in the West Indian herpetofauna, based on data in Hedges (1996). otides = 9.05 substitutions. In addition, the simulations in- dicate that at this rate about 44% of the lineages will cross the speciation threshold within the 107 years of the simulation interval, with a mean time to speciation of about 3.2 my (i.e., six migration-fixation intervals; see Table 1 for simulation results). For an island population of TV females, the probability that the mtDNA haplotype of a single female colonist will become fixed in the population by drift is UN, and the average time in generations required for fixation (the coalescence time) is N (Hartl and Clark 1997). For example, for a population size of N = 10,000 females, the probability of fixation of a single mtDNA haplotype is 0.0001 (104) per generation. Thus, to achieve a rate of migration-fixation of M = 2.1 X 10~6, females from the source population would have to immigrate to an island population at a rate of m = 2.1 per hundred years, assuming one year per generation. This rate of migration is consistent with the rapid spread of several species (shiny cowbird Molothrus bonairiensis and bare-eyed thrush Turdus nudigenis) through the Lesser An- tilles during the 1900s (Bond 1956) and the continuous dis- tributions without genetic differentiation of many recent col- onists across the islands (Raffaele et al. 1998; Ricklefs and Bermingham 1999). However, it is not consistent with gaps in the distributions of older (>1 my) lineages (Ricklefs and Cox 1972; Ricklefs and Bermingham 1999), implying that migration rates do not remain constant through time, partic- ularly up to the average time to speciation of 3.2 my indicated by our simulations. Nor is it consistent with the absence of multiple colonization events to single islands in the archi- pelago following achievement of the speciation threshold. The oldest colonization times in the Hawaiian avifauna occur at d = 0.10, which we presume corresponds to about 5 my, close to the age of the oldest of the present-day large islands (Kauai, 5.1 my; Wagner and Funk 1995). Thus, a convenient calibration in our 10,000-step simulation model is 2000 time steps per my (my-1), and the fitted constants are thus equivalent to a nucleotide divergence rate of 2.2% my-1 and a migration-fixation rate (M) of 0.9 my-1. The migration-fixation rate seems unrealistically high in this case. The average mtDNA sequence divergence of Hawaiian land bird lineages is 4.7%, which corresponds to about 2.35 my, or an average colonization rate (inverse of the waiting time to colonization) among successful colonists of 0.42 my-1. Because migration-fixation rates for neutral alleles by genetic drift are on the order of the migration (? colonization) rate (m) divided by population size, a migration-fixation rate with- in even several orders of magnitude of the estimated colo- nization rate is unrealistic. Thus, although a speciation-di- vergence model fits the observed data closely, the model probably is not applicable. This is not surprising for such a SPECIATION AND DIVERGENCE 1671 TABLE 6. Simulation statistics for a speciation-divergence model of colonization times of reptiles and amphibians of the West Indies. Colonization times are drawn from a uniform random distribution over the 10,000 time steps of the simulation. f? age of lineages (time steps since colonization); ts, time steps to speciation; d, the divergence time in my. Lineages t, SD t, SD d SD Undifferentiated Differentiated 540 460 3931 6416 2757 2445 2941 2229 0.20 25.4 0.45 15.9 remote island group that receives potential colonists at very long intervals. However, the same inconsistency applies to the birds of the Lesser Antilles. The estimated colonization rate in a mass-extinction model for the Lesser Antilles was 33 species my * years (Ricklefs and Bermingham 2001). If the 21 species having divergence times less than 0.025 represented the entire source pool for recent colonization, the colonization rate per species would be about 1.5 my-1, and less if the source pool were larger. Thus, the migration-fixation rate of 2.1 my * years is too high by comparison. The only way to reconcile this discrep- ancy in the speciation-divergence model is too assume very high rates of migration, with most species reaching the ar- chipelago soon after islands were available for colonization (e.g., Fig. 4). Of course, this does not explain the absence from the archipelago of so many of the source-area species or gaps in the distribution of older Lesser Antillean endemics. The estimated migration-fixation rate under the speciation- threshold model for the West Indian herpetofauna also seems very high. The colonization rate in the herpetofauna as a whole (73 endemic lineages over 65 my) is close to one per my, and the colonization rate per lineage is probably on the order of 0.01-0.1 my-1, which makes a migration-fixation rate of 4.6 my ^ several orders of magnitude too high, es- pecially considering that populations of many reptiles and amphibians on the larger islands might number in the tens of millions (Ricklefs and Lovette 1999). Even when the col- onization of nonendemic lineages is spread over 5 my, and the speciation threshold (ds) is increased from 2.4 to 10 or 20 my, the estimated migration-fixation rates of 0.34 and 0.13, respectively, are still high and the speciation threshold is unrealistically long. Although dispersal of individuals (m) to an established island population of size N is undoubtedly much more likely than an initial colonization event, it is unlikely that mIN, the estimated probability of fixation of a migrant mitochondrial allele, would approach the migration- fixation rates estimated under any of these sets of parameter values. The Speciation Threshold Although the speciation-threshold model can approximate a particular distribution of divergence values, the threshold value itself must also be realistic. Where the distribution of genetic divergence values show an obvious break, as in the case of the Lesser Antillean land birds, the model speciation threshold is clearly defined and offers little room for ad- justment. Does an mtDNA distance of 0.025 (2.5%) represent enough genetic divergence to prevent reproduction through either premating or postmating mechanisms? Differences be- tween populations of the same "species" on different islands range up to 7%. We surveyed 32 monophyletic lineages of passerine birds within the Lesser Antilles, 30 of which were represented by two or more island populations (Ricklefs and Bermingham 2001). Of these, 19 exhibited Tamura-Nei dis- tances greater than 0.5% (d = 0.005) between at least one pair of islands, which exceeds more than 97% of wifhin- population genetic distances. Twelve species had interisland divergences exceeding 1%, and 8 exceeded 2%. Genetic divergence between geographically distinct pop- ulations within continental species (incipient species, per- haps) ranged up to 8.5% (average 2.5 ?2.1% SD, n = 26; Avise and Walker 1998). Between continental "species" in the same genus, many of them sympatric, surveys of mtDNA genetic distances have shown ranges between 1.6 and 7.3% (4.4 ? 1.9% SD, n = 11; Seutin et al. 1993) and between 0.4 and 10.9% (5.1 ? 3.0% SD, n = 35; Klicka and Zink 1997). Among allopatric "species" of the thrasher genus Toxostoma. genetic distances averaged 5.2% (3.1 SD, n = 11 pairwise comparisons), and among sympatric species, d av- eraged 9.3% (1.1 SD, n = 6; Zink et al. 1999). Darwin's finches (Geospizinae) of the Galapagos Islands exhibit smaller genetic distances between sympatric species (<0.7% within Geospiza, n = 5; <1% within Camarhynchus, n = 4; Sato et al. 1999). Indeed, Zink (2002) seriously sug- gested that patterns of genetic variation do not reject the hypothesis that all Geospiza belong to a single species. Even between recognized genera, however, genetic distances are modest (Platyspiza-Certhidia, 3.9%; Camarhynchus-Platys- piza, 2.6%). The Darwin's finches are exceptional in that many species hybridize readily (P. R. Grant and B. R. Grant 1996, 1997a; B. R. Grant and P. R. Grant 1998); reproductive isolation is effected by song discrimination rather than re- lying on genetic incompatibility (B. R. Grant and P. R. Grant 1996; P. R. Grant and B. R. Grant 1997b). The data generally suggest that genetic distances between full species average about twice the speciation threshold of 2.5% mtDNA sequence divergence used to fit the speciation- divergence model to birds of the Lesser Antilles. However, genetic distance varies considerably among named species and some exhibit genetic distances of 2% or even less. In addition, the accumulation of neutral mitochondrial nucleo- tide substitutions at the point of reproductive isolation might be lower among island populations where selection for eco- logical and behavioral diversification in species-poor faunas could be strong. This is implied by the rapid morphological diversification of some bird lineages in the Galapagos and Hawaiian Islands (e.g., Lovette et al. 2002), which appears to be associated with secondary sympatry of sister species with relatively little genetic divergence (Grant 1998). If speciation of Lesser Antillean populations from source populations were prevented by a high rate of migration, one would expect secondary invasions of islands following spe- 1672 R. E. RICKLEFS AND E. BERMINGHAM ciation events that did occur. However, there are no cases in the Lesser Antilles of two or more closely related species of colonists on any single island that were sequentially derived from the same source population (Raffaele et al. 1998, E. Bermingham and R. E. Ricklefs, unpubl. data). Two cases might represent multiple invasions of distinct genetic hap- lotypes of the same species from one source to a single island, but these require further investigation. The population of the house wren {Troglodytes aedon) on Grenada harbors two mtDNA haplotypes differing by 4.1% and <0.1% from pop- ulations in Venezuela and Trinidad (E. Bermingham and R. E. Ricklefs, unpubl. data). However, the two mainland ref- erence sequences came from different localities, and so they may represent different source populations, as we have seen in the Caribbean grackle (Quiscalus lugubris). Nonetheless, it would appear that an mtDNA genetic distance of 4.1% was below the speciation threshold in this case, unless two cryptic species of house wrens occur on Grenada. The population of the yellow-bellied elaenia (Elaeniaflav- ogaster) on Grenada has two haplotypes that are 0.1 and 0.7% distant from contemporary haplotypes in the source popu- lation on Trinidad. The higher level of genetic divergence occasionally occurs within populations and we do not know whether additional sampling of the Trinidad population might reveal the more distant haplotypes there as well. Regardless of these details, the high colonization rates required by the speciation-divergence model should be accompanied by fre- quent multiple colonization of islands from the source and a rapid buildup of sympatric species through adaptive radiation within the archipelago. Except for the endemic radiation of four species of thrashers (Mimidae; Hunt et al. 2001: mini- mum genetic divergence = 0.104) and two hummingbirds of the genus Eulampis (genetic divergence = 0.069), this has not happened in the Lesser Antilles. Indeed, the endemic radiations suggest a higher speciation threshold than 0.025. Geographic Structure With respect to archipelagoes such as the Lesser Antilles, the speciation-divergence model also predicts that islands closest to the source would be most similar genetically to the source population when the migration rate is high. If time has been sufficient for speciation between the first island and the source population, there should also have been enough time for speciation between the second island and the first. In the present model, colonization, which is likely to be gov- erned by the migration rate m, is rapid compared to the rate of differentiation. Thus, all ecologically suitable islands would have populations soon after the colonization process begins, as observed in several recent cases and inferred from the lack of genetic differentiation among many widespread species. Because migration, divergence, and speciation occur independently across each gap in the distribution of a species, deep genetic divergence is as likely to arise between two islands as between the first island and the mainland source of colonists. This never happens. There is not a single case in the Lesser Antilles in which the divergence between two islands is greater than that between any one island and the source population. Reconciling this observation with the spe- ciation-divergence model requires a much higher rate of mi- gration between islands than between the continent and the first island in the chain. This might be the case in the Lesser Antilles because the islands are closer to each other than the first and last islands in the archipelago are to either the South American continent or the Greater Antilles, and because is- land populations tend to have higher densities in a broader range of habitats (Ricklefs and Bermingham 1999), poten- tially making dispersal to other islands more likely. However, the implied high rate of interisland movement is not consis- tent with the genetic divergence of island populations and gaps in the distributions of many species across the archi- pelago. Conclusions The data and model fits presented in this analysis do not provide a clear choice between models for the accumulation of species of small land birds in the Lesser Antilles. Speci- ation-divergence, heterogeneous colonization, and mass-ex- tinction models all provide excellent fits to the data. However, a speciation threshold of 2.5% mtDNA sequence divergence is probably too low for birds in reasonably diverse avifaunas. Variation in the speciation threshold itself would reduce the abruptness of the change in slope of the species accumulation curve, contrary to the pattern observed in the data and re- produced by a mass-extinction model. The strongest points of evidence disfavoring the speciation-divergence model for Lesser Antillean birds are (1) the lack of multiple coloni- zation events where island species have passed the speciation threshold, (2) the absence of species in which island-island genetic distances exceed the source-archipelago distance, and (3) the extinction of island populations without recolo- nization from within the archipelago or the source population. Infrequent phases of secondary expansion of tax a within the Lesser Antilles (Ricklefs and Bermingham 2001) suggest that high colonization rates are transient and therefore inconsis- tent with the continued migration presumed by the speciation- threshold model. Any ambiguity concerning mechanism represented by the lineage accumulation curve for the Lesser Antillean avifauna disappears in cases where colonization potential is consid- erably reduced, as in remote islands or groups with poor dispersal ability. When there is little potential for migration to prevent divergence, one does not expect a heterogeneous distribution of genetic divergence values between source and island populations. In this context, we examined data for the avifauna of the remote Hawaiian Islands and the herpetofauna of the West Indies, the latter presumably built up through infrequent colonization by rafting. Moreover, migration-fix- ation rates in the West Indian herpetofauna should be par- ticularly low owing to the large population sizes of reptiles and amphibians on islands in the Greater Antilles. In both cases, the estimated rates of migration for the spe- ciation-divergence model are too large, by several orders of magnitude, to be plausible. The Hawaiian avifauna does not exhibit marked heterogeneity of genetic divergence values, and a simple colonization-extinction model described the data adequately, as did a mass-extinction model with a moderate- sized event (S = 0.40 at d = 0.02 mtDNA sequence diver- gence). The West Indian herpetofauna exhibits a striking SPECIATION AND DIVERGENCE 1673 change in the slope of the lineage accumulation curve caused by assigning a young age to the large number of undiffer- entiated taxa in the islands. It is unlikely that improved ge- netic resolution of the relative colonization times would change the pattern markedly or that the fitted migration-fix- ation constant of the speciation-threshold model would be- come more acceptable. Because the speciation-divergence hypothesis remains marginally plausible for birds of the Lesser Antilles, addi- tional studies on species formation and on the more detailed phylogeographic history of the avifauna are priorities. In par- ticular, both high and low source-island divergence values among independent molecular markers within populations would be expected of the speciation-divergence model, whereas uniformly low divergence in recent colonists would favor transient phases of colonization followed by rapid ge- netic isolation of island populations. One conclusion from the analyses presented here, which is independent of the particular model adopted, is that barring mass extinctions the background extinction rate of lineages in archipelagoes is very low. In all cases, in the absence of mass extinctions, estimated average persistence times of lin- eages are at least half the total span represented by the range of colonization times. As a result, a substantial portion of the total colonization history of a group within a region is preserved in the living descendants of the original colonists. This means that much of the material needed to evaluate historical hypotheses concerning the accumulation and main- tenance of diversity on islands is available. ACKNOWLEDGMENTS K. Johnson, J. Lichstein, T. Simons, two anonymous re- viewers, and Associate Editor G. Wallis provided valuable comments on the manuscript. 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