Theor Ecol (2012) 5:181–194DOI 10.1007/s12080-010-0107-8

ORIGINAL PAPER

Do fitness-equalizing tradeoffs leadto neutral communities?

Annette Ostling

Received: 2 December 2010 / Accepted: 10 December 2010 / Published online: 6 January 2011© Springer Science+Business Media B.V. 2010

Abstract Neutral theory in ecology is aimed at describ-ing communities where species coexist due to similari-ties rather than the classically posited niche differences.It assumes that all individuals, regardless of speciesidentity, are demographically equivalent. However,Hubbell suggested that neutral theory may describeeven niche communities because tradeoffs equalizefitness across species which differ in their traits. Infact, tradeoffs can involve stabilization as well as fitnessequalization, and stabilization involves different dy-namics and can lead to different community patternsthan neutral theory. Yet the important question re-mains if neutral theory provides a robust picture of allfitness-equalized communities, of which communitieswith demographic equivalence are one special case.Here, I examine Hubbell’s suggestion for a purelyfitness-equalizing interspecific birth–death tradeoff, ex-panding neutral theory to a theory describing thisbroader class of fitness-equalized communities. In par-ticular, I use a flexible framework allowing examinationof the influence of speciation dynamics. I find that thescaling of speciation rates with birth and death rates,which is poorly known, has large impacts on communitystructure. In most cases, the departure from the predic-tions of current neutral models is substantial. This work

Electronic supplementary material The online versionof this article (doi:10.1007/s12080-010-0107-8) containssupplementary material, which is availableto authorized users.

A. Ostling (B)Ecology and Evolutionary Biology,University of Michigan, 830 N. University Ave.,Ann Arbor, MI 48109-1048, USAe-mail: aostling@umich.edu

suggests that demographic and speciation complexitiespresent a challenge to the future development and useof neutral theory in ecology as null model. The frame-work presented here will provide a starting point formeeting that challenge, and may also be useful in thedevelopment of stochastic niche models with speciationdynamics.

Keywords Life history tradeoff · Species abundancedistribution · Macroecology · Competition ·Coexistence · Stochastic community models

Introduction

Hubbell (2001) challenged a key assumption of com-munity ecology, namely that the coexistence of speciesrequires species niche differences. Hubbell proposedthat species often coexist in communities instead dueto their similarities. He proposed a “neutral theory”in which individuals of different species are ecologi-cally identical, and hence speciation, dispersal limita-tion, and demographic stochasticity are the primarydeterminers of community composition and structure.Despite mixed empirical success (McGill 2006), thisneutral theory works surprising well given its simplicity.

Hence, some have argued the community patternpredictions tested are insensitive to whether speciescoexist through similarities or differences (e.g. Bell2001). Hubbell himself suggested that neutral theorymay even describe many niche communities (Hubbell2001; Levine 2002). He argued that when species differ,tradeoffs tend to equalize fitness across individualsof different species, and that this fitness equalizationis essentially what neutral theory assumes. However,

182 Theor Ecol (2012) 5:181–194

Hubbell’s original argument is not completely accu-rate. Tradeoffs can involve stabilizing mechanisms aswell as fitness-equalizing mechanisms (Chesson 2001;Adler et al. 2007). Under stabilization, species invadefrom low abundance (i.e., there is a rare species ad-vantage). To most, this is the hallmark of true nichedifferences between species. Under fitness equaliza-tion, species’ abundances are just as likely to go up asdown from any starting abundance. Stabilization leadsto different community properties than the neutralcase, just as frequency-dependent selection leads todifferent population genetic structure than the neutralcase (Neuhauser 1999), although since the differencemay be small at high diversity Chishom (2010) andstabilization may lead to the same range of patterns asneutrality Chave et al. (2002), whether the differencecan be empirically distinguished is unclear. Thus, neu-tral theory does not in general describe communitieswith stabilizing tradeoffs, i.e., it does not describe nichecommunities, and may still be useful as a null model (ora “null theory” (Harte 2004)) for tests for niche assem-bly, although some novel approaches may be required.

However, it is an open question whether communi-ties with purely fitness-equalizing tradeoffs are well de-scribed by current neutral theory (Chave 2004). Insteadof equal fitness, Hubbell’s original neutral theory takesthe more stringent assumption that all individuals haveequal birth and death rates (i.e. they are “demograph-ically equivalent”). But species vary in birth and deathrates, with “ecological equivalence” (if it exists) arisingbecause those species with high birth rates also havehigh death rates, leading to equal fitness across species.Reproductive investment comes at the cost of adultsurvival. Growing evidence points to interspecific varia-tion and tradeoffs in reproductive investment and adultsurvival (e.g. Resnick 1985; Gunderson 1997; Saetherand Bakke 2000; Forbis and Doak 2004; Moles et al.2004). Once one considers the entire age structure ofpopulations, one realizes there is a great deal of de-mographic complexity to consider (Roff 1992; Metcalfand Pavard 2006). So for neutral theory to describeany real communities, or at least provide a null modelwhose rejection is evidence of niche assembly ratherthan other complexities, it must accurately describecommunities where species vary in demographic ratesand achieve “similarity” through tradeoffs that lead toequal fitness.

Recently, ecologists have begun studying caseswhere species differ in birth and death rates but not infitness (Ostling 2004; Etienne et al. 2007; Lin et al. 2009;Allouche and Kadmon 2009) (in contrast to models

where species vary in both demographic rates andfitness (Zhang and Lin 1997; Yu et al. 1998; Zhou andZhang 2009)). These recent studies produced mixedresults about whether fitness-equalized life history vari-ation matters for structural properties of the commu-nity and perhaps for that reason have gone largelyunnoticed by most ecologists interested in neutral the-ory. Here, I use a more flexible birth–death tradeoffmodel to consider how assumptions about speciationdynamics influence the community. In particular, thescaling of speciaton rate with birth and death rateshas been shown to vary (e.g. Marzluff and Dial 1991;Owens et al. 1999; Liow et al. 2008), and I demonstratethat this scaling has a critical influence on the ap-portionment of abundance and species richness acrosslife history strategies, and on the total species richnessand species abundance distribution. In one case, theseproperties are consistent with the predictions of currentneutral models, but in all others there are importantdifferences. Hence, I explain how the discrepanciesbetween previous studies arose and show that thereare many more possibilities for the structure of fitness-equalized communities than even those studies consid-ered. I also provide a new framework for predictingthose varied structures, and hence a new starting pointfor the development of neutral models for communi-ties. Elements of the framework I develop may also beuseful in the development of stochastic niche modelswith speciation dynamics.

The model

In this section, I first consider the dynamics of any com-munity of differing species and provide a differentialequation for the probability of a given configurationof the whole community. I then specify this differentialequation for the case where species differ only in birthand death rates, but are equal in fitness (i.e., the caseof a fitness-equalizing birth–death tradeoff). Next, Ipresent a “non-interactive” approximation to the dy-namics of this birth–death tradeoff community, whichshould apply well when the community size is large.Finally, I specify the speciation dynamics part of mytradeoff model in detail. I then obtain a differentialequation for the probability that a species of a givenlife history type has a given abundance at a given time.From this differential equation, I am able to derivepredictions for various equilibrium properties of thecommunity.

Theor Ecol (2012) 5:181–194 183

Dynamics of joint distribution when speciesdiffer in type

In this paper, I focus on the properties of a “meta-community”, i.e., the large pool of species that mightdisperse to any local community, and ignore the poten-tial influence of dispersal limitation within the meta-community (Chave and Leigh 2002; Rosindell andCornell 2007). Hence the only process adding speciesin the model here is speciation. I will refer to thismetacommunity simply as a “community”. I begin bywriting down a differential equation for the distribu-tion, P(n1, n2, . . . , nS, t), defined as the probability forthe community to have the configuration {n1, . . . , nS},where nk is the abundance of species k, and the time t isthe time since species 1 speciated. Note some importantthings which are implicit to this configuration. Firstly,each species has a type vector Tk associated with it,where multiple species can have the same type vector(there is no reason in principle why a given set oftrait values could not arise more than once in multiplespecies. Secondly, I set the time to 0 when species 1speciates, and the species listed in the configuration{n1, . . . , nS} are all species who were either in existenceat the time that species 1 speciated, or arose since then.Clearly, the number of species in the list increases overtime and as the list grows, the probability associatedwith configurations where abundance values early inthe list are zero will tend to be higher. Finally, thespecies other than species 1 are listed in the order inwhich they speciated (species 1 is an exception becausesome of the other species in the list may have been inexistence at the time species 1 speciated).

Although this approach to specifying the configura-tion of the community may seem complex, it’s ad-vantage is that it enables us to handle the speciationdynamics in a very straightforward way. In particular,in deriving the species abundance distribution, we donot need to approximate speciation as immigration inthe infinite community size limit as has often beendone by developers of neutral theory (Volkov et al.2003; Etienne et al. 2007). In particular, I will use thealternative approach (Vallade and Houchmandzadeh2003) of expressing the species abundance distributionin terms of the marginal probability distribution fora species to have abundance n a time t after it arosethrough speciation. Placing the species determining thetime variable t at the beginning of the configurationwill enable me to use the simplest notation in derivingthe differential equation for this marginal probabilitydistribution.

The differential equation for P(n1, n2, . . . , nS, t) is

dP(n1,. . ., nS, t)dt

=∑

k

lknk+1 P(n1,. . ., nk+1,. . ., t)

+∑

k

gknk−1 P(n1,. . ., nk−1,. . .t)

−∑

k

(lknk

+gknk

)P(n1,. . ., nk,. . ., t)

+∑

k

ρ(nk+1, Tk, TS)

×δnS,1 P(n1,. . ., nk+1,. . ., nS−1, t)

−∑

k

∑

TS+1

ρ(nk, Tk, TS+1)P(n1,. . ., nk,. . ., nS, t). (1)

The first three lines in Eq. 1 reflect events in whicha species loses or gains an individual through birthand death events. (Note the lk

n and gkn reflect loss and

gain rates, respectively.) The last two terms reflectspeciation events, in which I assume one species losesan individual when it speciates and the new speciesarises with one individual (reflected in the presence ofnk + 1 and δnS,1 in the second to last term). I use thisassumption for consistency with the widely used versionof neutral theory in ecology, but it could be generalized(Allen and Savage 2007; Haegeman and Etienne 2009).These terms involve ρ(nk, Tk, TS), which is the rate atwhich a species of type Tk with abundance nk givesrise to a daughter species of type TS. The last termalso involves a sum over possible types for the daughterspecies.

Fitness-equalizing birth–death tradeoff

In the rest of the paper, I explore a more specificmodel captured by Eq. 1 that has a fitness-equalizinginterspecific birth–death tradeoff. I build this into amodel that has “zero-sum” dynamics analogous to theMoran model in population genetics (Hubbell 2001;Blythe and McKane 2007). Under “zero-sum” dynam-ics, each death event is immediately followed by a birthevent. Although other approaches have been used forneutral models in both the population genetics andcommunity contexts most seem to lead to similar resultsin the completely neutral case (Blythe and McKane2007; Haegeman and Etienne 2008) (although the morerealistic case of density-dependent population or com-munity regulation is relatively unexplored), and hence I

184 Theor Ecol (2012) 5:181–194

take this approach as a good starting point for studyingthe impact of introducing fitness-equalized life historyvariation.

Specifically, I define the transition rates lkn and gk

n inEq. 1 as:

lkn = dkn

(∑k′ b k′nk′ − b kn∑

k′ b k′nk′

)(2)

gkn =

(∑

k′dk′nk′ − dkn

)(b kn∑

k′ b k′nk′

), (3)

where b k and dk are the per-capita birth and deathrates, respectively, of species k. In Eq. 2, the first factorreflects the rate at which individuals of species k die,and the second the probability that the replacing indi-vidual is of a species other than k. Similarly, in Eq. 3,the first factor reflects the rate at which individuals ofspecies other than k die, and the second the probabilitythat the replacing individual is from species k. Since themodel is essentially only keeping track of adult individ-uals, the birth rate b k is really the rate of recruitmentinto the adult stage for species k. Furthermore, the rateb k reflects the inherent reproductive potential of thespecies k. Specifically, it is the per-capita recruitmentrate species k would obtain in a completely available,favorable habitat. Equations 2 and 3 assume that theseper-capita rates in an open, favorable habitat accuratelyreflect the relative likelihood of species capturing asingle available patch for which they are competing. Inthis model, the type of each species is specified by avector of length two: Tk = (b k, dk).

I assume species vary and tradeoff in their per-capita“birth” and “death” rates, or more exactly in their per-capita rates of adult recruitment and adult survival.Specifically, I assume that species have equal fitness,where I define equal fitness as the case where the rateof loss of individuals from each species is equal to therate of gain, regardless of the abundance of the speciesor the overall configuration of the community, i.e., thecase where

lkn = gk

n for all k and all {n1, . . . n, . . . , nS}. (4)

This leads to the following condition on the birth anddeath rates:

b k = c dk for all k. (5)

In words, equal fitness requires that each species has itsbirth rate are proportional to their death rate accordingto the same constant, c. This constant reflects how

closely balanced recruitment and mortality of any ofthe species would be in an open spaced environment.Most species would produce more adults per unit timethan would die per unit time if given plenty of favorablehabitat, so c > 1 is expected. I take the assumptionof equal fitness across species from here on, in whichcase I no longer need a vector Tk to specify the typeof each species, and can instead use one parameterTk = dk = b k/c.

Note that equal fitness will cause species’ averageabundances to be constant in time in the absence ofspeciation dynamics. Consider the expectation value ofthe abundance of species 1 at time t, E[n1(t)] (notingthat any species can be thought of as species 1), whichcan be obtained from the joint distribution by

E[n1(t)] =∑

configs

n1 P(n1, . . . , nS, t) (6)

where∑

configs is the sum over all possible configura-tions of the community. This includes a sum over thepossible values for the number of species in the commu-nity configuration, integration over the possible types(presuming b k and dk are continuous variables) foreach species (except for species 1) and a sum over thepossible abundances for all the species. Under equalfitness, species’ abundances will on average be neitherincreasing nor decreasing in the absence of speciationdynamics (i.e., in the absence of the last two terms ofEq. 1):

dE [n1(t)]dt

= E[(

g1n1

(t) − l1n1

(t))] − speciation terms

= 0 − speciation terms,

if l1n1

= g1n1

for all {n1, . . . , nS}. (7)

Note that eventually every species will either go extinctor fix in the community (i.e., have an abundance equalto the total community size), but the relative probabil-ities of these events under equal fitness should be suchthat the average abundance of each species is fixed atone after it arises with one individual through specia-tion, just as in the completely neutral model (Blytheand McKane 2007). Furthermore, note that the addi-tion of speciation dynamics causes each species to onaverage decline in abundance over time, and eventuallygo extinct.

Non-interactive approximation

To derive community properties from the complex dy-namics in Eq. 1 for the fitness-equalizing birth–death

Theor Ecol (2012) 5:181–194 185

tradeoff case, a useful approximation is to assumeabundances n of the focal species are small comparedwith the total number of individuals in the community,N. This approximation has been shown to work wellin the completely neutral case (Alonso and McKane2004; Chave 2004; Etienne et al. 2007). Under thisassumption, Eqs. 2 and 3 can be written as:

lkn = dkn

(1 − b kn

〈b〉 N

)≈ dkn = Tkn (8)

gkn = (〈d〉 N − dkn)

(b kn

〈b〉 N

)≈ b kn

〈d〉〈b〉 = Tkn (9)

Hence under this approximation the species are essen-tially “non-interactive”, meaning that the probability ofloss or gain of an individual of any species through birthor death is independent of the abundance of the otherspecies. This approximation allows consideration of thebirth–death dynamics of each species separately, a keystep towards decoupling the complex dynamics in Eq. 1.

Speciation dynamics

The complexity of speciation dynamics (i.e., the last twoterms in Eq. 1) must also be simplified for the tradeoffmodel to be tractable. However, because here I useEq. 1 as my starting point, I incorporate more flexibilityin speciation dynamics than previous approaches tocommunity tradeoff models. There are two aspects ofthe speciation dynamics I will consider: (1) how the rateat which a species speciates depends on its life historystrategy, and (2) how quickly new species of a given lifehistory strategy arise through speciation.

The first of these, the dependence of the rate atwhich a species speciates on its life history type T, isone component of the function ρ(n, T, Tk), defined asabove in Eq. 1 as the rate at which species of type Tand abundance n give rise to species of type Tk. In theanalysis that follows, I take this function to have theform:

ρ(n, T, Tk) = f lossT ν n ρ ′(T, Tk) (10)

In Eq. 10, f lossT ν n reflects the rate at which the species

of abundance n and type T speciates, where ν isthe community-average per-capita speciation rate, andρ ′(T, Tk) reflects the probability for a species of type Tk

to arise from a parent species of type T that is speciat-ing. Hence, I am taking the simplification that the rateat which a species speciates is determined entirely byits abundance and life history and not the abundance

and types of the rest of the species in the community.In particular, I take the rate at which a species speciatesto be linearly proportional to its abundance. This isconsistent with most neutral models used in ecology,but variation from this assumption does influence pre-dictions (Etienne et al. 2007). More importantly to thefocus of this paper, I have used a dimensionless scalingfactor f loss

T to encapsulate how the rate at which aspecies speciates depends on its life history type T,i.e., how it depends on its birth or death rate. Here, Iallow this factor to have any dependence on T. Previousstudies (Ostling 2004; Etienne et al. 2007; Lin et al.2009; Allouche and Kadmon 2009) implicitly assumeda form for f loss

T .Studies point to faster rates of molecular evolution

among taxa with faster life history strategies (Bromhamet al. 1996; Smith and Donoghue 2008). However, therate at which a species speciates may or may not havea similar relationship with life history, depending onwhether it is determined primarily by the rate of geneticdivergence, the likelihood for geographic isolation,or the prevalence of ecological opportunity, or allthree (Futuyma 1997; Webster et al. 2003). Fossilrecord and phylogenetic studies point to a variety ofrelationships between speciation rate or diversificationrate (speciation rate minus extinction rate) and lifehistory (Marzluff and Dial 1991; Owens et al. 1999;Liow et al. 2008).

This leaves open a wide variety of possibilities, andthe likelihood of a great deal of system specificity, forthe scaling of the rate at which a species speciates withits life history, f loss

T . If the rate of molecular evolutiondoes determine speciation rate, one would expect f loss

Tto be linear in T, since the rate of molecular evolution isinversely proportional to generation time. However, ifother factors dominate, or if they influence speciationrates in concert with the rate of molecular evolution,then f loss

T could increase, remain constant or decreasewith T (and the increase or decrease could be linear ornon-linear).

I now turn to the second key aspect of speciationdynamics, how quickly new species of a given life his-tory type T arise through speciation. I will derive anexpression for this dependence in terms of f loss

T andthen demonstrate that this dependence can be differentthan the scaling of the rate at which a species speciatesencapsulated in f loss

T . Take pT(t) to be the rate at whichspecies of type T are arising at time t. This rate can becalculated from the distribution P(n1, . . . , nS, t) by:

pT(t) =∑

configs

∑

k

ρ(nk, Tk, T)P(n1, . . . , nS, t). (11)

186 Theor Ecol (2012) 5:181–194

Using Eq. 10 and converting the sum over species kinto an integral over types T ′ this expression for thespeciation rate becomes

pT(t) =∫ Tmax

Tmin

dT ′ f lossT ′ ν ρ ′(T ′, T) NT ′(t), (12)

where NT ′(t) is the expected total abundance acrossspecies of type T ′ at time t, i.e., NT ′(t) = ∑

k δ(Tk − T ′)E[nk(t)] (See Appendix S1 in the Electronic supple-mentary material). So pT(t′) depends on the currentdistribution of abundance across types in the com-munity (NT ′(t)), how the rate at which they speciatedepends on their type ( f loss

T ′ ), and the degree of corre-lation between parent and daughter types (ρ(T ′, T)).It is the large t behavior of pT(t) that matters forequilibrium community properties, i.e., what matters ispT = limt→∞ pT(t), the behavior some long time afterany particular starting configuration of the communityor particular speciation event. In particular, what mat-ters here is how pT scales with T. For convenienceI define the dimensionless scaling factor f gain

T = pTη

νNwhere η is the range of possible T values of the com-munity, N and ν are defined as above. The relationshipbetween the scaling factors f loss

T and f gainT is then:

f gainT =

∫ Tmax

Tmin

dT ′ f lossT ′ ρ ′(T ′, T)

NT ′

N/η, (13)

where NT ′ = limt→∞ NT ′(t).One can see from Eq. 13 that f gain

T can be differentthan f loss

T . In particular, consider the relationship be-tween these two scaling factors in two extremes for therelationship between parent and daughter life historytypes, ρ ′(T ′, T). Firstly, if parent and daughter types areequal, i.e., if ρ ′(T ′, T) = δ(T ′ − T), the historical rela-tive abundance of different types in the community willbe maintained (see Appendix A) and will determinethe equilibrium NT ′ . Hence any historical bias in NT ′

will cause f gainT to differ from f loss

T . Secondly, if the re-lationship between daughter and parent species’ traitsis very weak, i.e., if ρ ′(T ′, T) is relatively independentof T ′, then f gain

T is simply equal to ρ ′(T ′, T) and henceis unrelated to f loss

T . A weak relationship like this mightarise because the daughter type T is heavily determinedby fundamental physiological or genetic constraints, oreven ecological opportunity (there is the possibility thatniche space availability has had a larger influence onthe speciation dynamics of the community than it iscurrently having on the birth–death dynamics, or onemay want to separate the evolutionary- and ecological-

time-scale influences of niche space). Ultimately, themodel is completely specified by specifying ρ ′(T ′, T),f lossT ′ (which together determine NT ′ (see Appendix A)

and f gainT ), and, if ρ ′(T ′, T) = δ(T ′ − T), the historical

value of NT ′ . However, it is complex to solve for NT ′ insituations other than the two extremes just mentioned,and all of the community properties I study here areultimately just dependent on f loss

T and f gainT . Hence later

in this paper when I study the community propertiespredicted by this model, I take the approach of studyingsome cases delineated by the form of f loss

T and f gainT , and

providing very basic insight into how these cases couldarise by considering them in the two extremes of therelationship between parent and daughter traits.

Species abundance distribution for each type

Taking the “non-interactive” approximation, and mod-elling speciation dynamics as described above, the fol-lowing differential equation for the probability for aspecies of type T to have abundance n at time t, PT(n, t)can be obtained either by intuition or by summing Eq. 1for the joint distribution over the configurations ofspecies other than species 1 (the species that speciatedat time t = 0) (see Appendix S2):

dPT(n, t)dt

= wTn+1(t)PT(n + 1, t) + zT

n−1(t)PT(n − 1, t)

− (wTn (t) + zT

n (t))PT(n, t). (14)

where

wTn = lT

n + f lossT νn (15)

and I have now used T in place of k in labeling lkn and

gkn for species 1. Note that there can be multiple species

of type T present, and that Eq. 14 would capture thebehavior of any of them.

The equilibrium of the distribution PT(n, t) is ob-vious without solving Eq. 14. On average, all speciesare declining in abundance because they are loosingindividuals to speciation events, eventually all speciesgo extinct, and the equilibrium of PT(n, t) is δn,1 re-gardless of the value of T. But it is not just the equilib-rium of PT(n, t) that determines the species abundancedistribution but its value at all times. To derive thespecies abundance distribution (SAD) of species oftype T, meaning the expected number of species oftype T with abundance n at time t (which we’ll writeas ST(n, t)), we must write down an expression for it in

Theor Ecol (2012) 5:181–194 187

terms of the behavior of PT(n, t) at all times, and thenapply that expression to Eq. 14 to obtain a differentialequation for it. Ultimately we will not need to knowthe full time-dependent behavior of PT(n, t) to get atthe equilibrium SAD, but the approach of initially writ-ing it in terms of the time-dependent behavior enablesus to solve for the SAD without approximating specia-tion as the infinite community size limit of immigrationdynamics (the approach taken by Volkov et al. (2003)and Etienne et al. (2007), and others). The relationshipbetween the SAD of each type and PT(n, t) is Valladeand Houchmandzadeh (2003):

ST(n, t) =∫ t

0dt′ pT(t′)PT(n, t − t′)

+ terms from species present at t = 0. (16)

Equation 16 integrates over the possible times t′ thateach type T species might have arisen in the commu-nity. Specifically it integrates over the rate at which typeT species are arising at time t′, pT(t′), multiplied by theprobability that a species of type T has n individualsat time t − t′ after it arose through speciation. Thisintegral, when added to terms from type T species thatare present at time t = 0 that now have abundance n,gives the SAD of each type.

I’ll consider the large t behavior of the SAD, i.e.,the equilibrium ST(n, t) would approach a long timeafter any starting configuration of the community. Inthat limit the terms from species initially present in thecommunity are negligible, and the time dependence ofthe speciation rate is also negligible. At large t the SADof each type can be approximated by

ST(n, t) ≈ pT

∫ t

0dt′ PT(n, t − t′), (17)

where pT = limt′→∞ pT(t′). Carrying out the integra-tion in Eq. 17 on both sides of Eq. 14 leads to (seeAppendix S3 in the Electronic supplementary materialfor details):

dST(n, t)dt

= wTn+1 ST(n + 1, t) + zT

n−1 ST(n − 1, t)

− (wTn + zT

n ) ST(n, t) + δn,1 pT (18)

The equilibrium solution of Eq. 18 is (see SI in theElectronic supplementary material for details):

ST(n) = f gainT

T/ 〈T〉ν

〈T〉Nη

((1 + f loss

TT/〈T〉

ν〈T〉

)−1)n

n(19)

In Eq. 19, 〈T〉 is the average life history type acrossindividuals of the community, i.e., 〈T〉 = (1/N)

∑T

ST(n)nT. Although 〈T〉 actually cancels out of Eq. 19,I introduce it for conceptual value, to emphasize de-pendence on the speciation rate per average death rate,ν/ 〈T〉. Note that Eq. 19 is simply a log-series distrib-

ution, i.e., ST(n) = αT(xT )n

n with xT = (1 + f lossT

T/〈T〉ν

〈T〉 )−1

and αT = f gainT

T/〈T〉ν

〈T〉Nη

, so the scaling factors f lossT and

f gainT in comparison to linear scaling with birth and

death rates, i.e., T/ 〈T〉, along with the speciation rateper average death rate ν/ 〈T〉 and the total communitysize N are all that are needed to determine the proper-ties of the community.

I now use Eq. 19 to calculate equilibrium expecta-tions for a variety of community structure properties.Firstly, the expected total number of individuals acrossall species of life history strategy T, NT , is:

NT =∞∑

n=1

ST(n) · n = αTxT

1 − xT= f gain

T

f lossT

Nη

(20)

Next, the total number of species of life history strategyT, ST , follows:

ST =∞∑

n=1

ST(n) = −αT log(1 − xT)

= f gainT

T/ 〈T〉ν

〈T〉Nη

log

⎛

⎝1 + f loss

TT/〈T〉

ν〈T〉

f lossT

T/〈T〉ν

〈T〉

⎞

⎠ (21)

Finally, the average abundance of a species of lifehistory strategy T directly follows from these twoproperties:

〈nT〉 = NT

ST= 1

f lossT

T/〈T〉ν

〈T〉 log

(1+ f loss

TT/〈T〉

ν〈T〉

f lossT

T/〈T〉ν

〈T〉

) (22)

Sensitivity of predictions to speciation dynamics

Here, I use Eqs. 19–22 to examine the sensitivity ofthese community properties to the scaling of speciationrates with birth and death rates as reflected in f loss

T

and f gainT . Firstly, through study of these expressions

for the community properties one can see when theyare the same as the naive predictions from currentneutral models with equal demographic rates across all

188 Theor Ecol (2012) 5:181–194

individuals of all species. ST(n) and ST , the averagenumber of type T species with abundance n and theaverage total number of species of type T, respectively,reduce to the completely neutral prediction only whenf gainT = f loss

T = T/ 〈T〉. In that case, Eq. 19 is the logseries with α = ν

〈T〉Nη

and x = (1 + ν〈T〉 )

−1, which is thecompletely neutral prediction for the species abun-dance distribution of a randomly chosen fraction 1/η

of the species in a community with an average deathrate of 〈T〉 (see Appendix S4 in the Electronic supple-mentary material). NT , the total abundance across allspecies of type T, is independent of T, and in that senseconsistent with the completely neutral case, in a widervariety of circumstances, namely as long as the twoscaling factors f gain

T and f lossT depend on T in the same

way. Finally, 〈nT〉 is independent of T, and in that senseconsistent with the completely neutral case, wheneverf lossT = T/ 〈T〉. Hence, it is only sensitive to the scaling

of the rate at which species of type T speciate.

Next, I plot the expected form of the properties inEqs. 20–22 (Fig. 1), as well as the total species richnessand total species abundance distribution (SAD) of thecommunity (Fig. 2) under five cases for the form of f loss

T

and f gainT that demonstrate a wide range of behavior

in community structure (see Box 1 for a list of thecases and insight into how each one might arise). I usecommunity size N = 1 million, life history parameterT ranging over integer values 1–100, and a speciationrate per average death rate of 5 × 10−4. Restriction tointeger values of T simplifies the calculation and isanalogous to binning predictions on the T axis as wouldbe done for real data. The two order of magnituderange in T is comparable to the range of seed outputseven within gap-specialist species in tropical forests(R. Condit, personal communication). These plots arebased on Eqs. 19–22 and hence the non-interactiveapproximation. However, I verified for case 2 that thepredicted community properties are indistinguishable

0

5,000

10,000

15,000

20,000

NT

Case 1

0

5,000

10,000

15,000

20,000

Case 2

0

5,000

10,000

15,000

20,000

Case 3

0

50,000

100,000

150,000

200,000

Case 4

0

5,000

10,000

15,000

20,000

Case 5

0

20

40

60

S T

0

200

400

600

800

0

500

1000

1500

0

200

400

600

800

0

20

40

60

0 20 40 60 80 100

T

0

150

300

450

< n

T >

0 20 40 60 80 100

T

0

150

300

450

0 20 40 60 80 100

T

0

5,000

10,000

15,000

0 20 40 60 80 100

T

0

150

300

450

0 20 40 60 80 100

T

0

150

300

450

tradeoff neutral

Fig. 1 Apportionment of abundance and species richness acrosslife history types in the tradeoff model. Community structureproperties are shown under the five different cases of the scalingof speciation rates with life history. Shown for each case arethe expected equilibrium values of total abundance across allspecies of a given life history type, NT = ∑

n ST (n)n (top graphs,based on Eq. 20, but also implicit to the speciation dynamicsassumptions of each case), the number of species present in thecommunity with a given life history type, ST = ∑

n ST (n) (middlegraphs, based on Eq. 21), and the average abundance of speciespresent with a given life history type, 〈nT 〉 = NT/ST (bottomgraphs, based on Eq. 22). I used a community size N = 1 million,

life history parameter T ranging over integer values 1–100, and aspeciation rate per average death rate of 5 × 10−4. Solid lines aretradeoff model properties; dashed red lines are naive expectationsfrom neutral theory (i.e., equal apportionment of abundance andspecies richness across types). Case 1: f loss

T and f gainT linearly

proportional to T. Case 2: f lossT and f gain

T independent of T.

Case 3: f lossT and f gain

T decreasing linearly with T. Case 4: f lossT

linearly proportional to T, but f gainT independent of T. Case 5:

f lossT independent of T, but f gain

T linearly proportional to T. SeeBox 1 for insight into how each case might arise

Theor Ecol (2012) 5:181–194 189

1 1000 2000 3000 4000Rank

1

10

100

1,000

10,000

100,000A

bund

ance

neutraltradeoff cases 5,2,3

neut

ral

0

2000

4000

6000

8000

10000

Spec

ies

Ric

hnes

s

52

3

trade

off

2

3

5

Fig. 2 Species abundance distribution and species richness inthe tradeoff model. The left graph compares rank abundancecurves (RACs) (calculated by drawing without replacement theabundance n for each of S species with probability S(n)/S =∑

T ST (n)/S), and the right graph species richness, of the tradeoffand neutral models for communities of the same parametervalues and speciation rate scaling cases as specified in the captionof Fig. 1. To remove drastic differences in the RACs caused bydifferences in species richness rather than relative abundance,speciation rates were decreased in the tradeoff RACs so thatspecies richness matched the neutral case. Tradeoff RACs reflectmore unevenness, and tradeoff species richness tends to behigher, than the neutral case (except for tradeoff model cases 1and 4 which match neutral predictions and are not shown). Notethat the abundance of the most dominant species (i.e., the speciesof rank 1, the left intercept on the graph) is highest in case 3, nexthighest in case 2, then case 5, and lowest in the neutral case

from those predicted by simulation of a Moran-style or“zero-sum” model (see Appendix S5 in the Electronicsupplementary material). In addition, I verified thatthe same qualitative differences between the tradeoffand completely neutral cases arise when the communitysize N, per-capita speciation rate ν, and range in Tare varied from the values used to generate the plotsshown here.

These plots show that different forms for the scalingfactors f loss

T and f gainT lead to very different apportion-

ment of abundance and species richness across lifehistory strategies (Fig. 1), many of which are quitedifferent from a naive application of neutral theory toa community containing different life history strate-gies (i.e., equal apportionment across all strategies).Furthermore, in the presence of a fitness-equalizingtradeoff, total species richness can be much higher andthe total species abundance distribution much moreuneven than in the completely neutral model (see SI inthe Electronic supplementary material for completelyneutral model calculations) (Fig. 2). Note in particularthat the plots in Fig. 1 verify the direct analysis of theequations for the community properties made above, inthat ST is independent of T only in case 1, where both

scaling factors are linear in T, NT is independent of T incases 1–3, where the two scaling factors take the sameform, and 〈nT〉 is independent of T in cases 1 and 4,where f loss

T is linear in T.These effects can to some degree be understood in-

tuitively. Consider in particular case 2, where f lossT and

f gainT are constant with respect to T. In this case the total

abundance across all species of given life history type ishomogeneous across types (top graph in Fig. 1). Thismakes sense because there is equal fitness across lifehistory types and f loss

T = f gainT means each life history

type is losing and gaining individuals through speciationevents at the same rate. However, in this case there is adeclining relationship between species richness and lifehistory strategy (middle graph in Fig. 1). This is becausespecies of the faster life history types go extinct morequickly due to larger fluctuations in abundance fromdemographic stochasticity. This means that for eachlife history type, the same total abundance (NT = N/η)is spread over fewer species (ST) for the faster lifehistory types, leading to higher abundance per extantspecies (〈nT〉) for those faster life history types (bottomgraph for case 2 in Fig. 1). Although this result makesintuitive sense, it is at the same time surprising. Even ina community in which the different types present havethe same fitness, there can be a relationship betweenabundance and type! Given this relationship, it makessense that the total SAD tends to be more uneven thanthe completely neutral case (Fig. 2), but why the speciesrichness is higher than in the neutral case is not obvious.

These results explain why Etienne et al. (2007) andAllouche and Kadmon (2009), who looked at a birth–death tradeoff model, but essentially assumed f gain

T =f lossT = T/ 〈T〉, found no difference between the SAD

of their fitness-equalized birth–death tradeoff modeland that of the completely neutral case. Two other stud-ies of a birth–death tradeoff model (Ostling 2004; Linet al. 2009) reported a relationship between 〈nT〉 andT, which will arise when f loss

T is not linear in T. Surpris-ingly, the dynamics described in these studies seem toinstead imply an f loss

T linear in T. (They each describethe speciation process as occurring in a fraction ν of thedeath events, in which case species with higher deathrates will be proportionally more likely to speciate.)However, one of these studies was my own (Ostling2004), and in that study the actual simulations used aslightly different speciation algorithm to improve com-putation time (I mention this very briefly in the study).Namely they carried out Nν speciation events afterevery N death events, where N is the metacommunitysize. Species were not selected for speciation accordingto their death rate, but instead independent of their

190 Theor Ecol (2012) 5:181–194

Box 1 Speciation rate scaling cases in Figs. 1 and 2.

There are two key aspects of the speciation dynamics that determine the community properties under this fitness-equalized tradeoffmodel (see Eq. 19): f loss

T , which describes how the rate at which species speciate (and loose and individual to the new species) scaleswith their birth or death rate (or their life history type T = d = b/c where b and d are the birth and death rates respectively and theconstancy of c across species in this equality enforces equal fitness); and f gain

T , which describes how the rate at which new species arisescales with their birth or death rate (or life history type T). In Figs. 1 and 2 , I selected five cases for the form of these two scaling factorsthat demonstrate a range of behavior in community properties, but there are likely many biologically plausible possibilities. Here Iprovide some very basic insight into how each case could arise in the two extremes for the relationship between parent and daughtertraits: a very strong relationship or a very weak one. A very strong relationship is when ρ′(T ′, T) = δ(T ′ − T) where ρ′(T ′, T) is theprobability density for a species of type T to arise from a parent species of type T ′ that is speciating. Under a strong relationship, therelationship between f loss

T and f gainT is determined by any historical biases in the relative abundance of types (meaning the relative

total abundance across all species with that type: NT = limt→∞∑

k δ(Tk − T)E[nk(t)]) (See Appendix A). In particular, f gainT is

simply the product of the historical bias in the relative abundance of types and f lossT , i.e., f gain

T = NT, historical f lossT (see Eq. 13), so

these two scaling factors will have the same form whenever the historical relative abundance of types is uniform. A very weakrelationship is when ρ′(T ′, T) independent of T ′. Under a weak relationship, f gain

T is simply equal to the dependence of ρ′(T ′, T)

on T, which may be determined by fundamental mechanisms influencing speciation (e.g. physiological or genetic constraints, orecological opportunity). See Appendix B for calculation of constants of proportionality in the scaling factors f loss

T or f gainT in each

case based on consistency conditions.

Case 1: f lossT and f gain

T linearly proportional to T.Linear f loss

T could arise if the rate at which species speciate is largely determined by rate of molecular evolution, which should scale

linearly with the life history type T. This same form for f gainT will arise under a tight relationship between parent and daughter traits if

the historical relative abundance of types is uniform. Alternatively, it could arise under no relationship between parent and daughtertraits if there is some fundamental bias for species with faster life history strategies to arise in a given community. For example,ecological opportunity might favor species with faster life history strategies if there is a lot of disturbance.Case 2: f loss

T and f gainT independent of T.

An f lossT that is independent of T could arise if the rate at which species speciate is largely determined by the opportunity for

geographic isolation, and if that opportunity is relatively similar across life history strategies. f gainT could then be the same as f loss

Tunder a tight relationship between parent and daughter traits and a uniform historical relative abundance of types, or under a weakrelationship between parent and daughter types if there is no bias is towards a given life history in the fundamental mechanismsgoverning the rate at which different life histories arise through speciation.

Case 3: f lossT and f gain

T decreasing linearly with T.An f loss

T that decreases with T could arise for example if the rate at which species speciate is largely determined by the opportunityfor geographic isolation, and if species with bigger body sizes have greater opportunity for this, perhaps because they are more wideranging. f gain

T could have the same scaling as f lossT under to a strong parent–daughter-type relationship due to no historical bias in the

relative abundance of types, or under a weak parent–daughter-type relationship due to for example greater ecological opportunityfor species with larger body sizes because the environment is relatively undisturbed.

Case 4: f lossT linearly proportional to T, but f gain

T independent of T.As in Case 1, linear f loss

T could arise if the rate at which a species speciates is largely determined by the rate of molecular evolution,

which should scale linearly with life history type T. The different, constant behavior of f gainT could then arise under a strong parent–

daughter type relationship if there is a historical bias in the relative abundance of types towards slower life history types, perhapssimply because the community happened to be first colonized by slow life history strategies due to prior environmental conditions,or under a weak parent–daughter relationship and no life history bias in the mechanisms governing which types of species are likelyto arise through speciation.

Case 5: f lossT independent of T, but f gain

T linearly proportional to T.As in Case 2, an f loss

T that is independent of T could arise if the rate at which species speciate is largely determined by the opportunityfor geographic isolation, and if that opportunity is relatively similar across life history strategies. The different, linear behavior inf gainT could then arise under a strong parent–daughter relationship if there is a historical bias towards faster life history strategies

(they might be more likely to be the first colonists of a community), or under a weak parent–daughter type relationship due to forexample ecological opportunity favoring faster life history strategies arising through speciation due to high disturbance rates.

death rate. Essentially the simulation took f lossT to be

independent of T. The result was an approximatelylinear relationship between 〈nT〉 and T, as expected ac-cording to Eq. 22, as well as an SAD differing from theoriginal neutral model. Lin et al. (2009) may have used

a similar simulation short cut but just didn’t report it.In the context of my more flexible speciation dynamicsframework in this paper, it is clear that these previousstudies explored only a small number of the manypossibilities for the structure of tradeoff communities.

Theor Ecol (2012) 5:181–194 191

Discussion

Species differ and tradeoff in demographic rates likeadult recruitment and survival (e.g. Gunderson 1997;Saether and Bakke 2000; Forbis and Doak 2004; Moleset al. 2004). Hence neutral theory’s strict assumptionof demographic equivalence is clearly violated. Canneutral theory provide robust insight into communitieswhere species have the same fitness but differ in de-mographic rates? Here I have expanded neutral theoryto a theory of communities with a fitness-equalizing“birth–death” tradeoff to resolve this fundamentallyimportant issue. I find that the structure of these typesof communities depends on how speciation rates scalewith birth and death rates, and that in many cases thisstructure can vary substantially from the predictions ofcurrent neutral theory.

Since its beginnings, ecologists have criticized neu-tral theory for its strict assumption of demographicequivalence, but there has been only limited explo-ration of the implications of demographic variation forcommunity structure. A key problem with early stud-ies (Zhang and Lin 1997; Yu et al. 1998) was that theyallowed for demographic variation without enforcinga tradeoff across species. Hence, the phenomena theyfound seemed due to the resulting fitness differencesacross species, which are counter to the spirit of neu-tral theory and not so clearly justifiable by empiricalwork except in special cases. More recent studies havefocused on the fitness-equalized case (Ostling 2004;Etienne et al. 2007; Lin et al. 2009; Allouche andKadmon 2009). (Note that although what constitutesequal fitness in a stochastic context can be subtle insome model frameworks (Parsons and Quince 2007;Parsons et al. 2008; Turnbull et al. 2008), these recentstudies and my own here do indeed consider the caseof equal fitness.) However, these recent studies havediffered in their conclusions. My work here identifiesthe key difference between them: their assumptionsabout how speciation rates scale with life history. Fur-thermore it shows that the speciation dynamics casesthese studies considered are just two of many possibil-ities, across which community structure patterns varyconsiderably.

These results indicate that differences between realcommunities and the predictions of current neutraltheory might arise due to demographic variation acrossspecies and the scaling of speciation rates with demog-raphy rather than the presence of stabilizing mecha-nisms. The uncertainty in our knowledge of the scalingof speciation rates means neutral theory is currently oflimited use as a null model for tests for stable coexis-tence. Empirical analysis testing for stabilization must

either be based on patterns that are insensitive to de-mographic variation, or consider the potential influenceof demographic variation and speciation dynamics. Thisrealization adds to a growing awareness that speciationdynamics can have an important influence on com-munity structure and must be factored into tests ofcommunity assembly theory (Allen and Savage 2007;Etienne et al. 2007; Haegeman and Etienne 2009). It ispossible that the effect of demographic variation acrossspeciation might be incorporated simply through somesort of effective parameters (although the speciationrate per average death rate is multiplied by differentscaling factors in different places where it appears inEqs. 19–22 and so this may not be so simple). Buteven if the original neutral theory fits are valid throughthe use of effective parameters, consideration of demo-graphic variation will be vital if we aim to verify thatneutral theory predicts the correct patterns for realis-tic values of the speciation rate and metacommunitysize.

Although information about speciation dynamicsmay be difficult to obtain, the results of this studysuggest that if it can be obtained, the signature of sta-bilizing mechanisms may be clearer than expected. Inparticular, stabilization and fitness equalization mightbe distinguishable through patterns of apportionmentof abundance and species richness across types likethose examined here. For example, ecologists mightinfer the means of coexistence in a forest from the rel-ative abundance and diversity of gap-specialist versusshade tolerant species. If speciation rates are ho-mogeneous across these differing life history strate-gies, fitness equalization would predict communitypatterns as shown for Case 2 above, with higherabundance per species among species with faster lifehistory strategies. This prediction could be directlycompared with observed relationships between abun-dance and life history traits. Although it is possibleto get a wide variety of species richness and speciesabundance patterns from both neutral and niche mod-els under uncertainty in total speciation and dispersalrates (Chave et al. 2002), trends in abundance withtraits may provide greater power to distinguish modelseven under such parameter uncertainty if at least thescaling of speciation rates can somehow be inferred.Note however that life history tradeoffs are typicallymore complex than modeled here, involving differencesin the size and age-dependence of demographic rates.Hence in order to develop an empirically useful nullmodel of fitness-equalized communities, a frameworklike the one here might need to be combined withrecently developed size structured neutral models(O’Dwyer et al. 2009).

192 Theor Ecol (2012) 5:181–194

Acknowledgements I thank R. Condit, J.M. Levine, J. Harte,D. Alonso, A.J. McKane, E.P. White, C. Moritz, S.P. Hubbell,D. Goldberg, and A.P. Allen, for helpful conversations relatedto this work and Adnrew Noble and Bart Haegeman for theirthorough reviews. This work was partially supported by the Classof 1935 Endowed Chair at UC Berkeley, by the EPA Scienceto Achieve Results (STAR) Graduate Fellowship, and by theUniversity of Michigan Elizabeth C. Crosby Research Fund.This work also benefited from my participation in the Santa FeInstitute Complex Systems Summer School, the Institute for The-oretical Physics Program on Physics and Biological Information,and the Unifying Theories in Ecology Working Group at theNational Center for Ecological Analysis and Synthesis.

Appendix A: NT at equilibrium

Here, I show that limt→∞ NT(t) is determined by f lossT

and ρ(T ′, T). To see this, first note that an alternativeway of expressing the expected total abundance acrossall species of type T, NT(t) (which we originally definedas NT(t) = ∑

k δ(Tk − T)E[nk(t)] is

NT(t) =∑

n

ST(n, t) · n (23)

where ST(n, t) is the expected number of species of typeT with abundance n at time t and comes from Eqs. 16and 17 and take this sum over the differential equationfor ST(n) (Eq. 16 from the main text) in order to obtaina differential equation for NT . The result is

dNT(t)dt

= pT(t) − ν f lossT NT(t) (24)

= ν

∫ Tmax

Tmin

dT ′ f lossT ′ ρ ′(T ′, T) NT ′(t)

− f lossT NT(t) (25)

So at equilibrium

f lossT NT =

∫ Tmax

Tmin

dT ′ f lossT ′ ρ ′(T ′, T) NT ′ (26)

Note that if parent and daughter species traits areexactly equal, i.e., if ρ ′(T ′, T) = δ(T ′ − T) then any NT

and f lossT can provide a solution to this equation, so any

historical influences on NT will be preserved, and f gainT

as defined by Eq. 13 can be different than f lossT if history

produced a bias in NT (i.e., an NT that depends on T).If instead the relationship between parent and daugh-ter species is very weak (i.e., ρ ′(T ′, T) is independentof T ′), then NT = ρ ′(T ′, T)/ f loss

T (and as discussed inthe main text f gain

T is simply ρ ′(T ′, T)). The integralequation Eq. 26 is more complex when the parent and

daughter species traits are related by not exactly. It isalso useful (in particular for Appendix B) to note thatat equilibrium NT can be expressed as

NT = f gainT

f lossT

Nη

. (27)

So by specifying the model parameters f lossT and f gain

Tone essentially specifies the equilibrium value of NT .This expression for NT makes it obvious that one willhave an inhomogeneous NT at equilibrium in caseswhere f gain

T is different in form from f lossT .

Appendix B: Constants of proportionalityfor cases 1–5

Here, I provide some details of the five different spe-ciation rate scaling cases used in the main text. In par-ticular, I show how I calculated the specific parametervalues of the speciation rate scalings from constraintson the total size of the community and total specia-tion rate, and how the average death rate could becalculated in each case. The two constraints I used tocalculate proportionality constants of the scalings off lossT and f gain

T with T are:

1.∑

T NT = N2.

∑T pT = νN

where NT is the expectation value of the total numberof individuals of type T, and pT is the rate at whichspecies of type T are arising in the community. Notethat here I am using sums over the possible death ratesT because in these example cases I took T to range overinteger values. I did this for convenience and becausethe output is then more analogous to a real community,where abundance and diversity would be measuredwithin discrete bins of the life history characteristicsof the species. Also, note that a second constraint onemight imagine on the speciation rate, namely on thetotal rate at which species are speciating leads to thesame condition on the speciation rate scaling factors asconstraint (2) above on the total rate at which speciesare arising.

These two constraints can be translated into con-straints on the scaling factors f loss

T and f gainT as follows.

Firstly, as shown in the previous section the equilibriumexpectation for NT is

NT = f gainT

f lossT

Nη

. (28)

Theor Ecol (2012) 5:181–194 193

Table 1 Constraints on scaling factor parameters for the five cases in Fig. 1 of the main text

Case Scaling factors Constraints 〈T〉1 f loss

T = b T, f gainT = b ′T (1) b ′ = b , (2) b ′ = 1/T 〈T〉 = T

2 f lossT = a, f gain

T = a′ (1) a′ = a, (2) a′ = 1 〈T〉 = T

3 f lossT = a − b T, f gain

T = a − b T (1) Satisfied, (2) a = 1 + b T 〈T〉 = T

4 f lossT = b T, f gain

T = a (1) b/a = T−1, (2) a = 1 〈T〉 =(

T−1)−1

5 f lossT = a, f gain

T = b T (1) b/a =(

T)−1

, (2)b =(

T)−1 〈T〉 = T2/T

Here, 〈T〉 refers to the average death rate across individuals in the community, whereas T = ∑T T is the mean of all possible death

rate types in the community, and T−1 and T2 are the mean values of T−1 and T2, respectively, across all possible death rate types inthe community

The first constraint above then becomes:

1.1

η

∑T

f gainT

f lossT

= 1

The second constraint can be written in terms of thespeciation rate scaling factors simply by using pT =f gainT

νNη

. The result is:

2.1

η

∑T f gain

T = 1

Table 1 below shows the scaling factor parametervalues resulting when each of these two constraints isapplied to the five different speciation rate scaling casesin Fig. 1 of the main text. It also shows the expectedaverage death rate across individuals in the communityfor each case, which I calculated according to

〈T〉 = 1

N

∑

T

NT T (29)

The information in Table 1, plus the rage of values forT (which I took to be from 1 to 100) are all that isneeded to generate predictions for the apportionmentof abundance and species richness across life historytypes using Eq. 17–20 of the main text.

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