null models chapter 7

8/13/2019 Null Models Chapter 7

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CO OCCURRENCE

COM MU NITY ASSEMBLY

A pervasive theme in community ecology is that the species composition of acomm unity is governed by deterministic assembly rules (Cody and Diam ond1975; Case and Diamond 1986). These rules emph asize the impo rtance ofinterspecific interactions in determ ining which specie s are found in a particularassem blage (Drake 199 0). We have already exam ined two such rules in thisbook: (I ) species that overlap too much in phenology, resource use, or otherniche dimensions cannot coexist (C hapters 4 and 5 ; (2) spec ies that d o coexistmust differ in body size or trophic morphology by a critical minimum thatallow s them to exploit different resources (Ch apter 6).

In this chapter, we con sider related assem bly rules that predict the presenceand ab sence of particular species rather than their sizes or patterns of resourceutilization. The significance and even the reality of such assembly rules havebeen widely debated in community ecology. Proponents have argued for theimp ortanc e of resource exploitation (Diam ond 197 5), com petitive hierarchies(Gilpin et al. 1 986 ), and priority effects (Drake 1991) in producing asse mb lyrules. Critics have co mp lained that man y of the rules are trivial tautologies thatlack predictive power (Connor and Simberloff 1979 and that the evidence forconsistent patterns of community structure, much less for assembly rules, ishardly compelling (Wiens 1 980 ; Wilson 1991a).

Laboratory stu dies have provided the strongest evide nce for assemb ly rules.For exa mp le, Gilpin et al. (1 986 ) thoroughly explored com petitive interactionsamon g 28 species of Drosophila. They varied the level of food resources, thelength of the experiments, and, most importantly, the initial combinations anddensities of species. Com petitive interactions were strong and predictable, andfew species persisted until the end of the experiments. For example, experi-me nts that started with 10 specie s alwa ys ended with few er than four. With 10initial species, there were 2 1,024 different possible species combinations,


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but fewer than a dozen of these persisted. These results contrast with nature,wh ere dozens of Dro-osophila spec ies may co-occur sym patrically.

Field experim ents also provide som e evidence for assembly rules. Abele(1984) experimentally supersaturated Poc illopo ra coral heads with com -ponent sp ecies of deca pod crustaceans. Agonistic interactions and predationby voracious wrasses caus ed the fauna to relax to a predictable speciesnumber and composition for coral heads of a given size. Cole (1983)described assembly rules for five ant species on sm all man grove islets in theFlorida Keys. Two primary species (Crem atog aster ashm eidi and Xeno-myrmex floridan us) could never be introduced successfully on very sm allislands, perhaps because of frequent flooding. In con trast, the minim umisland size occu pied by two secondary species (Pseudomyrm ex elon gatusand Zacryptocercus varians) was set by the presence of primary species,which were superior competitors. The two primary species also did notcoexist and form ed a classic checkerboard pattern (Dia m ond 1975), inwhich only one of the two species, but not both, occurred on an island.Experimental transplants and behavioral arena e xperimen ts confirmed thataggressive interactions between workers prevented coexistence and thateither primary species as an island resident could repu lse an invader. Thus,observed species comb ination s could be predicted on the basis of island sizeand com petit ive relationships am ong colonizers.

However, the debate over assembly rules has not been concerned with suchdetailed experimental studies. Rather, the controversy has been over whetherassembly rules can be inferred from nonexperimental data, specifically fromcombinations of coexisting species, usually on islands. Such data are conve-niently summ arized in a presence-absence matrix, which form s the fundam en-tal unit of study in many analyses of community ecology and biogeography(McCoy and Heck 1987 .Null models have been a useful tool for evaluatingpattern in such matrices and for revealing the extent to which sp ecies combina-tions can be predicted on the basis of simple models of island colonization(Simberloff 197th).

PRESENCE ABSENCE MATRICESPresence-absence matrices summarize data on the occurrence of a group ofspecies at a set of sites (Table 7.1 . The sites may range in size from 160-mm 2vegetation quadrats (Watkins and Wilson 1992) to large oceanic islands (Dia-m ond 1975) or entire continents (Smith 1983). Similarly, the set of speciesanalyzed may be restricted to ecologically similar congeners (Graves and


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Table 7.1P r e s e n c e - a b se n c e m a t r i x f o r f i v e fish t a x a i n 2 8 A u s t r a li a n s p r in g s

N u m b e r o fsp r ingsG o b y x x x x x x x x x x x x x x x x x x x x x x x x x x x x 2 8G u d g e o n x x x x x x x x x x x x x x x x x x x 1 9Ca t f i s h x x x x x x x x x x x x x x 1 4H a r d y h e a d x x x x x x x x x 9P e r c h x x x x x x x 7N u m b e r of spe c ie s 5 5 5 5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1Each row is a different taxon and each column is a different site = spring . Each represents the occurrence ofa taxon in a particular spring. Springs are ordered by species richness. Goby =Chlamydo~obius p.; GudgeonMogurnda mo~urnda; atfish = Neosilurus sp.; Hardyhead = Craterocephalus sp.; Perch = Leiopotheraponunicolor Note the nearly perfect pattern of nestedness. From Kodric-Brown and Brown 1993).


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Gotelli 1993) or to ecological or trophic guilds (Heatwole and Lev ins 1972), orit may encom pass entire avifaunas (Co nnor and Simberloff 1979).

In an r c matrix, each row of the matrix represents a different sp ecies andeach column represents a different site (Simberloff and Connor 1979). Eachcell in the matrix a,) contains a 0 or a 1, denoting, respectively, the absence orpresence of species i on site j. Th e row sum R, represents the total number ofoccurre nces of each species; it ranges fro m a minimu m of I to a maxim um ofc the number of sites censused . Similarly, the colum n sum C, gives the num berof species censused on site j and ranges from a minimum of to a maxim um ofr the num ber of species recorded. Th e grand matrix sum N represents the totalnumber of site occurre nces.

ASSUMPTIONS UNDERLYING THE ANALYSISOF PRESENCE ABSENCE MATRICESBefore discussing the analysis of such matrices, it is important to consider theassumptions implicit in the choice of the sampling universe and hence in thedim ensions of the matrix. T he decision of which species to analy ze is critical.If ecologically diverse assem blages are analyzed together, sp ecies interactionsmay not be apparent because of the dilution effect (Diam ond and Gilpin1982), in which many species will be compared that are not interacting withone another. Or, as Grant and Abb ott (198 0) have m ore colorfully expressed it,the analysis is in dan ger of throwing the baby out with the bathwa ter, or, mo respecifically, dro wn ing the baby in a tub that is too deep. On the other hand, ifthe analysis is restricted to very few species, the sam ple sizes .m ay simply betoo small to reveal any significant patterns, no matter what test is used (Biehland Matthews 1984 .

Equally important is the choice of which sites to include and which toexclude from the analysis. An implicit assumption in the analysis of presence-abse nce ma trices is that individual islands serv e as replicates to reveal repeatedpatterns of spec ies co-occurrence. But if the islands differ in size or suitabilityfor coloniza tion, imp ortant adjustments t o the null mo del are necessary. Indeed,if the islands vary too mu ch in area, habitat, and isolation, it may be invalid toassume they share a comm on colonization history. In stead, it may be necessaryto tailor sourc e pools for each individual island (G raves and Gotelli 1983) or tocontrol for island differences statistically (Sc hoe ner and Adler 1991). Wesuspect that many of the island archipelagoes that have been used for matrixanalysis, such as the West Indies, probably should be analyzed on an island-by-island basis, rather than as replicates within an archipelago.


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Published presence-absence matrices almost never include empty rows orcolumns, that is, sites with no species or species that occur on no sites.However, these degenerate arrangem ents may be very impo rtant in assessingnonran dom ness of species on the sites they d o occur on (Reddin gius 1983). Forexam ple, suppose the source pool fo r a pair of islands contains many speciesthat never occu r on either island. In this case, null mo dels that use the presence-absence matrix to construct the sampling universe will overestimate the ex-pected number of shared species (Wright and Biehl 1982). We believe that theinitial dec isions about the dimensionality of the matrix may e mo re importantin determ ining the results than the mechanics of the null mod el used.

There are other limitations of presence-absence matrices. By their verynature, these matrices do not include any information on absolute or relativeabundances of species (Haila and JLvinen 1981). If abundances are known,more powerful null models of com mu nity assembly can b e built (Gotelli et al.1987; Graves and Gotelln 1993). The analysis of presence-absence matricesassum es that mec hanism s controlling co-occu rrence are reflected in the dom i-nant or easily census ed life-history stage (Pearson 198 6). Bu t in many ass em -blages, larval dispersal (Roug hgarde n et al. 1988) and interactions am ong earlylife-history stage s (Wilbur 1988) are responsible fo r com mu nity patterns in theadult stage. Presence -absence matrices also require that residency criteria beunambiguous (Connor and Simberloff 1978). But it is sometimes unclearwhether a species is present on an island. Breedin g status may be difficult toestablish, and nonbreeding species may have significant ecological effects onresidents (Lynch and Johnson 1974; Simberloff 1976a). Finally, presence-absence matrices assume that all sites have received equal sampling effort,which may not be true even for well-studied archipelagoes (Connor and Sim-berloff 1978).

In spite of these limitations, the analysis of presence -abse nce matrices holdsthe promise of revealing general patterns of con~muni tystructure (Pearson1986). In some c ases, prescriptions for conservation biology hav e been mad eon the basis of presence-absence matrices (Patterson 198 7), so it is especiallyimportant that their analysis be on solid ground.

TW O MOD ES OF N LYSIS

Using the terminology of num erical taxonom y (Sneath and Sokal 1973), thereare two modes of analysis for a presence-absence matrix (Simberloff andConnor 1979). The Q-mode analysis assesses the similarity of different col-umns, revealing how similar sites are in the species they contain. In R-mode


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analysis, we compare the rows of the matrix and ask how similar species are inthe set of islands they occupy. Both m odes of analysis are relevant to the issueof assembly rules, although the Q-m ode analysis has its roots in biogeograp hicstudies of faunal similarity (McC oy and Heck 1987).

Q MO DE ANALYSES IN BIOGEOGRAPHYGiven a list of species for two sites, how can we quan tify the degree of similaritybetween the sites? The sites will obviously be most similar if they share identicalspecies lists and most dissimilar if they share no species. In analyses of biogeogra-phy, a plethora of correlated indices has been used to quantify site similarity(Simpson 1960; Cheetharn and Hazel 1969; Jackson et al. 1989). For example,Jaccard's (1 908) index scales similarity to range from 0 to I

N and N are the number of species present on each site, and Nc is the numberof species common to both sites. As with measures of species diversity (Chapter 2)and niche overlap (Chapter 4 , these indices are sensitive to sample size-thevalue of the index depends as much on the total species richness as it does on thenumber of shared species. Although some indices have been derived from proba-bility theory (Goodall 1966, 1974; Baroni-Urbani and Buser 1976), most lackan underlying statistical distribution. Conseq uently, it is difficult to say whethera particular value of an index is statistically significant.

If such indices are to be used at all, they mus t be combined with sim ulationsto assess the degree to which the index is unusually large or small (Wolda1981). Rice and B elland (19 82) used this approach to ex am ine the sim ilarity ofmoss floras in five lithophysiographic regions of B o rn e Bay, Newfoundland.They com bined the species lists from all five regions to generate an aggregatespecies pool, and weighted each species by its number of occurrences (one tofive). Without su ch weighting, the analysis would have assum ed that all specieshave equivalent probabilities of dispersal and persistence. From the weightedspecies pool, they drew randomly the observed number of species found ineach region, and then calculated Jaccard's index between all possible pairs ofsites. The simulations were repeated 20 times to generate 95 confidenceintervals for Jaccard's index. The result was that all the observed pairwisesimilarities were less than exp ecte d, and half of the combinations were signifi-cantly less than expected (Figure 7.1). This suggests that the floral regions a reindeed distinct, although if the regions were originally delineated solely on thebasis of these species lists, the test would be circular.


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Figure 7.1. Mean and 95 confidence intervals for randomized values of Jaccard's co -efficient of similarity between all possible pairs of five floristic provinces. The triad ol'vertical lines represents different source pool designations for mosses of Newfound-land. The dashed line is the observed sim ilarity between pairs of provinces. From Riceand Belland 1982),with permission .

In a biogeographic context, faunal provinces are delineated by grouping manysites together on the basis of painvise similarities (Pielou 1979a). A variety ofcluster analyses are available for this purpose, w hich hierarchically g roup sites onthe basis of similarity (Jackson et al. 1989). How ever, all of these methods as sum ethat the resulting groups are biologically meaningful. Because groupings can bemade for sets of random numbers, some method is needed for distinguishinggroups that are more similar than ex pected by chance. Strauss 1982) describedrandomization techniques to ensure that, whatever the clustering algorithm, theclusters are actually more similar than expected by chance. In an evolutionarycontext, similar bootstrapping m ethods have been used to assess the significance ofphylogenetic reconstructions (Felsenstein 1985).

Q MO DE ANALYSES N E C OL OGYIn ecological studies, pairwise similarities have also been used to relate faunalsimilarities to other site characteristics. For example, Terborgh (1973a) re-


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gressed pairwise similarities for West Indian avifaunas against interislanddistances. He concluded that island size and position exp lained between 80 and93 of the variation in avifaunal com position, and attributed the rema inder tohabitat differences and species interactions. Pow er (1 975 ) regressed interislandsimilarities of Gala pag os bird and plant species against physical charac teristicsof the islands. He concluded that bird similarity among islands was explainedby plant similarity, but that plant similarity was explained by interisland dis-tance, along with island area and elevation. For Galapagos finches, Ab bott et al.(1977) also used similarity indices to evalu ate the effects of floral com positionon avifaunal composition. All of these analyses are problematic because thesimilarity indices are a d hoc and because the painvise points in the regressionanalysis are not independen t of one another.A Simple Colonization Model Null Hypothesis 0)In an unpublished presentation, Johnson (1974; cited in Simberloff 1978a)pioneered a different approach. He used the number of shared species as asimple index of similarity between sites and then asked what the expectednumber of shared species is under the simplest colonization model (NullHypothesis 0). If two islands with m and n species, respectively, are co lonizedrandom ly by a pool of P equiprobable species, the expected number of sharedspecies E , , ) s

E mn P (7.2)with a variance (from C onn or and Simberloff 1978) of

As a standardized index of site similarity, Connor and Simberloff (1978)recommended

(Observed,, E , , . , ) / E (7.4)For the 29 G alapagos Islan ds, there are 40 6 island pairs. O f these, ther29observed num ber of shared species exceed'ed [he expected in 369 cases. Con nor

and S imberloff (1978) obtained sim ilar results a t the generic level fo r Galap a-gos plants, and Simberloff (1978a) found that all pairs of nine mangrove


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islands in the Florida K eys shared mo re insect species than expected if specieshad equ al colonization probabilities. Note th at this hypothesis do es not assumethat all islands are identical, because differences in species richness amongislands are m aintained in this model.A Sm all-Island Lim itation M odel Null Hyp othesis I)Johnson (1974) noted that habitat availability m ight be responsible for the factthat most sites shared more species than expec ted comp ared with Null Hypo th-esis 0. In particular, species may b e missing from small islands if these islandslack critical habitat. A modified protocol also placed species randomly andequiprobably, but the species pool for each island w as comp osed of only thosespecies found on islands of that size or smaller. In this case, the expectednumb er of shared spec ies is mn/P,, wh ere P, is the num ber of species in the poolof the larger island (m n). Incorporating this small-island limitation improv edthe fit of the G alapago s plant data to the expec ted values: 285 of the 40 6 pairsof islands shared more species than expected. If species number (rather thanisland siz e) characterized the low er minim um, 259 of the 406 pairs of islandsshared more species than exp ected.Limits at the uppe r size end are a lso possible. Diam ond (1975 ) hypothesizedthat certain supertramp species were restricted to species-po or com mu nitiesby diffuse competition. b incorpo rate this con straint into the null m odel, thespecies pool w ould con sist of all species that occurred o n islands of a particularsize or larger. The exte nt to which the probab ility of oc curr enc e of a species isrelated to comm unity size, island area, or other site attributes is the incidencefunction of the species (Diamond 1975). Incidence functions may serve asrealistic constraints in null models of species co-occurrence. They may alsorepresent a type of assembly rule that can be tested against other colonizationmodels, as we describe later in this chapter.A Nonrando m Dispersal M odel Null Hyp othesis 11)Null Hypothesis 0 is unrealistic, in part because it assumes species are identi-cal. But even if colonization were stochastic, species would be expected toocc ur at different freque ncie s on islands because they differ in their abilities todisperse and persist. Dispersal and persistence abilities are probably a functionof many species-level attributes, including population size (Terborgh and Win-ter 1978) and variability (K arr 1982 a), body size (Pimm et al. 1988 ), andgeographic range (G raves and Gotelli 1983; Jablonski 1986 ). W hen speciesoccurren ce probabilities can be measured independently, they can be a pow er-


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ful tool in null model analyse s. But for most archipelago es, this informa tion islacking. On e must either assum e (unrealistically) that species occurrence prob-abilities are equal or somehow estimate those probabilities from the matrixitself.

The occurrence distr ibution (Con nor and Simberloff 1978) has beenused to weigh t species, so that the probability of oc curre nce of a species isproportional to i ts f requency of occurrence (Abele and Patton 1976). Manycritics hav e objected that this procedure is circular-the incidence data areused to estim ate probabilities of occ urren ce, which are then used to test thepattern of co-occurrence (Grant and Abbo tt 1980 ; Diam ond and Gilpin1982; Case and Sidell 1983) . However , constraining marginal totals doesnot uniquely determine the pattern of co-occurrence in the data (Manly1991). As in co ntingenc y table analysis (Fienberg 1980 ), som e arrange-ments of the data may be highly improbable given a set of marginal con-straints (Connor and Simberloff 1983 ) , and this should be true whether ornot the marginals themselves are influenced by competition (Simberloff1978a) . As we shall see, the consequences of marginal constraints on theresolution of co-occurrence patterns depend greatly on whether the con-straints are absolu te or probabilistic.Analytic expressions for Null Hypothesis I1 are not known, so the ex-pected nu mb er of shared spec ies mu st be determine d with a s imulation. Insuch a s imu lation, species are drawn from an ag gregate source pool for thearchipelago , with probabili t ies weighted by species occurrences. As in NullHypotheses and 1 the observed number of species on each is land ismaintained as an absolute constraint. For the Galapagos plants, 338 of the4 6 is land pairs had more shared species than ex pected under Null Hy poth-esis I (Con nor and Simberloff 1978) . Johnso n (1974) performed a mixed-model simulation that simultaneously incorporated both the small-islandlimitation (Null Hyp othesis I ) and the weighted species pool (Null Hyp oth-esis 11 . For the Ga lapagos plant da ta, this was the only analysis in w hichthere were too man y island pairs with few er shared species than expected .However , only about 5 of the pairwise comparisons were statisticallysignificant (Fig ure 7.2).

From this result, Simberloff (1978a) concluded that the mixed mo del mightnot be far from an accurate description of colon ization in this archipelago. Healso reasoned that an archipelago structured by competition should show apredominance of island pairs with fewer shared species than expected. How-ever, he warned that shared species number is a weak statistic for detectingdiffuse competition, because it says nothing about taxonomic or ecologicalrelationships of the spec ies that do co-occur.


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Equiprobable Small island Weighted Mixe dlimitation

Null modelFigure 7 2 Four nu ll models of shared plant species between pairs of Galapagos Is-lands. For each model, the left-hand bar represents the number of island pairs thatshared more species than expected and the right-hand bar represents the num ber of is-land pairs that shared fewer species than expected. The hatched region represents thenumber of pairs that deviated at the 5 level of significance. Equiprobable is themodel in which all species had equal colonization probabilities. Small-island limita-tion allowed for equiprobable colonization but restricted species occurrences to theminimum island size observed. Weighted assigned colonization probabilities foreach species proportional to the number of island occurrences. Mixed model in-cluded both the small-island limitation and weighted colonization probabilities. If spe-cies occurrences were independent of one another, the height of the two bars would beapproximately equal, and roughly 2.5% of each bar would be shaded. Note the pre-dominance of island pairs that shared more species than expected for the first threenull m odels. Data from Simberloff (l9 78 a).

Criticisms of Ecological Q M ode AnalysisWright and Biehl (1982) a rgued tha t sha red- i s land ana lyses may not r evea lcompe t i t ive e f fec t s , because many of the pa invi se i s l and compar i sons a rebe tween i s l ands tha t have the same spec ie s se t s . Consequent ly , the sha red-spec ies ana ly s is wo uld fa i l to de tec t a high ly s ignif icant checkerboard dis t r i-bu t ion in which a pa i r o f s pe c i es ne ve r oc c u r toge t he r on t he s a m e is l a nd( D i a m ond 1975). T h e p r ob le m i s no t a s s e ve r e if m or e t ha n a s i ng l e pa ir o fs pec ie s oc c u r s i n a c he c ke r boa r d . W i t h a n a s s e m b l a ge c om po s e d o f s e ve r alc om pe t i t ive gu i l d s , s om e s ha r ed s pe c ie s pa t te r ns m a y be i m pr oba b l e , al t hough


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it will still be difficult to detect competition from these sorts of data (Connorand Simberloff 1984).

Ham ill and Wright (1988) pointed out two other complications with Q -modeanalysis. First, the number of species pairs should be tallied above and belowthe simulated median, not the simulated mean (see also Hendrickson 1981).Second , because the pairs of islands are not independent, it is not appropriate toask whether more than 5 of the pairs are significantly different from theexpectation. They recommended examining shared species among randomlyselected, independent pairs of islands, rather than among all possible pairs.These island pairs could be com pared to randomizations for each, rather than toa single randomization of the entire presence-absence matrix. With these re-finements, Hamill and Wright (1988) found a good agreement between ob-served and expected number of shared species for the Galapagos avifauna. Incontrast, Connor a nd S imbe rloff s (1978) analyses suggested a sign ificantexcess of shared species am ong islands. Neither analysis implicated a role forinterspecific competition in the distribution of the Galapagos avifauna, al-though these studies did not consider body size differences among coexistingspecies (see Chapter 6 .

We think that using Q-mode analysis to detect competition is a case of theright answer to the wrong question. Q-mode analysis should be used as abiogeograp hic tool to group sites on the basis of sim ilarity in species com posi-tion, rather than to infer spec ies interactions on the basis of dissimilarity. Th eanalysis is trivial for a single pair of species, but for larger assemblages,Equation 7.4 is a natural index for classifying pairs of sites on the basis ofspecies similarity. Of course, such an analysis depends on the assum ption of acommon species pool and on whether small island constraints and speciesweighting are retained, but these assumptions are implicit in any similarityanalysis. By generating an explicit null expectation and allowing for increas-ingly complex colonization models, Q-mode analysis provides a reliable frame-work for comparing species composition of sites (McCoy and Heck 1987 . Ifthe question is one of species interactions, then an R-mode analysis, whichcom pares sp ecies in the sites they sh are, is more app ropriate.

SSEMBLY RU LESMissing Species Com binationsMost analyses of presence-absence matrices have examined the number ofshared sites for a set of species. These R -mo de analyses have em phasize d the


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detection of nonran dom patterns of species association. Two broad classes ofinteractions can lead to distinctive species combinations. (1) Interaac tions ofspecies and sites These include site characteristics, such as island area , isola-tion, and habitat availability, and species characteristics, such as dispersal,persistence, a nd habitat affinities. Even if species colo nize sites independentlyof on e another, interactions be tween species and sites may lead to characteristicspecies combinations. (2) Interactions between species Com petition, preda-t ion, and mutualism between part icular species pairs and among largergroups of species will also lead to nonrando mness. Biotic interactions maylead to forbidden com binations of species (Diamo nd 1975) and also tocomb inations of species that are unusually s table and o ccur more frequentlythan expected.

The search for assembly rules of community organization is predicated onthe assum ption that biotic interactions are strong enou gh to produce discerniblepatterns. The null mo del approach has been t o build colonization mod els thatincorporate only species-site interactions (autecological factors) and to com-pare their predictions to the empirical data. T he goal is to provide a baseline for-recognizing nonrandom patterns caused by species interactions. Because avariety of forces can lead to either positive or negative species associations(Schluter 1984), it is a challenge to properly frame a nd interpret null m odels forassessing species interactions. Even if a pattern can be shown by a null modelanalysis to be nonra ndom , it may not be possible to determ ine the underlyingcause (Simberloff and C onno r 198 1 .Statistical Tests for Missing Species om binationsTh e pioneering work of E. Chris Pielou (Pielou and Pielou 1968; Pielou 1972 a)on the R-mode analysis of species associations was ignored and uncited forover a decade (Simberloff and Con nor 1981). Pielou and Pielou (1968) notedthat for presence-absence matrices with spe cies = row s), there are 2 possiblespecies combinations (including the co mbin ation of none of the r species beingpresent). With enough replicated sites, the observ ed and e xpected frequen ciesof these com binations can be com pared w ith a chi-squared test. Th is com pari-son holds the row totals of the presence-absence matrix constant. The nullhypothesis is that the observed frequency of different species combinations isno different than expected by chan ce, given that so me sp ecies in the collectionare common and others are rare.

Unfortunately, this test is rarely practical, because the number of speciescombinations is usually very large comp ared to the number of sites. Pielou andPielou (1968) instead advocated a simulation procedure in which each species


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is placed random ly and independently in sites. Fro m these simulations, on e canobtain the exp ected num ber of different species combinations.

The observed number of combinations may be less than this expectationeither because (1) som e sites are uniformly b etter than o thers for all species, orbecause (2) species associate nonrandomly with on e another or with differentsites. In either case, there will be fewer co mb inations of species than expectedunder the null model. By carefully comparing the number of occupied siteswith the curve generated by the null model, it may be possible to distinguishbetween segregative and nonsegregative association. Pielou a nd Pielou (1 968)did this fo r 1 3 collections of insect and spid er faun a associated w ith olyperusfungus brackets. T he num ber of combinations did not differ from the expectedin five collections. In the remaining eight, there were fewer associations thanexpe cted, although there was evide nce for segregative association in only threeof the samples.

Two restrictions ar e impo rtant in Pielou and Pielou s (1968) analysis. First,all sites are assum ed equiprobable. If this assumption can be relaxed, it may beeasier to reveal the effects of species interactions. S econ d, the analysis requiresan estim ate of the freque ncy of unoccupied sites. Unfortunately, these data arenot usually collected, an d ad hoc procedures f or dealing with em pty sites (e.g.,Siegfried 1976) may not be valid (Simberloff and Con nor 1981).

Whittam and Siegel-Causey (1981a) expanded o n Pielou an d Pielou s (1968)method and used a contin gen cy table analysis to test for species associations inAlaskan seabird colonies. Rather than looking just at the number of speciescombinations, Whittam and Siegel-Causey (1981a) used a series of hierarchi-cal log linear m odels to tease ap art positive and negative spec ies associations.Observed species combinations differed in frequency from those expectedunder a m odel of independen t assortm ent. M odels that best fit the data includedmostly positive two-way associations between species (Figure 7.3 . Higher-order interactions were uncommon, and positively associated pairs of speciestended to overlap in diet. These sophisticated analyses derive their strengthfrom very large sample sizes [20 species, 902 colonies = sites), and 19 dietcategories] and prov ide consid erable insight into species interactions.

Pielou (1972a) developed a second type of R-mode analysis. She analyzedthe variance of the row sums in the presence-absence matrix, that is, thevariance in the frequency of occurrence of each species. If the species aredistributed independently of one another, then the row sums will behave as aseries of random variables, with a covariance of zero. As described in Chap-ter 5, the ratio of variance in total species number in sites to the sum of thevariances of individual species provides a simple test for positive or negativecovariance in a presence-absence matrix. The variance ratio was d erived inde-


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LOWERGG TP1

MIXED

CLIFF IITbM RfC

BUR ROW

Figure 7.3. Complex species associations in Alaskan seabird colonies, as detected bycontingency table analysis. For each of five habitat types, the diagram summarizes sta-tistically significant interactions. Solid line= positive painvise interaction; dashed line= negative painvise interaction; heavy line = negative three-way interaction. GG =Glaucous Gull (Larus hyperhoreus); TP = Tufted Puffin (FI-ater-culacirrhatu); HP =Homed Puffin (Fratercula cornirulata); GwG Glaucous-winged Gull (Larusglaucescens); PG = Pigeon Guillemot (Cepphus columba); CrA = Crested Auklet(Aethia cristatellu); PA = Parakeet Auklet (Cyrlorrhynchus psittucula); LA= LeastAuklet (Aethia ~~ us il l a) ;A = Cassin s Auklet (Ptychoramphus a1eutic.u.~);WAWhiskered Auklet (Aethia pygmaea); DcC = Double-crested Cormorant (Phalacro-corax auritus); RfC Red-faced Cormorant (Phalacrocorax ur-ile); PC = PelagicCormorant Pha1acrocoraxpelagicu.s);RIK Red-legged Kittiwake (Rissu hrevi-rostris); BIK = Black-legged Kittiwake (Rissa tridactyla); CM = Common Murre(Uria aalge); TbM = Thick-billed Murre (Uria lomvia); FSP= Fork-tailed StormPetrel (Oceanodromafurcata); LSP = Leach s Stonn Petrel (Oceanodromaleucorhoa); AM Ancient Murrelet (Synthilihot-amphusantiquus). From Whittamand Siegel-Causey 198 a), with permission.


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168 Chapter

pendently by Jarvinen (1979) in the analysis of temporal fluctuations and bySch luter (1982) in the analysis of spatial distributions.

Th e variance ratio and related tests do not focus on assem bly rules per se ormissing species comb inations, but on th e average amou nt of association be-tween each species and all others in the matrix. Schluter (1984) analyzedseveral published presence-absence matrices with the ratio test an d found thatpositive covariation, rather than neutral or negative co-occurrence, was themost common pattern. However, the variance ratio test assumes that speciesnumber per site can vary freely, which m ay acco unt for the predom inance ofpositive aggregations.

Simberloff and Connor (1981) thoroughly reviewed the li terature onmissing species combinat ions. The y used M axwell-Boltzmann sta t ist ics tocalculate the probability of observing a particular number of missing spe-cies combinat ions in an archipelago. The M axwell-Boltzmann analysis as-sumes that each combination of species is an equiprobable group ofcolonists. An alternative analysis assumes that the probability of a particu-lar combination is proportional to the product of the colonization prob abil-i t ies of the component species. Thus, combinat ions that include common= widespread) species are more probable than combinat ions with rarespecies. This proportional weighting alw ays raises the observed tail proba-

bility. If some combinations are more likely than others, the number ofcombinat ions expected under the null model decreases.

Simberloff and Connor (1981) investigated claims that missing speciescombinations were nonrandom for assemb lages of plants (A bbott 1977), birds(Siegfried 1976; Abbott et al. 1977), and mam mals (Grant 1972b; M Closkey1978; King and Moors 1979). Few significant patterns emerged from theseanalyses-in mo st case s, the observed num ber of missing species com bina -tions was about what one w ould expect, given a random sam ple of all speciesin the archipelago.

One notable exception was the Gal5pagos finches, which had fewer multi-species combinations than expected. With six species of Geospiza there are9 20 possible three-species com binations. But only tw o of the 2 0 possiblecombinations were found on the islands 0 = 0.000002; weighted p = 0.001).On the other hand, the observed number of one- and tw o-species com binationsdid not differ from the null model, and the significance of the multispeciespatterns hinged upon which species and islands were included in the sourcepool. Nevertheless, the presence of very few species combinations of Galapa-gos finches is consisten t with the eviden ce of nonrandom bod y siz e patterns inthis assemblage (see Chapter 6 .


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Diamond s Assembly RulesDiamond (1975) popularized the study of community assembly rules in adetailed treatise on the distribution of 141 land-bird species on New Guineaand its satellite islands in the Bismarck Archipelago. Diamo nd (1975) sum ma -rized many years of his own field studies on species-area relationships, in-cidence functions, species combinations, and resource use patterns in thisarchipelago. Although Diamond (1975) discussed the importance of factorssuch as dispersal, habitat availability, and chance colonization, he emphasizedthat interspecific competition within groups of related species (ecological guilds)was the most important determinant of observed species combinations.Diamond (1975) codified and generalized his findings in a list of sevenrules of com mun ity assem bly:

I I f one considers all the comhinations that can be formed @om a group o f relatedspecies, only certain ones of these comhinations exist in nature.2 . These permissible combinations resist invaders that would transform them into aforbidden comhination.3 A c,omhination that is stal>le n a large or species-rich island may he unsrahle ona small or species-poor island.4 On a small or species-poor island a comhination may resist invaders that wouldhe incorporated on a larger or more species-rich island.5 Some pairs of species never coexist, either by themselves or as part of a largercomhination.6 . Some pairs of species that form an unstable combination by themselves may formpart of a stable larger comhination.7 . Corzversely, ome comhinations that are omposed entirely ofstuhle sub-combinations are themselves unstable.

The publication of D iam ond's (1975) rules touched off an acrimonious debateover null models that has spanned more than 2 years in the ecological litera-ture [see Wiens (1989) for an even-handed review of the controversy]. In thissection, we review the null models that have been used to test Diamond's(1975) rules. and the various claim s and counterclaim s that have followed. W ethink that all of the null models proposed to date contain one or more seriousflaws. Later in this chapter. we propose a hybrid model that combines the bestfeatures of each and se em s to overcom e the m ost serious deficiencies.


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170 hapter 7

First, we need to consider the nature of the assembly rules. Wiens (1989)noted that D iamond 's (1 975 ) rules are statements of pattern and d o not explic-itly describe mechan isms, although they do require that comm unities be in anequilibrium state. Nevertheless, the use of terms such as forbidden c om bina -tions clearly implicates the role of com petitiv e interactions and implies thatthe patterns are nonrandom and would not be expected in the absence ofspecies interactions.

It has been very difficult to make these rules operational. O n the o ne hand,the rules as stated are so broad that they wou ld be difficult to apply to real data,much less test them (Haefner 1988b). Null model analyses have contributedconsiderably to the study of asse mb ly rules by forc ing the issue of preciselywhat patterns m ust be established in order to docum ent a competitively struc-tured comm unity. On the other hand, Diamond's (1975 ) ow n interpretation ofthese rules for the Bismarck Archipelago seems so detailed, anecdotal, andnonstatistical that it resembles more an historical narrative than a test ofcom mu nity assembly. We agree with Simberloff (1978 a) that one is left withthe uneasy feeling that the rules lack predictive power, in that all the data arerequired before any prediction can be made.Th e onnor and Simberloff ProcedureConnor and Simberloff (1979) launched an aggressive attack on Diamond's(1975) assembly rules. They argued that Rules 2, 3, 4, 6, and 7 were eithertautologies or restatements of other rules. Connor and Simberloff (1979) didtest Rules 1 and 5 with a null model analysis. Rules 1 and 5 are identical, exceptthat Ru le 1 mentione d related species and Rule 5 was restricted to spec iespairs. R ule 5 described a ch eckerb oard pattern of species occu rrences, which isperhaps the simplest and most clear-cut of D iamond's (1975 ) assembly rules(Grav es and Gotelli 1993). Th e requirement of a com plete distributional ch eck-erboard is especially stringent: two species might have exclusive distributionson 9 9 islands, but if they occurred together on a single island, the pair wouldnot be scored as a true checkerboard. D iamo nd (1975 ) presented seven exam -ples of pairs of closely related, ecologically similar species that were distrib-uted as checkerb oards in the Bismarck A rchipelago.The checkerboard patterns (Figure 7.4 were striking and provided thestrongest evidence for Diam ond's (1975 ) hypothesis. For two of the exam ples(Macropygia c uckoo -dove s and Pach yce ph ala flycatchers), the probability thatthe species within each genus formed a distributional checkerboard [theshared islands analysis of W righ t and Biehl (1982)J wa s unusually sm all

(Connor and Simberloff 1979).


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147 E 150E 153 EI I

Checkerboard Distribution Small M M r o ~ g i a uckm-dovesM M. mrkinlayE. ngrirostrirnelther

G b ? ~y

Figure 7.4. Checkerboard distnbution of two Macropygia cuckoo-dove species in the Bis-marck Archipelago. Each island supports only one of the two species. M Macropygiamackinlayi;N =Mucropygiu nigrir ostris; = neither species present. From Diamond(1975). Reprinted by permission of the publisher from Ecology and Evolution ofConvnunities.M . L Cody and J. M. Diamond (eds). Cambridge, Mass.: The BelknapPress of Harvard University Press. CopyrightO 1975 by the President and Fellows of Har-vard College.

But even these examples may not be so clear-cut. Macr-opygia nigt-imstt-isoccurred m ostly on large, species-rich islands, whereas its putative com petitor(M. mackinlayi) occurred mostly on small and medium-sized islands with fewspecies. If these incidence constraints were not a direct result of competitiveinteractions, then the resulting checke rboard may be du e to independent co lo-nization by each species of different-sized islands (Wiens 1989). Moreover,Diamond's (1975) Figure 2 of this example (and also Diamond's (1975)Figure 22 for the genu s Ptilinopu s) did not show all of the empty islands thatsupported neither species. When these were incorporated into the calculation,the p value was ma rginally nonsignificant (Co nnor and Simberloff 1983 ).

However, Co nnor and Simberloff (197 9) were not satisfied with the demo n-stration that some pairs of species showed unusually exclusive distributions.The y argued that, with (I:') 9,8 70 possible species pairs, it was not surpris-


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172 hapter

ing that seven pairs showed exclusive distributions. Malanson (1982) made asimilar point about checkerboard distributions of plant species in the hang inggardens of Zion National Park.

Because D iamond (1975) did not provide data on all guild designations forthis assemblage, we cannot say how improbable it is that all five pairs would beecologically similar congeners. In order to assert that species checkerboardsare a manifestation of community assembly rules, Connor and Simberloff(1979) required that the observed number of pairs, trios, etc. of species thatexhibit checkerboards be significantly greater than the ex pected num ber gener-ated by an appropriate null model.

Diam ond s (1975) Bismarck d ata were never published, so Connor andSimberloff (1979) instead used presence-absence data for West Indian birdsand bats, and New Hebridean birds. Conno r and S imberloff (1979) used a nullmodel that randomized an observed presence-absence matrix subject to thefollowing three constraints:

1. The row totals of the randomized matrix were m aintained.2. The column totals of the randomized matrix were maintained.3. For each row, species occurrences w ere restricted to those islandsfor which total species richness fell within the range occupied by

the species.Constraint (1) maintained differences between species in their frequency ofoccurrence. Con straint (2) maintained d ifferences am ong islands in the numberof species they contained. Constraint (3) maintained the observed incidencefunction for each species (it could not occur in assemblages larger or sm allerthan those observed). The sim ulation first ordered the row s of the matrix frommost comm on to least common species, and then random ly placed species onislands until all three constraints we re satisfied.

When the matrix was sparsely filled, this algorithm was satisfactory. But ifmany of the species were widespread = large row totals), it was sometimesimpossible to place species late in the simulation and still maintain the con-straints. In these case s, Co nno r and Simberloff (1979) repeatedly interchangedsubmatrices in the matrix so that row an d column sum s were not altered. Afterthe interchanges, new m atrices that were n onequivalent to the ob served m atrixwere retained to estimate the null distribution. More recent advances in ran-domization algorithms (Stone and R oberts 1990; Daniel Sim berloff, personalcomm unication) have overcome this problem.

Connor and Simberloff (1979) used 10 such randomizations to construct ahistogram of the mean and standard deviation of the number of pairs (or


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able7 2Null model analyses of co-occurrence in island faunas ~

Observed ExpectedNumber of number of number ofpossible checkerboard checkerboard

Taxon species pairs species pairs species pairs pNew Hebrides birds 99 1 0.90 >0.99West Indies birds 1,029 62 1 437.0


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1 7 4 hapter 7

N U M B E R O I S L N D S S H R E D

Figure 7.5. Observed and expected numbers of species pairs versus number of islandsshared for West Indies birds. The solid line is the expected value generated by a nullmodel that held row and column totals constant and maintained species area restric-tions. Dots are the observed values. Although the logarithmic scale suggests a good fitbetween the observed and expected values, the difference is highly significant0.01 and in the direction predicted by the competition hypothesis (more species pairsthat share zero islands than expected by chance). See also Table 7.2. From C onnor andSimberloff (l9 79 ), with permission.

expec ted da ta was good , and tha t the re fore D iamond s (1975) seven examp lesof checke rboa rds were not a compe l l ing demons t ra t ion of Assembly Rules Ia n d 5. Fo r b i rd s o f t he B i s m a r c k A r c h i pe l ago , and for the Wes t Ind ian b ird an dba t f aunas , Co nno r and S imber lof f (1979) sugges ted tha t a l lopa t ri c spec ia t ionand l imi ted d i spe rsa l o f s ing le - i s l and endemics were a l t e rna t ive hypotheses


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that did not invoke interspecific competition but might account for checker-board distributions.

The Connor and Simberloff (1979) model has been used in two additionaltests. Matthews (1982) analyzed the occurrence of 13 minno w species distrib-uted in six streams of the Ozark watershed. Although some species pairs thatnever co-occurred in watersheds were morphologically and ecologically sim-ilar, the observed num ber of check erboa rd pairs matched the prediction s of thenull model. How ever, Matthews s (1982) analysis was based on a binomialdistribution, sampling with replacement, whereas a more appropriate analysisuses the hypergeometric dx stribution, sam pling w ithout replacem ent (Biehl andM atthews 1984). Jackson et al. (1992) used the Co nnor and Simberloff (1979)mod el to analyze five presence-absence m atrices fo r Ontario lake fish and a lsofound no evidence of nonrandomness.

riticisms of the onnor and Simberloff ProcedureConno r and S imberloff s (1979) analysis provoked several critiques (Alatalo1982; Diam ond and Gilpin 1982; Gilpin and D iamond 1982, 1984; Gilpin et al.1984 ) and subsequent rebuttals (C onno r and S imberloff 1983 , 1984; Gilpin etal. 1984 ; Simberloff and CJonnor 1984 ). He re, we sum marize the most im port-an t of the criticisms:

I The dilution effect Because Connor and Simberloff (1979) analyzedconfam ilial groups or entire avifaunas, com petitive effects were not apparent.Diamond s (1975) choice of exam ples suggested that the ecological guild wasthe correct unit of analysis for revealing competitive effects. However, guildsmust be established a priori by criteria that are independent of the co-occur-rence data being tested (Co nnor and S imberloff 1983). De lineating guilds is notan easy task (JaksiC and M edel 1990; Simberloff and D ayan 1991), and guilddesign ations clearly affect the outcom e of null model tests.

For example, Graves and Gotelli (1993) tested the significance of checker-board distributions in mixed-species flocks of Amazonian forest birds. Therewas no evidence of unusual patterns for the entire assemblage of flockingspecies, or for species grouped into ecological foraging guilds. Only when theanalysis was restricted to congeneric species within feeding guilds was thereevidence of unusual check erboa rd distributions. Even at this level, the patternswere statistically significant only for null models based on abundance andpopulation structure, rather than presence-absence data (Table 7.3). Vuilleu-m ier and Simberloff (1980) also found that co-occurrence patterns of Andeanbirds were affected by the designation of ecological and taxonomic guilds in


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Table 7 3Observed and expected numbers of perfect checkerboard distributions among pairs of Amazonian bird species in mixed species flocks

Null model

FlockGuild13

67

SPEC ABUN DEMOLevel n Obs. Exp. P Exp. Exp. P

Fisher s combined probabilities test(df 14)


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GeneraMonasaXiphorhynchusPhilydorAutomolus 3XenopsThamnomanesMyrmotherula 7HylophilusFisher s combined probabilities testdf 16)

Three null models were tested: (1) SPEC (randomization of presence-absence matrices), (2) ABUN (randomization of abundance data), and (3) DEMO(randomization of abundance data with intraspecific demographic constraints). These analyses were camed out at three hierarchical levels: (1) entireflocks, (2) foraging guilds, and 3) congeneric groups within feeding guilds. Guild designations: 1 arboreal gleaning insectivores; 2 arboreal sallyinginsectivores; 3 arboreal dead-leaf-searching insectivores; bark interior insectivores; 5 superficial bark insectivores; 6 arboreal omnivores; 7arboreal frugivores. Note that significant checkerboards were consistently revealed only for congeners within the same feeding gui ld. From Graves andGotelli (1993).


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178 Chapter

the source pool. Although the designation of guilds is a crucial step in theanalysis of co-occu rrence matrices, it is a procedure that is quite distinct fromthe null model randomizations themselves.

2 Effects of randomization constraints The three simultaneous constraintsimposed by C onno r and Simberloff (1 979) were severe and m ade it less likelythat the null hypothesis would be rejected. Fo r exam ple, relaxing the inci-dence constraint prevented the simulation for the New Heb rides matrix fromhanging up and revealed significant negative associations between species(Wilson 1987). For a similar model w ith only row and column sum constraints,Wilson et al. (1992) detected no evidence of nonrandom ness in the distributionof rock-pool algae, whereas both positive and negative associations wererevealed in the flora of islands in Lake Manapouri, New Zealand (Wilson1988).

However, if row and column totals are constrained, some, but not all,checke rboard distributions canno t be detected by the null model (Diamond andGilpin 1982; Connor and Simberloff 1984). In an empirical comparison ofseveral R-mod e analyses, the Conn or and S imberloff (1979) procedu re was theonly one that could not detect nonrandomness in presence-absence matrices forfish assemblages (Jackson et al. 1992). More formally, Roberts and Stone(1990) showed that the average number of shared islands among all speciespairs cannot change in a simulation if row and column totals are constrained.Somewhat paradoxically, matrices with mutually exclusive species pairs willalways contain som e species pairs that co-occur more than expec ted (Stone andRoberts 1992).

Th e sam ple variance of the num ber of shared islands reflects this pattern. Fo rthe New Hebrides bird data, this variance was significantly larger than ex-pected under the null model that held row and colum n totals co nstant, suggest-ing that some species pairs shared too many islands and others sh ared too few(Roberts and Stone 1990). Indices of checkerboardedness (Stone and Rob -erts 1990) and togetherness (Stone and Roberts 1992) also confirm ed thatthere were too many exclusive and aggregated pairs of species within con-familial subsets of the New Hebridean avifauna.

Som e authors (Grant and Abbott 1980 ; Colwell and Winkler 1984 ) haveclaimed that it is circular to constrain marginal totals, because the marginalsalso reflect interspecific com petition. This claim h as never been validated, butif it is true, then the Connor and Simberloff (1979) model will reveal onlycompetitive effects above and beyond those expressed in the row and columnsums. The hypothesis that competition sets the total number of island occur-rences for a particular spe cies (or the total num ber of species on a particular


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island) is quite distinct from the hypothesis of forbidden species com bination s,and deserves to be tested in its own right. There are many reasons besidescompetition that spe cies occur on many or few islands and that species numbervaries among islands. We agree with Connor and Simberloff (1983) that thesefactors need to be incorporated into null models, although imposing ab solutemarginal constrain ts may be to o severe a restriction.

Finally, the debate over marginal constraints also reflects the way in whichthe null models are interpreted. A s we noted in Chapter 1 one interpretation isthat the null mode l is simply a statistical randomization. T his interpretation iscons istent with the use of absolu te marginal co nstraints, beca use the question iswhether the co-occurrence patterns are nonrandom, given the observed sam-ple of species and islands. On the other hand, if the rando mization is view edas a model of community colonization in the absence of competition, themarginal constraints may not be appropriate. In a group of randomly colonizedarchip elago es, we would not expect each replicate of an island to have exa ctlythe same number of species, nor would we expect each species to occur onexactly the sam e numbe r of islands in the different archipelagoes. We developthese ideas later in this chapter.3 Significance tests Conno r and Simberloff (1979) comp ared the observed

and expected distributions with a chi-squared test. However, this may beinappropriate because of the nonlinear dependence imposed by the marginalconstraints. Even if the chi-squared test were appropriate, the number of de-grees of freedom used by Connor and Simberloff (1979) was roughly doublethe best fit value found empirically (Robe rts and Stone 1990 .The same criti-cism applies to Gilpin and Diam ond's (1982) null model, which also comparedobserved and expected d istributions of species pairs w ith a chi-squared test.The Wright and iehl ProcedureW right and B ieh l(1 98 2) suggested a shared-island test for detecting unusualspecies co-occurrences. For each species pair, they ca lculated the tail probabil-ity of finding the observed number of co-occurrences. The calculation isidentical to Co nnor and Sim berloff's (1978 ) shared species analy sis, but withthe rows and colum ns of the presence-absence m atrix transposed. Wright andBie hl(19 82) argued that an assem blage exhibited nonrandom ness if more than5% o f the species pairs shared more islands than expected (at the 5% signifi-cance level). By this test, the New Hebridean avifauna exhibited aggregation,because 8% of the species pairs shared significantly mo re islands than expectedby chance.


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180 Chapter

One advantage of the Wright and Biehl (1982) method is that it directlypinpoints particular species pairs that show aggregated or segregated distri-butions. However, assessing the statistical significance of all possible non-independent specie s pairs is problematic. E ven if the null hypothesis is rejectedfor more than 5 of the pairs, this pattern can be caused by non rando mn ess inthe distribution of only a few specie s (Connor and Simberloff 1983). Th e samecriticism applies to the shared species analysis of Connor and Simberloff(1978) and to G ilpin and Diamond s (1982) null model fo r R-mode analyses.An excess of species pairs that are significant at the 5 level does not meanthat the patterns are biologically meanin gful for all of these pairs.

A more serious problem is that Wright and Biehl s (1982 ) shared islandmodel assumes that all sites are equivalen t. Consequen tly, t confounds species-siteassociations with the effects of species interactions. The slight excess of NewHebrides species pairs that shared more islands than expected may simplyreflect differences in island suitability.The Gilpin and iamond Procedure

Gilpin and Diamond (1982) developed their own R-m ode a nalysis as an alter-native to the Con nor and Sim berloff (1979) approach . Gilpin and Diam ond s(1982) null model was based on the principles of contingency table analysis.For species i on island j they calculated the probability of occurrence as

where R , is the row total for spec ies i, C s the colum n total for island j and Nis the grand total for the presence-absence matrix. Next, they calculated theexpected overlap for each species pair by summing the product of theseprobabilities across all islands. Observed and expected overlaps for each spe-cies pair were standardized and then compared with a chi-squared test to astandard normal distribution. If the null hypothesis of independent placementwere true, the histogram of normalized devia tes would follow a normal distri-bution. Species pairs that showed unusual aggregation would appear in theright-hand tail of the distribution, and species pairs that showed unusual segre-gation w ould appea r in the left-hand tail.

For the New Hebridean birds, the observed distribution of deviates wassignificantly different from a standard norma l, and w eakly skew ed tow ards theright. The Gilpin and Diamond (1982) analysis did not reveal any significantnegative associations, in con trast to the results of Wilson (1987 ) and of S toneand Rob erts (1992). Returning to Diamond s (1975) original Bismarck data ,Gilpin and Diamond 1 982 found a strong excess of positive a ssociatio ns and


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a weak excess of negative associations. They examined the extreme speciespairs in detail and attributed positive associations to shared habitat require-ments, clumping of single-island endemics on large islands, shared geographicorigins, and shared distributional strategies. Negative associations were attrib-uted to competitive exclusion, differing distributional strategies, and differinggeographic origins. Compared to Diamond's (1975) original conclusions, thenull model interpretations of Gilpin and Diamond (1982) placed considerablyless emphasis on competitive interactions in producing community assemblyrules for the birds of the Bism arck A rchipelago.

Using the Gilpin and Diamond (1982) model, Jackson et al. (1992) alsofound mostly positive associations between species pairs of fishes in lakes.These positive associations were usually between pairs of cold-water specieswith similar habitat requirements. Negative associations were usually betweenpiscivorous fish species and their prey. Finally, McFarlane (1989) found anexcess of positive associations for the Antillean bat fauna and attributed thispattern to a large num ber of single-island end em ics in the northern Antilles.

Gilpin and Diamond's (1982) approach was important because it introducedthe idea that the marginal totals may represent expe ted values rather thanabsolute constraints In different runs of a stochastic model, we would notexp ect each island to support precisely the observed num ber of species, or eac hspecies to always occur with its observed frequency. In fact, putting a strictcap on the number of species that can occur on an island, as Connor and

Simberloff (1979) did, could be interpreted as a competitive limit to localspecies richness. In contrast, if islands behave as targets that are colonizedindependently by different species (Coleman et al. 1982), we would expectsome variance about the expected species num ber in our null m odel. Em piri-cally, a target mod el is consistent with the find ing that island area typicallyexplains only 50% of the variation in species number (Boecklen and Gotelli1984). Although Gilpin and D iam ond (1982) did not em phasiz e this point, theirapproach was an impo rtant first step toward incorporating this sort of variabil-ity into null m odels.

Unfortunately, there are several problems with the Gilpin and Diamond(1982) model. First, the formula for cell probabilities R,C,/N) is actually thecell expectation (Connor and Simberloff 1983), and consequently some of theexpected probabilities in a matrix can take on values 1.0. Even when acorrected formula is used, the contingency table model is not strictly correct,because it allows for multiple occurrences in eac h cell of the matrix, whereas apresence-absence matrix only contains zeros and ones. This may not be aproblem in practice, because M onte C arlo simulations of cell probabilities gavequalitatively similar results (appendix to Gilpin and Diamond 1982).


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182 hapter 7

Seco nd, the Gilpin and Diamo nd (1 982) model implicitly allows for degen-erate distributions: emp ty islands and missing species, which correspon d tomatrix colum ns or rows with all zero entries (Con nor and Sim berloff 1983 ). Inthe Gilpin and Diamond (1982 ) model, occasionally a given island would neverbe hit by any of the species. Unless the observed presenc e-absence matrixcontained the full sam pling universe of m issing species and em pty islands, theexpected values from this mod el would be biased. On the other hand , Haefner(198%) found that degenerate distributions introduced only a slight bias to-ward a ccepting the null hypothesis.

Finally, the Gilpin and D iamond (1982 ) model is susceptible to Type I error.Wilson (1 987) co nstructed a random presence -absence m atrix with a probabil-ity of occurrence of 0.33 for every cell. This matrix was significantly non-random by the Gilpin and Diamond (1982) model O 0.001), with a slightexcess of positive species associations. Gilpin and Diamond (1987) objectedthat such randomly constructed matrices may contain biological structure.Nevertheless, Wilson's (1987) analysis suggests that the Gilpin and Diamond(1982) model is probably not appropriate for the analysis of co-occurrencedata.Summary of the Controversy and a New pproachWe think the controversy o ver R-m ode analys is boils down to four issues:

1 Which species and which islands should he analyzed? It is surprising howlittle attention has been given to this point. The same data sets have beenanalyzed over and over with little discussion of source pools, colonizationpotential, and hab itat availability of islands (Terborgh 1981 Graves and Gotelli1983 . Some of these factors have been discussed post hoc, but they must beconsidered in a systematic fashion before any analysis is attem pted. Careful apriori selection of sites and species by ex plicit criteria is at least as im portant asthe particular null model used for analysis.

2 Which metric should be used? In other words, how do we properlyquantify nonrandomness and species associations in a presence-absence ma-trix? Th ere are many different kinds of structure in a prese nce-abse nce matrix,but for the purposes of recognizing species associations, we think th e followingfive metrics are m ost informative:

a The number of species combinations This is perhaps the most basicmeasure of community organization (Pielou and Pielou 1968). If assembly


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rules are operative, there should be few er species com binations observ ed thanexpected under a n approp riate null model.

b. The number of checkerboard distributions. Th e checkerboard distributionis the simplest and most clear-cut of Diamond's (1975) assembly rules. Itrepresents the strongest possible pattern of species repulsion.

c. The checkerhoardedness index q Stone and Roberts 1990) .This sta-tistic measures the overall tendency fo r species pairs in a m atrix to co-occur. Itmay reveal species pairs that associate negatively but d o not occu r in a perfectcheckerboard.

d. The togetherness index of Stone and Roberts 1992) .Both positive andnegative associations are possible in the sam e matrix, and this index m easuresthe tendency for species to co-occur.

e. Schluter's (1984) variance ratio. Negative co-occurrence as measuredby the variance rat io is not always equivalent to that measured by thecheckerboardedness index. This difference arises because the variance

ratio test does not constrain column totals . Different patterns of negativecovariat ion may be revealed by com paring the variance rat io to null modelpredictions.

3. Which simulation procedure should he used? We accept the logic ofCon nor and Simberloff (1979 ) that neither islands no r species are equiprob ableand that this should be reflected in the null mod el. How ever, their simultaneousconstraint of row and column totals was simply too severe. We prefer anapproach s in ~i la r o Gilpin and D iamond's (1982), which treats the row andcolumn totals as expectations; the actual row and column totals should beallowed to vary from on e simulation to the next. Although both the Connor andSimberloff (197 9) and the Gilpin and Diamo nd (1982) approaches are flawed,we think the following two simulation procedures are acceptable alternatives:

a. Row totals,fixed, column totals pmhahilistic. This model takes the ob-served frequency of oc currenc e of each sp ecies as a constrain t bu t treats eachsite as a target ; the probability of occu rrence of each spe cies at ea ch site isproportional t o the total num ber of species at that site. Th us, species num ber ineach site will vary somewhat from one simulation to the next, although therelative rankings of sites in their species richness w ill be m aintained on aver-age. Simberloff and G otelli (1984) used this procedure to ex am ine incidencefunctions, which will e described later. This protocol is the inverse of Co nnoran d Simberloff's (197 8) Null Hypo thesis I1 and Patterson and Atmar's (1986 )RANDOM algorithm, in which column totals were fixed and row totals wereprobabilistic.


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184 Chapter 7

b. Row and column totals probabilistic. A ess constrained null m ode lwou ld allow both the row and column totals to vary randomly. In this case,what is simu lated is not the placement of each individual species across theset of sites, but rather, the placemen t of the N species-occurrences acrossthe entire matrix. F or cell a,, in the m atrix, the initial prob ability of a "hit"in the sim ulation is the joint prob ability of selecting the species R, /N)andselecting th e site (C,/N). Th is cell proba bility is thus (R,c,IN~), wh ich cor-rects the error in Gilpin and D iamond (1982). In this simulation m ode l, themost l ikely occurrence will be of the most common species on the mostspecies -rich site, and the least likely occurre nce w ill be of the rarest specieson the mo st depauperate site.

One com plication is that these estimated probabilities will apply only to thevery first occurrence that is placed in the matrix. After the in itial placem ent, thedistribution of less probable com binations will be mo re "even" than predictedby these probabilities. Beca use sites can only be occupied once per species, theless probable sites are consecutively hit as the matrix fills up. If the marginalprobabilities are squared and then rescaled, the resulting marginal distributionsin the sim ulated matrix w ill more closely resem ble the original colum n and rowtotals (Bruce D. Pa tterson, personal com munication).

second complication is that both protocols, and particularly protocol b),could occasionally lead to degenerate distributions = empty rows or columns)in simulated matrices. As we noted earlier, degenerate matrices should not becompared to observed presence-absence matrices, unless the full samplinguniverse of sites and species is known. Simulated deg enerate matrices shouldbe either discarded from the analyses or have a single species occurrencerandomly repositioned to fill the empty row or column. Because all occur-rences can be easily placed by either algorithm a )or b), difficulties in fillingthe matrix and sim ultaneous ly maintaining row and colum n totals (Connor andSimberloff 1979) are not present.

This simulation protocol m ay be expanded so that marginal probabilities donot depend on row and colum n totals but on independently m easured attributesof species and islands. For exam ple, site probabilities could be set proportionalto island area (Colem an et al. 1982 ; Simberloff and G otelli 1984) rather thantotal species richness. Species probabilities could be scaled proportional todensity (Haila and Jarvinen 1981) or geographic range size (Grave s and Gotelli1983) rather than to the total number of oc currences for each species. This sortof analysis requires a good deal of biological insight and information aboutsites and species, but it avoids the circularity (and convenience) of usingmarginal sum s to estimate ce ll probabilities.


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4 How should the observed and simulated distributions be compared BothConnor and Simberloff (1979) and Gilpin and Diamond (1982) used a chi-squared test to decide statistical significance. This is problem atic, because n oneof the frequenc y classes in their analyses were independent of one another. Werecommend summarizing matrix patterns in a single metric (or five metrics ).In this way, statistical significa nce can be directly estimated by classicalrandom ization procedu res (Edging ton 1987)-the metric fo r the observed ma -trix can be directly com pared to the distribution of values from a large numb erof sim ulated matrices (usually 21 00 0) to estimate the probability value. This ismuch more straightforward than trying to compare a simulated with an ob-served distribution when the frequency classes of that distribution are notindepend ent of one another.

This approach also highlights an important difference in the strategy ofanalyzing presence-absence matrices. Connor and Simberloff (1978), W rightand Biehl (1982), and Gilpin and Diamond (1982) all advocated examiningdeviations of all possible pairs of islands or species and then attributing signif-icance to those pairs that show ed extr em e values. We think the re are statisticaland con ceptual difficulties in this. T he statistical difficulty is that none of thepairs are inde penden t of one another, so it is unclear w hich pairs are statisticallyand biologically significant. The conceptual difficulty is that this procedurecom es perilously close to "data-dredging" (Selvin and Stewart 1966). Theremay be a temptation to infer the hypothesis from em ergent patterns, rather thanexplicitly stating an a priori hypothesis and testing for it with the null model.Rather than examining the individual pairs to reveal potential interaction,we prefer to carefully sele ct spec ies and sites for this purpose ahea d of time,and then test whether there are nonrandom patterns in the matrix. In thisway, the patterns and the m echanisms that produce them are kept conceptu-ally distinct from one another. We cannot overemphasize that the selectioncriteria must not include the distributional patterns being testedOther Tests o ssembly RulesDiamond's (1975) approach to assembly rules emp hasized the particular com-bination of species within a guild that coexist and the intermeshing resourceutilizations that allow these sp ecies to persist. Other studies have expanded onDiamond's (1975) model and used null models to test for patterns of comm u-nity assembly. For exa mp le, Pulliam (1975) used estim ates of seed availability,bill sizes, and diets of wintering sparrow species to construct a "coexistencematrix" that predicted the bill sizes of species combinations that cou ld coexist.Th e initial application of the model w as successfu l, but add itional data for three


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186 Chapter 7Iite Arc the Basic Species poolConditions requirements Tor

existence met?Yes

heck anotherspecies from

the poolInsert species types

Are all speciesin pool testedfor insertion?

Apply dclction rulesto each species type

In community

Delete marked speciestypes and those without

opposrte sex

Species l~ s tfor current site

Repeat forall sites

Figure 7.6. Flow chart for null models used to test assembly rules of Greater AntilleanAnolis lizards. From Haefner, J . W. 1988b. Assembly rules for Greater AntilleanAn-ol s lizards. Competition and random models compared. Oecologia 74:55 1-565, Fig-ure 1. Copyright 1988 by Springer-Verlag GmbH Co. KG.

subseq uent years fit the mo del very poorly (Pulliam 19 83). null model thatincorporated habitat preferences of the species but did not include speciesinteractions fit the data about as well as the assembly rules, although somepatterns were still nonrandom with respect to this model.

Haefner s (1988b) an alysis of Anolis l izard coexistence in the GreaterAnti lles represents the most a mb it ious at tempt to test Diam ond s 1975)


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assembly rules against an ap propriate set of null m odels . Haefne r (1988 b)analyzed the occurrence of eight species of Anolis a t 11 Puer to Ricansample sites and asked what percentage of the observed species-site occur-rences could be successfully predicted by a suite of 20 ( ) different assem-bly mo dels (Fig ure 7.6). Random insert ion and deletion models placedspecie s on sites independently of one another, but w ith varying de grees ofhabitat aff inity and n iche requirements . These m odels co nsti tuted null hy-potheses of varying complexity with respect to competitive interactions. Incon trast, a set of simple deletion mod els removed spe cies acco rding torules of body s ize and niche overlap. Complex delet ion models remo vedspecies according to rules derived from Will iams 's (1972, 1983) ecom orphmodel.

Some of Haefner's (1988b) null models provided a good fit to the data,especially those that constrained row or column sum s and incorporatedhabitat affinities. As expected, the best-fitting models were the complexdelet ion models , which were cal ibrated to maximize the f i t to the PuertoRican data set . Simple delet ion models did not f i t the data very well ,suggesting that if competition structured this community, competitive as-ymm etries between species were probably com mo n. When these sam e mo d-els were applied to co -occurrence of Jamaican Anolis the data w ere best fitby a null model that maintained co lumn su ms, or that maintained row sumsand habitat affinities of species. As with the Pue rto Rican d ata, the fit of thecom plex deletion mod els was good and that of the s imple delet ion modelswas poor.

Although Haefner 's (1988b) results enhance our understanding of com-munity assembly, the interpretations are by no means clear-cut. On the onehand , the goo d fit of the d ata to several of the sim ple null models suggeststhat competi t ion is unimportant in com mun ity assembly. On the other hand,the ability of the calibrated com plex m odels to successfully predict comm u-nity structure of Jamaican nolis supports Williams's 1972, 1983) eco-morph model and im plicates com peti t ive interactions.

OTHER SSEMBLY RULESAlthough most of the controversy surrounding assembly rules has been overforbidden combinations of species and competitive interactions, there are othe rtypes of assembly rules that may dictate the organization of communities.These other assembly rules have also been addressed productively with nullmodels, and here w e review these approaches.


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islands, and that shifts in incidence functions in different archipelagoes re-flected different species poo ls and subsequent levels of competition.

In contrast. G ilpin and Diam ond (1981 explored the connection between theincidence function and the equilibrium theory of island biogeography. Theyviewed an assem blage as having n o com petitive interactions, with each speciesat a dynamic equilibrium in which colonizations and extinctions were bal-anced. In this scenario, the incidence function represents the fraction of timthat a species occup ies islands of a particular size class. From these incidencefunctions, Gilpin and Diamond (1981) inferred species-specific colonizationand extinction rates, and derived the com mu nity-level extinction and coloniza-tion curves of the MacArthur and Wilson (1967) equilibrium theory (seeChapter 8). Hanski (1992) successfully applied this model to the insular distri-bution of shrews in eastern Finland, where field data have corroborated theassumption of dynamic turnover (Peltonen and Hanski 1991).

Other interpretations of incidence functions are possible. The incidencefunction may reflect nothing m ore than the distribution of habitat types amo ngislands. Thus, h i g h s species may simply be habitat specialists, such as birdsthat are restricted to lakes, marshes, and high-elevation forest found only onlarge islands (Wiens 1989). Even in the absence of habitat specialization, theincidence function w ill reflect sampling variability, both in terms of the num berof islands cerisused and the num ber of islands sampled by the species (Taylor1991). Common and widespread species will have their incidence functionsshifted to the left because they will occur on a variety of different island sizes(Haila et a]. 1983). W hatever the cau se of the incidence function, environme n-tal stochasticity should lead to a shallower curve as occurrences become moreunpredictable with respect to ordered site characteristics (Schoener 1986b).

Null models clarify some of these differing interpretations by asking how anincidence function would appear in the absence of any structuring force.Whittam and Siegel-Causey (1981b) made the first attempt to analyze inci-dence functions statistically, fo r seabird colonies of Alaska. For an archipe lagoof islands that have been sorted into size classes, they constructed an x Ctable, where each species is a row and each column is a size class not anindividual island). The entries in the table were the number of islands of aparticular size class occup ied by a species. Contingency table analy sis was usedto assess statistical sign ificance, and standa rdized residuals pinpointed particu-lar cells in the table that con tributed positive or negative d eviations.

For Alaskan seabird colonies, the distributions of 20 species were highlynonrandom with respect to colony richness. Each species showed one or moresignificant deviations, and the sign of the deviations showed only one changein direction, running from small to large colonies (Figure 7.8). The analysis


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190 hapter

LL

0.2 W Figure 7.8. Statistical analysis of

colony, and they axis is the frac-0tion of species occurrences in colo-nies of a given size . Plus andminus signs indicate significantdeviations for a particular size

incidence functions for species inT m ? ? Alaskan seabird colonies. The x

class. Numbers indicate overallfrequency of occurrence of speciesin colonies of different size.ALL = all species studied. Otherspecies abbreviations as in Figure7.3. From Whittam and Siegel-Causey 1981b), with permission.

I I I I I I I I I axis is the number of species per

SPC

Colony r ichness

identified supertramp species, such as the Glaucous-winged Gull (Larus glau-cescens) and the Tufted Puffin (Lunda cirrh ata), which occurred less often thanexpected in large colonies and more often than expected in small colonies. Themore typical deviation was the high-S pattern, typified by several species of murresand auklets that were unusually present in large colonies and unusually absent fromsmall colonies. Finally, some species showed no significant pattern of deviation,suggesting that occurrences were random with respect to colony size.

There are two points to note about Wh ittam and Siegel-Causey s (1981 b)analysis of incidence functions. First, they defined incidence differently fromDiamond (1975). Whereas Diamond (1975) defined incidence as the proportion ofoccupied islands in a size class, Whittam and Siegel-Causey (1981b) used theproportion of the total occurrences of a species that fell in a particular size class.Secon d, unusual d eviations in the W hittarn and Siegel-C ausey (1981b) analysiswere measured relative to the distribution of all other species in the assem-blage. Thus, high-S species were those which occurred relatively more fre-quently on species-rich islands than did all other species. If all species showedan identical sup ertram p distribution, nonrandomness would no t be revealed.


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Incidence functions m ay have impo rtant implications for conserva tion biol-ogy. In particular, an incidence function analysis can be used to identifyunusual minim um area requirem ents for particular species. Sim ply scanning alist of occup ied sites may not be sufficient for this purpose, because even if aspecies were distributed randomly, it might be missing from some small sitesby chance. Simberloff and G otelli (1984 ) tested for unusually small minimumarea occurrences of plant species in five archipelagoes of remn ant prairie andforest patches in the North American prairie-forest ecotone. For each species,they ordered the patches on the basis of site area and then calculated the tailprobability of finding the observed number of smaller, unoccupied sites. Forthree of the five archipeliigoes, there were more species showing nonrandompatterns than expected by chance.

However, it might not be correct to conclude from this result that specieshave unusual minimum area requirements. In particular, this null modelassumes that all sites are equivalent. But even if species have no specialminimum area requirements, we expect large sites to be disproportionatelyoccupied. simulation model placed each species randomly in sites, with theprobability of occurrence being proportional to the area of the patch. Thesimulation w as repeated 10 times, an d then the cumulative distribution func-tions for the observed and expected site occupancy were com pared. The resultwas that far fewer of the species showed unusual deviations. In other words,nonran dom ncss in the occup ancy of sites could be effectively accoun ted for bydifferences in the areas of the sites. Nevertheless, the distributions of sevenplant spec ies in one of the assem blages were significantly nonrandom by thistest. Interestingly, none of these species sho wed an unusually large minimumarea requirement. Instead, they followed the supertramp pattern and oc-curred too frequently in small prairie and forest patches (Simberloff andGotelli 1984). W ilson (1988) tested for the occurrence of plant species w ithrespect to island area and also found only a few species that exhibitedsupertramp distributions.

Finally, Schoener and Schoener 1983) expanded Diamond's 1975) idea ofincidence functions beyond considerations of island area or species richness.They pointed out that sites can be ordered on any nu mber of criteria, and thenthe occurrence of species tested against this ordering. Haphazard sequen cesof presence and absence would suggest that a variable does not play a directrole in determining the occurrence of a spec ies. But if the seque nce is highlyordered, the occurrence of the species is predictable w ith respect to that factor(Figure 7.9). The M ann-Wh itney test is a simple nonparametric analysis ofthe degree of this ordering for any particular variable. Within an archipelago,each species displays the sam e total num ber of presences and ab sences, so the


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192 Chapter 7

A A A A A A A A A A A P P P P P P P P P P L I ~ ~ ~ ~nol~s ogret

P P P P P P P P PResldentb~rdDendro~copefech~oP P P P P P P P M ~ q r o n t b ~ r dDendrolca palmarum )sland arm

Figure 7.9. Occurrence sequence for resident species of 21 Bahamian islands. Islandsare ordered on the basis of area. A = absent; P = present; T = islands of equal area.Note that the resident lizard nolis sagrei has a perfectly ordered sequence the resi-dent bird Dendroicapetechia has a highly ordered sequence and the migrant birdDendroica palmarum has a more haphazard sequence. From Schoener and Schoener1983),with permission.

size of the Mann-Whitney statistic is a measure of the relative strength of theorderings Simberloff and Levin 1985).

Schoener and Schoener 1983) used this test to analyze their extensivedistributional data on 76 species of birds on 521 small is lands in theBahamas. Not only did they systematically census vertebrates on theseis lands, they also measured 54 variables that quantified area, isolat ion,habitat availability, and vegetation structure. The results indicated a veryhigh degree of ordering. Species occurrences were quite predictable, al-though different groups of species seemed to follow different assemblyrules . For exa mple, the occu rrence of l izards and resident birds w

null models chapter 7

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