analytically tractable model for community ecology with...

PHYSICAL REVIEW E 94 022423 (2016)

Analytically tractable model for community ecology with many species

Benjamin Dickens Charles K Fisher and Pankaj Mehtadagger

Department of Physics Boston University Boston Massachusetts 02215 USA(Received 2 February 2016 published 30 August 2016)

A fundamental problem in community ecology is understanding how ecological processes such as selectiondrift and immigration give rise to observed patterns in species composition and diversity Here we analyze arecently introduced analytically tractable presence-absence (PA) model for community assembly and we useit to ask how ecological traits such as the strength of competition the amount of diversity and demographicand environmental stochasticity affect species composition in a community In the PA model species are treatedas stochastic binary variables that can either be present or absent in a community species can immigrate intothe community from a regional species pool and can go extinct due to competition and stochasticity Buildingupon previous work we show that despite its simplicity the PA model reproduces the qualitative featuresof more complicated models of community assembly In agreement with recent studies of large competitiveLotka-Volterra systems the PA model exhibits distinct ecological behaviors organized around a special (ldquocriticalrdquo)point corresponding to Hubbellrsquos neutral theory of biodiversity These results suggest that the concepts ofecological ldquophasesrdquo and phase diagrams can provide a powerful framework for thinking about communityecology and that the PA model captures the essential ecological dynamics of community assembly

DOI 101103PhysRevE94022423

A central goal of community ecology is to understandthe tremendous biodiversity present in naturally occurringcommunities The observed patterns of species compositionand diversity stem from the interaction of a number of eco-logical processes Traditional models of community assembly(referred to as ldquonicherdquo models) emphasize the important roleplayed by competition and ecological selection in shapingcommunity structure [1ndash5] However due to the introductionof the neutral theory of biodiversity the past 15 years haveseen a renewed interest in the role of drift or stochasticity inshaping community assembly [6ndash9] In the neutral theory allspecies have identical birth and death rates so that all variationin species abundances is due entirely to random processesA complete theory of community assembly must take intoaccount other ecological processes such as immigrationand speciation in addition to selection and drift [1011]This has led to a renewed interest in using methods fromstatistical physics to understand the basic principles governingcommunity assembly [12ndash17]

One common approach for modeling community assemblyin complex communities is to consider generalized Lotka-Volterra models (LVMs) [131518] In generalized LVMsecological dynamics are modeled using a system of nonlineardifferential equations for species abundances Each species ischaracterized by a carrying capacitymdashie its maximal popu-lation size in the absence of other species Species interactionsare modeled using a matrix of ldquointeraction coefficientsrdquo Ingeneral it is extremely difficult to precisely measure theseinteraction coefficients [19] However for ecosystems withmany species we can overcome this difficulty by consideringa ldquotypical ecosystemrdquo for which species interaction matricesare drawn from a random matrix ensemble [18]

Historically LVMs emphasized the role of ecologicalselection and resource availability For this reason LVMs were

Present address Pfizer Cambridge MA 02139 USAdaggerpankajmbuedu

traditionally analyzed as deterministic ordinary differentialequations (ODEs) However several recent studies havemoved beyond deterministic ODE models to incorporate theeffects of immigration and stochasticity on ecological dynam-ics [1315] These recent studies on typical ecosystems havedemonstrated that communities can exhibit distinct ecologicalldquophasesrdquo (ie regimes with qualitatively different speciesabundance patterns) as ecological parameters such as immi-gration rates and the strength and heterogeneity of competitionare varied For example by numerically simulating stochasticdifferential equation-based implementation of a generalizedLVM Ref [13] showed that communities can exhibit a sharptransition between a selection-dominated regime dominatedby a single stable equilibrium and a drift-dominated regimewhere species abundances are uncorrelated and the ecologicaldynamics is well approximated by neutral models Theselection-dominated regime is favored in communities withlarge population sizes and relatively constant environmentswhereas the neutral phase is favored in communities withsmall population sizes and fluctuating environments SimilarlyRef [15] used a stochastic LVM to analyze a local communityof competing species with weak immigration from a staticregional pool and identified four distinct ecological regimesorganized around a ldquocritical pointrdquo corresponding to Hubbellrsquosneutral model [6]

Although LVMs are among the standard tools of theo-retical ecology they are difficult to analyze with analytictechniquesmdashespecially in the stochastic setting For thisreason Ref [13] introduced an immigration-extinction pro-cess [referred to as the presence-absence (PA) model] forcommunity assembly that attempts to capture the essentialecology of LVMs using a simpler model In the PA modelspecies are treated as stochastic binary variables that areeither present or absent in a local community A species maybecome extinct (ie absent) in the local community due tocompetitive exclusion and stochasticity but it can reappearin the community by immigrating from a regional speciespool In contrast to LVMs the PA model is amenable to

2470-0045201694(2)022423(14) 022423-1 copy2016 American Physical Society

BENJAMIN DICKENS CHARLES K FISHER AND PANKAJ MEHTA PHYSICAL REVIEW E 94 022423 (2016)

analytical arguments using techniques from statistical physicsrelated to the study of disordered spin systems For examplethe aforementioned sharp transition between the selection-dominated regime and the drift-dominated regime seen ingeneralized LVMs corresponds to the analog of the ldquofreezingtransitionrdquo in the PA model [13] Despite this previous workthe dynamics of the PA model are not fully understood andwarrant further study

In this work we address the extent to which the PA modelreproduces the qualitative behaviors and ecological regimesfound in more complicated LVMs In particular we use the PAmodel to analyze a local community of competing species withweak immigration from a static regional pool and we compareit to the ecological dynamics seen in numerical simulations ofthe generalized LVMs [15] We numerically simulate the PAmodel and examine ecological patterns of species abundancesto see how ecological processes such as selection drift andimmigration affect species abundance patterns We supplementthese results with analytic arguments We then compare andcontrast the results obtained from the PA model and LVMsFinally we discuss the implications of our results for modelingcomplex ecological communities

Throughout this paper for simplicity we employ someabuses of terminology In particular we refer to plots depictingecological behaviors as a function of ecological parametersas ldquophase diagramsrdquo We also refer to a special point inparameter space that separates three qualitatively distinctecological behaviors as a ldquocritical pointrdquo In the statistical andcondensed-matter physics literature these terms are definedbased on strict technical criteria that are not fulfilled by ourplots Our terminology is therefore not exact Indeed thebehaviors shown here are more akin to crossovers in statisticaland condensed-matter physics [2021]

I THE PRESENCE-ABSENCE (PA) MODEL

A Ecological motivation

The goal of introducing the PA model is to capture theessential ecological features present in LVMs in a simpleanalytically tractable model For the sake of tractability thePA model ignores species abundances and instead focuseson a simpler question is a species present or absent in thecommunity The basic idea behind the definition of the PAmodel is roughly speaking that the propensity of a speciesto be present or absent is determined by a quantity called itsldquoeffective carrying capacityrdquo The effective carrying capacityof a species i which generally depends on both environmentalfactors as well as the abundances of other species sets themaximum possible abundance of a species i in the presenceof the others Therefore species i may persist if its effectivecarrying capacity is positive whereas it will go extinct if itseffective carrying capacity is negative

To gain intuition about the role of effective carryingcapacities in the definition of the PA model it is helpfulto recount the results on species invasion in Lotka-Volterracommunities derived by MacArthur and Levins in their classic1967 paper [5] wherein effective carrying capacities playeda crucial role Species abundances x are modeled using asystem of ODEs of the form dxidt = λi + xifi(x) with λi

Carrying capacity in IsolationEffective carrying capacity

increases for mutualistic reactions Effective carrying capacity

decreases for competitive reactions

(a)

(b)Regional Species Pool Immigration Extinction

FIG 1 The presence-absence model (a) Species i has a carryingcapacity Ki in the absence of other species Species irsquos interactionwith species j is characterized by the interaction coefficient cij Depending on whether they are mutualistic or competitive theinteractions can increase or decrease the effective carrying capacityof species i (b) Species i can be present in the communitysi = 1 or absent from the community si = 0 Species can immigratefrom a regional community pool and go extinct in the communityproportional with a rate equal to the exponential of the ratio ofeffective carrying capacity to the stochasticity parameter ω

the rate of immigration and fi(x) the ecological fitness ofspecies i which is a function of the species abundances x Ingeneral fi(x) may be a complicated function due to nonlinearfunctional responses or other phenomena Regardless of theexact form of fi(x) the ecological fitness can always belinearized near an equilibrium point x lowast where the dynamicsare approximately described by LVM equations

In the LVM the ecological fitness fi(x) = Ki minus xi minussumj cij xj is a linear function of the carrying capacity Ki and

interaction coefficients cij which measure how the presenceof species j affects the growth rate of species i The interactioncoefficients cij lt 0 are negative when interactions withspecies j benefit the growth of species i cij gt 0 when speciesj competes with species i and cij = 0 if species i and j do notinteract (see Fig 1) We interpret fi(x) + xi = Ki minussum

j cij xj

as an effective carrying capacity Keffi (x) for species i and we

write dxidt = λi + xi[Keffi (x) minus xi] In general the effective

carrying capacity is a function of the abundances of all thespecies in the community

MacArthur and Levins [5] used the idea of an effectivecarrying capacity to ask whether a new species i couldinvade a community with species abundances x lowast Usinggraphical stability arguments they showed that species i caninvade successfully if its effective carrying capacity is positive[Keff

i (x lowast) = Ki minussumj cij x

lowastj gt 0] but it will be unsuccessful

if its effective carrying capacity is negative [Keffi (x lowast) =

Ki minussumj cij x

lowastj lt 0] Therefore the mean extinction time

of a species in the local community depends strongly onthe effective carrying capacity the time to extinction islong for Keff

i (x lowast) gt 0 and short for Keffi (x lowast) lt 0 Based on

these observations we hypothesize that the extinction rateof a species depends exponentially on the effective carryingcapacity This assumption is used in the definition of the PAmodel

022423-2

ANALYTICALLY TRACTABLE MODEL FOR COMMUNITY PHYSICAL REVIEW E 94 022423 (2016)

B Definition of the PA model

The PA model describes the probability that variouscollections of species will be present (or absent) in a localecological community which we assume is attached to a largeregional species pool containing S species We parametrizethe presence (or absence) of species i isin 1 S by a binaryrandom variable si where si = 1 if species i is present andsi = 0 if it is absent Therefore the state of the ecosystem isdescribed by the random vector s = (s1 sS) isin 01S Wedenote the probability distribution to observe a particular states at time t by Pt (s) The probability distribution is governed bya differential equation called a master equation which definesthe dynamics of the PA model

Prior to writing down the master equation we specifytwo kinds of rates First there is the rate at which speciesi immigrates into the local community from the regional poolie the rate at which si = 0 rarr 1

RIi (s) = λi

There is also the rate of an extinction event si = 1 rarr 0 REi (s)

given by

REi (s) = exp

(minus 1

ωKeff

i (s)

)

Keffi (s) = Ki minus

j=Ssumj=1

Kjcij sj

Here Keffi (s) represents the effective carrying capacity of

species i given that the state of the ecosystem is s Ki

denotes the carrying capacity of species i in the absence ofother species whereas cij denotes an interaction coefficientdescribing how species j influences the effective carryingcapacity of species i (with the convention that cii = 0) Thenumber ω parametrizes the impact of random noise on speciesextinction and is thus called the ldquonoise strengthrdquo The units oftime have been set so that the rate of extinction equals 1 in thelimit that ω rarr infin

With these rates the time evolution of Pt (s) is given by themaster equation

dPt (s)

dt=

i=Ssumi=1

[RE

i (s + ei)Pt (s + ei) minus RIi (s)Pt (s)

](1 minus si)

+ [RI

i (s minus ei)Pt (s minus ei) minus REi (s)Pt (s)

]si

(1)

where ei denotes the vector whose ith component is unity andall other components are zero

C Choosing carrying capacities and interaction coefficients

The ecological dynamics of the PA model depend on thechoice of carrying capacities and interaction coefficients Foran ecosystem with S species this involves specifying S2

parameters Deriving all of the parameters describing thedynamics of a real community from observations is a dauntingtask for ecosystems with many species (S 1) However it ispossible to make progress by analyzing a ldquotypicalrdquo ecosystemwhere the interaction coefficients and carrying capacities aredrawn randomly from an appropriate probability distribution[18]

For simplicity we restrict our analysis to purely competitivespecies interactions cij gt 0 We draw interaction coefficientsindependently for each pair i = j from a γ distribution withmean μcS and variance σ 2

c S

pc(cij ) = 1

θkcc (kc)

ckcminus1ij exp

(minuscij

θc

)

where denotes the Gamma function and

kc = μ2c

Sσ 2c

θc = σ 2c

μc

The 1S scaling of the mean and variance of cij is necessaryto prevent pathological behaviors when S becomes large Inphysics terminology this ensures a well-defined thermody-namic limit [22]

The carrying capacities are also drawn independently froma log-normal distribution with mean μK and variance σ 2

K

pK (Ki) = 1

KizK

radic2π

exp

(minus 1

2z2K

[ln(Ki) minus lK ]2

)

where

lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)

These choices of probability distributions ensure that boththe interaction coefficients and the carrying capacities arestrictly positive while simultaneously allowing for analyticcalculations Considering typical ecosystems for which Ki andcij are random variables circumvents the proliferation of freeparameters by reducing the number of relevant parametersfrom S2 to four namely the means and variances of the inter-action coefficients and carrying capacities μc σc μK and σK

D Relation to island biodiversity

The PA model describes the dynamics of a well-mixedisolated community of competing species with weak immigra-tion from a static regional pool For this reason the model iswell-suited for discussions in the context of island biodiversityIsland biodiversity the study of the species richness andecological dynamics of isolated natural communities [1023]has played an important role in the development of theoreticalecology For example it was a precursor to Hubbellrsquos neutraltheory [6] The success of the neutral theory of biodiversity andbiogeography [69] at explaining patterns in biodiversity hasresulted in a vigorous debate on the processes underlying com-munity assembly and in particular on the relative importanceof selection and stochasticity in shaping ecological dynamicsand species abundance patterns [91624ndash32] Overall insularcommunities provide a tractable arena for studying the effectsof selection and stochasticity while minimizing the effectof other ecological processes such as complicated dispersalphenomena

The PA model allows one to study the roles of selectionand stochasticity within the context of island biodiversityIn particular one may use the PA model to describe thedynamics of a local island community of competing specieswith weak immigration from a static regional pool ie anearby mainland This situation was recently analyzed using

022423-3


FIG 2 (a) Mean number of species M (b) the freezing parameter α and (c) the composite order parameter C(Mα) (see the main text fordefinitions) computed from numerical dynamics simulations with S = 20 μK = 100 λ = 001 ω = 1 and Ki = μK for all i The dynamicsexhibit three distinct regimes the coexistence regime (CR) with C(Mα) minus1 the partial coexistence regime (PCR) with C(Mα) 0 andthe noisy regime (NR) C(Mα) 1 (I) (II) and (III) illustrate the typical dynamics of the CR PCR and NR

LVMs and found to exhibit distinct regimes of ecologicaldynamics and species abundances centered around a specialcritical point corresponding to Hubbellrsquos neutral theory ofbiodiversity [15] Inspired by earlier work showing that the PAmodel can reproduce the sharp transitions between a nichelikeselection-dominated regime and a neutral-like drift-dominatedregime [13] we numerically simulated the PA model to testwhether it can reproduce the basic phenomenology seen inmuch more complicated LVMs This is discussed in the nextsection

II NUMERICAL SIMULATIONS

To see if the PA model can reproduce the basic behaviorsexhibited by more complicated LVMs [1315] we numericallysimulated the PA model dynamics We found that the dynamicsof the PA model can be classified into three broad regimes(see the bottom panels of Figs 2 and 3) a coexistence regime(CR) where all species are present a partial coexistence regime(PCR) where only a small fraction of species are stably presentin the community and a noisy regime (NR) where all speciesfluctuate between being present and absent over small timescales Using order parameters measured in the numericalsimulations we summarized our findings by constructingphase diagrams for these ecological regimes As seen inRef [15] the regimes organize themselves around a specialldquocriticalrdquo point corresponding to Hubbellrsquos neutral theory Wediscuss simulation details and results in this section

A Simulation details

In all simulations we assume that all species have the sameimmigration rate λi = λ and we ask that ω |ln(λ)| μK where μK denotes the average value of all carrying capacitiesRoughly speaking this assumption assures that the probabilityfor a species to be present or absent is determined primarilyby the weakness or strength of that speciesrsquo extinction rateexp[minusKeff

i (s)ω] Thus in our simulations the propensity fora species to survive in the local community is determinedby its interactions with other species and its environment asdescribed by the effective carrying capacity Keff

i (s) ratherthan its immigration ability

Numerical simulations of the PA model master equationwere performed using Gillespiersquos algorithm [33] To comparewith the results of [15] we started by simulating the PAmodel for the case in which all species have the same carryingcapacities (μK = 100σK = 0) For each choice of (μcσc)30 random realizations of cij rsquos were independently drawnfrom a γ distribution of mean μcS and variance σ 2

c SWe took S = 20 λ = 001 and ω = 1 in these simulationsFor each realization PA model dynamics were simulated forT = 400 000 units of PA model time with s(t) sampled intime steps of τs 2000 In addition to heterogeneity in theinteraction coefficients we wanted to investigate the effectof the heterogeneity in the carrying capacities of speciesThus we also performed simulations with random carryingcapacities for each choice of (μcσK ) in which 30 randomrealizations of the Ki vector were independently drawn from

022423-4


FIG 3 (a) The mean number of species M (b) the freezing parameter α and (c) the composite order parameter C(Mα) (see the maintext for definitions) computed from numerical dynamics simulations with S = 20 μK = 100 λ = 001 ω = 05 and cij = μcS fixed forall i For this choice of parameters σKμK = 09 the PA model never reaches the NR until μc S The dynamics exhibit two regimes thecoexistence regime (CR) with C(Mα) minus1 and the partial coexistence regime (PCR) with C(Mα) 0 (I) illustrates the typical dynamicsof the CR whereas (II) and (III) illustrate the typical dynamics of the PCR

a log-normal distribution of mean μK = 100 and varianceσ 2

K with S = 20 λ = 001 and ω = 05 Dynamics weresimulated for each realization for T = 400 000 units of PAmodel time

The numerical phase diagrams in Figs 2 and 3 exhibitedfor μc gt 1 artifacts of finite-size effects inherent in numericalsimulation To smooth out these artifacts without destroyingthe essential features of the plotsmdashspecifically their behaviornear μc = 1 and σcK = 0mdashthe μc direction of all order-parameter plots was convolved with a variable Gaussian kernelof increasing variance equal to μc We expect that a similarsmoothing would be achieved by increasing both S and thenumber of random realizations of species but this quicklybecomes computationally expensive

B Order parameters for ecological dynamics

We constructed phase diagrams for the PA model to sum-marize our findings about its ecological dynamics Simulationsrevealed three regimes of qualitatively distinct dynamicsthe CR the PCR and the NR These three regimes can bedistinguished between by measuring two order-parameter-likequantities from numerical simulation data the mean number ofspecies and a ldquofreezing parameterrdquo To define these quantitiesit is necessary to introduce two kinds of averages timeaverages which we denote as 〈middot middot middot 〉 and averages over randomdraws of the ecological parameters cij or Ki which we denoteby [middot middot middot ]av

Define the mean number of species present in the commu-nity as

M =i=Ssumi=1

[〈si〉]av (2)

It is a ldquomeanrdquo in two senses it is the number of speciesaveraged over both time and random draws of speciesparameters Intuitively M tells us whether or not the PA modelis exhibiting the coexistence regime In particular we expectthat M = S in the CR and M lt S otherwise Inspired by thetheory of disordered systems we also define the ldquofreezingparameterrdquo

α = 4

S

i=Ssumi=1

([〈si〉2]av minus [〈si〉]2

av

)

α is proportional to the sum of the variances of 〈si〉 overrandom draws of species traits and it tells us whether or notthe PA model is in the PCR Our intuition for this quantity isas follows In the PCR we expect dominant species to emergethat stay in the ecosystem for almost all time If species i issuch a dominant species then we have 〈si〉 1 On the otherhand nondominant species in the PCR will remain absentfrom the ecosystem for almost all time hence a nondominantspecies j in the PCR will satisfy 〈sj 〉 0 Moreover the subsetof species that are dominant depends on the random draw ofspecies parameters cij and Ki For this reason if the variance in

022423-5


cij and Ki is appreciable then the variance in which species aredominant will also be appreciable in particular the variancein 〈si〉 over random draws of species traits will be maximaland we expect α 1 On the other hand if the PA model isexhibiting either CR or NR then no species is dominating overthe others (recall that in the CR all species are present and inthe NR all species are fluctuating between present and absent)regardless of the random draw of cij rsquos and Kirsquos In this casefluctuations in cij rsquos and Kirsquos over random draws will not leadto a nonzero variance in 〈si〉 and the freezing parameter willbe close to zero α 0 Thus α measures whether or not thereare dominant species in the ecosystem For this reason weexpect that α 1 in the PCR and α 0 otherwise The nameldquofreezing parameterrdquo comes from the interpretation that inthe PCR the ecosystem appears ldquofrozenrdquo or stuck for almost alltime in a configuration in which dominant species are presentand nondominant species are absent In this sense we can saythat α measures whether or not the ecosystem is ldquofrozenrdquo

To summarize we expect that M = S and α 0 in the CRM lt S and α 1 in the PCR and M lt S and α 0 in theNR thus between these two quantities we can distinguishbetween all three regimes from numerical simulation data (seeFigs 2 and 3)

To compare with phase diagrams obtained by analyticcalculations and through other models it is useful to define acomposite order parameter C(Mα) whose value distinguishesall three dynamical regimes We ask that it satisfies C(Mα) minus1 in the CR C(Mα) 0 in the PCR and C(Mα) 1 in theNR Moreover C(Mα) should be continuous and monotonicin both M and α Many functions satisfy these properties Herewe choose

C(Mα) = (1 minus α)gSγ (M)

where

gSγ (x) =

γminusx

Sminusγif x gt γ

γminusx

γif x lt γ

When S = 20 and γ = 19 C(Mα) will be negative wheneverM gt γ = 19 and positive if M lt γ = 19 With this defini-tion C(Mα) satisfies the desired properties

C The PA model phase diagrams

To understand the effect of competition on the dynamicsof the PA model we constructed a phase diagram as afunction of the mean strength of competition (μc) and thecompetition diversity (σc) when all species have identicalcarrying capacities (σK = 0) The results are shown in Fig 2As expected the average number of species present inthe community (M) decreases with increasing competitionFurthermore for uniform interaction coefficients (σc = 0)all species are present in the community until a criticalcompetition strength μc = 1 after which species start goingextinct The middle panel in the figure shows the freezingparameter (α) which measures whether a subset of species areconsistently in the environment Notice that this occurs aroundμc 1 when the interaction coefficients are heterogeneous

Taken together these numerical observations suggest thatthe competitive CR is favored when the mean competitionstrength is low whereas the NR-type dynamics are favored

when competition is very strong This is consistent withour intuition that a species i should tend to be present ifKeff

i (s) gt 0 or tend to be absent if Keffi (s) lt 0 To see this note

that Keffi (s) = μK minus μK

sumj =i cij sj in this case Therefore

Keffi (s) gt 0 for all i precisely when competition is sufficiently

low namely when the CR occurs On the other hand weexpect to see Keff

i (s) lt 0 in the presence of some specieswhen mean competition is high which is when the NR isobserved In the NR the dynamics appear to be dominatedby stochasticity and drift because species quickly go extinctafter immigrating into the community This can be explainedby the fact that negative carrying capacities Keff

i (s) lt 0 seta fast extinction rate At intermediate levels of competitionμc 1 and in the presence of heterogeneity in the interactioncoefficients the ecological dynamics are characterized bypartial coexistence where only a subset of species remainspresent in the community This PCR is consistent with thestatement that Keff

i (s) lt 0 for some species namely the absentones and Keff

i (s) gt 0 for the present species Thus in the PCRregime some species are more fit for the environment thanothers leading to reproducible species abundance patterns

We also examined the effect of heterogeneity in carryingcapacities on the dynamics of the PA model To do this weconstructed phase diagrams as a function of the carryingcapacity diversity (in units of the mean carrying capacityie σKμK ) and the mean competition strength for uniforminteraction coefficients assuming that σc = 0 (see Fig 3) Theresulting phase diagram once again exhibits three phases withthe NR favored when there is strong competition and the CRfavored when competition is weak

A striking aspect of the phase diagrams is that the dynamicalregimes organize themselves around a special point in the PAmodel parameter space where μc = 1 and σcK = 0 Just asin generalized LVMs [15] we can identify this point withHubbellrsquos neutral theory To see this note that all speciesare equivalent with respect to their ecological traits such asimmigration rates carrying capacities and competition coef-ficients at this point Moreover the intraspecies competitiondescribed by Ki balances the interspecies competition givenby

sumj =i Kj cij sj More precisely if all species are present in

the community (si = 1 for all i) then the effective carryingcapacity of every species is zero Keff

i (s) = 0 This holdsbecause cij = 1S for all pairs i = j so

Keffi (s) = μK

⎛⎝1 minus 1

S

sumj =i

si

⎞⎠ = 1

S 0

for S 1 These are precisely the conditions characterizingHubbellrsquos neutral model [6ndash9] Small perturbations aroundthis Hubbell point can lead to qualitatively different speciesabundance patterns and dynamical behaviors

III ANALYTIC RESULTS

To better understand our numerical simulations we per-formed analytic calculations on the PA model Recall that aspecies i will tend to persist in the local community providedthat its extinction rate exp[minusKeff

i (s)ω] is small (much lessthan 1) and it will tend to go extinct if its extinction rate is large

022423-6


(much bigger than 1) It follows that a species will probablypersist if it tends to have a positive effective carrying capacitywhereas it will go extinct quickly if its effective carryingcapacity tends to be negative This basic observation suggestscriteria for classifying the three dynamical regimes analyticallyin terms of effective carrying capacities We emphasize thatthis discussion is relevant only when μKω is large Sincethe noise strength ω simply provides the units in whichto measure Keff

i (s) we may take ω = 1 here without lossof generality and ask that μK is taken ldquolargerdquo μK = 100 issufficient for our purposes as in our numerical simulations

To provide precise criteria for the three regimes in terms ofKeff

i (s) we introduce a quantity

κi = κi(c K) = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc

where vart [middot middot middot ] denotes a time-variance and ldquohotcrdquo standsfor terms proportional to the ldquohigher-order time cumulantsrdquoof Keff

i (s)μK In statistics language κi is proportional tothe cumulant-generating function of minusKeff

i (s)μK evaluatedat unity where the averages are over time fluctuations of thespecies We sometimes employ the notation κi(c K) to remindourselves that κi depends on randomly drawn cij and Ki

Our basic intuition about κi can be summarized as fol-lows The above equation shows that κi equals the mean〈Keff

i (s)〉 minus some cumulant terms that represent theldquotypical fluctuationsrdquo of the effective carrying capacity Denotethe sum of these cumulant terms by δKeff

i so that κi =〈Keff

i 〉 minus δKeffi and note that δKeff

i is positive-definite due toJensenrsquos inequality By comparing 〈Keff

i 〉 and δKeffi we obtain

important information about how often Keffi (s) will be positive

or negative and by extension whether or not we expect speciesi to survive or go extinct For example if 〈Keff

i 〉 gt δKeffi

then we expect that Keffi (s) will tend to fluctuate in the

positive real line hence species i will persist On the otherhand if 0 lt 〈Keff

i 〉 lt δKeffi then Keff

i (s) fluctuates betweenbeing positive and negative hence species i fluctuates betweenstates of probable persistence and probable extinction Ifinstead 〈Keff

i 〉 lt 0 and 〈Keffi 〉 lt δKeff

i then Keffi (s) is almost

always negative and species i is almost always absent unlessit attempts to immigrate This intuition suggests the followingcriteria for the PA model ecological regimes

(i) Coexistence regime (CR) The CR occurs whenκi(c K) gt 0 for all i isin 1 S In this regime all speciestend to coexist stably in the local community because theireffective carrying capacities tend to fluctuate in the positivereal line permitting all species to survive As we willshow in the Appendixes B 1 and C 1 this criterion reducesapproximately to a simpler criterion the CR occurs whenKi(s) gt 0 for all i isin 1 S when all species are present

(ii) Noisy regime (NR) The NR occurs when κi(c K) lt 0for all i isin 1 S In this regime the effective carryingcapacities of species are either fluctuating between beingpositive and negative or they are fluctuating in the negative realline In either case no species should persist in the ecosystemfor a very long time because negative effective carryingcapacities do not permit them to survive As a result all species

fluctuate between being present and absent over small timescales yielding dynamics that appear to be noise-dominated

In the Appendix B 3 we show that this criterion has a naturalinterpretation in terms of robustness to a typical fluctuation inspecies abundances Denote the average number of species inthe local community by M Define a new quantity

13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av

with δKeffi = 〈Keff

i 〉 minus κi a ldquotypical fluctuationrdquo of the effec-tive carrying capacity for species i In the Appendix B 3we show that the system will be in NR if Ki(s) lt 0 for alli isin 1 S whenever M + 13 species are present where M

is the mean species abundance defined in Eq (2) In particularwhen the number of species fluctuates a threshold fluctuation13 above the mean no species should persist for very longbecause their carrying capacities become negative

(iii) Partial coexistence regime (PCR) This occurs whenκi(c K) lt 0 for a fraction of species i isin 1 S andκi(c K) gt 0 for other species i In this regime we expectdominant species to emerge in the community namely thosewith positive κi(c K) and remain in the ecosystem for almostall time Meanwhile those species with negative κi(c K) willattempt and fail to invade going extinct quickly and fluctuatingbetween presence and absence

Given these criteria one can use a mean-field-theoreticapproach to analytically calculate M and a phase diagram forthe PA model (see the Appendix A for details) The resultsare shown in Fig 4 To determine the phase boundary of theCR we calculated the probability PCR that κi(c K) gt 0 whenM = S species are present in the community given randomdraws of cij or Ki (see the Appendixes) This is plotted in theblue region of the left-hand side of the analytic phase diagramsin Fig 4 To determine the boundary of the NR we calculatedthe probability PNR that κi(c K) lt 0 The results of thesecalculations are shown in the red region of the right-hand sideof the analytic phase diagrams in Fig 4 The PCR occurs inthe white region where neither the CR nor the NR is probable

In the NR the statistics of species abundances appear tobe ldquoneutralrdquo (ldquostatistically neutralrdquo in the language of [13])Thus we assumed that species are statistically independentand we neglected contributions from the heterogeneity in cij

and Ki The latter assumption implies that an equilibriumprobability distribution will be reached as t rarr infin and thatall species have the same mean value m = 〈si〉 The formerassumption allows us to write a species probability distributionQ(s) = prodi=S

i=1 Qi(si) in equilibrium that factorizes into indi-vidual probability distributions Qi(si) We approximate thesemarginal distributions Qi(si) as Gaussian distributions withmean m and variance σ 2

m = m(1 minus m) by neglecting cumulantsof si of higher order than the variance This yields the sameanswer one would obtain by neglecting the moments of cij

and 1SKj of third and higher order Similar approximations

were also employed to compute the CR phase boundary butno Gaussian approximation was needed See the Appendixesfor details

The agreement between analytic and numerical results isremarkable The mean-field calculations of M agree with

022423-7


FIG 4 (a) Left panel Mean number of species M computed analytically with λ = 001 ω = 1 and μK = 100 in the absence of speciesheterogeneity Middle and right panels For comparison we display M computed numerically using the data shown in Fig 2 (middle panel) andFig 3 (right panel) (b) and (c) Probability to exhibit the CR (blue) or the NR (red) as calculated analytically Long-dashed curves mark wherethe probability to exhibit a regime is P = 098 whereas dotted curves indicate a probability of P = 090 In (b) each cij is independentlydrawn from a γ distribution of mean μcS and variance σ 2

c S while Ki = μK is fixed for all i (σK = 0) In (c) each Ki is drawn independentlyfrom a log-normal distribution of variance σ 2

K and fixed mean μK whereas cij = μcS for all i = j (σc = 0) In (b) and (c) λ = 001 ω = 1and μK = 100

numerical simulations even for moderately large values ofσc and σK Despite our approximations there is a surprisingagreement between our analytic phase diagrams and thenumerical phase diagrams in Figs 2 and 3 Strikingly in thecase of heterogeneous Ki the NR is very small in both analyticor numerical calculations where S = 20 Overall our resultssuggest that we can capture the essential ecology of the PAmodel by thinking about the means and typical fluctuations ofthe effective carrying capacities of individual species

IV DISCUSSION

We analyzed the binary presence-absence (PA) model forcommunity assembly first introduced in Ref [13] The PAmodel describes an immigration-extinction process in whichspecies are treated as stochastic binary variables that can eitherbe present or absent in a community Species immigrate tothe community from a regional species pool Once in thelocal community a species competes for resources until itbecomes locally extinct due to competition and stochasticityHere we investigated the effects of heterogeneous competitioncoefficients and carrying capacities on the ecological dynamicsin large ldquotypicalrdquo communities We found that the PA modelexhibits three distinct regimes a coexistence regime (CR)where all species are present in the community a noisy regime

(NR) where all species quickly go extinct after immigratingto the community leading to neutral-like dynamics and apartial coexistence regime (PCR) where a broad distributionof effective carrying capacities leads to a few dominant speciesthat remain present in the community most of the time

These three regimes all converge at a special point (calledthe Hubbell point) in parameter space corresponding to theneutral theory of biodiversity where all species are identicaland their dynamics are uncorrelated The Hubbell point playsan analogous role to a quantum critical point in the phasediagrams of systems that exhibit phase transitions [21] Inthe absence of heterogeneity in the interaction coefficients orcarrying capacities (ie σc = 0 and σK = 0) the Hubbell pointseparates a selection-dominated regime where all species arepresent and the dynamics look fairly deterministic from adrift-dominated regime where selection is not important forthe dynamics For nonzero σc and σK the effect of the Hubbellpoint manifests itself in the existence of the partial coexistenceregime wherein a subset of the species are always presentin the community due to selection while the dynamics ofthe remaining species are dominated by noise This suggeststhat the neutral theory of biodiversity plays a special rolein understanding ecological dynamics perhaps as much ascritical points play an important role in the theory of phasetransitions One interesting question worth investigating is

022423-8


whether ideas such as universality and critical exponents [20]can also be exported to this ecological setting

Despite its simplicity the PA model is able to reproduce thequalitative behaviors of more complicated generalized Lotka-Volterra models (LVMs) For example our phase diagram forthe PA model in Fig 2 is almost identical to the phase diagramof the LVM obtained using numerical simulations in Ref [15]Nevertheless while the PA model exhibits three regimes theLVMs were found to exhibit four Both the PA model andLVMs exhibit coexistence and partial coexistence regimesrespectively at low and intermediate levels of competitionHowever instead of a noisy regime at high competitionRef [15] identified a ldquodisordered phaserdquo and a ldquoglasslikephaserdquo The disordered phase appears to be analogous toour noisy regime That is there is no longer a fixed set ofresident species that are always present instead there is aconstant turnover in the community composition The glasslikephase which appears at levels of competition greater thanthat of the disordered phase is characterized by occasionalnoise-induced transitions between a few equilibria such thatfor each equilibrium only a few dominant species are presentWe did not identify this behavior in the PA model Thisdiscrepancy is likely due to the simplified dynamics in thePA model that ignores species abundance distributions Thusthe PA model provides a compromise between complexity andinterpretability given that it is amenable to analytic techniquesIn the future it will be interesting to investigate whether the PAmodel can reproduce more complicated behaviors of LVMsespecially when extended to a spatial setting For exampleit will be interesting to see if cyclic dominance betweenspecies can lead to specific types of pattern formation suchas propagating fronts [3435]

The idea of an effective carrying capacity plays a centralrole in the PA model The importance of this quantity wasalready noted in the early works of Macarthur and Levins [5]The effective carrying capacity essentially sets the extinctiontime in the local community and it measures how susceptiblea species is to stochastic events that can cause it to die outOur analytic calculations demonstrate that a mean-field-likepicture based on effective carrying capacities is sufficientto reproduce the numerical phase diagram This suggeststhat in large communities with many species the effect ofdifferent ecological processes can be understood by askinghow they change the effective carrying capacity for a typicalspecies configuration This is similar in spirit to recent work inthe theoretical ecology literature [36] These simplificationssuggest that the behaviors of large ecosystems with manyspecies may differ significantly from the behavior of smallsystems with a few species

In this work we limited ourselves to considering purelycompetitive interactions in a spatially well-mixed populationwith low immigration rates from a regional species pool It willbe interesting to generalize these results to the case in which theinteraction coefficients can be mutualistic or even hierarchical[37] Another important avenue for future research is to askhow the introduction of spatial structure affects the mean-fieldpicture In particular it will be interesting to understand ifthe phase diagram of the PA model is still organized aroundHubbellrsquos neutral theory and if the PA model can reproducethe species-area relationships seen in real ecosystems [38]

ACKNOWLEDGMENTS

This work was partially supported by a Simons Investigatorin the Mathematical Modeling of Living Systems and aSloan Research Fellowship to PM BD also acknowledgesthe Boston University Undergraduate Research OpportunitiesProgram for partial funding

APPENDIX A MEAN-FIELD APPROXIMATION ANDCALCULATING THE MEAN SPECIES ABUNDANCE

We can analyze the PA model in the coexistence regime(CR) and the noisy regime (NR) using mean-field theory(MFT) In MFT the true distribution of species is approxi-mated by an equilibrium variational distribution Q(s) thatfactorizes over species

Q(s) =i=Sprodi=1

Qi(si)

For the PA model where si isin 01 the mean-field variationaldistribution takes the form

Qi(si) = mδsi 1 + (1 minus m)δsi 0 (A1)

where δsi 1 is the Kronecker delta function and m is avariational parameter that measures the probability of aspecies being present 〈si〉 = m Notice that we use the sameparameter of m for all i

In the coexistence regime we know that all species arepresent so that m 1 For this reason in the CR the mean-fieldvariational Ansatz is well approximated by

QCRi (si) = δsi 1 (A2)

In the noisy regime we approximate Qi(si) by a Gaussiandistribution with mean m and variance σ 2

m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)

QNRi can be thought of as an approximation to the full

variational distribution Qi where we have ignored higher-order cumulants beyond the variance Since m 1 in the NRthis is expected to be a good approximation

These mean-field variational Ansatzes are consistent withnumerical simulations of the CR and the NR that show thatthe heterogeneity of cij and Ki do not significantly modifythe dynamics in these regimes and species appear to bestatistically independent because the dynamics of differentspecies are uncorrelated in time

We would like to compute m for the variational distribution(A1) which requires some knowledge of the true equilibriumdistribution in the absence of heterogeneity (σc = σK = 0)In this case the dynamics approach a unique equilibriumdistribution as t rarr infin [13]

PPA(s) = 1

Zexp

(minusμK

ωU (s)

)

U (s) = minus[1 + ]i=Ssumi=1

si + μc

2S

sumi =j

sisj

022423-9


where = ω ln(λ)μK and Z is a normalization constantThe function U (s) is sometimes called the ldquointernal energyrdquoor just the ldquoenergyrdquo associated with the species configurations Due to ergodicity we reinterpret time averages 〈middot middot middot 〉 asaverages over the distribution PPA(s) In particular given afunction A = A(s) of the random variable s we have

〈A〉 = Trs A(s)PPA(s) = 1

ZTrs A(s) exp

(minusμK

ωU (s)

)

where Trs denotes the operation of summing over all possible2S configurations of s Since U (s) is invariant under theexchange i harr j of species indices it follows that 〈si〉 = 〈sj 〉for all i = j

With these observations we can use the functional form(A1) for Q(s) to calculate the mean species abundance M Todetermine the variational m we minimize the variational freeenergy

F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q

where 〈middot middot middot 〉Q denotes an average with respect to Q Since Q iscompletely specified by m we can express F [Q] as a functionF (m) of m One obtains

F (m)

S= minus [1 + ]m + μcm

2

2

+ ω

μK

[m ln(m) + (1 minus m) ln(1 minus m)]

A necessary condition for minimization is that ddm

F (m) = 0which yields

m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)

The mean species abundance is obtained by noting that M =Sm This is plotted in the main text

APPENDIX B PHASE DIAGRAM FOR HETEROGENEOUSINTERACTION COEFFICIENTS

In this section we set σK = 0 so that Ki = μK for all i isin1 S We seek to answer the following question givena choice of (μcσc) and a random draw of cij rsquos what is theprobability that the PA model is in the CR or the NR

1 Boundary of the coexistence regime

First we calculate the probability PCR that the system isin the CR We reintroduce the quantity

κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)

and we recall that the CR occurs when κi gt 0 for all i isin1 S Thus PCR is the probability that κi gt 0 To proceedwe must explicitly calculate κi in terms of cij Since weexpect our Ansatz (A2) to be valid near the CR we computethe time average in Eq (B1) with respect to QCR

i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠

which we can rewrite as

1

μK

κi = 1 minussumj =i

cij

This equation has a simple interpretation Namely the CRoccurs when the effective carrying capacity of every speciesis positive in the presence of all species Thus PCR is justthe probability that

sumj =i cij lt 1 Since each cij is drawn

independently from a γ distribution of mean μcS and varianceσ 2

c S it follows that yi = sumj =i cij is γ -distributed with mean

μc and variance σ 2c at leading order in large S the explicit

probability distribution is

py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc

PCR is the probability that yi lt 1 ie

PCR = P (yi lt 1) =int 1

0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)

where is the Gamma function and γ is the lower incompletegamma function Given (μcσc) this formula can be used tocalculate PCR numerically resulting in Fig 4

2 Boundary of the noisy regime

Now we find the probability PNR that a random draw of cij rsquoscauses the PA model to exhibit the NR This is the regime whereκi lt 0 for all i isin 1 S hence PNR is the probability thatκi lt 0 Using the Ansatz (A3) to compute the time average inEq (B1) one getslang

exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m

[sjminusm]2 + cij sj

)

022423-10


Performing the Gaussian integrals yieldslangexp

(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠

This can be rewritten as

1

μK

κi = 1 minus msumj =i

cij minus 1

2σ 2

m

sumj =i

c2ij (B2)

We further make the approximationsumj =i

c2ij μc

sumj =i

cij (B3)

yielding the expression

1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij

Now define a new random variable yi =(m + 1

2μcσ2m

)sumj =i cij so that PNR is the probability

that yi gt 1 Then to leading order in S yi is γ -distributedwith mean

(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that

PNR 1 minus P (yi lt 1) = 1 minus 1

(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)

3 Alternative interpretation of the NR boundary

As mentioned in the main text PNR has an alternativeinterpretation namely the probability that Keff

i (s) lt 0 whenat least M + 13 species are present where M = sumi=S

i=1 [〈si〉]av

is the mean number of species and

13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av

is a threshold fluctuation in the number of species above themean

We justify this interpretation now First M is givenby Sm where m satisfies Eq (A4) To compute 13 wenote that δKeff

i = 〈Keffi 〉 minus κi Given that 〈Keff

i 〉 = μK minusμKm

sumj =i cij we use our approximate expression for κi to

obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij

Thus averaging over cij rsquos yields [δKeffi ]av = 1

2σ 2mμ2

cμK fromwhich we get

13 = 12Sμcσ

2m

When M + 13 species are present we have

Keffi (s) = μK minus μK

sumj =i

cij sj = μK minus μK

j=M+13sumj=1

cij

where it is understood that cii = 0 The quantitysumj=M+13

j=1 cij

is to leading order in S a γ -distributed random variable ofmean 1

S(M + 13)μc and variance 1

S(M + 13)σ 2

c a propertyshared by 1

S(M + 13)

sumj =i cij at leading order Thus the

probability that Keffi (s) lt 0 is equal to the probability that

μK minus μK1S

(M + 13)sum

j =i cij lt 0 Plugging in our expres-sions for M and 13 we see that this is equal to the probabilitythat

κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0

which is exactly PNR obtained above

APPENDIX C PHASE DIAGRAM FOR HETEROGENEOUSCARRYING CAPACITIES

Now set σc = 0 so that cij = μcS for all i isin 1 SWe calculate the probabilities PCR and PNR that given(μcσKμK ) and a random draw of Kirsquos the PA model isin the CR or the NR


To compute PCR we again employ our mean-field varia-tional Ansatz (A2) Using this Ansatz one getslang

exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠

which we can rewrite using (B1) as

κi = Ki minus μc

S

sumj =i

Kj

This is simply Keffi (s) when all species are present and PCR

is the probability that this is positive Since we draw each Ki

independently from a log-normal distribution pK with meanμK and variance σ 2

K we may apply the central limit theoremto 1

S

sumj =i Kj in the limit S rarr infin In particular 1

S

sumj =i Kj

approaches a Gaussian distribution of mean μK and varianceσ 2

KS plus some terms of higher order in 1S Thus as S rarrinfin

1

S

sumj =i

Kj μK

which is to say that

κi Ki minus μcμK

022423-11


when S is very large Therefore PCR is the probability that Ki gt μcμK or

PCR P (Ki gt μcμK ) =1 minusint μcμK

0dKi pK (Ki) = 1 minus

(log(μcμK ) minus lK

zK

)

where is the normal cumulative distribution function and

lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)


We compute the probability PNR that κi lt 0 using the Gaussian variational distribution (A3) To do so one evaluates theappropriate Gaussian integralslang

exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j

For large S we can replace 1S

sumj =i Kj and 1

S

sumj =i K2

j by the expectation value Ki and K2i respectively

1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K

Substituting these expressions yields

κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]

For simplicity of notation define δ = 12S

μc

(1 + σ 2

K

μ2K

)σ 2

m The probability that κi lt 0 is

PNR P (Ki lt μcμK [m + δ]) =int μcμK [m+δ]

0dKi pK (Ki) =

(log(μcμK [m + δ]) minus lK

zK

)

where lK and zK are defined as aboveOnce again PNR has an interpretation as the probability that Keff

i (s) lt 0 when at least M + 13 species are present where M =sumi=Si=1 [〈si〉]av and 13 = 1

μcμK

sumi=Si=1 [δKeff

i ]av Using a calculation analogous to the one presented in the preceding Appendix B 3it is straightforward to show that 13 = Sδ

APPENDIX D COMPUTING PNR DIRECTLY FROM Q

Here we show for the sake of completeness that one can compute κi for the NR using the mean-field Qi(si) = mδsi 1 + (1 minusm)δsi 0 without the Gaussian approximation (A3) to the variational distribution We first restrict ourselves to the case in whichonly the carrying capacities are heterogeneous (σc = 0) One can obtain the same answer as in the preceding Appendix C 2 by

022423-12


taking the large-S limit To see this we expand κi into powers of 1S

One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n

where we employed the series expansions of both ln(1 + x) and exp(x) minus 1 Now neglecting terms of third order or higher in 1S

we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]

Using the identity σ 2m = m(1 minus m) we obtain

κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j

which is exactly what we obtained using (A3)A similar expansion can be employed in the case of heterogeneous cij but the limit is more delicate Higher-order terms cannot

be neglected as S rarr infin because in this limit the momentsum

j =i cnij does not vanish for any positive integer n However these

higher-order moments can be neglected in the limit μc σ 2c (provided that S μ2

cσ2c ) which according to our numerical

simulations is consistent with the NR This follows from the form of the moment-generating function of γ distribution

M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]

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(2013)[13] C K Fisher and P Mehta Proc Natl Acad Sci USA 111

13111 (2014)[14] E Kussell and M Vucelja Rep Prog Phys 77 102602 (2014)[15] D A Kessler and N M Shnerb Phys Rev E 91 042705

(2015)

[16] S Azaele S Suweis J Grilli I Volkov J R Banavar and AMaritan arXiv150601721

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Konkel C L Larson K E Nelson A Qu L B Schook et alProc Natl Acad Sci USA 109 9692 (2012)

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022423-13


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(2007)

022423-14


analytical arguments using techniques from statistical physicsrelated to the study of disordered spin systems For examplethe aforementioned sharp transition between the selection-dominated regime and the drift-dominated regime seen ingeneralized LVMs corresponds to the analog of the ldquofreezingtransitionrdquo in the PA model [13] Despite this previous workthe dynamics of the PA model are not fully understood andwarrant further study

In this work we address the extent to which the PA modelreproduces the qualitative behaviors and ecological regimesfound in more complicated LVMs In particular we use the PAmodel to analyze a local community of competing species withweak immigration from a static regional pool and we compareit to the ecological dynamics seen in numerical simulations ofthe generalized LVMs [15] We numerically simulate the PAmodel and examine ecological patterns of species abundancesto see how ecological processes such as selection drift andimmigration affect species abundance patterns We supplementthese results with analytic arguments We then compare andcontrast the results obtained from the PA model and LVMsFinally we discuss the implications of our results for modelingcomplex ecological communities

Throughout this paper for simplicity we employ someabuses of terminology In particular we refer to plots depictingecological behaviors as a function of ecological parametersas ldquophase diagramsrdquo We also refer to a special point inparameter space that separates three qualitatively distinctecological behaviors as a ldquocritical pointrdquo In the statistical andcondensed-matter physics literature these terms are definedbased on strict technical criteria that are not fulfilled by ourplots Our terminology is therefore not exact Indeed thebehaviors shown here are more akin to crossovers in statisticaland condensed-matter physics [2021]

I THE PRESENCE-ABSENCE (PA) MODEL

A Ecological motivation

The goal of introducing the PA model is to capture theessential ecological features present in LVMs in a simpleanalytically tractable model For the sake of tractability thePA model ignores species abundances and instead focuseson a simpler question is a species present or absent in thecommunity The basic idea behind the definition of the PAmodel is roughly speaking that the propensity of a speciesto be present or absent is determined by a quantity called itsldquoeffective carrying capacityrdquo The effective carrying capacityof a species i which generally depends on both environmentalfactors as well as the abundances of other species sets themaximum possible abundance of a species i in the presenceof the others Therefore species i may persist if its effectivecarrying capacity is positive whereas it will go extinct if itseffective carrying capacity is negative

To gain intuition about the role of effective carryingcapacities in the definition of the PA model it is helpfulto recount the results on species invasion in Lotka-Volterracommunities derived by MacArthur and Levins in their classic1967 paper [5] wherein effective carrying capacities playeda crucial role Species abundances x are modeled using asystem of ODEs of the form dxidt = λi + xifi(x) with λi

Carrying capacity in IsolationEffective carrying capacity

increases for mutualistic reactions Effective carrying capacity

decreases for competitive reactions

(a)

(b)Regional Species Pool Immigration Extinction

FIG 1 The presence-absence model (a) Species i has a carryingcapacity Ki in the absence of other species Species irsquos interactionwith species j is characterized by the interaction coefficient cij Depending on whether they are mutualistic or competitive theinteractions can increase or decrease the effective carrying capacityof species i (b) Species i can be present in the communitysi = 1 or absent from the community si = 0 Species can immigratefrom a regional community pool and go extinct in the communityproportional with a rate equal to the exponential of the ratio ofeffective carrying capacity to the stochasticity parameter ω

the rate of immigration and fi(x) the ecological fitness ofspecies i which is a function of the species abundances x Ingeneral fi(x) may be a complicated function due to nonlinearfunctional responses or other phenomena Regardless of theexact form of fi(x) the ecological fitness can always belinearized near an equilibrium point x lowast where the dynamicsare approximately described by LVM equations

In the LVM the ecological fitness fi(x) = Ki minus xi minussumj cij xj is a linear function of the carrying capacity Ki and

interaction coefficients cij which measure how the presenceof species j affects the growth rate of species i The interactioncoefficients cij lt 0 are negative when interactions withspecies j benefit the growth of species i cij gt 0 when speciesj competes with species i and cij = 0 if species i and j do notinteract (see Fig 1) We interpret fi(x) + xi = Ki minussum

j cij xj

as an effective carrying capacity Keffi (x) for species i and we

write dxidt = λi + xi[Keffi (x) minus xi] In general the effective

carrying capacity is a function of the abundances of all thespecies in the community

MacArthur and Levins [5] used the idea of an effectivecarrying capacity to ask whether a new species i couldinvade a community with species abundances x lowast Usinggraphical stability arguments they showed that species i caninvade successfully if its effective carrying capacity is positive[Keff

i (x lowast) = Ki minussumj cij x

lowastj gt 0] but it will be unsuccessful

if its effective carrying capacity is negative [Keffi (x lowast) =

Ki minussumj cij x

lowastj lt 0] Therefore the mean extinction time

of a species in the local community depends strongly onthe effective carrying capacity the time to extinction islong for Keff

i (x lowast) gt 0 and short for Keffi (x lowast) lt 0 Based on

these observations we hypothesize that the extinction rateof a species depends exponentially on the effective carryingcapacity This assumption is used in the definition of the PAmodel

022423-2





RIi (s) = λi


given by

REi (s) = exp

(minus 1

ωKeff

i (s)

)


j=Ssumj=1

Kjcij sj





dPt (s)

dt=

i=Ssumi=1

[RE


](1 minus si)

+ [RI


]si

(1)






c S

pc(cij ) = 1

θkcc (kc)

ckcminus1ij exp

(minuscij

θc

)


kc = μ2c

Sσ 2c

θc = σ 2c

μc



K

pK (Ki) = 1

KizK

radic2π

exp

(minus 1

2z2K

[ln(Ki) minus lK ]2

)

where

lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)





022423-3












022423-4









M =i=Ssumi=1

[〈si〉]av (2)


α = 4

S

i=Ssumi=1


av

)


022423-5






where

gSγ (x) =

γminusx

Sminusγif x gt γ

γminusx

γif x lt γ




















Keffi (s) = μK

⎛⎝1 minus 1

S

sumj =i

si

⎞⎠ = 1

S 0





022423-6







langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc












i 〉 gt δKeffi












13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av













022423-7






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14





RIi (s) = λi


given by

REi (s) = exp

(minus 1

ωKeff

i (s)

)


j=Ssumj=1

Kjcij sj





dPt (s)

dt=

i=Ssumi=1

[RE


](1 minus si)

+ [RI


]si

(1)






c S

pc(cij ) = 1

θkcc (kc)

ckcminus1ij exp

(minuscij

θc

)


kc = μ2c

Sσ 2c

θc = σ 2c

μc



K

pK (Ki) = 1

KizK

radic2π

exp

(minus 1

2z2K

[ln(Ki) minus lK ]2

)

where

lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)





022423-3












022423-4









M =i=Ssumi=1

[〈si〉]av (2)


α = 4

S

i=Ssumi=1


av

)


022423-5






where

gSγ (x) =

γminusx

Sminusγif x gt γ

γminusx

γif x lt γ




















Keffi (s) = μK

⎛⎝1 minus 1

S

sumj =i

si

⎞⎠ = 1

S 0





022423-6







langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc












i 〉 gt δKeffi












13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av













022423-7






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14












022423-4









M =i=Ssumi=1

[〈si〉]av (2)


α = 4

S

i=Ssumi=1


av

)


022423-5






where

gSγ (x) =

γminusx

Sminusγif x gt γ

γminusx

γif x lt γ




















Keffi (s) = μK

⎛⎝1 minus 1

S

sumj =i

si

⎞⎠ = 1

S 0





022423-6







langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc












i 〉 gt δKeffi












13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av













022423-7






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14









M =i=Ssumi=1

[〈si〉]av (2)


α = 4

S

i=Ssumi=1


av

)


022423-5






where

gSγ (x) =

γminusx

Sminusγif x gt γ

γminusx

γif x lt γ




















Keffi (s) = μK

⎛⎝1 minus 1

S

sumj =i

si

⎞⎠ = 1

S 0





022423-6







langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc












i 〉 gt δKeffi












13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av













022423-7






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14






where

gSγ (x) =

γminusx

Sminusγif x gt γ

γminusx

γif x lt γ




















Keffi (s) = μK

⎛⎝1 minus 1

S

sumj =i

si

⎞⎠ = 1

S 0





022423-6







langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc












i 〉 gt δKeffi












13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av













022423-7






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14







langexp

(minus 1

μK

Keffi (s)

) rang

= langKeff

i (s)rangminus μK

2Vart

[1

μK

Keffi (s)

]+ hotc












i 〉 gt δKeffi












13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av













022423-7






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14






IV DISCUSSION




022423-8






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14






ACKNOWLEDGMENTS




Q(s) =i=Sprodi=1

Qi(si)







m = m(1 minus m)

QNRi (si) = 1radic

2πσ 2m

exp

(minus 1

2σ 2m

[si minus m]2

) (A3)





PPA(s) = 1

Zexp

(minusμK

ωU (s)

)


si + μc

2S

sumi =j

sisj

022423-9




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14




ZTrs A(s) exp

(minusμK

ωU (s)

)



F [Q] = 〈U (s)〉Q + ω

μK

〈ln Q(s)〉Q


F (m)


2

2

+ ω

μK




m = 1

2+ 1

2tanh

(μK

2ω[1 + minus μcm]

) (A4)






κi = minusμK ln

langexp

(minus 1

μK

Keffi (s)

) rang (B1)


i (si) = δsi 1

We obtain

langexp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus1 +

sumj =i

cij sj

⎞⎠˛

QCR

= eminus1prodj =i

〈ecij sj 〉QCRi

= eminus1prodj =i

ecij

= exp

⎛⎝minus1 +

sumj =i

cij

⎞⎠


1

μK


cij







py(yi) = 1

θky

y (ky)y

kyminus1i exp

(minus yi

θy

)


ky = μ2c

σ 2c

θy = σ 2c

μc



0dyi py(yi)= 1

(μ2

c

σ 2

c

)γ (μ2c

σ 2c

μc

σ 2c

)




exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminus1prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj

times exp

(minus 1

2σ 2m


)

022423-10



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14



(minus 1

μK

Keffi

) rangQNR

= exp

⎛⎝minus1 + m

sumj =i

cij + 1

2σ 2

m

sumj =i

c2ij

⎞⎠


1

μK


cij minus 1

2σ 2

m

sumj =i

c2ij (B2)


c2ij μc

sumj =i

cij (B3)


1

μK

κi 1 minus(

m + 1

2μcσ

2m

)sumj =i

cij


2μcσ2m



(m + 1

2μcσ2m

)μc and variance

(m + 1

2μcσ2m

)σ 2

c It follows that


(

μ2c

σ 2c

[m + 1

2μcσ 2m

])

times γ

(μ2

c

σ 2c

[m + 1

2μcσ

2m

]μc

σ 2c

)




i=1 [〈si〉]av


13 = 1

μcμK

i=Ssumi=1

[δKeff

i

]av






obtain

δKeffi = lang

Keffi

rangminus κi = 1

2σ 2

mμcμK

sumj =i

cij


2σ 2mμ2


13 = 12Sμcσ

2m



sumj =i


j=M+13sumj=1

cij


j=1 cij



S(M + 13)σ 2


S(M + 13)



μK minus μK1S

(M + 13)sum


κi μK minus μK

(m + 1

2μcσ

2m

)sumj =i

cij lt 0






exp

(minus 1

μK

Keffi

) rangQCR

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

QCR

= exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj

⎞⎠


κi = Ki minus μc

S

sumj =i

Kj





S


S

sumj =i Kj



1

S

sumj =i

Kj μK


κi Ki minus μcμK

022423-11






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14






zK

)


lK = ln

⎛⎝ μ2

Kradicμ2

K + σ 2K

⎞⎠ zK =

radicln

(1 + σ 2

K

μ2K

)



exp

(minus 1

μK

Keffi

) rangQNR

=intRS

ds QNR(s) exp

(minus 1

μK

Keffi

)

= eminusKiμK

prodj =i

1radic2πσ 2

m

int +infin

minusinfindsj exp

(minus 1

2σ 2m

[sj minus m]2 + μc

SμK

Kjsj

)

= exp

⎛⎝minus 1

μK

Ki + mμc

SμK

sumj =i

Kj + 1

2σ 2

m

μ2c

S2μ2K

sumj =i

K2j

⎞⎠

Therefore

κi = Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j


sumj =i Kj and 1

S

sumj =i K2


1

S

sumj =i

Kj μK1

S

sumj =i

K2j μ2

K + σ 2K


κi Ki minus μcμK

[m + 1

2Sμc

(1 + σ 2

K

μ2K

)σ 2

m

]


μc

(1 + σ 2

K

μ2K

)σ 2



0dKi pK (Ki) =


zK

)



μcμK

sumi=Si=1 [δKeff




022423-12



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14



One has

langexp

(minus 1

μK

Keffi

) rangQ

=˝

exp

⎛⎝minus 1

μK

Ki + μc

SμK

sumj =i

Kj sj

⎞⎠˛

Q

= eminusKiμK

prodj =i

langexp

(μc

SμK

Kjsj

) rangQ

= eminusKiμK

prodj =i

[m exp

(μc

SμK

Kj

)+ (1 minus m)

]= exp

⎛⎝minus Ki

μK

+sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]⎞⎠

yielding

κi = Ki minus μK

sumj =i

ln

[1 + m exp

(μc

SμK

Kj

)minus m

]= Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[ infinsuml=1

1

l

(μcKj

SμK

)l]n


we get

κi Ki minus μK

sumj =i

infinsumn=1

1

n(minus1)n+1mn

[μcKj

S+ 1

2

(μcKj

SμK

)2 ]n

Ki minus μK

sumj =i

[m

μcKj

SμK

+ m(1 minus m)1

2

(μcKj

SμK

)2 ]


κi Ki minus mμc

1

S

sumj =i

Kj minus 1

2σ 2

m

μ2c

SμK

1

S

sumj =i

K2j







M(x) = exp

[minus 1

S

μ2c

σ 2c

ln

(1 minus σ 2

c

μc

x

)]










(2015)












022423-13









(2007)

022423-14









(2007)

022423-14

analytically tractable model for community ecology with...

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