ECOGRAPHY 24: 341–351. Copenhagen 2001

Latitudinal gradients in diversity: real patterns and random models

Patricia Koleff and Kevin J. Gaston

Koleff, P. and Gaston, K. J. 2001. Latitudinal gradients in diversity: real patternsand random models. – Ecography 24: 341–351.

Mid-domain models have been argued to provide a default explanation for the bestknown spatial pattern in biodiversity, namely the latitudinal gradient in speciesrichness. These models assume no environmental gradients, but merely a randomlatitudinal association between the size and placement of the geographic ranges ofspecies. A mid-domain peak in richness is generated because when the latitudinalextents of species in a given taxonomic group are bounded to north and south,perhaps by a physical constraint such as a continental edge or perhaps by a climaticconstraint such as a critical temperature or precipitation threshold, then the numberof ways in which ranges can be distributed changes systematically between thebounds. In addition, such models make predictions about latitudinal variation in thelatitudinal extents of the distributions of species, and in beta diversity (the spatialturnover in species identities). Here we test how well five mid-domain models predictobserved latitudinal patterns of species richness, latitudinal extent and beta diversityin two groups of birds, parrots and woodpeckers, across the New World. Whilst bothgroups exhibit clear gradients in richness and beta diversity and the general trend inspecies richness is acceptably predicted (but not accurately, unless substantial empir-ical information is assumed), the fit of these models is uniformly poor for betadiversity and latitudinal range extent. This suggests either that, at least for these data,as presently formulated mid-domain models are too simplistic, or that in practice themid-domain effect is not significant in determining geographical variation in diver-sity.

P. Koleff (p.koleff@sheffield.ac.uk) and K. J. Gaston, Biodi6ersity and MacroecologyGroup, Dept of Animal and Plant Sciences, Uni6. of Sheffield, Sheffield, U.K. S10 2TN.

The species richness of most higher taxa increases to-wards lower latitudes. A general explanation for thispattern has been much sought after, and a diverse arrayof mechanisms have been postulated and discussed (e.g.Pianka 1966, Stevens 1989, Currie 1991, Rohde 1992,1997, 1998, 1999, Rosenzweig 1992, 1995, Jablonski1993, Gaston 1996, Blackburn and Gaston 1996a,Rosenzweig and Sandlin 1997, Shepherd 1998, David-owitz and Rosenzweig 1998, Kaufman and Willig 1998,Willig and Lyons 1998, Gaston and Blackburn 2000).These have dwelt heavily on such issues as speciationand extinction rates, energy availability, environmentalcomplexity and stability, and patterns of competitionand predation. Colwell and Hurtt (1994) argue that theanswer may lie in something far simpler. They observethat null models that assume no environmental gradi-

ents, but merely a random latitudinal association be-tween the size and placement of the geographic rangesof species predict a peak of species richness at tropicallatitudes. This occurs because when the latitudinal ex-tents of species in a given taxonomic group arebounded to north and south, perhaps by a physicalconstraint such as a continental edge or perhaps by aclimatic constraint such as a critical temperature orprecipitation threshold, then the number of ways inwhich ranges can be distributed changes systematicallybetween the bounds. Thus, whilst species with theirlatitudinal midpoints at the midway point between thebounds can extend a little or a long way before thesebounds are encountered, those with midpoints close tothe bounds can only extend a little way before thisoccurs.

Accepted 25 September 2000

Copyright © ECOGRAPHY 2001ISSN 0906-7590Printed in Ireland – all rights reserved

ECOGRAPHY 24:3 (2001) 341

In fact, such mid-domain models generalise to predicta central peak in species richness along any spatialgradient with hard bounds at either end which preventspecies from extending their distributions any further(e.g. elevational and depth gradients; Rahbek 1997,Pineda and Caswell 1998). However, to date, suchmodels have been little tested even for latitudinal gradi-ents. Lees (1996) argued that latitudinal variation inspecies richness in Madagascar of the 50% most wide-spread species of mycalesine butterflies correspondedwell with a mid-domain model (see also Lees et al.1999), and Willig and Lyons (1998) found that a nullmodel of random determination of the limits of speciesranges explained a substantial amount of variation inthe latitudinal pattern of species richness of marsupialsand bats in the New World. In the latter case, as thedomain was constricted from all of the continental NewWorld to the distributional limits of all New Worldmarsupials or even to the smallest domain within which95% of all marsupial species occur, the pattern inspecies richness predicted by the null model fits moreaccurately the empirical data (Colwell and Lees 2000).

Two other predictions of the mid-domain modelswith regard to latitudinal gradients have also been littleexplored. First, a random latitudinal association be-tween the size and placement of geographic rangesresults in a greater average latitudinal extent of speciesoccurring at low latitudes than at high ones (Colwelland Hurtt 1994, Lyons and Willig 1997). This is con-trary to Rapoport’s rule, which states that latitudinalextents become smaller towards lower latitudes (Rapo-port 1982, Stevens 1989). Colwell and Hurtt (1994)suggest that some documented cases of latitudinal in-creases in latitudinal extent may have arisen becauseper-species sampling effort declines with species rich-ness, that latitudinal range is correlated with samplesize, and that as a result the latitudinal extents oftropical species have been markedly underestimated.Whilst the conclusion is undoubtedly correct for manygroups of species, this seems an unlikely explanation forany observed latitudinal increases in latitudinal extentfor groups of organisms such as birds, whose basicranges are reasonably well understood (at least at theresolution at which most macroecological analyses areconducted) even in the tropics. In any case, the contra-diction may not be important, because the evidence forRapoport’s rule is rather poor (Smith et al. 1994,Rohde 1996, Gaston et al. 1998, see also Ruggiero andLawton 1998, Hecnar 1999). What are wanting at thispoint, however, are direct comparisons of real latitudi-nal patterns in latitudinal extents, and the patternspredicted by mid-domain models (but see Lyons andWillig 1997).

Second, mid-domain models should make predictionsabout the form of latitudinal patterns in beta diversity,the extent of turnover (replacement) in species identities(Whittaker 1972) from one latitudinal band to another.

There is wide recognition that these patterns are centralto an understanding of latitudinal variation in biodiver-sity, but they have been remarkably poorly explored(but see Wilson and Shmida 1984, Willig and Sandlin1992, Harrison et al. 1992, Blackburn and Gaston1996b, Gaston and Williams 1996, Willig and Gannon1997). In part this is because of differences in themeasures of beta diversity which have been employedand differences in the form of turnover examined (e.g.calculated within or between latitudinal bands), butmay also have arisen through the lack of a clear nullpattern against which to compare those observed. Al-though mid-domain models may provide these predic-tions, they have not explicitly been formulated, andremain essentially untested.

In this paper, we conduct empirical tests of thelatitudinal patterns in species richness, latitudinal rangeextent, and beta diversity predicted by mid-domainmodels. We use data on the distributions of two groupsof birds, parrots (Psittacidae) and woodpeckers (Pi-cidae), in the New World (the land mass with thewidest continuous latitudinal extent). These taxa havedifferent overall latitudinal extents, are moderately spe-ciose and have distributions that are well understood.

Methods

Empirical data

Data on the geographical distribution of all 146 speciesof parrots and 121 species of woodpeckers occurring inthe New World were obtained from Juniper and Parr(1998) and Winkler et al. (1995), respectively. Thenortherly and southerly limits of the ranges of eachspecies were recorded within 5° bands of latitude, aresolution at which the distributions of all the speciesare sufficiently well known, and sufficiently accuratelydocumented by these sources, for present purposes.

Null models

Five mid-domain null models were constructed, the firstthree being those of Colwell (1999), see also Colwelland Hurtt 1994). In each case, either the latitudinalextent of the range, the latitudinal midpoint of thisextent, or both, expressed in 5° bands of latitude, werechosen at random from a defined distribution for eachof the 146 species of parrots or the 121 species ofwoodpeckers. Following Lyons and Willig (1997; seealso Colwell and Lees 2000), the hard boundaries to thelatitudinal extent of each group were regarded as beingformed by the 5° bands within which fell the mostnortherly and the most southerly occurrence of a spe-cies in the group. Thus, for parrots the bounds were at30°N and 60°S, between which lie 18 bands of 5°

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latitude, and for woodpeckers the bounds were at 70°Nand 55°S, between which lie 25 such bands.

Model A – within the hard boundaries to the latitu-dinal extent of the group, one of all the permissiblecombinations of the latitudinal midpoint and latitudinalextent was chosen at random, with replacement, foreach of the species (model 1 of Colwell 1999).

Model B – within the hard boundaries to the latitu-dinal extent of the group, for each species the latitudi-nal midpoint was chosen at random, with replacement,from all possible midpoints, and then the latitudinalextent was chosen at random, with replacement, fromthe possible extents for ranges with such a midpoint(model 2 of Colwell 1999).

Model C – within the hard boundaries to the latitu-dinal extent of the group, for each species the latitudi-nal extent was chosen at random, with replacement,from all possible extents, and then the latitudinal mid-point was chosen at random, with replacement, fromthe possible midpoints for ranges with such an extent(model 3 of Colwell 1999).

These three models make minimal assumptions aboutthe actual distributions of species of parrots and wood-peckers, including the maximum observed range extentof any one species (see also Rahbek 1997). This couldbe as high as the full span of the domain; the mostwidely distributed parrot species spans 11 of the 18possible latitudinal bands, whilst the most widely dis-tributed woodpecker species spans 12 of the 25 possiblebands. In this sense, these models are probably as closeto being null mid-domain models as is reasonable.

Model D – within the hard boundaries to the latitu-dinal extent of the group, for each species the latitudi-nal extent was chosen at random, without replacement,from all observed extents, and then the latitudinalmidpoint was chosen at random, with replacement,from the possible midpoints for ranges with such anextent.

Model E – within the hard boundaries to the latitu-dinal extent of the group, for each species the latitudi-nal midpoint was chosen at random, withoutreplacement, from all observed midpoints, and then thelatitudinal extent was chosen at random, with replace-ment, from the possible extents for ranges with such amidpoint.

Model D is the only model that explicitly makesassumptions about the maximum observed range ex-tent, since it takes into account empirical latitudinalextents. Model E again makes no assumptions aboutthe maximum observed range extent (but see Discus-sion), although not all possible midpoints within thedomain are represented by the observed data.

Each model was iterated 1000 times, separately forparrots and for woodpeckers, to determine the pre-dicted values of species richness, latitudinal extent andbeta diversity. In all cases the 95% confidence limits ofthe predicted values were very small relative to the

variation between models, and between the models andthe real data, and thus only the mean values for all1000 iterations are given.

For each model and separately for each of the twobird groups, for each latitudinal band the predictedspecies richness was determined as the number of spe-cies with latitudinal extents overlapping that band.

Latitudinal range extent was calculated as the differ-ence between the extreme northern and southern lati-tudes where a species is found (for empirical data evenif they have a discontinuous distribution). For eachmodel and separately for each of the two bird groups,species were grouped by the midpoint of this latitudinalextent, and the average extent determined for each suchmidpoint. These averages provided the basis for com-parison between predicted and observed values.

Predicted beta diversity was determined between eachcontiguous pair of 5° latitudinal bands across the gradi-ent, using the measures described below, from the sameiterations that species richness and latitudinal extentwere obtained, separately for parrots and forwoodpeckers.

Measures of beta diversity

A number of measures have been proposed for quan-tifying beta diversity across transects (e.g. Whittaker1960, Cody 1975, Routledge 1977, Wilson and Shmida1984, Magurran 1988, Harrison et al. 1992). It is notclear that any single measure has logical priority, andthus five measures were considered here (their proper-ties have been discussed by Wilson and Shmida 1984and Blackburn and Gaston 1996b):

(i) bw= (S/a)−1 (1)

where S is the overall number of species recorded intwo contiguous latitudinal bands, and a is the averagenumber of species in each of these bands (Whittaker1960).

(ii) b−2= (S/amax)−1 (2)

where amax is the number of species in the more spe-ciose of two contiguous latitudinal bands (Harrison etal. 1992; the division through by (N−1) is ignored, ashere this is always unity). Note that when a:amax,bw:b−2.

(iii) bj=1− (amax/S) (3)

bj was suggested by Williams (1996) as a means ofavoiding over-estimation of small absolute amounts ofturnover in partly occupied neighbourhoods, but withthe advantage of being an index with clear maximumvalues, from 0 to 1, where 0 implies no turnover. As

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implemented here this measure gives the same patternof results as b−2, although absolute values are different(correlations between the results for the two indices, forparrots and woodpeckers respectively, were r=0.995,pB0.001, n=17 and r=0.99, pB0.001, n=24). Wereport only the results for b−2.

(iv) bc= �g+ l�/2 (4)

where g is the cumulative gain in species between twocontiguous bands and l is the cumulative loss in speciesbetween the same contiguous latitudinal bands (Cody1975). This is an intuitive index to evaluate change inspecies composition along habitat gradients.

(v) bt= �g+ l�/2a (5)

This modification of bc, proposed by Wilson andShmida (1984), generated exactly the same results as bw

and so these are not presented (see also Wilson andShmida 1984, Blackburn and Gaston 1996b). bw and bt

produce the same results only when two localities arecontrasted, because of a very simple relationship be-tween these measures when comparing two sites, �g+l�=2S−2a.

Results

Species richness

For the present data, random models A and C predicta broadly parabolic pattern of change in species rich-ness along a latitudinal gradient, with a peak at themid-domain, whilst models B and D predict less pro-nounced variation but with richness still peaking in themore central regions, and declining symmetrically to-wards the ends of the domain (Fig. 1), as with Colwelland Hurtt’s models (1994). Model E, which considersempirical data for the latitudinal midpoints, predicts apeaked pattern of richness that relative to the othermodels is skewed to the north for parrots and to thesouth for woodpeckers (Fig. 1). Levels of richness at

any given point on the gradient are higher for model Cthan E or A, and generally higher for C, E and Acompared with B and D.

The actual richness of parrots and woodpeckerstends to peak somewhat south of the equator (between0° and 10°S), and to decline both north and south ofthis region (Fig. 1). Relationships between the values ofspecies richness at each latitude predicted by the ran-dom models and those observed were all statisticallysignificant (Table 1). In most cases, except for model E(parrots, r2=0.92; woodpeckers r2=0.85), the coeffi-cients of determination were low to moderate, althoughfor woodpeckers for models A and C 72–75% ofvariation in observed species richness was explained.However, in the case of parrots the observed peak inrichness is further north than that predicted by modelsA and C and in woodpeckers further south. In bothcases, the pattern of species richness is more peakedthan predicted by any of the models A to D. Speciesrichness is consistently overestimated by models C andA, and at its peak is underestimated by models B andD. In short, only model E provides a reasonable predic-tion of the slope of the species richness gradient forboth taxa.

Latitudinal extent

The three fully stochastic mid-domain models A, B,and C, and model E, predict a strongly peaked relation-ship between the latitudinal extent of a species and thelatitudinal midpoint of this extent, with the peak at themidpoint of the latitudinal gradient (Fig. 2). The pre-dictions of these models differ chiefly in the linearity ofthe declines in latitudinal extent away from this peak.Model D, which used the data for the actual latitudinalextents of species, predicts on average the same latitudi-nal extent across the domain.

For both taxa, statistically significant correlationswere found between observed latitudinal extents atdifferent latitudes and values predicted by all ra-ndom models, except for model D for woodpeckers(Table 1). Correlations were positive for parrots and

Table 1. Coefficients of correlation between the observed values of species richness (SR), average latitudinal range extent (LRE),and beta diversity (bw, b−2, bc) at different latitudes and the values predicted for those latitudes by each of the five randommid-domain models. For species richness n=18 for parrots and n=25 for woodpeckers, for latitudinal range extent n=35 forparrots and n=49 for woodpeckers, and for measures of beta diversity n=17 for parrots and n=24 for woodpeckers. pB0.01,**pB0.005, *** pB0.001.

Parrots Woodpeckers

SR bwLRE b−2 bcb−2bwLRESRbc

0.1390.728***−0.183−0.528***0.867***−0.2260.306−0.2760.532**0.712***Random ARandom B 0.632** 0.521** −0.256 −0.279 −0.536 0.770*** −0.532*** −0.208 −0.694*** −0.723***

−0.683***Random C 0.698*** 0.510** −0.288 0.292 −0.527 0.850*** −0.549*** −0.167 0.768***−0.102 0.479 0.712***0.682** 0.518** −0.335 0.328Random D 0.537 0.780*** −0.283−0.199 0.703*** 0.506*0.959***Random E 0.526** −0.291 0.721** 0.741*** 0.921*** −0.522**

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Fig. 1. Observed andpredicted relationshipsbetween latitude and speciesrichness for (a) parrots(observed data: r2=0.748,n=18, pB0.001) and (b)woodpeckers (observed data:r2=0.863, n=25,pB0.0001). Confidencelimits are not shown for therandom models as they arevery close to the meanvalues presented. Latitudesin the northern hemisphereare arbitrarily designated asnegative.

negative for woodpeckers. Indeed, the actual meanlatitudinal extents of neither woodpeckers nor parrotsshow a strong peak around the middle of the domainas models A, B, C and E predict (Fig. 2). Nor do

they show the roughly constant mean extents thatmodel D predicts. Rather, both groups exhibit rathercomplex latitudinal patterns of latitudinal rangeextent.

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Fig. 2. Observed andpredicted relationshipsbetween latitude andmean latitudinal extentfor (a) 146 species ofparrots and (b) 121species of woodpeckers.Confidence limits are notshown for the randommodels as they are veryclose to the mean valuespresented. For theobserved data barsindicate 91 SD.Latitudes in the northernhemisphere are arbitrarilydesignated as negative.

The relationship between latitudinal extent and lati-tude was different for each bird group. Consideringthe whole region, parrots show a statistically signifi-cant correlation (r=0.541, n=26, pB0.01), while nosuch result was found for the woodpeckers (r= −0.1999, n=34, ns). An inverse Rapoport effect wasobserved for the parrots of North America (r=0.622,

n=12, pB0.05), while for the southern hemisphereno significant relationship was found (r=0.163, n=15, ns). On the other hand, a Rapoport effect seemsto be evident for the woodpeckers in North America(r= −0.632, n=19, pB0.001), but not in the south-ern hemisphere (r= −0.0023, n=16, ns) (see Black-burn et al. 1998).

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Beta diversity

The different measures of beta diversity capture differ-ent features of spatial turnover in the identities ofspecies, and thus the mid-domain models make differ-ent predictions as to how they will change with latitude(Fig. 3). All five models predict that bw will declinetowards the centre of a latitudinal gradient (Fig. 3a, d),models B and C that bc will do likewise (Fig. 3c, f), andmodel B that b−2 will also do so (Fig. 3b, e). ModelsA, C and D predict that b−2 will peak towards thecentre of such a gradient (Fig. 3b, e), and model D thatbc will do so (Fig. 3f).

Although some correlations between observed andpredicted values of beta diversity across latitudes werestatistically significant (Table 1), none of the fully ran-dom models predicts well the observed patterns forparrots or woodpeckers for any of the measures (Fig.3a–f). The two random models based on empirical datado not predict bw, the most widely used measure of betadiversity. Model D also does not predict well the realpatterns of b−2 or bc for both taxa (Fig. 3c, f). ModelE seems to enhance the prediction of b−2 and bc forparrots, but not for woodpeckers (Table 1, Fig. 3b, c, e,f). For both bird groups, the latitude at which betadiversity was at its peak was approximately the samefor all measures, at ca 15°N. For bw the latitudinalgradient was approximately bimodal for parrots, with asecondary peak at 35°S, whilst for woodpeckers therewas no simple pattern. For b−2 and bc diversity de-clines away from the peak level to both the north andsouth, resulting in a highly asymmetrical pattern forparrots and a more symmetrical one for woodpeckers.

Beta diversity is connected with the idea of overlap-ping species distributions, i.e. beta diversity as an in-verse function of average range size (Routledge 1984,Harrison et al. 1992). Correlations between beta diver-sity and mean latitudinal extent across latitudinal bandswere not significant for parrots (bw, r= −0.315, n=16, ns; b−2, r= −0.261, n=16, ns; bc, r= −0.263,n=16, ns), but for two of the measures were statisti-cally significant for woodpeckers (bw, r= −0.3445,n=24, ns; b−2, r= −0.742, n=23, pB0.001; bc,r= −0.505, n=24, pB0.05). This relationship wassignificant for the woodpeckers in North Americaalone, with all three indices (bw, r= −0.828, n=11,pB0.005; b−2, r= −0.829, n=11, pB0.005; bc, r=−0.755, n=11, pB0.01), but not in the southernhemisphere.

Discussion

Both parrots and woodpeckers show fairly typical gra-dients of increasing species richness towards low lati-tudes, with quite smooth patterns of change at theresolution employed here (Fig. 1). In both cases the

peak in richness is somewhat to the south of theequator, as has been observed to be a general patternfor bird species in the New World (Blackburn andGaston 1996a, b, c, Blackburn et al. 1998). This resultsin the groups showing asymmetrical latitudinal patternsof richness about the equator, skewed towards thesouth in parrots and to the north in woodpeckers (Fig.1).

Neither of the two avian groups displays a simpleRapoport effect (Fig. 2). This conforms to the growingbody of evidence that, contrary to the assertions ofStevens (1989), a pattern of increasing latitudinal extenttowards higher latitudes is not a general one (Rohde1996, Gaston et al. 1998, Gaston and Chown 1999).Indeed, for both parrots and woodpeckers some of thelargest latitudinal extents are possessed by species withlatitudinal midpoints that lie close to the equator. De-spite such poor support, the Rapoport effect has at-tracted much attention. This is probably in large partbecause the primary mechanism proposed to explainsuch a pattern is intuitively appealing. The climaticvariability hypothesis postulates that towards higherlatitudes individual organisms need to be able to with-stand a wider range of climatic conditions and as aresult the species to which they belong can becomemore widely distributed (Stevens 1989). Gaston andChown (1999) have recently argued that this mecha-nism may indeed play a significant role in determininglatitudinal variation in the latitudinal extents of species,but what is important is not simply the variability ofclimate but the interplay between this variability andmean climatic conditions. This interaction predicts thatat low latitudes species may have relatively broad ex-tents, that these will tend to decline in the subtropicsand will then increase again towards higher latitudes.Because of hemispherical differences in patterns of cli-matic variation this effect is expected to be more con-spicuous in the northern hemisphere (Gaston andChown 1999). This pattern is indeed approximatelywhat is observed here for woodpeckers, which exhibit abroadly increasing mean latitudinal extent northwardsfrom ca 12.5°N (Fig. 2). Although parrots have alimited distribution in the northern hemisphere in theNew World the decline in mean latitudinal extentnorthwards from the equator may reflect the samegeneral pattern (Fig. 1).

There have been rather few published analyses oflatitudinal gradients in beta diversity (Gaston andWilliams 1996). Moreover, those that do exist havebeen conducted in sufficiently different ways (particu-larly with regard to the measures employed and meth-ods of analysis) as to render comparisons between themdifficult (but see Wilson and Shmida 1984 and Black-burn and Gaston 1996b). The comparisons betweenparrots and woodpeckers are thus revealing. In bothcases, the peak zone of turnover (using all measures)lies at ca 15°N. Blackburn and Gaston (1996b) found asimilar result for New World birds in general, using

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b−2. In this region the borders of Mexico, Belize,Guatemala, Honduras, and Nicaragua all cluster, and itmarks a zone of substantial floristic and faunisticchange (e.g. Leith and Werger 1989, Anon. 1992, Gauld

and Gaston 1995, Williams 1996, Ortega and Arita1998). The humid, lowland rainforest typifying much ofequatorial South America reaches its northern limitaround southern Mexico, and northern forests likewise

Fig. 3. Observed and predicted relationships between latitude and beta diversity for (a–c) parrots and (d–f) woodpeckers.Latitude is averaged between each pair of bands between where beta diversity is evaluated (a) bw (r2=0.001, n=17, ns), (b) b−2(r2=0.413, n=17, ns), (c) bc (r2=0.452, n=17, pB0.005), (d) bw (r2=0.023, n=24, ns), (e) b−2 (r2=0.001, n=24, ns), (f)bc (r2=0.022, n=24 ns). Confidence limits are not shown for the random models as they are very close to the mean valuespresented. Latitudes in the northern hemisphere are arbitrarily designated as negative.

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attain their southern range limit around this point.Northern conifers, for example, extend no further souththan Honduras and western Nicaragua, and tropicalforest does not extend north of 22°N in Mexico. Manybird species show similar range limits. Likewise, thehigh values of turnover obtained with bw at ca 30°Sseem to be coincident with the juxtaposition of thebiogeographic regions identified, the Guayano-Brazilianand the Andean-Patagonian, by the analysis of spatialpatterns of mammalian species (Ruggiero et al. 1998).

Whilst, as outlined above, some of the latitudinalpatterns of diversity of parrots and woodpeckers seemreadily interpretable in other terms, the logic behindmid-domain models as predictors of such patterns isintuitively attractive. The latitudinal ranges of the spe-cies in almost any group are inevitably bounded byabiotic or biotic factors. Nonetheless, for the two datasets examined here, in the main the models do notprovide good predictors of observed patterns. The prin-cipal exceptions are the fit of model E to the patterns ofspecies richness and, arguably, to some of the patternsof beta diversity. This model uses the observed mid-points of the species in each group of birds, but drawstheir latitudinal extents at random from all possibleextents in each case.

The distribution of midpoints might be expected apriori to provide a reasonable predictor of latitudinalpatterns of species richness, and combined with thelimitations of the random choice of possible extents, thefit of model E to the observed pattern of speciesrichness is perhaps unsurprising. Certainly there arereasonably good correlations between the observednumber of midpoints for each latitudinal band and theobserved number of species with distributions overlap-ping these bands (parrots, r=0.857, n=18, pB0.001;woodpeckers: r=0.899, n=25, pB0.001). The best fitof model E to observed patterns of beta diversity is formeasures b−2 and bc, and is substantially poorer thanthe fit to patterns of species richness (see Figs 1 and 3).However, again, the possible departures from the ob-served pattern may be sufficiently constrained in thismodel that this similarity should be anticipated. Thepoor prediction by model E of latitudinal patterns oflatitudinal range extent (Fig. 2) suggests that, nonethe-less, there are elements of the determination of latitudi-nal gradients in diversity which this modelfundamentally fails to capture.

Colwell and Hurtt (1994) showed that limiting themaximum range size to the empirical maximum reducesthe mid-domain peak in their null models, which mightimprove the fit of the species richness prediction formodels A, C and E. Although, incorporating yet moreinformation in the null model might be a matter fordiscussion, we examined the effect of using the maxi-mum range observed on the fit of model E (model E%).The predicted and observed patterns were better corre-lated for both parrots and woodpeckers for species

richness (r=0.986 and r=0.967 respectively, Fig. 1),and for bc (r=0.869 and r =0.905), but not for bw

(r= −0.155 and r= −0.063) or b−2 (r=0.708 andr=0.691, Fig. 3) and not for latitudinal extent (r=0.411, r= −0.270, Fig. 2).

In short, the results presented here suggest that mid-domain models tend to provide better predictions ofobserved patterns of diversity the more tightly specifiedthey are by observed data (e.g. actual latitudinal mid-points), begging the common question of the point atwhich a null model is so tightly specified in this sensethat it ceases to be strictly null.

In general, there are some obvious concerns aboutthe formulation of mid-domain models. Of these, fore-most is determining where the hard boundaries mightlie which provide the constraints on the positioning ofranges of different extent. When thought of in terms ofphysical boundaries, such as the edges of land massesthis is fairly straightforward. However, for probablymany groups of organisms, as with those consideredhere, the latitudinal extents of species do not extend asfar as these limits. Following Lyons and Willig (1997),we have used the most northerly and southerly occur-rences of any species in a group to define the positionof the hard boundaries for the mid-domain models.However, there is a difficulty in this approach. If wewere to repeat this exercise for the subfamilies of eachgroup we would find that in at least some cases thepositions of the most northerly and southerly occur-rences differed from that of the family as a whole. Wemight then perhaps legitimately analyse these groupsseparately on the grounds that the positions of the hardboundaries determining the patterns of diversity dis-played were themselves different, generating a differentset of predicted patterns. But, presumably one couldrepeat this argument for progressively smaller andsmaller taxonomic units, until such time as each specieswas being modelled separately (and the null distributionwas the real one). Moreover, there seems to be nological cut-off point at which one stops looking atsmaller taxonomic units, unless the boundaries sta-bilised or one took some pragmatic approach based onsample size. This begs the fundamental question ofwhether, with the exception of physical boundaries, onecan recognise bounds to the occurrence of species in agroup in the way in which mid-domain models demand.

A second concern about mid-domain models as theyhave been formulated to date (Colwell and Hurtt 1994,Willig and Lyons 1998), and in common with manyother considerations of the relationship between speciesrichness and the geographic ranges of species (e.g.Stevens 1989), is that they consider only a single dimen-sion of a species’ spatial distribution (but see Colwelland Lees 2000). The longitudinal component is ignored.Constructing two-dimensional mid-domain modelswould in most cases not be simple, particularly giventhe large and complex latitudinal variation in land area

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Table 2. Results of multiple regression analyses of the rela-tionships between species richness observed in a given latitudi-nal band (dependent variable) and the richness predicted byeach of the five random mid-domain models and the land areaof the latitudinal band. r2 is the coefficient of determinationfor the full model (including both predicted species richnessand area), P1 indicates the significance of an F-test testing fora significant increase in explained variance when area is addedto a model with predicted species richness already as anindependent variable, and P2 indicates the significance of anF-test testing for a significant increase in explained variancewhen predicted species richness is added to a model with areaalready as an independent variable. *pB0.01, **pB0.005,***pB0.0001.

WoodpeckersParrots

r2 P1 P2 r2 P2P1

Random A 0.693*** * ***ns 0.765*** nsRandom B 0.701*** ** ns 0.595*** ns ***Random C 0.695*** * ns 0.733*** ns ***Random D 0.695*** ** ***ns 0.610*** nsRandom E 0.936*** ns *** 0.906*** * ***

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Acknowledgements – We are grateful to Ana Rodrigues forhelp in programming random models, and to Tim Blackburnand Robert K. Colwell for discussion and comments on themanuscript. P.K. is funded by CONACyT (51822/122128),and K.J.G. is a Royal Society University Research Fellow.

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