The Pennsylvania State University

The Graduate School

Department of Physics

STATISTICAL PHYSICS AND ECOLOGY

A Thesis in

Physics

by

Igor Volkov

c© 2005 Igor Volkov

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

August 2005

ii

The thesis of Igor Volkov has been reviewed and approved∗ by the following:

Jayanth Banavar

Distinguished Professor and Head of Physics

Thesis Adviser

Chair of Committee

Bryan Grenfell

Alumni Professor of the Biological Sciences

Julian Maynard

Distinguished Professor of Physics

Peter Schiffer

Professor of Physics

∗Signatures are on file in the Graduate School.

iii

Abstract

This work addresses the applications of the methods of statistical physics to prob-

lems in population ecology. A theoretical framework based on stochastic Markov pro-

cesses for the unified neutral theory of biodiversity is presented and an analytical solution

for the distribution of the relative species abundance distribution both in the large meta-

community and in the small local community is obtained. It is shown that the framework

of the current neutral theory in ecology can be easily generalized to incorporate sym-

metric density dependence. An analytically tractable model is studied that provides

an accurate description of β-diversity and exhibits novel scaling behavior that leads to

links between ecological measures such as relative species abundance and the species

area relationship. We develop a simple framework that incorporates the Janzen-Connell,

dispersal and immigration effects and leads to a description of the distribution of relative

species abundance, the equilibrium species richness, β-diversity and the species area re-

lationship, in good accord with data. Also it is shown that an ecosystem can be mapped

into an unconventional statistical ensemble and is quite generally tuned in the vicinity

of a phase transition where bio-diversity and the use of resources are optimized. We also

perform a detailed study of the unconventional statistical ensemble, in which, unlike in

physics, the total number of particles and the energy are not fixed but bounded. We show

that the temperature and the chemical potential play a dual role: they determine the

average energy and the population of the levels in the system and at the same time they

act as an imbalance between the energy and population ceilings and the corresponding

iv

average values. Different types of statistics (Boltzmann, Bose-Einstein, Fermi-Dirac and

one corresponding to the description of a simple ecosystem) are considered. In all cases,

we show that the systems may undergo a first or a second order phase transition akin to

Bose-Einstein condensation for a non-interacting gas. We discuss numerical schemes for

studying the new ensemble. The results of simulations are presented and are in excellent

agreement with theory.

v

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Neutral Theory and Relative Species Abundance . . . . . . . . . . . 7

2.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Splitting of a species and peripheral isolate speciation . . . . . . . . 17

2.3 Neutrality and stability of forest biodiversity – comment on paleodata

analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 3. Density dependence as an explanation of tree species abundance and

diversity in tropical forests . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Is our approach a neutral theory? . . . . . . . . . . . . . . . . . . . . 37

3.3 Relationship between the zero sum rule and our approach . . . . . . 40

3.4 Utility of relative species abundance data for elucidating biological

mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 4. Spatial scaling relationships in ecology . . . . . . . . . . . . . . . . . 48

vi

Chapter 5. Spatial patterns of ecological communities: α-β diversity and species-

area relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 6. Organization of ecosystems in the vicinity of a novel phase transition 71

6.1 Sketch of the derivation . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 78

Appendix. A novel ensemble in statistical physics . . . . . . . . . . . . . . . . . 83

A.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2 Theoretical and numerical results for systems with different statistics 94

A.2.1 Boltzmann Statistics (Figures A.1, A.2 and A.3) . . . . . . . 94

A.2.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 94

A.2.1.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . 95

A.2.2 Fermi-Dirac Statistics (Figures A.4, A.5, A.6 and A.7) . . . . 96

A.2.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.2.2.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . 97

A.2.3 Bose-Einstein Statistics (Figures A.8 - A.14) . . . . . . . . . 97

A.2.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 97

A.2.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . 98

A.2.4 Ecological case (Figures A.15 and A.16) . . . . . . . . . . . . 98

A.2.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 98

A.2.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . 99

A.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

vii

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

viii

List of Tables

3.1 Maximum likelihood estimates of the density-dependant symmetric model

and dispersal limitation model[83] parameters (upper table) and com-

parison between the models (lower table) for the six data sets of tropical

forests. In the six plots coordinated by the Center for Tropical Forest

Science of the Smithsonian (http://www.ctfs.si.edu), we considered trees

with diameter at breast height ≥ 10 cm. S is the number of species, J is

the total abundance and θ1 and θ2 are the biodiversity parameters in the

dispersal limitation model[83] and equation (3.1) respectively (note that

θ2 is a function of c, x and S and both models have the same number of

fitting parameters). The comparison of the models was carried out with

the likelihood ratio test[3, 30, 40]. The lower table presents deviance

(twice the difference in the log-likelihoods L1 and L2) between the two

models and the corresponding P -value of the χ2-distribution with one de-

gree of freedom. The main result is that the dispersal limitation model

and the simple symmetric density dependent model presented here are

statistically comparable to each other in their ability to fit the tropical

forest data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Scaling exponents for d = 1, 2, 3 determined from the scaling collapse of

the RSA and SAR plots (Figs. 4.1 and 4.2). . . . . . . . . . . . . . . . 50

ix

List of Figures

2.1 Data on tree species abundances in 50 hectare plot of tropical forest

in Barro Colorado Island, Panama taken from Condit et al.[21]. The

total number of trees in the dataset is 21457 and the number of distinct

species is 225. The red bars are observed numbers of species binned into

log(2) abundance categories, following Preston’s method[75]. The first

histogram bar represents〈φ1〉

2 , the second bar〈φ1〉

2 +〈φ2〉

2 , the third bar

〈φ2〉2 + 〈φ3〉+

〈φ4〉2 , the fourth bar

〈φ4〉2 + 〈φ5〉+ 〈φ6〉+ 〈φ7〉+

〈φ8〉2 and

so on. The black curve shows the best fit to a lognormal distribution

〈φn〉 = Nn exp(− (log2 n−log2 n0)2

2σ2) (N = 46.29, n0 = 20.82 and σ =

2.98), while the green curve is the best fit to our analytic expression

Eq.(2.14) (m = 0.1 from which one obtains θ = 47.226 compared to the

Hubbell[45] estimates of 0.1 and 50 respectively and McGill’s best fits[66]

of 0.079 and 48.5 respectively.) . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The red bars represent the numbers of species derived from Eq.(2.26)

binned into log(2) abundance categories, following Hubbell’s method[45].

The first histogram bar represents 〈φ1〉, the second bar 〈φ2〉 + 〈φ3〉, the

third bar 〈φ4〉 + 〈φ5〉 + 〈φ6〉 + 〈φ7〉 and so on. Here θ = 40 and p = 40. 22

x

2.3 Simulations of neutral dynamics showing the divergence among sites. a,

Lottery model, where ten species with identical parameters and responses

to stochasticity compete for space. b, Results for a single species shown

for eight different sites. Abundances diverge with the random accumula-

tion of changes at each site. c, Variance among sites increases over time

owing to accumulation of the random changes in abundance shown in

b, as does (d) the coefficient of variation, CV. In c and d, the middle

line indicates the median, and the dashed lines bound 90% of simulated

values. This figure and the caption are taken from Ref. [18] . . . . . . . 25

3.1 Fits of density-dependant symmetric model (red line) and dispersal lim-

itation model[83] (blue circles) to the tree species abundance data from

the BCI, Yasuni, Pasoh, Lambir, Korup and Sinharaja plots, for trees

≥ 10 cm in stem diameter at breast height (see Table 3.1). The frequency

distributions are plotted using Preston’s binning method as described in

Ref. [83]. The numbers on the x-axis represent Preston’s octave classes. 35

3.2 Plot of r(n) derived from Eq.(3.1) versus n for the six data sets of tropical

trees. For large values of n, rn asymptotes at a value slightly less than

1. The BCI data (cyan circles) at small n is almost invisible since it

coincides with the Pasoh dataset (red circles). . . . . . . . . . . . . . . . 36

xi

3.3 Upper panel: Plot of φn versus n for the model in Ref.[83] with the

biodiversity parameter θ = 50, the immigration parameter m = 0.1 and

population of 20, 000. Middle panel: Binned tree species abundance data

from the upper panel. The frequency distributions are plotted using

Preston’s binning method (right panel) and the method from Hubbell’s

book (left panel). Bottom panel: Graph of rn vs. n deduced from the

RSA data using the exact formula described above: rn = n+1n

〈φn+1〉〈φn〉

. . 47

4.1 Left column: plots of the normalized RSA for d = 1, 2, 3 with ν =

0.001, 0.003, 0.01, 0.03, 0.1 (d = 2 plot also shows the results for ν =

0.0001, 0.0003, 0.3). Right column: plots of the data collapse yielding a

measure of the exponents a and b in Table 4.1. . . . . . . . . . . . . . . 52

4.2 Left column: plots of the SAR for d = 1, 2, 3 with ν = 0.001, 0.003, 0.01, 0.03, 0.1.

Right column: plots of the data collapse yielding a measure of the expo-

nent z in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

xii

5.1 Relative Species Abundance plots for a metacommunity with JM =

11400 individuals. Here 〈φn〉 is the number of species with population n,

S denotes the total number of species, ν is the speciation rate and R is the

Janzen-Connell length scale. The two mean field cases are well-described

by the Fisher log-series (thin blue line fits). The three cases with ν = 0.1

(dashed lines) lead to overlapping plots. The Janzen-Connell effects are

not important in these cases because of the few individuals per species.

For the three cases with ν = 0.005, there is a pronounced internal mode

with the behavior at intermediate values of R being distinct from the

R = 0 and ∞ cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Beta Diversity data along with the best fits using Equation (1) for plots in

(a) Panama (R = 46m, γ−10

= 68m, γ−11

= 210km and c0 = 120m) and

in (b) Ecuador-Peru (Yasuni) (R = 86m, γ−10

= 69m, γ−11

= 23, 500km

and c0 = 19m). The Janzen-Connell effect pushes conspecific individuals

further away from each other and thus the probability function F declines

more steeply within the zone of its operation than at larger distances as

in the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xiii

5.3 Scaling collapse of the species-area relationship (SAR) plots. We present

the species-area relationships for metacommunities of two sizes, three

values each of the Janzen-Connell effective distance, R, and 6 values of

the speciation rate ν (0.1, 0.05, 0.01, 0.005 and 0.001). Note that, when

one uses the scaling variables S/Az and νA, the curves for the different

speciation rates become superimposed. Red: no Janzen-Connell effect

(R = 0). Green: local Janzen-Connell effect (R = 10). Blue: infinite

range (R = ∞) effect. The black points correspond to the bigger size

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Equilibrium snapshots of two metacommunities with no Janzen-Connell

effect (R = 0, ν = 0.001, S = 2670, left panel) and Janzen-Connell

effect (R = 10, ν = 0.0001, S = 2206, right panel). Even though there

is patchiness in both cases, the spatial distribution of species is quite

distinct depending on whether the Janzen-Connell effects are operational

or not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xiv

6.1 Comparison of the results of computer simulations of an ecosystem with

theory. We consider a system with 100, 000 energy levels with εk = k2/3,

k = 1..100, 000, corresponding to d = 1/2. We work with a constant

Emax (the figure shows the results for several values of Emax) and con-

sider a dynamical process of birth and death. We have verified that the

equilibrium distribution is independent of the initial condition. At any

given time step, we make a list of all the individuals and the empty

energy levels. One of the entries from the list is randomly picked for

possible action with a probability proportional to the total number of

entries in the list. Were an individual to be picked, it is killed with 50%

probability or reproduced (an additional individual of the same species

is created) with 50% probability provided the total energy of the system

does not exceed Emax. When an empty energy level is picked, specia-

tion occurs with 50% probability and a new individual of that species is

created provided again the energy of the system does not exceed Emax.

With 50% probability, no action is taken. This procedure is iterated un-

til equilibrium is reached. The effective temperature of the ecosystem is

defined as the imbalance between Emax and the average energy of the

system (Eq. (6.1)). The figure shows a plot of the effective temperature

of the ecosystem deduced from the simulations. The circles denote the

data averaged over a run of 109 time steps with the last 500 million used

to compute the average temperature while the solid line is the theoretical

prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xv

6.2 Phase transition in an ecosystem with Nmax = NM = 65 and d = 1/2.

The dashed and solid curves are plots of theoretical predictions of〈N〉NM

and∂〈E〉∂T

TminEM

respectively versus scaled temperature T/Tmin, where

Tmin = 11.6 and EM = 1000. The data points denote the results of

simulations. ∂〈E〉/∂T is a quantity analogous to the specific heat of a

physical system and has the familiar λ shape associated with the super-

fluid transition in liquid helium[34]. It was obtained in the simulations

as the derivative of the interpolated values of 〈E〉. The continuous phase

transition is signaled by the peak in ∂〈E〉/∂T (and the corresponding

drop in 〈N〉) on lowering the temperature and occurs in the vicinity of

the temperature Tmin (the transition temperature moves closer to Tmin

as the system size increases). . . . . . . . . . . . . . . . . . . . . . . . . 82

A.1 The results of the simulations of the novel ensemble with Boltzmann

statistics. r = 100, Nmax = ∞, εk = k2/3, k = 1..1000. The solid line

denotes the theoretical prediction. . . . . . . . . . . . . . . . . . . . . . 102

A.2 The results of the simulations of the novel ensemble with Boltzmann

statistics. r = 100. εk = k2/3, k = 1..1000, Nmax = 35, Tmin ≈ 0.62.

Here Cv = ∂〈E〉/∂T is the specific heat of a system. The peak in the

specific heat occurs at the phase transition. The solid line denotes the

theoretical prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A.3 Boltzmann Statistics. r = 1, d = 1, Nmax = 106. . . . . . . . . . . . . . 104

xvi

A.4 The results of the simulations of the novel ensemble with Fermi-Dirac

statistics. r = 100, Nmax = ∞, εk = k3/2, k = 1..1000. The solid line

denotes the theoretical prediction. . . . . . . . . . . . . . . . . . . . . . 104

A.5 The results of the simulations of the novel ensemble with Fermi-Dirac

statistics. r = 100, εk = k3/2, k = 1..1000, Nmax = 65, Tmin ≈ 115.

The peak in the specific heat occurs at the phase transition. The solid

line denotes the theoretical prediction. . . . . . . . . . . . . . . . . . . . 105

A.6 Plot of 〈nk〉 versus εk for the system with Fermi-Dirac statistics. r = 100,

Nmax = ∞, Tmin = 3.1, εF = 14.2, εk = k2/3. The solid line denotes

the theoretical prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A.7 Fermi Statistics. r = 1, d = 1, Nmax = 106. . . . . . . . . . . . . . . . . 106

A.8 The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 1, Nmax = ∞, εk = k0.5, k = 1..100, 000 (left), εk = k3/2,

k = 1..1000(right). The solid line denotes the theoretical prediction. . . 107

A.9 The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 1. Left panel: εk = k0.5, k = 1..100, 000, Nmax = 70,

Tmin ≈ 4.87. Right panel: εk = k3/2, k = 1..1000, Nmax = 275,

Tmin ≈ 115. The peak in the specific heat occurs at the phase transition.

Note the absence of a phase transition when d is negative. The solid line

denotes the theoretical prediction. . . . . . . . . . . . . . . . . . . . . . 108

xvii

A.10 The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.33/0.67. Nmax = ∞. Left panel: εk = k3/2, k =

1..1000. Right panel : εk = k2/3, k = 1..10000. The solid line denotes

the theoretical prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.11 The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.33/0.67. Left panel: Nmax = 45, εk = k3/2, k =

1..1000, Tmin ≈ 545. Right panel: Nmax = 16, εk = k2/3, k = 1..10000,

Tmin ≈ 7.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.12 The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.67/0.33, Nmax = ∞, εk = k2/3, k = 1..1000. Solid

lines and circles represent theory and simulations, respectively. Note

the deviation from the theory as the system approaches the “infinite”

temperature ε1/ ln(r). The solid line denotes the theoretical prediction. 111

A.13 The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.67/0.33, Nmax = 50, εk = k2/3, k = 1..1000 . . . . . . 111

A.14 Bose Statistics. r = 1, d = 1, Nmax = 106. . . . . . . . . . . . . . . . . 112

A.15 The results of the simulations of the model of an ecosystem with r = 1,

Nmax = ∞. Left panel: εk = k3/2, k = 1..1000. Right panel : εk =

k2/3, k = 1..1000. The solid line denotes the theoretical prediction. . . . 112

xviii

A.16 The results of the simulations of the model of an ecosystem with r = 1.

Left panel: Nmax = 200, εk = k3/2, k = 1..1000, Tmin ≈ 400. Right

panel: Nmax = 10, εk = k2/3, k = 1..1000, Tmin ≈ 3.6. The peak in

the specific heat occurs at the phase transition. Note that the similarity

between this figure and Fig. A.9 for Bose-Einstein statistics. . . . . . . . 113

xix

Acknowledgments

I am deeply indebted to my adviser, Prof. Jayanth Banavar, for his continuous

support, kind advice and patient guidance through my graduate career. I also would like

to thank Prof. Amos Maritan for his invaluable help and collaboration. I am grateful to

Marek Cieplak, Steve Hubbell and Tommaso Zillio for all their input and collaboration.

I also wish to thank the members of my thesis committee for their encouraging and

valuable comments on my dissertation. I would like to express my sincere gratitude to

Prof. Serguei Zavtrak for his tremendous help during my studies at the Belarus State

University. Finally, this work would be impossible without the support and help of Oleg,

Tanya and significant others.

1

Chapter 1

Introduction

An ecological community represents a formidable many-body problem – one has

an interacting many body system with imperfectly known interactions and a wide range

of spatial and temporal scales. In tropical forests across the globe, ecologists recently

have been able to measure certain quantities such as the distribution of relative species

abundance (RSA), the species area relationship (SAR), and the probability that two

individuals drawn randomly from forests a specified distance apart belong to the same

species (also called β-diversity). In order to make theoretical progress toward under-

standing the relationships among these measures, it is useful to compare the data with

the predictions of tractable models which begin to capture some of the features that are

known to be important in tropical forest tree communities. The performance of the the-

ory confronted by the data should help guide the development of more realistic models

of communities and the processes that assemble them.

A traditional way of describing an ecological community is to assume that the

species differ one from each other in their response to the available resources and in-

teractions with other species. Such differences (known as niches) lead to the situation

where each species is specialized in utilizing it’s partition of the resources and has a

unique role in the functioning of the ecosystem. The total number of species in a par-

ticular ecosystem which defines the biodiversity is thus determined by the variety of the

2

available resources and the ways the species can utilize them. The mathematical models

that seek to capture the niche structure of the ecosystems are, therefore, very complex

and need to rely on a lot of empirical assumptions.

Recently, it has been suggested that niche approach may not be adequate for the

description of the particular ecological communities. For example, there are hundreds

of tree species in the tropical forests that coexist in the environment with a very few

available resources (sun energy, water, soil). It is argued that such a systems cannot

have enough niches to accommodate such a large number of species.

An alternative approach to the description of the community structure, which

is called neutral theory, has been recently proposed by Hubbell[45]. In this model the

population dynamics of the ecosystem is purely stochastic and there are no differences

between species, so that each individual in an ecosystem, regardless of the species it

belongs to, has the same chances of giving birth or dying. Hence, there are no direct

interactions between the species – all the interactions arise from the limited size of the

community. In the neutral model, the species diversity is maintained by the processes of

speciation, when a completely new species emerges.

Unlike niche theory, neutral models use a very small number of inputs, such as the

birth and death rates, and allow for an elegant mathematical formulation of the problem

in the form of a stochastic master equation for a one-step Markov process[82]. Since all

the species are equivalent, one can consider a mean field approach, where a single species

is treated against the backdrop of the others. While neutral theory is controversial in

ecology, such approaches have been employed in physics for a long time. Indeed, one

can draw analogies between the “neutral” treatment of individuals in the ecosystem and

3

the concept of the ideal gas in physics, where the interactions between the particles are

neglected. Similar to the ideal gas, neutral theory can serve as a “null” model for studies

of ecosystems. Starting with the neutral framework one can begin to incorporate the

differences and interactions among species thus finally approaching niche theory. In this

thesis we demonstrate how the neutral model, with very simple modifications, can be

used successfully in the analysis of community patterns.

We begin with the study of the theory of island biogeography[63] which asserts

that an island or a local community approaches an equilibrium species richness as a

result of the interplay between the immigration of species from the much larger meta-

community source area and local extinction of species on the island (local community).

Hubbell[45] generalized this neutral theory to explore the expected steady-state distri-

bution of relative species abundance (RSA) in the local community under restricted

immigration. In Chapter 2 we present a theoretical framework for the unified neutral

theory of biodiversity[45] and an analytical solution for the distribution of the RSA both

in the metacommunity (Fisher’s logseries) and in the local community, where there are

fewer rare species. Rare species are more extinction-prone, and once they go locally

extinct, they take longer to re-immigrate than do common species. Contrary to recent

assertions[66], we show that the analytical solution provides a better fit, with fewer free

parameters, to the RSA distribution of tree species on Barro Colorado Island (BCI)[21]

than the lognormal distribution[75, 65].

Explaining recurrent patterns in the commonness and rarity of species in ecolog-

ical communities has been a central goal of community ecology for more than half a

century[35, 75]. In Chapter 3 we show that the framework of the current neutral theory

4

in ecology[83, 45, 7, 67, 8, 81, 42, 3] can be easily generalized to incorporate symmetric

density dependence[51, 22, 17, 16]. One can calculate precisely the strength of the rare

species advantage that is needed for an explanation of a given RSA distribution. In

Chapter 2 we demonstrated that a mechanism of dispersal limitation also fits RSA data

well[83, 45]. Here we compare fits of the dispersal and density dependence mechanisms

for the empirical RSA data on tree species in six New and Old World tropical forests and

demonstrate that both mechanisms offer sufficient and independent explanations. We

suggest that RSA data by themselves cannot be used to discriminate among these expla-

nations of RSA patterns[31] – empirical studies will be required to determine whether

RSA patterns are due to one or the other mechanism, or to some combination of both.

Another major scientific challenge is to explain the very high levels of tree di-

versity in many tropical forests. One aspect of this challenge is to understand the

evolutionary origin and maintenance of this diversity on large spatial and temporal

scales[69]. Another is to understand how such extraordinarily high tree diversity can

be maintained on very local scales in some tropical forests. For example, there are over

a thousand tree species in a 52 ha plot in Borneo (Lambir, Sarawak). Numerous mech-

anisms have been proposed to explain tropical tree species coexistence on local scales;

many of these hypotheses invoke density- and frequency dependent mechanisms. Two

of the most prominent of these hypotheses are the Janzen-Connell hypothesis[51, 22]

and the Chesson-Warner hypothesis[17]. The Janzen-Connell hypothesis is that seeds

that disperse farther away from the maternal parent are more likely to escape mortality

from host-specific predators or pathogens. This spatially structured mortality disfa-

vors the population growth of locally abundant species relative to uncommon species

5

by reducing the probability of species’ self-replacement in the same location in the

next generation. The Chesson-Warner hypothesis is that a rare-species reproductive

advantage arises when species have similar per capita rates of mortality but reproduce

asynchronously, and there are overlapping generations. Processes that hold the abun-

dance of a common species in check inevitably lead to rare species advantage because

the space or resources freed up by density-dependent deaths are then exploited by less

common species. Therefore, among-species frequency dependence is the community level

consequence of within-species density dependence, and thus they are two different mani-

festations of the same phenomenon. There is accumulating empirical evidence that such

density- and frequency dependent processes may play a large role in maintaining the

diversity of tropical tree communities[5, 47, 37, 20, 38, 89].

In Chapter 4 we present an analytically tractable model that provides an accurate

description of β-diversity and exhibits novel scaling behavior that leads to links between

ecological measures such as relative species abundance and the species area relationship.

A simple framework presented in Chapter 5, incorporates the Janzen-Connell,

dispersal and immigration effects and leads to a description of the distribution of relative

species abundance, the equilibrium species richness, β diversity and the species area

relationship, in good accord with data.

An ecological community consists of individuals of different species occupying

a confined territory and sharing its resources[63, 65, 60, 88]. One may draw parallels

between such a community and a physical system consisting of particles. One of the

most marvelous phenomena in physics is Bose-Einstein condensation[10, 27, 28] (BEC)

in which a system of a conserved number of indistinguishable particles, on cooling to very

6

low temperatures, abruptly occupies the lowest possible energy state. BEC is related to

superfluidity[53, 71] and superconductivity[52, 6] and has been observed in dilute gases

of alkali atoms[4, 23]. In Chapter 6 we show that an ecosystem can be mapped into

an unconventional statistical ensemble and is quite generally tuned in the vicinity of a

phase transition where bio-diversity and the use of resources are optimized. Strikingly

this transition is analogous to BEC but in a classical context.

In the Appendix we present a detailed derivation of the unconventional statistical

ensemble, in which, unlike in physics, the total number of particles and the energy

are not fixed but bounded. We show that the temperature and the chemical potential

play a dual role: they determine the average energy and the population of the levels

in the system and at the same time they act as an imbalance between the energy and

population ceilings and the corresponding average values. Different types of statistics

(Boltzmann, Bose-Einstein, Fermi-Dirac and one corresponding to the description of a

simple ecosystem) are considered. In all cases, we show that the systems may undergo

a first or a second order phase transition akin to Bose-Einstein condensation for a non-

interacting gas. We discuss numerical schemes for studying the new ensemble. The

results of simulations are presented and are in excellent agreement with theory.

7

Chapter 2

Neutral Theory and Relative Species Abundance

2.1 General theory

The neutral theory in ecology[45, 8] seeks to capture the influence of speciation,

extinction, dispersal, and ecological drift on the RSA under the assumption that all

species are demographically alike on a per capita basis. This assumption, while only

an approximation[24, 80, 86], appears to provide a useful description of an ecological

community on some spatial and temporal scales[45, 8]. More significantly, it allows

the development of a tractable null theory for testing hypotheses about community

assembly rules. However, until now, there has been no analytical derivation of the

expected equilibrium distribution of RSA in the local community, and fits to the theory

have required simulations[45] with associated problems of convergence times, unspecified

stopping rules, and precision[66].

The dynamics of the population of a given species is governed by generalized birth

and death events (including speciation, immigration and emigration). Let bn,k and dn,k

represent the probabilities of birth and death, respectively, in the k-th species with n

individuals with b−1,k = d0,k = 0. Let pn,k(t) denote the probability that the k-th

species contains n individuals at time t. In the simplest scenario, the time evolution of

8

pn,k(t) is regulated by the master equation[11, 13, 33]:

dpn,k(t)

dt= pn+1,k(t)dn+1,k + pn−1,k(t)bn−1,k − pn,k(t)(bn,k + dn,k) (2.1)

which leads to the steady-state or equilibrium solution, denoted by P :

Pn,k = P0,k

n−1∏

i=0

bi,k

di+1,k, (2.2)

for n > 0 and where P0,k can be deduced from the normalization condition∑

n Pn,k = 1.

Note that there is no requirement of conservation of community size. One can show

that the system is guaranteed to reach the stationary solution (2.2) in the infinite time

limit[82].

The frequency of species containing n individuals is given by

φn =S∑

k=1

Ik, (2.3)

where S is the total number of species and the indicator Ik is a random variable which

takes the value 1 with probability Pn,k and 0 with probability (1 − Pn,k). Thus the

average number of species containing n individuals is given by

〈φn〉 =S∑

k=1

Pn,k . (2.4)

The RSA relationship we seek to derive is the dependence of 〈φn〉 on n.

9

Let a community consist of species with bn,k ≡ bn and dn,k ≡ dn being indepen-

dent of k (the species are assumed to be demographically identical). From Eq.(2.4), it

follows that 〈φn〉 is simply proportional to Pn, leading to

〈φn〉 = SP0

n−1∏

i=0

bidi+1

. (2.5)

We consider a metacommunity in which the probability d that an individual dies

and the probability b that an individual gives birth to an offspring are independent of

the population of the species to which it belongs (density independent case), i.e. bn = bn

and dn = dn (n > 0). Speciation may be introduced by ascribing a non-zero probability

of the appearance of an individual of a new species, i.e. b0 = ν 6= 0. Substituting the

expressions into Eq.(2.5), one obtains the celebrated Fisher logseries[35] :

〈φMn

〉 = SMP0b0b1...bn−1

d1d2...dn= θ

xn

n, (2.6)

where M refers to the metacommunity, x = b/d and θ = SMP0ν/b is the biodiversity

parameter (also called Fisher’s α). We follow the notation of Hubbell[45] here. Note that

x represents the ratio of effective per capita birth rate to the death rate arising from

a variety of causes such as birth, death, immigration and emigration. Note that in the

absence of speciation, b0 = ν = θ = 0, and, in equilibrium, there are no individuals in the

metacommunity. When one introduces speciation, x has to be less than 1 to maintain a

finite metacommunity size JM =∑

n n〈φn〉 = θx1−x .

10

We turn now to the case of a local community of size J undergoing births and

deaths accompanied by a steady immigration of individuals from the surrounding meta-

community. When the local community is semi-isolated from the metacommunity, one

may introduce an immigration rate m, which is the probability of immigration from the

metacommunity to the local community. For constant m (independent of species), im-

migrants belonging to the more abundant species in the metacommunity will arrive in

the local community more frequently than those of rarer species.

We study the dynamics within a local community following the mathematical

framework of McKane et al.[67], who studied a mean-field stochastic model for species-

rich assembled communities. In our context, the dynamical rules[45] governing the

stochastic processes in the community are:

1) With probability 1 − m, pick two individuals at random from the local com-

munity. If they belong to the same species, no action is taken. Otherwise, with equal

probability, replace one of the individuals with the offspring of the other. In other words,

the two individuals serve as candidates for death and parenthood.

2) With probability m, pick one individual at random from the local community.

Replace it by a new individual chosen with a probability proportional to the abundance of

its species in the metacommunity. This corresponds to the death of the chosen individual

in the local community followed by the arrival of an immigrant from the metacommunity.

Note that the sole mechanism for replenishing species in the local community is immigra-

tion from the metacommunity, which for the purposes of local community dynamics is

treated as a permanent source pool of species, as in the theory of island biogeography[63].

11

These rules are encapsulated in the following expressions for effective birth and

death rates for the k-th species:

bn,k = (1 − m)n

J

J − n

J − 1+ m

µkJM

(1 − n

J

), (2.7)

dn,k = (1 − m)n

J

J − n

J − 1+ m

(1 − µk

JM

)n

J, (2.8)

where µk is the abundance of the k-th species in the metacommunity and JM is the

total population of the metacommunity.

The right hand side of Eq.(2.7) consists of two terms. The first corresponds

to Rule (1) with a birth in the k-th species accompanied by a death elsewhere in the

local community. The second term accounts for an increase of the population of the k-

th species due to immigration from the metacommunity. The immigration is, of course,

proportional to the relative abundance µk/JM of the k-th species in the metacommunity.

Eq.(2.8) follows in a similar manner. Note that bn,k and dn,k not only depend on the

species label k but also are no longer simply proportional to n.

Substituting Eq.(2.7) and Eq.(2.8) into Eq.(2.2), one obtains the expression[67]

Pn,k =J !

n!(J − n)!

Γ(n + λk)

Γ(λk)

Γ(ϑk − n)

Γ(ϑk − J)

Γ(λk + ϑk − J)

Γ(λk + ϑk)≡ F (µk), (2.9)

where

λk =m

(1 − m)(J − 1)

µkJM

(2.10)

12

and

ϑk = J +m

(1 − m)(J − 1)

(1 − µk

JM

). (2.11)

Note that the k dependance in Eq.(2.9) enters only through µk. On substituting

Eq.(2.9) into Eq.(2.4), one obtains

〈φn〉 =

SM∑

k=1

F (µk) = SM 〈F (µk)〉 = SM

∫dµρ(µ)F (µ). (2.12)

Here ρ(µ)dµ is the probability distribution of the mean populations of the species in the

metacommunity and has the form of the familiar Fisher logseries (in a singularity-free

description[35, 76])

ρ(µ)dµ =1

Γ(ε)δε exp(−µ/δ)µε−1

dµ, (2.13)

where δ = x1−x . Substituting Eq.(2.13) into the integral in Eq.(2.12), taking the limits

SM → ∞ and ε → 0 with θ = SM ε approaching a finite value[35, 76] and on defining

y = µ γδθ , one obtains our central result, which is an analytic expression for the RSA of

the local community:

〈φn〉 = θJ !

n!(J − n)!

Γ(γ)

Γ(J + γ)

∫ γ

0

Γ(n + y)

Γ(1 + y)

Γ(J − n + γ − y)

Γ(γ − y)exp(−yθ/γ)dy, (2.14)

where Γ(z) =∫∞0

tz−1e−tdt which is equal to (z − 1)! for integer z and γ =m(J−1)

1−m .

As expected, 〈φn〉 is zero when n exceeds J . The computer calculations in Hubbell’s

book[45] as well as those more recently carried out by McGill[66] were aimed at estimating

〈φn〉 by simulating the processes of birth, death and immigration.

13

One can evaluate the integral in Eq.(2.14) numerically for a given set of parame-

ters: J , θ and m. For large values of n, the integral can be evaluated very accurately and

efficiently using the method of steepest descent[70]. Any given RSA data set contains

information about the local community size, J , and the total number of species in the

local community, SL =∑J

k=1〈φk〉. Thus there is just one free fitting parameter at one’s

disposal.

McGill asserted[66] that the lognormal distribution is a “more parsimonious” null

hypothesis than the neutral theory, a suggestion which is not borne out by our reanalysis

of the BCI data. We focus only on the BCI data set because, as pointed out by McGill[66],

the North American Breeding Bird Survey data are not as exhaustively sampled as the

BCI data set, resulting in fewer individuals and species in any given year in a given

location. Furthermore, McGills analysis seems to rely on adding the bird counts of 5

years at the same sampling locations even though these data sets are not independent.

Figure 2.1 shows a Preston-like binning[75] of the BCI data[21] and the fit of

our analytic expression with one free parameter (11 degrees of freedom) along with a

lognormal having three free parameters (9 degrees of freedom). Standard chi-square

analysis[74] yields values of χ2 = 3.20 for the neutral theory and 3.89 for the lognormal.

The probabilities of such good agreement arising by chance are 1.23% and 8.14% for

the neutral theory and lognormal fits, respectively. Thus one obtains a better fit of the

data with the analytical solution to the neutral theory to BCI than with the lognormal,

even though there are two fewer free parameters. McGill’s analysis[66] on the BCI

data set was based on computer simulations in which there were difficulties in knowing

when to stop the simulations, i.e. when equilibrium had been reached. It is unclear

14

whether McGill averaged over an ensemble of runs, which is essential to obtain repeatable

and reliable results from simulations of stochastic processes because of their inherent

noisiness. However, simulations of the neutral theory are no longer necessary, and all

problems with simulations are moot, because an analytical solution is now available.

The lognormal distribution is biologically less informative and mathematically less

acceptable as a dynamical null hypothesis for the distribution of RSA than the neutral

theory. The parameters of the neutral theory or RSA are directly interpretable in terms

of birth and death rates, immigration rates, size of the metacommunity, and speciation

rates. A dynamical model of a community cannot yield a lognormal distribution with

finite variance because in its time evolution, the variance increases through time without

bound. However, as shown by Sugihara et al.[79], the lognormal distribution can arise

in static models, such as those based on niche hierarchy.

The steady-state deficit in the number of rare species compared to that expected

under the logseries can also occur because rare species grow differentially faster than

common species and therefore move up and out of the rarest abundance categories due

to their rare species advantage[15]. Indeed, it is likely that several different models (e.g.

an empirical lognormal distribution, niche hierarchy models[79] or the theory presented

here) might provide comparable fits to the RSA data (we have found that the lognormal

does slightly better than the neutral theory for the Pasoh data set[64], a tropical tree

community in Malaysia). Such fitting exercises in and of themselves, however, do not

constitute an adequate test of the underlying theory. Neutral theory predicts that the

degree of skewing of the RSA distribution ought to increase as the rate of immigration

into the local community decreases. Dynamic data on rates of birth, death, dispersal and

15

immigration are needed to evaluate the assumptions of neutral theory and determine the

role played by niche differentiation in the assembly of ecological communities.

Our analysis should also apply to the field of population genetics in which the

mutation-extinction equilibrium of neutral allele frequencies at a given locus has been

studied for several decades [55, 32, 54, 85, 57, 56].

16

1 2 4 8 16 32 64 128 256 512 1024 20480

5

10

15

20

25

30

35

BCI Plot

Number of individuals (log2 scale)

Num

ber

of s

peci

es

Fig. 2.1. Data on tree species abundances in 50 hectare plot of tropical forest inBarro Colorado Island, Panama taken from Condit et al.[21]. The total number oftrees in the dataset is 21457 and the number of distinct species is 225. The red bars areobserved numbers of species binned into log(2) abundance categories, following Preston’s

method[75]. The first histogram bar represents〈φ1〉

2 , the second bar〈φ1〉

2 +〈φ2〉

2 , the third

bar〈φ2〉

2 +〈φ3〉+〈φ4〉

2 , the fourth bar〈φ4〉

2 +〈φ5〉+〈φ6〉+〈φ7〉+〈φ8〉

2 and so on. The black

curve shows the best fit to a lognormal distribution 〈φn〉 = Nn exp(− (log2 n−log2 n0)2

2σ2)

(N = 46.29, n0 = 20.82 and σ = 2.98), while the green curve is the best fit to ouranalytic expression Eq.(2.14) (m = 0.1 from which one obtains θ = 47.226 compared tothe Hubbell[45] estimates of 0.1 and 50 respectively and McGill’s best fits[66] of 0.079and 48.5 respectively.)

17

2.2 Splitting of a species and peripheral isolate speciation

The master equation approach outlined above lends itself straightforwardly to

the consideration of several distinct modes of speciation. The case considered above

corresponds to a point mutation wherein a new species has just one individual in it. It

is straightforward to write down similar master equations for other interesting cases. As

an illustration, we consider here the master equation for a case in which we allow for the

possibility that a species splits into two different species.

Let φn be the number of species with population n. The total number of species

is fixed to be S =∑∞

n=0φn. Let P(~φ, t) be the probability that at time t we have the

configuration ~φ = (φ0, φ1, φ2, ...), i.e. φ0 species with zero individuals, φ1 species with

one individual and so on. Pn, the probability that a given species has population n, is

Pn = 〈φn〉/S ≡ 1

S

∑

~φ

P(~φ)φn. (2.15)

The evolution equation is of the form:

∂P(~φ)

∂t=∑

~φ′

[W (~φ, ~φ′)P(~φ′) − W (~φ′, ~φ)P(~φ)]. (2.16)

The transition rate W (~φ, ~φ′) denotes the probability that configuration ~φ′ is con-

verted into ~φ and is affected by three processes: birth, death and speciation. For the

birth process in a species with population n, at a rate bn, only two φ’s are influenced:

Wbirth(~φ, ~φ′) =∑

n

φ′nbnΓn,n+1(~φ, ~φ′), (2.17)

18

where Γi,j(~φ, ~φ′) is equal to zero unless φi = φ′

i− 1, φj = φ′

j+ 1 and φk = φ′

k, k 6= i, j,

in which case Γi,j takes on the value 1.

Likewise, for the death process occurring in a species with population n with rate

dn we have:

Wdeath(~φ, ~φ′) =∑

n

φ′ndnΓn,n−1(

~φ, ~φ′) (2.18)

As in our earlier work, we will assume that dn = bn = 0 when n 6 0.

We now turn to splitting of a species and peripheral isolate speciation. Let us

postulate that a species with population k can lose k−n > 0 individuals with a transition

rate Wn,k. Also, a species with population k = 0 can obtain n > 0 individuals with a

transition rate Wn,k=0. Thus

Wsplit(~φ, ~φ′) =

∑

n,k

Wn,kΓk,n(~φ, ~φ′). (2.19)

Note that the Wn,k’s are zero when n > k 6= 0. This corresponds to forbidding a jump

in the population of an already existing species.

On averaging Eqn.(2.16) (by multiplying both sides by φn and summing over ~φ)

we obtain the following evolution equation:

∂Pn∂t

=∑

~φ,~φ′

W (~φ, ~φ′)P(~φ′)(φn − φ′n) (2.20)

which results in

19

∂Pn∂t

= Pn+1dn+1 + Pn−1bn−1 − Pn(bn + dn) +∞∑

k=0

Wn,kPk −∞∑

k=0

Wk,nPn. (2.21)

Choosing the transition rates as Wi,j = νWi,jΘ(j − i) + µWi,0δj,0, where Θ(k) = 1 if

k > 0 and zero otherwise, we get

∂Pn∂t

= Pn+1dn+1 + Pn−1bn−1 − Pn(bn + dn) +

µWn,0P0 + ν∞∑

k=n+1

Wn,kPk − µδn,0

∞∑

k=0

Wk,0P0 − νn−1∑

k=0

Wk,nPn. (2.22)

Let us consider the biological situation in which the birth and death rates are much

bigger than those for speciation or splitting. In equilibrium,∂Pn∂t = 0 and assuming that

both µ and ν are much less than 1, one can obtain1 the expression for Pn:

Pn = µP0

n∑

i=1

1

di

n−1∏

k=i

bkdk+1

∞∑

j=i

Wj,0 + O(µν), n > 0. (2.23)

The above equation underscores the fact that when one has any peripheral isolate

speciation, the random splitting term corresponding to Mayr’s allopatric speciation is

necessarily of higher order and can be neglected safely i.e. no qualitatively new behavior

results when one has both random splitting and peripheral speciation (including point

mutation). Note that the simulations carried out by Hubbell[45] considered the special

1An easy way to obtain this solution is to solve∑n

k=0

∂Pk∂t

= 0 rather than ∂Pn∂t

= 0.

20

case in which he excluded peripheral speciation but one may question the biological

validity of such an assumption.

After taking the limits ν → 0, S → ∞, µ → 0 such that Sµ/b1 → θ, the expression

for 〈φn〉 simplifies to

〈φn〉 = θ

n∑

i=1

b1di

n−1∏

k=i

bkdk+1

∞∑

j=i

Wj,0. (2.24)

This is our central result.

Let us consider the frequency-independent case and choose the birth/death rates

to be bn = bn, dn = dn, where b/d = x < 1. Then

〈φn〉 = θxn

n

n∑

i=1

x1−i

∞∑

j=i

Wj,0. (2.25)

A knowledge of 〈φn〉 allows one to determine Sobs =∑∞

n=1〈φn〉 – the “observable” num-

ber of species, namely the number of species with non-zero population, and∑∞

n=1n〈φn〉

– the total population of the community.

Let us consider a peripheral isolate speciation mechanism in which one could have

two parallel processes. The first is a splitting process in which a species with population

n ≥ p loses p = const > 0 individuals, namely, Wi,n = nδn−i,p. The factor n accounts

for the fact that the species undergoing fission is chosen with probability proportional

to its population. The speciation process leads to the creation of a new species with

exactly p individuals, i.e. Wi,0 = δi,p. On solving Eqn.(2.24) one obtains

21

〈φn〉 = θx

1 − x

[1 − xn

nΘ(p − n − 1) +

1 − xp

xpxn

nΘ(n − p)

], (2.26)

where Θ(k) = 1 if k > 0 and zero otherwise (see Figure 2.2).

When p = 1 the above formula reduces to the well-known Fisher log-series: 〈φn〉 =

θxn/n, n > 0.

Note that one can obtain an exact analytical solution of Eqn.(2.22) in equilibrium

for arbitrary ν. Strikingly, one obtains the same form of the solution for 〈φn〉 as in

Eqn.(2.26) except that x is replaced by y with x = y[1+ν(1−yp)/(1−y)]. As expected,

x = y, when ν = 0.

22

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5x=0.1

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25x=0.4

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70x=0.7

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250x=0.9

1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000x=0.99

0 5 10 150

200

400

600

800

1000

1200x=0.999

Fig. 2.2. The red bars represent the numbers of species derived from Eq.(2.26) binnedinto log(2) abundance categories, following Hubbell’s method[45]. The first histogrambar represents 〈φ1〉, the second bar 〈φ2〉 + 〈φ3〉, the third bar 〈φ4〉+ 〈φ5〉+ 〈φ6〉+ 〈φ7〉and so on. Here θ = 40 and p = 40.

23

2.3 Neutrality and stability of

forest biodiversity – comment on paleodata analysis

The unified neutral theory of biodiversity[45] provides a dynamic null hypothesis

for the assembly of natural communities and has proved to be useful for understanding

the influence of speciation, extinction, dispersal, and ecological drift on some spatial and

temporal scales. Recently, Clark and McLachlan[18] have argued that neutral drift is

inconsistent with paleodata. We show that their analysis is based on a misunderstanding

of neutral theory and that their data alone cannot unambiguously test its validity.

Hubbell’s approach[45] builds on the theory of island biogeography[63], which

asserts that an island or a local community approaches a steady-state species richness

at equilibrium between immigration of species from the much larger metacommunity

source area and local extinction of species on the island or in the local community.

Quite generally, the dynamics of the population of a given species is governed by birth

and death events. The Fisher log-series distribution is obtained in the metacommunity

when the per-capita birth and death rates are independent of the species and speciation

is introduced by ascribing a non-zero probability of the appearance of an individual of a

new species[83]. The characteristic time scale for random evolutionary drift accompanied

by speciation and natural selection is, of course, much longer than the 200 generation

period studied in Ref.[18] and during this time interval, the metacommunity distribution

of relative species abundance ought to be essentially constant.

We now turn to an analysis of the consequences of the assumption that the fossil

pollen record analyzed in Ref.[18] is from 8 local sites. The neutral theory[45, 83] for

24

a local community considers the dynamics of birth and death events accompanied by

a steady flow of individuals from the surrounding metacommunity. When the local

community is semi-isolated from the metacommunity, one may introduce an immigration

rate, which is the probability of immigration from the metacommunity to the local

community. For constant immigration rate (taken to be independent of the species),

immigrants belonging to the more abundant species in the metacommunity will arrive

in the local community more frequently than those of rarer species. For the time scale

of 200 generations, the metacommunity provides a fixed backdrop for the action within

the local community and the key point is that not all species are equivalent during

this limited time scale in the metacommunity – there are some that are more abundant

than others. Indeed, in the local community, unlike in the metacommunity, the original

relative species abundance is recovered after a perturbing influence.

Figure 1 of the Clark and McLachlan paper[18] (reproduced here as Figure 2.3)

is misleading. It is based on simulations of neutral dynamics within a lottery model

and claims to show the divergence among sites with time. One can prove rigorously for

their model, which does not admit extinction, that the variance does not increase indef-

initely with time but levels off on reaching equilibrium. Thus their data merely shows

an approach to equilibrium starting from a zero variance initial condition. Their state-

ment that “drift results in rapid accumulation of among-site variance which is readily

identifiable and would continue to increase until much of the diversity was lost through

extinction” is confused. Apart from the fact that the variance does not grow indefinitely,

the time scale for equilibration in their simulations is vastly different from the evolution-

ary time scales associated with speciation in the metacommunity. Indeed, biodiversity

25

Fig. 2.3. Simulations of neutral dynamics showing the divergence among sites. a,Lottery model, where ten species with identical parameters and responses to stochas-ticity compete for space. b, Results for a single species shown for eight different sites.Abundances diverge with the random accumulation of changes at each site. c, Varianceamong sites increases over time owing to accumulation of the random changes in abun-dance shown in b, as does (d) the coefficient of variation, CV. In c and d, the middle lineindicates the median, and the dashed lines bound 90% of simulated values. This figureand the caption are taken from Ref. [18]

26

is maintained in equilibrium in neutral theory due to the balance between extinction of

species and speciation.

More importantly, the simplified lottery model used as a benchmark does not

capture the basic premise of the unified neutral theory of biodiversity[45], that not all

species are equivalent in the local community because of their unequal abundances in the

metacommunity. Thus the “strong evidence for stabilizing forces in the paleo-record”[18]

is not inconsistent with neutral drift.

27

Chapter 3

Density dependence as an explanation of

tree species abundance and diversity

in tropical forests

3.1 General theory

Density and frequency dependence are familiar mechanisms in population biology,

but it is surprising how rarely their consequences for species diversity and relative species

abundance in communities have been discussed [see Chapter 3 of Ref. [45]]. Here we show

that these mechanisms are sufficient to explain quite precisely the species abundance

patterns in six tropical forest communities on three continents.

The neutral theory of biodiversity provides a convenient theoretical framework

for linking community diversity patterns to the fundamental mechanisms of population

biology (e.g., birth, death and migration) and speciation[45]. The celebrated statisti-

cal distribution for relative species abundance, Fisher’s logseries[35], can be shown to

arise directly from the stochastic equations of population growth under neutrality at the

speciation-extinction equilibrium. More significantly, Fisher’s logseries arises when the

birth and death rates are density independent[83]. According to the theory, the mean

number of species with n individuals, 〈φn〉, in a community at the stochastic speciation-

extinction equilibrium takes the general form: 〈φn〉 = SP0

⟨∏n−1i=0

bi,kdi+1,k

⟩

k, where

〈. . .〉k represents the arithmetic average over all species, S is the average number of

species present in the ecosystem, P0 is a constant, and bi,k and di,k are birth and death

28

rates for the k-th species with i individuals. Here we have subscripted the birth and

the death rates for arbitrary species k to indicate that these rates could, in principle, be

species-specific for an asymmetric community. In contrast, “... symmetry occurs at the

species level when no change in community dynamics or the fates of individuals occurs

upon switching the species of any two given populations in the community. Any given

population behaves as it would previously, despite is new species label, and its effects

on other populations remain the same, regardless of their species labels.” (P. Chesson,

personal communication).

We note that what is important in determining the mean number of species,

〈φn〉, are not the absolute rates of birth or death but their ratio, ri,k =bi,k

di+1,k. Indeed,

〈φn〉 is proportional to 〈r1,kr2,k . . . rn−1,k〉k. This formulation is sufficiently general to

represent communities of either symmetric or asymmetric species. Such a situation could

arise, for example, from niche differences or from differing immigration fluxes resulting

from the different relative abundances of the species in the metacommunity. Hereafter,

however, we consider only the symmetric case of a community of non-interacting species

with identical vital demographic rates. For large community size, this formulation is

equivalent to the case of zero-sum dynamics studied by Hubbell[45].

Let us define rn =bn

dn+1

n+1n , where the factor n+1

n is chosen to obtain rn = x for

the Fisher log-series 〈φn〉 ∝ xn/n. In Fisher’s case, rn does not change with population

density and is an intraspecific parameter that measures the relative vital rates of birth

and death of a population. In order to obtain intraspecific density dependence, rn

becomes a function of the population density n. Within our framework,〈φn+1〉〈φn〉

=

nn+1 rn.

29

We now introduce the modified symmetric theory that captures density depen-

dence (rare species advantage or common species disadvantage). In the modified the-

ory, rn will be a decreasing function of abundance, thereby incorporating density de-

pendence. The equations of density dependence in the per capita birth and death

rates for an arbitrary species of abundance n are:b(n)n = b ·

[1 +

b1n + o

(1n2

)]and

d(n)n = d ·

[1 +

d1n + o

(1n2

)], for n > 0 as the leading terms of a power series in (1/n),

b(n)n = b ·

∑∞l=0

bln−l and

d(n)n = d ·

∑∞l=0

dln−l, where bl and dl are constants. This

expansion captures the essence of density-dependence by ensuring that the per-capita

birth rate-death rate ratios decrease and approach a constant value for large n. This

happens because the higher order terms are negligible. Note that the quantity that con-

trols the RSA distribution is the ratio bn/dn+1. Thus the birth and death rates, bn

and dn, are defined up to multiplicative factors f(n + 1) and f(n) respectively, where

f is any arbitrary well-behaved function. One expects that the per capita birth rate

or the fecundity goes down as the abundance increases whereas the mortality ought to

increase with abundance. Indeed, the per capita death rate can be arranged to be an

increasing function of n, as observed in nature, by choosing an appropriate function f

and adjusting the birth rate appropriately so that the ratio bn/dn+1 remains the same.

For example, the choice f(n) = n/(n+c) yields a constant per capita death rate dn = dn

and a fecundity which decreases with increasing abundance.

This mathematical formulation of density dependence may seem unusual to ecol-

ogists familiar with the logistic or Lotka-Volterra systems of equations, in which density

dependence is typically described as a polynomial expansion of powers of n truncated

30

at the quadratic level. However, this classical expansion is not valid in our context be-

cause the range of n is from 1 to an arbitrarily large value, not to some fixed carrying

capacity. Therefore an expansion in terms of powers of (1/n) is more appropriate. For

this symmetric model, noting that 〈φn〉 = SP0∏n−1

i=0bn

dn+1, one readily arrives at the

following relative species-abundance relationship:

〈φn〉 = θxn

n + c, (3.1)

where x = b/d and for parsimony, we have made the simple assumption that b1 = d1 = c.

The biodiversity parameter, θ, is the normalization constant which ensures that the

average number of species in the community is S and is given by θ = S 1+ccx F−1(1 +

c, 2+c, x), where F (1+c, 2+c, x) is the standard hypergeometric function. The parameter

c measures the strength of the symmetric density dependence in the community, and it

controls the shape of the RSA distribution. Note that when c → 0 (the case of no density

dependence), one obtains the Fisher log-series. In this case, as shown in Ref. [83], θ

captures the effects of speciation.

We now show how Eq.(3.1) estimates the strength of symmetric density depen-

dence that is consistent with the observed RSA distributions of tree species in six large

tropical forest plots on three continents: Barro Colorado Island (BCI), Panama, Yasuni

National Park, Ecuador, Pasoh Forest Reserve, Peninsular Malaysia, Korup National

Park, Cameroon, Lambir Hills National Park, Sarawak, Malaysia and Sinharaja World

Heritage Site, Sri Lanka. These sites plots are part of a global network of large plots

managed by the Center for Tropical Forest Science of the Smithsonian Tropical Research

31

Institute. These New and Old World tropical forests have had long separate ecological

and evolutionary histories, but despite these different histories, the symmetric theory

with density dependence fits each of the RSA distributions very well. Figure 3.1 shows

the fits of Eq.(3.1) and the dispersal limitation model[83] to the data sets of tree abun-

dance data collected from the six permanent plots of tropical forest. These plots are 50

ha except Lambir (52 ha), Yasuni (25 ha) and Sinharaja (25 ha). The results in Table

3.1 and Figure 3.1 show that the RSA data of tree species in these plots are equally well

described both by the density dependent model and the dispersal limitation model[83].

The rare species advantage is illustrated in Figure 3.2 and is of the same order

of magnitude in the different forests. The key quantity that controls the RSA is the

birth to death ratio rn defined above. The curves in Figure 3.2 were derived from the

parameters in Table 3.1, which in turn were obtained from the empirical RSA data using

the maximum likelihood method. At stochastic steady state, community size (mass

balance) is maintained by the slow rate of decline in common species (at large n in

Figure 3.2) exactly balanced by the growth of rare species, and by the very slow input

of new species by speciation.

Several important ecological insights result from this new theory. First, we have

shown that an assumption of asymmetric density dependence, for example, postulating

different carrying capacities for each species, is not necessary to explain patterns of rel-

ative species abundance, at least in these 6 tropical forests: a much simpler symmetric

hypothesis is sufficient. Second, we have shown that the population sizes that exhibit

rare species advantage consistent with the observed RSA data are all quite small. The

32

transition to a Fisher log-series-like value for x = b/d which is slightly less than replace-

ment occurs at what would be considered low tree species populations densities in these

forests (< 1 tree/ha). Third, the theory shows why density dependence is scale depen-

dent and must give way to density independence at large spatial scales on which we have

proved that Fisher’s log-series must apply[83]. This means that density dependence has

a characteristic length scale in these tropical forests, above which the strength of density

dependence must necessarily diminish.

Finally, we have demonstrated that symmetric density dependence gives an equally

sufficient mechanistic explanation for RSA patterns in addition to and independent of

dispersal limitation[83]. In Table 3.1, we show the fits of the two mechanisms to the RSA

data from the 6 forests, from which it is clear that both mechanisms yield fits that cannot

be distinguished statistically in quality. However the ecological explanation that accom-

panies each of these mechanisms is very different. According to the dispersal mechanism,

the explanation for the lower frequency of rare species compared to species of middling

abundance is that rare species are more extinction-prone, and when they go extinct in a

community, they take longer to re-immigrate than common species do. According to the

density dependence mechanism, on the other hand, the reduced steady-state frequency

of rare species arises because populations of rare species grow differentially faster into

higher abundance categories due to a rare-species advantage. An important conclusion

is that one cannot deduce the mechanisms causing a particular RSA pattern from RSA

data alone. Because these mechanisms are not mutually exclusive, it must be left to

empirical research to uncover the relative contributions of each mechanism to observed

RSA patterns. However, we do note one distinction between the two mechanisms. The

33

dispersal limitation mechanism generally implies that one is considering a local commu-

nity into which immigration is possible. However, the density dependence mechanism

can apply equally well to local communities or to the metacommunity. If one general-

izes “immigration” to include speciation events, however, then the “dispersal limitation”

mechanism can apply to the the metacommunity as well.

34

Plot S J θ1 m θ2 c xBCI, Panama 225 21457 48.1 0.09 47.5 1.80 0.9978Yasuni, Ecuador 821 17546 204.2 0.43 213.2 0.51 0.9883Pasoh, Malaysia 678 26554 192.5 0.09 189.5 1.95 0.9932Korup, Cameroon 308 24591 52.9 0.54 53.0 0.24 0.9979Lambir, Malaysia 1004 33175 288.8 0.11 301.0 2.02 0.9915Sinharaja, Sri Lanka 167 16936 27.3 0.55 28.3 0.38 0.9983

Plot L1 L2 Deviance P -valueBCI, Panama −314.0 −315.0 2.0 0.16Yasuni, Ecuador −301.0 −303.6 5.2 0.02Pasoh, Malaysia −363.7 −365.3 3.2 0.07Korup, Cameroon −322.3 −323.1 1.6 0.21Lambir, Malaysia −390.5 −391.2 1.4 0.24Sinharaja, Sri Lanka −258.9 −258.5 0.8 0.37

Table 3.1. Maximum likelihood estimates of the density-dependant symmetric modeland dispersal limitation model[83] parameters (upper table) and comparison between themodels (lower table) for the six data sets of tropical forests. In the six plots coordinatedby the Center for Tropical Forest Science of the Smithsonian (http://www.ctfs.si.edu),we considered trees with diameter at breast height ≥ 10 cm. S is the number of species,J is the total abundance and θ1 and θ2 are the biodiversity parameters in the dispersallimitation model[83] and equation (3.1) respectively (note that θ2 is a function of c, xand S and both models have the same number of fitting parameters). The comparisonof the models was carried out with the likelihood ratio test[3, 30, 40]. The lower tablepresents deviance (twice the difference in the log-likelihoods L1 and L2) between the two

models and the corresponding P -value of the χ2-distribution with one degree of freedom.The main result is that the dispersal limitation model and the simple symmetric densitydependent model presented here are statistically comparable to each other in their abilityto fit the tropical forest data.

35

0 5 100

10

20

30

BCI

0 5 10 150

20

40

Korup

0 5 100

50

100

150

Lambir

0 5 100

50

100

Pasoh

0 5 100

50

100

150

Yasuni

0 5 100

10

20

Sinharaja

Fig. 3.1. Fits of density-dependant symmetric model (red line) and dispersal limitationmodel[83] (blue circles) to the tree species abundance data from the BCI, Yasuni, Pasoh,Lambir, Korup and Sinharaja plots, for trees ≥ 10 cm in stem diameter at breast height(see Table 3.1). The frequency distributions are plotted using Preston’s binning methodas described in Ref. [83]. The numbers on the x-axis represent Preston’s octave classes.

36

100

101

102

103

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

n

n

BCIKorupYasuniLambirPasohSinharaja

r

Fig. 3.2. Plot of r(n) derived from Eq.(3.1) versus n for the six data sets of tropicaltrees. For large values of n, rn asymptotes at a value slightly less than 1. The BCI data(cyan circles) at small n is almost invisible since it coincides with the Pasoh dataset (redcircles).

37

3.2 Is our approach a neutral theory?

Let us address the issue of what constitutes neutrality. We draw from Peter

Chesson’s definition (private communication and references below): “Symmetry at the

individual level means that the species that an individual belongs to is irrelevant to

its fate or the fates of its off spring. An individual’s species could be changed without

affecting the individual in any other way, and without affecting the impact that the

given individual has on other individuals, regardless of species. This form of symmetry,

symmetry at the individual level, is the actual definition of neutrality applicable to the

models in the recent monograph on neutral theory by Steve Hubbell.” This definition,

while simple and concise, may lead to ambiguity when applied to particular models. As

an example, let us consider the model proposed in McKane et al[67]. In that model birth

and death rates are given by the following expressions:

gn = C∗(1 − µ)n

N

N − n

N − 1+

µ

S

(1 − n

N

)(3.2)

and

rn = C∗(1 − µ)n

N

N − n

N − 1+

µ

S(S − 1)

n

N, (3.3)

where N is the total population, S is the total number of species and µ is the immigration

rate.

It is clear from the description of the dynamical rules governing the evolution that

the system is neutral. Indeed, at each time step one picks a random individual, removes

38

(kills) it and places a new one which is either an offspring of another randomly chosen

individual or an immigrant (for simplicity, let us assume that every species has an equal

chance of receiving an immigrant at each time step).

One the other hand, it is also clear from the above equations that the per-capita

birth and death rates, gn/n and rn/n, are density dependant. Thus it is also possible

to view this model as a non-neutral one in which the fate of each individual depends on

the population of the species it belongs to and thus changing an individual’s species will

affect it’s probabilities of giving birth or dying.

Thus the question arises: is the system described in McKane et al[67] neutral

or not? One cannot answer this question merely by inspecting the governing equations.

Rather one needs information on the basis for the dynamical rules of the system. Indeed,

per capita density dependance in and of itself does not preclude the system from being

neutral.

(Interestingly, the more complete model of McKane et al[67] is not neutral because

it relies on the community interaction matrix. When the matrix elements are chosen

randomly, one ends up with a model that can be viewed as being neutral, which they

studied using a mean field type of approach.)

Similarly, consider a simplified case in which the birth and death rates are given

by

bn = b(n + c) (3.4)

39

and

dn = d(n + c) (3.5)

with c > 0.

Note that equation for the birth rate does not necessarily preclude a “neutral”

interpretation. For example, consider a system with the following dynamics: at each

time step either a birth or a death event happens and the probabilities for these events

are constant and proportional to b and d, respectively. For a given event one has two

options (with the probability for each option also fixed): either one chooses a random

individual or an individual from a randomly chosen species. The first option corresponds

to the linear term in the rhs of the above equations and the second option corresponds

to the constant term. With these rules the dynamics of the birth events in the system

is neutral (see the discussion of the model proposed in McKane et al (2000) above).

While the dynamical rules for the birth rates (Eq. (3)) are biologically meaning-

ful (the constant term may be assigned to immigration processes), the presence of the

constant term in the death process does not have a simple ecological basis. Also, in the

form given above, the per capita death rate decreases as n increases, which is in conflict

with our expectations. This unrealistic form of the death rate does not invalidate our

model because the RSA distribution that we obtained is controlled not by the individual

birth and death rates but by the ratio bn/dn+1. Thus the birth and death rates, bn

and dn, are defined up to multiplicative factors f(n + 1) and f(n) respectively, where

f is any arbitrary well-behaved function. One expects that the per capita birth rate

40

or the fecundity goes down as the abundance increases whereas the mortality ought to

increase with abundance. Indeed, the per capita death rate can be arranged to be an

increasing function of n, as observed in nature, by choosing an appropriate function f

and adjusting the birth rate appropriately so that the ratio bn/dn+1 remains the same.

For example, in a limiting case, the choice f(n) = n/(n + c) yields a constant per capita

death rate dn = dn and a fecundity which decreases with increasing abundance. Thus

the resulting formulae for the actual birth and death rates may not necessarily have a

simple interpretation and may very well be non-neutral in character.

3.3 Relationship between the zero sum rule and our approach

There is a departure in our model from the original “zero-sum dynamics” proposed

in Hubbell’s book[45]. The total population is not held fixed in our approach; instead,

we allow the total population to fluctuate and fix it’s average value by choosing the ratio

b/d < 1. A value of b/d = 1 would lead to an exploding population. The competition

between the species for the finite available resources yields an effective b/d < 1 thereby

placing a constraint on the average total population. Our approach is analogous to that

used in statistical physics (see Ref. [43], pp.193-199). Indeed, the zero-sum assumption

corresponds to the microcanonical ensemble, whereas our model corresponds to the grand

canonical ensemble. In the former ensemble, the total number of particles is fixed; in the

latter the average total number of particles is fixed by introducing a chemical potential

which is a measure of the resources needed for the introduction of a new particle into

the system. By adjusting the value of the chemical potential, one can fix the average

41

total population. In our model, the parameter x is analogous to the chemical potential

(more details in a different context are presented in Volkov et al[84]).

Consider an ecosystem consisting of S species with populations of n1, n2, . . . , nS

individuals. We assume that there is a zero sum rule in effect which constrains the total

population to be exactly N .

Let P (n) denote the probability that a given species contains n individuals. Then

the multivariate distribution is given by the following expression:

P (n1, n2, ...nS) =1

Q

S∏

k=1

P (nk)δ(N − n1 − n2 − ... − nS), (3.6)

where the δ function imposes the zero sum rule and the normalization factor (also referred

to as a partition function) is given by

Q =∑

n1,n2,...,nS

S∏

k=1

P (nk)δ(N − n1 − n2 − ... − nS)

=∑

n1,n2,...,nS

S∏

k=1

P (nk)

∫

γ

dz

2πez(N−n1−n2−...−nS)

=

∫

γ

dz

2πezN

[ ∞∑

n=0

P (n)e−zn

]S

=

∫

γ

dz

2πeg(z), (3.7)

where g(z) = zN − f(z)S and

e−f(z) =∞∑

n=0

P (n)e−zn. (3.8)

42

and the contour γ is parallel to the imaginary axis with all it’s points having a fixed real

part x (i.e. z ∈ γ ⇔ z = x + iy,−∞ < y < +∞ ). The integral is independent of x

provided x is positive[70].

In order to estimate the above integral we will use the method of steepest descent

(see Ref. [70], pp.434-443). The basic idea is that if g(z) is a function with a steep

maximum, a result valid in the large community size limit, the dominant contribution

to the integral arises from z-values in the vicinity of this maximum. Let us denote the

value of z at which the maximum occurs by µ. We thus expand g(z) around µ + i0 so

that g′(µ + i0) = N − f ′(µ + i0)S = 0 and keep the first two non-zero terms (again an

approximation justified for a large system):

Q =

∫ ∞

−∞dy

2πeg(µ)−g′′(µ)y2/2+... ≈ eg(µ)

√g′′(µ)

=eµN−f(µ)S√

−f ′′(µ)S(3.9)

From Eq.(3.8) one can show that

f ′(µ) =

∑∞n=0

nP (n)e−µn

∑∞n=0

P (n)e−µn ≡ 〈n〉 (3.10)

is the average number 〈n〉 of individuals per species. One can therefore determine µ by

solving the equation 〈n〉 = N/S.

Likewise

f ′′(µ) = 〈n〉2 − 〈n2〉 < 0. (3.11)

43

Finally, Eq.(3.7) can be rewritten as

Q =∑

n1,n2,...,nS

S∏

k=1

P (nk)e−µnk (3.12)

so that the “effective” probability becomes Peff (nk) = P (nk)e−µnk .

Thus the effect of considering all species simultaneously constrained by the zero

sum rule is entirely equivalent in the large community size limit to treating the species

as being effectively independent of each other but with their abundances controlled by

the quantity µ, the chemical potential.

Indeed, the zero-sum assumption and our approach of fixing the average total

population do lead to the same result for large systems. For example, the distribution

proposed in Ref. [81], which correctly describes Hubbell’s zero-sum metacommunity

model, converges to the Fisher log-series in the limit of large metacommunity size JM

and k ≪ JM :

〈φk〉M =θΓ(JM + 1)Γ(JM + θ − k)

Γ(JM + 1 − k)Γ(JM + θ)≈ θ

xk

k, (3.13)

where x = e−θ/JM / 1.

One can carry out a self-consistency check that the approximation k ≪ JM is

not unreasonable. Consider, for example, the BCI island data, given approximately by

φn = 47.50.9978n

n+1.8 . The total population of the community is more than 20, 000 and the

total number of species is 225. The number of species with a population of more than

2000 individuals is found to be 0.11, which is effectively zero after discretization. Thus

44

the most abundant species is expected to have a population less than around 10% of

the total population in accord with the empirical observation that the most abundant

species has 1717 trees.

There is a clear advantage, in some instances, to our approach. The complexity

of the analysis is greatly reduced. Indeed, by not imposing a zero-sum constraint (i.e.

by allowing the population of each species to run from zero to infinity), one can treat

the species completely independently but with a controlling chemical potential. Our

assumption of fixing the average population is more biologically meaningful because

ecosystems do undergo fluctuations in their abundance. It is important to note, however,

that the full analysis with zero sum rule is specially valuable for understanding the

behavior of species poor communities[3] (for example communities whose biodiversity

number θ is less than 1).

In simple terms there are two entirely equivalent ways of studying a large commu-

nity. One could require that the total number of individuals is fixed at all times so that

the growth in population of one species can only occur at the expense of the number of

individuals of another. Alternatively, one could consider the different species completely

independently and control their average abundance by an effective birth/death rates ra-

tio less than one. The striking result is that two approaches yield the same behavior for

large community size.

45

3.4 Utility of relative species abundance data for elucidating biological

mechanisms

Consider the dispersal limitation model studied by Volkov et al. [83] with the

biodiversity parameter θ = 50 and the immigration parameter m = 0.1. The top panel

of Figure 3.3 shows the RSA data for this model obtained through the analytic expres-

sion. The middle panel shows two different binnings of the same data underscoring

the sensitivity of the shape of the curve on the binning scheme employed. The bottom

panel shows a graph of rn vs. n deduced from the RSA data using the exact formula:

rn = n+1n

〈φn+1〉〈φn〉

, where φn represents the number of species with abundance n. As-

sume now that we were given the RSA data in the top panel and asked what we could

learn from it. Clearly the dispersal limitation model from which the data was derived

provides a perfect description of the data and one may be tempted to conclude that the

mechanism underlying the data is in fact dispersal limitation with the selected model

parameters. However, a symmetric density dependence mechanism yielding rn vs. n in

the bottom panel would also yield exactly the same RSA data. Furthermore, rn does not

constrain the individual birth and death rates but only their ratio. This provides even

greater flexibility in the range of models and mechanisms that are able to fit the data

exactly. One would expect that the RSA data from six forests would contain additional

information which one might use to identify the biological mechanisms underlying them.

However, as shown in Figure 3.1 and the fitting parameters in Table 3.1, both dispersal

limitation and symmetric density dependence provide plausible explanations for all six

46

forests. Thus exercises of estimating the quality of data fits are useful for ruling out mod-

els or parameter values which do not provide a good description of the data. The more

important task of elucidating the underlying biological mechanisms will need additional

empirical data such as data on dynamics or multivariate probability distributions.

47

100

101

102

103

0

10

20

n

⟨φn⟩

1 2 3 4 5 6 7 8 9 100

20

40

1 2 3 4 5 6 7 8 9 10110

20

40

100

101

102

103

0.5

1

1.5

n

r

Fig. 3.3. Upper panel: Plot of φn versus n for the model in Ref.[83] with the biodiver-sity parameter θ = 50, the immigration parameter m = 0.1 and population of 20, 000.Middle panel: Binned tree species abundance data from the upper panel. The frequencydistributions are plotted using Preston’s binning method (right panel) and the methodfrom Hubbell’s book (left panel). Bottom panel: Graph of rn vs. n deduced from the

RSA data using the exact formula described above: rn = n+1n

〈φn+1〉〈φn〉

.

48

Chapter 4

Spatial scaling relationships in ecology

In this chapter, we take a first step towards the development of an analytically

tractable model that, despite its simplicity, leads to a remarkably accurate quantitative

description of β-diversity in two different tropical forests. It also indicates the existence of

novel scaling behavior, revealing previously unexpected relationships between β-diversity,

RSA, and SAR. The model we study is a version of the well-known voter model [41] which

has been applied to a variety of situations in physics and ecology [26, 78]. The results

presented here were obtained in collaboration with Tommaso Zillio, a doctoral student

in SISSA, Italy.

Let us begin with a phenomenological equation for the time evolution of F (r, t),

the probability density that two randomly drawn individuals separated from each other

by r at time t are of the same species (for simplicity, the system is assumed to be

translationally invariant and F depends only on the separation distance):

F (r, t) =(∇2F (r, t) − γ2F (r, t)

)+ u + sδ(r). (4.1)

where the first term on the right hand side represents dispersal or diffusion; the second is

a decay term arising from the invasion of other species due to speciation or immigration

from a surrounding metacommunity, whose coefficient, γ2, could generally be a function

of r; the u > 0 term allows for the possibility that, in the steady state, F could attain a

49

non-zero value at large distances (as would be expected when the system is dominated

by the effects of immigration [83]); and the last term recognizes that when r is zero, one

has the same species by definition.

The above equation was written down based on general considerations, but it

can also be derived, for example, from the following microscopic model: consider a

hypercubic lattice in d dimensions with each site representing a single individual. At

each time step an individual chosen at random is killed and replaced, with probability

1 − ν with an offspring of one of its nearest neighbors or, with probability ν, with an

individual of a completely new species, not already present in the system throughout

its history. This last process is called speciation and the parameter ν is called the

speciation rate. The special case with ν = 0 has been thoroughly studied (see, for

example, [58, 36, 61, 62, 25]) and on a finite lattice results in a stationary state with just

a single species (monodominance). The more general version with ν 6= 0 has also been

studied [26, 14], but is not as well understood. The equation for F t+1~x

may be written

as

F t+1~x

= F t~x− 2

NF t

~x+

1 − ν

Nd

d∑

µ=1

(F t

~x+µ+ F t

~x−µ

), (4.2)

where ~x denotes the vector separation between two individuals of the same species, N

is the number of sites in the lattice, the sum runs over the basis vector set and one has

the boundary condition F t0

= 1. In steady state, one obtains (with periodic boundary

conditions):

F~x =

∫−π<pi<π

dd~p

(2π)dei~p·~x 1

l(~p)

∫−π<qi<π

dd~q

(2π)d1

l(~q)

, (4.3)

50

d a b z1 0.5 1.05 0.032 0.87 1.2 0.33 0.95 1.21 0.5

Table 4.1. Scaling exponents for d = 1, 2, 3 determined from the scaling collapse of theRSA and SAR plots (Figs. 4.1 and 4.2).

where

l(~p) = −1 +1 − ν

d

d∑

µ=1

cos(pµ). (4.4)

In d = 1

Fr = e−r/ξ , (4.5)

where r ≡ |~x| and the correlation length

ξ =

(ln

1 − ν

1 −√

ν(2 − ν)

)−1

(4.6)

decreases monotonically with increase of ν. In the continuum, one recovers an equation

of the form (4.1) with F (r, t) ≡ F t~x

a−d, γ2 = 2dν/a2, u = 0, and the time has been

rescaled as t → 2dt/a2, where a is the lattice parameter. Note that γ has dimensions

of an inverse length and is proportional to the square root of ν, a fact that we will use

later. The solution of the continuum equation is

F (r) =sγd−2

(2π)d2

(γr)2−d2 K2−d

2

(γr) + u/(γ2), (4.7)

51

where Kµ(x) is the modified Bessel K function of order µ [1] and r = |~x|. We have

carried out extensive simulations on a square lattice and on a hexagonal lattice with

periodic boundary conditions in both cases and have verified that the results are in

excellent accord with the analytic solution and independent of the microscopic lattice

used.

Within the context of the same model, we turn to an investigation of other quanti-

ties of ecological interest, notably relative species abundance (RSA) and the species-area

relationship (SAR). We begin by noting that in the limit ν → 0

∑

~x

F~x ∼

ν−d/2, d < 2

ν−1, d > 2

(4.8)

Physically,

∑

~x

F~x =〈n2〉〈n〉 ∼ ν−a, (4.9)

with

a = min(1, d/2), (4.10)

where n is the number of individuals of a species. The first equality in Eq.(4.9) follows

from the observation that∑

~x F~x = 1N

∑~x,~y P~x,~y = 1

N

∑k fkk2 where P~x,~y is the prob-

ability of finding individuals of the same species at sites ~x and ~y, fk is the number of

species with k individuals, and∑

k fkk = N by definition. Thus one obtains the depen-

dence of a characteristic measure of 〈n〉, the mean number of individuals per species, on

52

Fig. 4.1. Left column: plots of the normalized RSA for d = 1, 2, 3 with ν =0.001, 0.003, 0.01, 0.03, 0.1 (d = 2 plot also shows the results for ν = 0.0001, 0.0003, 0.3).Right column: plots of the data collapse yielding a measure of the exponents a and b inTable 4.1.

53

ν and the identification of a scaling variable nνa. The second scaling variable Aνd/2

follows from dimensional analysis because, as stated before, the speciation length scale is

inversely proportional to the square root of ν. One is therefore led to postulate a scaling

form for the RSA, the number of species f(n, ν,A) with n individuals in an area A (or

volume, if d = 3) when the speciation rate is ν

f(n, ν,A) = n−β f(nνa, Aνd/2). (4.11)

From this equation, one can compute the total number of species S ≡∑n f(n, ν,A)

and a measure of the total area A ≡∑n n f(n, ν,A) (recall that A is equal to the total

number of individuals) as

S ∼ νa(β−1)∫ ∞

νadx x−β f(x,Aνd/2), (4.12)

A ∼ νa(β−2)∫ ∞

νadx x1−β f(x,Aνd/2). (4.13)

The dependence of S in Eq.(4.12) on the area A defines the SAR, i.e. the number

of species S(ν,A) present in an area A when the speciation rate is ν. One can immediately

deduce a scaling form for the SAR:

S(ν,A) = Az S(Aνd/2). (4.14)

54

101

102

103

100

102

A

S

10−1

100

101

102

103

100

102

Aνd/2

S A

−z

102

103

100

102

A

S

10−2

100

102

100

102

Aνd/2

S A

−z

101

102

103

104

100

102

A

S

10−4

10−2

100

102

104

10−1

100

101

102

Aνd/2

S A

−z

d=1

d=2

d=3

d=1

d=2

d=3

Fig. 4.2. Left column: plots of the SAR for d = 1, 2, 3 with ν =0.001, 0.003, 0.01, 0.03, 0.1. Right column: plots of the data collapse yielding a measureof the exponent z in Table 4.1.

55

In a regime in which S is constant, the number of species scales as a power of the area

with an exponent z. As we shall see in d = 1, z is close to zero but S is a nearly linear

function of its argument leading to an effective exponent of 1.

Finally, let us define another auxiliary function which we call the RSA distri-

bution of species (which is simply a normalized version of the RSA) as φ(n, ν,A) ≡

f(n, ν,A)/S(ν,A) which we postulate to scale as

φ(n, ν,A) = n−bφ(nνa, Aνd/2). (4.15)

This defines the exponent b. Detailed simulations suggest that φ is only weakly dependent

on its second argument Aνd/2.In order to find a relationship between the exponent z

from Eq.(4.14) and the exponents a and β of Eq.(4.11), one needs to consider carefully

the convergence of the integrals in Eq.(4.12) and Eq.(4.13) to assess the role played by

the lower bound of the integrals.

In the limit of nνa → 0, let us postulate that f(x, y) ∼ xcg(y) for x → 0. A

straightforward analysis leads to three distinct scenarios:

1. 1 + c − β > 0: this leads to b = 1 and

z =

0, d 6 2

d−2d , d > 2

(4.16)

56

2. −1 6 1 + c − β < 0: this leads to 1 < b < 2 and the scaling relation

a(2 − b) =d

2(1 − z) (4.17)

3. 1 + c − β < −1: this leads to b = β − c > 2, c = −max(1, d/2) and z = 1.

In order to select between these three cases, we have carried out extensive simula-

tions with hypercubic lattices of various sizes in d = 1, 2, 3. A series of simulations with

fixed size and varying speciation was used for the determination of the RSA fraction of

species (L = 200 for d = 2 and L = 100 for d = 1, 3, L being the side of the hypercube

used). Another series of simulations, varying both the speciation rate and L was carried

out to determine the SAR curves. In this case we consider as the value of S(ν,A) the

mean number of species in a simulation with speciation rate ν on a hypercubic lattice of

size A = Ld.

In order to carry out the collapse, we used the automated procedure described

in [9]. Applying this procedure to the data on the RSA fraction of species obtained by

simulation we were able to obtain the values of the exponents a and b (see Table 4.1,

and Fig.4.1).

Strikingly, in all dimensions b was found to be in the interval (1, 2), confirming

the validity of the second case above. The scaling exponents in Table 4.1 approximately

satisfy Eq.(4.17) . Figure 4.2 shows a collapse plot which confirms the scaling postulates

above. The biggest deviation from our theory can be found in the value of a in d = 2, the

upper critical dimension for diffusive processes. Our results are suggestive of logarithmic

57

corrections but we were not able to find the exact nature of these. Interestingly, our

scaling relation seems to hold even for this case.

58

Chapter 5

Spatial patterns of ecological communities:

α-β diversity and species-area relationship

The spatial model presented in this chapter is based on neutral theory in ecology

[45, 8, 83] and provides a useful starting point for the quantitative study of the role

of interactions and niche differences between species. In tree communities, the spatial

density dependence is known as the Janzen-Connell effect1 [51, 22], which refers to the

negative effect of nearby individuals of a given species on the probability of recruitment,

growth and survival of new individuals of the same species in the community. Note that

the Janzen-Connell effect may be considered as a non-neutral component of otherwise

neutral theory. Indeed, the symmetry on the individual level, which is a key assumption

of neutrality, is broken in the Janzen-Connell mechanism – survival probabilities for

the offsprings depend on the conspecific density and can be regarded as intraspecific

interactions.

1In celebrated papers published more than three decades ago, Janzen [51] and Connell [22]independently hypothesized that tropical forest tree diversity would be enhanced by an inter-action between seed dispersal and density- dependent seed predation. If most seeds fall nearmaternal parents but are killed in the vicinity of the parent by predators and pathogens, and ifthe probability of seed or seedling survival (avoidance of predation) increases with the distancea seed is dispersed from the parent, then tree species will have a reduced probability of replacingthemselves in the next generation in exactly the same location, opening recruitment sites forother tree species in the forest. This spatially structured density dependence will cause a reduc-tion in the probability that two nearby trees are the same species expected from dispersal alone.Over the last two decades, empirical research on large mapped plots of tropical forest have pro-vided increasing evidence for the existence, strength and pervasiveness of Janzen-Connell effectsin species-rich tropical tree communities [87, 38, 46, 2].

59

The community is modeled by a space-filling community of trees of different

species with the following dynamics. A tree, selected at random, is replaced with an

immigrant from the metacommunity (with probability of immigration m) or with an

offspring of one of the neighboring trees (with probability 1 − m) chosen at random.

The metacommunity is assumed to be very large compared with the RSA represented

by the Fisher log-series. The Janzen-Connell effect is captured by introducing a survival

probability for the new tree, which decreases linearly with increase in the number of

conspecific trees in a neighborhood of spatial extent R around the tree. The special case

with R = 0 leads to a simpler model with just the effects of dispersal.

Figure 5.1 shows the results of computer simulations of our model on a hexagonal

lattice with periodic boundary conditions (i.e. a torus to eliminate edge effects) with

the total number of trees equal to JM = 11400. The results we obtain are universal

in that the qualitative behavior is independent of the specific lattice employed or on

the precise shape of the neighborhood around a tree. We depict four cases, Case (1)

the “mean field” approximation, in which choice of the offspring species depends on the

metacommunity relative species abundances and there is no dispersal limitation; Case

(2) in which there are no Janzen-Connell effects (R = 0), but dispersal is limited; Case

(3) in which there are Janzen-Connell effects, but they operate globally (R = ∞); and

case (4) which has intermediate range Janzen-Connell effects (R = 10). In Cases (2) -

(4), we limited dispersal in one time step to between adjacent lattice hexagons. We also

examined the effect of varying the speciation rate.

Case (1), the mean-field approximation with no dispersal limitation and no Janzen-

Connell effects, is fit quite precisely by the Fisher logseries, φn = θxn/n, where θ ∝ JMν

60

(thin blue lines through the black solid and dashed lines). One of the most important

general results is that the Fisher logseries (the mean field case) is supplanted by an RSA

distribution having a mode at intermediate species abundances as soon as one includes

symmetric density dependence in the model. The resulting distribution bears a striking

similarity to ecological data in tropical forests[45]. The distribution of RSA for case (2),

R = 0, and case (3), R = ∞, are very similar – when R is very large, the survival of an

offspring does not depend on the density of its species in its immediate neighborhood and

thus it is similar to the case in which there are no Janzen-Connell effects at all (R = 0).

The most pronounced modal value arises in Case (4), the case with an intermediate

Janzen-Connell length scale, and reflects the fact that the Janzen-Connell density de-

pendence causes community-level frequency dependence (rare species advantage). This

frequency dependence in turn causes species to have more similar relative abundances

at equilibrium.

When the speciation rate, ν, is very small, so that θ ≪ 1 (not shown in Fig. 5.1),

the RSA has a maximum for n = JM and the system approaches monodominance.

The larger the metacommunity, the smaller the value of ν must be in order to obtain

monodominance. Because our metacommunity simulations are for relatively small JM

by comparison with natural communities, the value of ν is chosen to be relatively large.

For rapid speciation, the distribution of RSA is no longer strongly dependent on the

value of R. This is because at high rates of speciation, large numbers of species are

present at steady state in the metacommunity, most of which are at low abundance. In

the numerical case studied, a value of ν = 0.1 results in a mean species abundance of

61

only about 3 individuals, which almost completely eliminates density-dependent Janzen-

Connell effects. Nevertheless, there is still an interior mode of the RSA distribution at

small values of n. For low rates of speciation (for example, ν = 0.005 in our simulation),

the mode of the RSA distribution moves to larger values of n and one observes the

Janzen-Connell effect. The position of the mode is independent of the value of R, but

the peak becomes broader with an increase in R.

We now turn to an analysis of the patterns of beta diversity in tropical tree

communities in Panama and Ecuador-Peru reported by Condit et al. [21], who studied

the probability that two trees separated by a distance vector ~x were of the same species.

The problem is spherically symmetric for the tropical forest and the solution depends

only on |~x| = r. Our model is amenable to exact solution [90] in the continuum limit.

Detailed computer simulations on a lattice are in excellent accord with the continuum

results.

The probability that two trees separated by r belong to the same species, F (r),

is given by

F (r) =

c0K0(γ0r) + c1I0(γ0r) r < R;

c2K0(γ1r) r > R,

(5.1)

where I0 and K0 are modified Bessel functions, R is the distance over which the Janzen-

Connell effects are operational, γ1 ∝ √ν and γ0 are inverse length scales in the outer

and inner domains respectively, and c0, c1 and c2 are constants. Two of these constants

are chosen to ensure the continuity of F and its derivative at r = R. Thus there are four

adjustable parameters, R, γ0, γ1 and c0.

62

The best fits to the beta diversity patterns for both Panama and Yasuni are

excellent2 (Fig. 5.2), although there are some interesting small but systematic positive

residuals in both sites at scales of 0.1 to 1km. In the case of Panama, the theory yields

an estimate of the effective radius of Janzen-Connell effects of 46m, whereas for Yasuni

in Amazonian Ecuador, which is richer in tree species than Panama, the theory yields

an estimate of 86m. These estimates are quite reasonable and plausible at least in the

case of Panama (to date no studies of Janzen-Connell effects have been completed at

Yasuni). Independent measurements of the strength and spatial range of Janzen-Connell

effects have been made in a 50ha permanent plot in the tropical forest on Barro Colorado

Island (BCI) in Panama [48], in which 240, 000 individually tagged trees and saplings

of > 300 species have been mapped, identified to species, and followed over the past

two decades [19]. We have used seed traps and seedling plots to measure seed dispersal

and mortality during the seed-to-seedling transition [49]. Janzen-Connell effects are

not only pervasive among species in the tree community [87], but also quite strong in

many species, particularly in the patterns of mortality of seeds and just germinated

seedlings [38]. Moreover, these mortality patterns are persistent throughout the sapling

and subadult stages of many BCI tree species [46, 2]. Janzen-Connell effects probably

persist throughout the 40 − 120 year duration of the juvenile stages, depending on the

tree species [44]. The effects are currently detectable to an average distance of 10− 20m

from conspecific adults, but in some species the effects are significant to 50m or more

[47]. We expect that as the duration of the BCI Forest Dynamics Project increases, the

2It is interesting to note that the inability of Hubbell’s theory [45] to account for the betadiversity patterns has led to the suggestion [21, 39] that neutral theory ought to be rejected.

63

spatial range over which these significant Janzen-Connell effects are detected will also

increase.

Our simulations of the species-area relationship (SAR) were carried out on two

metacommunities, one with dispersal limitation, and the other with both dispersal lim-

itation and symmetric Janzen-Connell effects. We used a hexagonal lattice with peri-

odic boundary conditions in both cases, we varied the metacommunity size (11, 400 and

46, 000 sites respectively) and we ran the simulations out to 20, 000 death-birth events

on average per site to ensure that equilibrium was reached. We selected 100 sites in the

lattice at random, calculated the SAR, taking each of these sites as the center, and then

we averaged the SAR results over all 100 sites. We calculated SARs for three values of

R (R = 0, R = 10, and R = ∞). We also varied the speciation rate, ν: 0.1, 0.05, 0.01,

0.005 and 0.001.

When plotted in a conventional log-log plot of S versus A, one does obtain three

phases. At short length scales, one has non-universal behavior which depends on the

details of the lattice. At intermediate length scales, one has a power law with an effective

exponent which increases. Finally, in the third phase, there is a linear dependence of S

on A.

Except for very small metacommunity sizes, the species area curves exhibit the

triphasic behavior in a conventional species-area log-log plot (not shown). At short

length scales, one has non-universal behavior which depends on the details of the lattice.

At intermediate length scales, one has a power law with an effective exponent which

increases. Finally, in the third phase, there is a linear dependence of S on A.

64

Strikingly, the data obtained for different areas and speciation rates exhibit a

scaling collapse (see Figure 5.3) consistent with

S ∼ AzFR(νA), (5.2)

where, for large argument the R-dependent scaling function FR(w) ∼ w1−z. The data

confirm the theoretical expectation of a characteristic area per species which is inversely

proportional to the number of species or equivalently the inverse of ν. The effective

values of z for R = 0, 10 and ∞ are 0.28, 0.45 and 0.36 respectively and are in excellent

accord with observational data at intermediate spatial scales. Note that the shallowest

species area curves arise when the Janzen-Connell effects are operational.

Figure 5.4, panels A and B, are snapshots of the species distribution of the two

metacommunities in equilibrium. Each color represents a different species. We chose the

speciation rates so that there is approximately the same number of species (about 110) in

both metacommunities. There are interesting differences between the two landscapes: in

Panel B, with strong Janzen-Connell effects, all of the species patches are approximately

of the same size (approximately equal abundances), whereas in Panel A, the patch sizes

vary considerably (no Janzen-Connell effects). In Panel B, two competing processes affect

the size of the patch: birth/dispersal and Janzen-Connell effects. Because they give rare

species a growth advantage, Janzen-Connell effects reduce the rate of extinction, thereby

increasing the steady-state diversity in the metacommunity. This diversifying effect was

previously reported by Chave et al. [15] and Hubbell and Lake [50].

65

We have shown that the shape of the distribution of relative species abundance in

local communities and the metacommunity is affected by the strength of density depen-

dence, as are steady-state species-area relationships. Questions about the importance

of Janzen-Connell effects in explaining tropical forest tree diversity have been raised

[59] after the discovery of similar Janzen-Connell effects in temperate forests [72]. Our

theory bears on this debate. It shows that Janzen-Connell effects not only influence

the α diversity of tropical forest tree communities on local spatial scales, but also that

they increase species persistence times (times to extinction) and thereby increase the

steady-state species richness at the speciation-extinction equilibrium in the entire meta-

community. However, to our knowledge no data yet exist on the spatial range of such

effects in temperate forests.

In summary, we have demonstrated that the simple model presented here is a

good foundation for the development of a unified understanding of several outstanding

problems in biodiversity. In spite of turning off the interactions and our treatment of

all species in a symmetric manner, our model does an excellent job of capturing the key

features of the observations. This does not mean that an ecosystem is symmetric and

non-interacting. What it does suggest is that the interactions act on top of and in addi-

tion to the demographic processes in the non-interacting system and lead to systematic

but relatively small second-order changes in the statistical properties of the interacting

ecosystem. For example, our non-interacting model produces patchy species distributions

[15] so that the instantaneous pattern of spatial organization of the species in our model

resembles the effects of spatial heterogeneity. In other words, one cannot discern from

the inspection of a “snapshot” whether the heterogeneity has been incorporated into the

66

model or not. Indeed, this patchiness can be thought of as the backdrop provided by

our simplified model, which, in turn, is stabilized by heterogeneous niche effects.

67

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

n

n ⟨φ

n⟩ /

S

MF, ν=0.1, S=2915MF, ν=0.005, S=342R=0, ν=0.005, S=412R=0, ν=0.1, S=3182R=10, ν=0.005, S=626R=10, ν=0.1, S=3258R=∞, ν=0.005, S=441R=∞, ν=0.1, S=3191

Fig. 5.1. Relative Species Abundance plots for a metacommunity with JM = 11400individuals. Here 〈φn〉 is the number of species with population n, S denotes the totalnumber of species, ν is the speciation rate and R is the Janzen-Connell length scale. Thetwo mean field cases are well-described by the Fisher log-series (thin blue line fits). Thethree cases with ν = 0.1 (dashed lines) lead to overlapping plots. The Janzen-Connelleffects are not important in these cases because of the few individuals per species. Forthe three cases with ν = 0.005, there is a pronounced internal mode with the behaviorat intermediate values of R being distinct from the R = 0 and ∞ cases.

68

0.01 0.1 1 10 100 1000 100000

0.01

0.02

0.03

0.04

0.05Panama

r (km)

F0(r

)

within 50 ha plot mean from pairs of single hatheory

0.01 0.1 1 10 100 1000 100000

0.01

0.02

0.03

0.04

Yasuni

r (km)

F0(r

)

within 25 ha plotmean from pairs of single hatheory

Fig. 5.2. Beta Diversity data along with the best fits using Equation (1) for plots

in (a) Panama (R = 46m, γ−10

= 68m, γ−11

= 210km and c0 = 120m) and in (b)

Ecuador-Peru (Yasuni) (R = 86m, γ−10

= 69m, γ−11

= 23, 500km and c0 = 19m). The

Janzen-Connell effect pushes conspecific individuals further away from each other andthus the probability function F declines more steeply within the zone of its operationthan at larger distances as in the data.

69

10−3

10−2

10−1

100

101

102

103

100

101

102

103

ν A

S /

Az

R=0, z=0.28R=10, z=0.46R=∞, z=0.35

Fig. 5.3. Scaling collapse of the species-area relationship (SAR) plots. We presentthe species-area relationships for metacommunities of two sizes, three values each of theJanzen-Connell effective distance, R, and 6 values of the speciation rate ν (0.1, 0.05,0.01, 0.005 and 0.001). Note that, when one uses the scaling variables S/Az and νA, thecurves for the different speciation rates become superimposed. Red: no Janzen-Connelleffect (R = 0). Green: local Janzen-Connell effect (R = 10). Blue: infinite range(R = ∞) effect. The black points correspond to the bigger size system.

70

Fig. 5.4. Equilibrium snapshots of two metacommunities with no Janzen-Connell effect(R = 0, ν = 0.001, S = 2670, left panel) and Janzen-Connell effect (R = 10, ν = 0.0001,S = 2206, right panel). Even though there is patchiness in both cases, the spatialdistribution of species is quite distinct depending on whether the Janzen-Connell effectsare operational or not.

71

Chapter 6

Organization of ecosystems in

the vicinity of a novel phase transition

An ecological community consists of individuals of different species occupying

a confined territory and sharing its resources[63, 65, 60, 88]. One may draw parallels

between such a community and a physical system consisting of particles. In this chapter,

we show that an ecosystem can be mapped into an unconventional statistical ensemble

and is quite generally tuned in the vicinity of a phase transition where biodiversity and

the use of resources are optimized.

Consider an ecological community, represented by individuals of different species

occupying a confined territory and sharing its common resources, whose sources (for

example, solar energy and freshwater supplies) depend on the area of the territory, its

geography, climate and environmental conditions. The amount of these resources and

their availability to the community may change as a result of human activity and natural

cataclysms such as climate change due to global warming, oil spills, deforestation due to

logging or volcanic eruptions.

Generally, species differ from each other in the amount and type of resources

that they need in order to survive and successfully breed. One may ascribe a positive

characteristic energy intake per individual, εk, of the k-th species (0 < ε0 < ε1 < . . . ),

which ought to depend on the typical size of the individual[77, 12, 68, 73]. The individuals

72

of the k-th species play the role of particles in a physical system housed in the energy

level εk .

We make the simplifying assumption that there is no direct interaction between

the individuals in the ecosystem. Our analysis does not take into account predator-prey

interactions but rather focuses on the competition between species for the same kind of

resources.

In ecology, unlike in physical systems where one has a fixed (average) number

of particles and an associated average energy of the system, one needs to define a new

statistical ensemble. One may define the maximum amount of resources available to the

ecological community to be Emax. The actual energy used by the ecosystem, 〈E〉, can

be no more than Emax

Emax = T + 〈E〉 (6.1)

and, strikingly, as shown later, the energy imbalance, T > 0, plays a dual role. First, for

a given Emax, the smaller the T , the larger the amount of energy utilized by the system.

Thus, in order to make best use of the available resources, a system seeks to minimize this

imbalance. Second, T behaves like a traditional ‘temperature’ in the standard ensembles

in statistical mechanics[34] and, in an ecosystem, controls the relative species abundance.

73

6.1 Sketch of the derivation

Consider the joint probability that the first species has n1 individuals, the second

species has n2 individuals and so on:

Peq(n1, n2, ...) ∝∏

k

P (nk)Θ(Emax −S∑

j=1

εjnj), (6.2)

where S is the total number of species (we assume that S ≫ 1) and Θ(u) is a Heaviside

step-function (defined to be zero for negative argument u and 1 otherwise) which ensures

that the constraint is not violated. Here, P (n) represents the probability that a species

has n individuals in the absence of any constraint and is the same for all species. Note

that P (n) does not have to be normalizable – one can have an infinite population in the

absence of the energy constraint.

As in statistical mechanics[34], the partition function (the inverse of the normal-

ization factor), Q, is obtained by summing over all possible microstates (the abundances

of the species in our case):

Q =∑

{nk}Peq(n1, n2, ...). (6.3)

On substituting Eq. (6.2) into Eq. (6.3) and representing the step-function in the integral

form one obtains

Q =∑

{nk}

∏

k

P (nk)

∫

γ

dz

2πizez(Emax−

∑Sj=1

εjnj) =

∫

γ

dz

2πizezEmax−

∑k h(zεk), (6.4)

74

where e−h(β) =∑

n e−nβP (n) and the contour γ is parallel to the imaginary axis with

all its points having a fixed real part z0 (i.e. z ∈ γ ⇔ z = z0 + iy,−∞ < y < +∞ ).

The integral is independent of z0 provided z0 is positive[70].

We evaluate the integral in Eq. (6.4) by the saddle point method[70] by choosing

z0 in such a way that the maximum of the integrand of Eq. (6.4) occurs when y = 0:

Emax − 1

z0=∑

k

εkh′(z0εk), (6.5)

where the prime indicates a first derivative with respect to the argument. Note that the

rhs is simply∑

k εk〈nk〉 with the average taken with the weight P (nk) exp[−z0εknk].

Comparing Eq. (6.5) with Eq. (6.1), one can make the identification z0 = 1/T

and therefore

Q ∝∑

{nk}

∏

k

P (nk) exp(−εknk/T ). (6.6)

This confirms the role played by the energy imbalance as the temperature of the ecosys-

tem. Indeed, the familiar Boltzmann factor is obtained independent of the form of

P (nk). Note that Eq. (6.6) with P (n) = 1 leads to a system of non interacting identical

and indistinguishable Bosons, whereas P (n) = 1/n! describes distinguishable particles

obeying Boltzmann statistics. It is important to note that terms neglected in the eval-

uation of the integral in Eq. (6.4) contribute for finite size systems but these vanish in

the thermodynamic limit.

In order to derive an expression P (n), consider the dynamical rules of birth, death

and speciation in an ecosystem. In the simplest scenario[45], the birth and death rates

75

per individual may be taken to be independent of the population of the species, with

the ratio of these rates denoted by x. Furthermore, when a species has zero population,

we ascribe a non-zero probability of creating an individual of the species (speciation).

Without loss of generality, we choose this probability to be equal to the per capita birth

rate.

One may write down a master equation for the dynamics[82, 83, 39] and show

that the steady state probability of having n individuals in a given species, P (n), is given

by the distribution:

P (n) = P (0)xn

n, n = 1, 2, 3, ... (6.7)

When x is less than 1, this leads to the classic Fisher log-series distribution[35] for the

average number of species having a population n, φ(n) ∝ P (n).

On substituting Eq. (6.7) into Eq. (6.6), one obtains

Q ∝∑

{nk}

∏

k

[x exp(−εk/T )]nk

nk, (6.8)

where the term [x exp(−εk/T )]nk/nk is replaced by 1 when nk = 0. Note that this

leads to an effective birth to death rate ratio equal to x exp(−εk/T ) < 1 for the k-th

species. In a non-equilibrium situation, such as an island with abundant resources and

no inhabitants, the ratio of births to deaths can be bigger than 1 leading to a build-

up of the population. In steady state, however, the deaths are balanced by births and

speciation (creation of individuals of new species).

76

It is interesting to consider an ecosystem with an additional ceiling on the total

number of individuals, Nmax, that the territory can hold. In analogy with physics, one

may define a chemical potential[34], µ 6 0, so that its absolute value is the basic energy

cost for introducing an individual into the ecosystem. Thus the total energy cost for

introducing an additional individual of the k-th species into the ecosystem is equal to

εk − µ – effectively, all the energy levels are shifted up by a constant amount equal to

this basic cost.

The chemical potential may also be defined as the negative of the ratio of the

energy imbalance to the population imbalance:

µ = −Emax − 〈E〉Nmax − 〈N〉 = − T

Nmax − 〈N〉 , (6.9)

where 〈N〉 is the average population. It follows then that

Nmax = − 1

ln(α)+ 〈N〉, (6.10)

where α = exp(µ/T ) 6 1. This equation has the same structure as Eq. (6.1). Inter-

estingly, the link between the population imbalance and the chemical potential can also

be established formally starting from Eq. (6.2), but with an additional ceiling on the

total number of individuals. The introduction of the ceiling on the population leads to

an additional suppression of the effective birth to death rate ratio which now becomes

αx exp(−εk/T ).

77

Following the standard methods in statistical mechanics[34], one can straight-

forwardly deduce the thermodynamic properties by taking suitable derivatives of F ≡

−T ln Q, the free energy:

F = −T∑

k

ln[1 − ln(1 − αx exp(−εk/T ))]. (6.11)

The average number of individuals in the k-th species, 〈nk〉 ≡ ∂F∂εk

, is given by

〈nk〉 =αxe−εk/T

[1 − αxe−εk/T ][1 − ln(1 − αxe−εk/T )]. (6.12)

In ecological systems, one would expect, in the simplest scenario, that there ought to be

a co-existence of all species in our model with an infinite population of each when there

are no constraints whatsoever or equivalently when Emax = Nmax = ∞. Noting that

α = 1 when Nmax = ∞, this is realized only when 1 − x exp(−εk/T ) = 0 for any k,

which, in turn, is valid if and only if T = ∞ and x = 1. We will restrict our analysis1 in

what follows to the case of x = 1.

Following the treatment in physics[34], we postulate that the number of energy

levels, or equivalently the number of species, per unit energy interval (the density of

states) is proportional to the area of the ecosystem and additionally scales as εd with

d > 0 in the limit of small ε. This is entirely plausible[77, 12, 68, 73] because one would

1Our model also shows interesting behavior for x values different from 1. When x < 1, thebirth attempts are fewer than the death attempts. In this case, the system is sparsely occupied.On the other hand, when x > 1, one finds, generally, for sufficiently large Emax (and Nmax = ∞)that the occupancy of the excited levels is small and independent of Emax with the populationof the ground state increasing proportional to Emax. This arises from Eq. (6.12): 〈n1〉 → ∞and 〈n2〉, 〈n3〉, ... are finite as T → ε1/ ln(x) from below.

78

generally expect the density of states to have zero weight both below the smallest energy

intake and above the largest intake and a maximum value somewhere in between.

In a continuum formulation, one obtains the following expressions for the average

energy 〈E〉 and population 〈N〉 of the ecosystem:

〈E〉 ≡S∑

k=1

εk〈nk〉 = Td+2I1(α) (6.13)

and

〈N〉 ≡S∑

k=1

〈nk〉 = Td+1I0(α), (6.14)

where

Im(α) =

∫ ∞

0

αe−t

[1 − αe−t][1 − ln(1 − αe−t)]td+m

dt. (6.15)

Note that, except for the second factor in the denominator, which is subdominant, these

integrals (I0(α) and I1(α)) are identical to those of a Bose system. The key point is that

(in a Bose system and here) they are both 0 when α = 0 and monotonically increase to

their separate finite maximum values at α = 1.

6.2 Results and conclusions

For a given Emax = EM , Eq. (6.1) can only be satisfied over a finite range of

temperatures. The temperature cannot exceed Tmax = EM , because 〈E〉 cannot be

negative. At this temperature, 〈E〉 = α = 〈N〉 = Nmax = 0 and the territory is bereft

of life. Also, the lowest attainable temperature, Tmin, corresponds to Nmax = ∞ and

79

satisfies the equation

EM = Tmin + Td+2min

I1(1) (6.16)

(recall that I1 is largest when α = 1).

The increase of 〈E〉 on decreasing T is counter-intuitive from a conventional

physics point of view. The simple reason for this in an ecosystem is that a decreasing

T corresponds to a decreasing energy imbalance (first term on rhs of Eq. (6.1) thereby

leading to a corresponding increase in the second term, which is 〈E〉. This increased

energy utilization, in turn, leads to an increase in the population of the community.

We have carried out detailed computer simulations (Figs. 6.1 and 6.2) of systems

with constraints on the total energy and the total population and find excellent accord

with theory. Fig. 6.1 shows the results of simulations corresponding to the case α = 1

with a constraint just on the total energy of the ecosystem.

A strong hint that there is a link between the behaviors of the ecosystem at Tmin

and the physical system of Bosons at the BEC transition is obtained by noting that both

these situations are characterized by α = 1 or equivalently a zero basic cost (µ = 0) for

the introduction of an individual or a particle into the system.

Let us set the value of Nmax equal to NM , the average population in a system with

Emax = EM and Nmax = ∞, and consider the effect of varying the temperature. As in

Bose condensation, one can identify (for d > 0) a critical temperature, Tc, as the lowest

temperature above which the first term on the rhs of Eq. (6.10) can be neglected[34].

At Tc, 〈N〉 ≈ Nmax = NM and α ≈ 1. Recall, however, that with only the energy

ceiling, at T = Tmin the ecosystem was characterized by α = 1 and 〈N〉 = NM . This

80

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16

Emax

Tm

in

Fig. 6.1. Comparison of the results of computer simulations of an ecosystem with

theory. We consider a system with 100, 000 energy levels with εk = k2/3, k = 1..100, 000,corresponding to d = 1/2. We work with a constant Emax (the figure shows the resultsfor several values of Emax) and consider a dynamical process of birth and death. Wehave verified that the equilibrium distribution is independent of the initial condition.At any given time step, we make a list of all the individuals and the empty energylevels. One of the entries from the list is randomly picked for possible action with aprobability proportional to the total number of entries in the list. Were an individualto be picked, it is killed with 50% probability or reproduced (an additional individualof the same species is created) with 50% probability provided the total energy of thesystem does not exceed Emax. When an empty energy level is picked, speciation occurswith 50% probability and a new individual of that species is created provided again theenergy of the system does not exceed Emax. With 50% probability, no action is taken.This procedure is iterated until equilibrium is reached. The effective temperature ofthe ecosystem is defined as the imbalance between Emax and the average energy of thesystem (Eq. (6.1)). The figure shows a plot of the effective temperature of the ecosystem

deduced from the simulations. The circles denote the data averaged over a run of 109

time steps with the last 500 million used to compute the average temperature while thesolid line is the theoretical prediction.

81

confirms that Tc = Tmin. The macroscopic depletion of the community population,

when T < Tmin is entirely akin to the macroscopic occupation of the ground state in

BEC (Fig. 6.2).

When T is larger than Tmin, the energy resources are sufficient to maintain the

maximum allowed population and 〈N〉 = NM . Physically, the state at Tmin corre-

sponds to a maximally efficient use of the energy resources so that any decrease in T

below Tmin inevitably leads to a decrease of the population and the biodiversity in the

community. The existence of this novel transition is quite general and is independent of

specific counting rules such as the ones used in the classic examples of distinguishable

and indistinguishable particles[34].

82

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

T/Tmin

Fig. 6.2. Phase transition in an ecosystem with Nmax = NM = 65 and d = 1/2.

The dashed and solid curves are plots of theoretical predictions of〈N〉NM

and∂〈E〉∂T

TminEM

respectively versus scaled temperature T/Tmin, where Tmin = 11.6 and EM = 1000.The data points denote the results of simulations. ∂〈E〉/∂T is a quantity analogousto the specific heat of a physical system and has the familiar λ shape associated withthe superfluid transition in liquid helium[34]. It was obtained in the simulations as thederivative of the interpolated values of 〈E〉. The continuous phase transition is signaledby the peak in ∂〈E〉/∂T (and the corresponding drop in 〈N〉) on lowering the temperatureand occurs in the vicinity of the temperature Tmin (the transition temperature movescloser to Tmin as the system size increases).

83

Appendix

A novel ensemble in statistical physics

The conventional grand canonical ensemble in physics describes two systems, one

of which (“the reservoir”) has many more degrees of freedom than the other (“the sys-

tem”). They are placed in contact with each other and allowed to exchange both energy

and particles. The average values of the energy and number of particles are controlled

by the temperature T and the chemical potential µ, respectively. The utility of such an

ensemble lies in the fact that it closely represents the conditions under which experiments

are often performed.

Recently, a theoretical scheme for the modification of the grand canonical ensem-

ble has been proposed[84] in the context of an ecosystem. In that ensemble, the system

consists of an infinite number of energy levels and is coupled with a reservoir. The total

energy and number of particles of both the system and the reservoir are fixed at some

predefined values Emax and Nmax, respectively. Again, the particles and the energy

associated with them are permitted to travel back and forth from the system to the

reservoir. It was found both theoretically and via simulations that, in equilibrium, the

temperature T played a dual role: it controlled the average energy 〈E〉 of the system

(as in the conventional approach) and at the same time it was equal to the imbalance

84

between Emax and 〈E〉. The chemical potential also had a dual role: apart from control-

ling the average number of particles 〈N〉, T/|µ| served as an imbalance between Nmax

and 〈N〉.

The above scheme arises naturally in the studies of the dynamics of ecosystems.

Indeed, as a first approximation, an ecosystem can be modeled as a community of non-

interacting individuals (particles) belonging to different species (energy levels). The

concept of finite Emax and Nmax arises from the limited resources available to an

ecosystem such as space, solar radiation and freshwater supplies.

In the simplest ecologically relevant scenario, the probabilities of the birth and

death events (arrival from / departure into the reservoir) in a given level can be chosen to

be density independent or proportional to the current population of the species. A non-

zero birth rate (speciation) is ascribed to unoccupied levels. This leads to a logarithmic

distribution[83] of the the number of individuals in each level.

Our earlier theoretical and computational studies of the model of the ecosystem

showed that in the absence of the constraints on the total population (Nmax = ∞)

an ecosystem may organize in the vicinity of a phase transition akin to Bose-Einstein

condensation. The transition is signalled by the macroscopic depletion of the population

below a critical temperature.

In this 2Appendix, we generalize our previous work by applying our model to sys-

tems obeying Bose-Einstein, Fermi-Dirac and Boltzmann statistics and carrying out both

theoretical and computational studies. We show that for a system with Bose-Einstein

statistics, the results are similar to the previously studied ecology case. Interestingly,

the systems obeying Fermi-Dirac or Boltzmann statistics also exhibit a first order phase

85

transition and, unlike the ecological or Bose-Einstein cases, this behavior is independent

of the density of states. Also, we expand our previous study of ecological systems and

demonstrate that depending on the value of the birth/death rate ratio there exist three

regimes with distinct behaviors.

The outline of the Appendix is as follows. In Section A.1, starting with the

one-step master equation, we provide a derivation of the partition function of the new

ensemble. For Boltzmann statistics, we demonstrate that the correct counting arises

naturally and the Gibbs paradox is averted. In Section A.2, we consider different types

of statistics and develop the numerical algorithms for simulations. Also, we present

the results of the simulations and compare them to theoretical expectations. Finally,

in Section A.3, we discuss the connections between our ensemble and those in classical

physics and consider a few examples of our model.

A.1 Theoretical framework

Consider S independent boxes in which balls (particles) can be inserted or re-

moved. We label the boxes (energy levels) using the numbers 1, 2, .., S. We postulate

that the dynamics of the balls is governed by simple, physically motivated rules. Our

goal will be to determine the steady state configuration of the system under these rules.

Let N represent the total number of balls in the system.

Let us postulate a constant death rate (or removal rate) per ball equal to d. The

rate of insertion (which may be thought of as a birth rate) of a ball into a given box may

be taken generally to be a function, b(n), of n, the number of balls within the box.

86

Let P (t;n) denote the probability that a given box contains n balls at time t.

The time evolution of P is regulated by the master equation[82]:

dP (t;n)

dt= P (t;n + 1)(n + 1)d +

P (t;n − 1)b(n − 1) − P (t;n)(nd + b(n)). (A.1)

The first (second) term on the right hand side corresponds to the removal (insertion)

of a ball from the box containing n + 1 (n − 1) particles leading to an enhancement of

the probability on the left hand side, whereas the last term corresponds to a depletion

of this probability. The stationary solution can be seen to satisfy detailed balance and

corresponds to an equilibrium situation[82] with

P (n) ∝n−1∏

m=0

b(m)

(m + 1)d. (A.2)

It follows that, when there are S boxes, all satisfying the same birth-death rules,

the unique equilibrium solution is

P (n1, n2, ...nS) =S∏

k=1

P (nk). (A.3)

One can readily work out other special cases of the framework we have presented.

If one chooses b(n) = b0(n + 1), one obtains a pure exponential distribution P (n) =

zn(1− z), where z = b0/d, which, in turn, leads to the Bose-Einstein distribution[43] for

non-degenerate energy levels, i.e. P (n1, n2, ...nS) = zN (1 − z)S . On the other hand, if

b(n) = 0 for any n greater than 0 and equal to b0 otherwise, we find P (n) = zn/(1 + z)

87

for n = 0 or 1 and zero for other values of n and the Fermi-Dirac distribution[43]

P (n1, n2, ...nS) = zN /(1+z)S , provided each of the ni’s is 0 or 1 and P (n1, n2, ...nS) = 0

is zero otherwise.

Note, that the same framework lends itself to the study of the species abun-

dance problem in ecology[45, 83, 84]. Consider the dynamical rules of birth, death

and speciation which govern the population of an individual species. In order to en-

sure that the community will not become extinct, speciation may be introduced by

ascribing a non-zero probability of the appearance of an individual of a new species, i.e

b(0) = b0 6= 0. If one chooses b(n) = b0n for n > 0 (this amounts to the assumption

that the birth rate per individual is constant), one obtains the logarithmic distribution

P (n) = [1 − ln(1 − z)]−1zn/n which, in turn, leads to the well-known Fisher log-series

distribution[35], i.e. 〈φn〉 = θzn/n, where θ = S/[1 − ln(1 − z)] and n > 0. Here, 〈φn〉

represents the average number of species (boxes) with population n.

If b(n) = b0 is taken to be constant, one finds the Poisson distribution P (n) =

e−zzn/n!. This leads to P (n1, n2, ...nS) ∝ zN /∏S

k=1nk!, which is the celebrated Boltz-

mann counting in physics in the grand canonical ensemble and where z plays the role

of a fugacity and N =∑

k nk. It is noteworthy that, unlike in conventional classical

treatments[43] in which one obtains an additional factor of N !, here one gets the correct

Boltzmann counting and one avoids the well-known Gibbs paradox in this scheme. If one

were to ascribe energy values εk to each of the boxes and enforce a fixed average total

energy, one would get the standard Boltzmann result that the probability of occupancy

of an energy level ε is proportional to e−βε, where β is proportional to the inverse of

the temperature.

88

Now let us assign the energy εk to the k-th level (box) so that 0 < ε0 < ε1 < . . . ,

and introduce the constraints Emax and Nmax on the total energy and population of

the system.

The partition function for the system with fixed Emax and Nmax may be written

as

Q =∑

{nk}

∏

k

P (nk)Θ(Emax − ε1n1 − ε2n2 − ...)

Θ(Nmax − n1 − n2 − ...), (A.4)

where Θ(x) is the unit step function, equal to 0 for x < 0 and 1 for x > 0.

Using the integral representation for Θ function[70] one can rewrite the above

equation in the following form:

Q =∑

{nk}

∏

k

P (nk)

∫

γ1

dz12πiz1

ez1(Emax−ε1n1−ε2n2−...)

∫

γ2

dz22πiz2

ez2(Nmax−n1−n2−...) =

∫

γ1

dz12πiz1

ez1Emax

∫

γ2

dz22πiz2

ez2Nmax∏

k

∞∑

n=0

P (n)e−(z2+z1εk)n, (A.5)

where the contours γ1,2 are parallel to the imaginary axis with all their points having

a fixed real part x1,2 (i.e. z1,2 ∈ γ1,2 ⇔ z1,2 = x1,2 + iy1,2, −∞ < y1,2 < +∞). The

integral is independent of x1,2 provided x1,2 is positive[70].

89

Let

e−f(x)

=

∞∑

n=0

P (n)e−xn

. (A.6)

Then

Q =

∫

γ1

∫

γ2

dz1dz2eg(z1,z2), (A.7)

where

g(z1, z2) = − ln(z1) − ln(z2) + z1Emax + z2Nmax −

∑

k

f(z2 + z1εk). (A.8)

In order to evaluate the integral in Eq.(A.7) we will apply the steepest descent method[70].

Let us expand g(z1, z2) about the point (x1 + i0, x2 + i0), where it is maximum:

g(z1, z2) ≈ g(x1, x2) − 1

2!gz21

(x1, x2)y21−

1

2!gz22

(x1, x2)y22− gz1,z2

(x1, x2)y1y2, (A.9)

where gzk11

,zk22

≡ ∂k1+k2g(z1,z2)

∂zk11

∂zk22

. Because g(x1, x2) is a maximum,

gz1(x1, x2) = gz2

(x1, x2) = 0. (A.10)

90

Substituting this expression into Eq.(A.7) and performing the integration one obtains

Q =eg(x1,x2)

√gz21

(x1, x2)gz22

(x1, x2) − g2z1,z2

(x1, x2)=

exp[Emax−µNmax

T −∑k f(

εk−µT

)]

√1 −∑k

ε2k+µ2

T 2f ′′(

εk−µT

) (A.11)

Here we omit the constant factor in the expression for Q and replace x1 and x2 by 1/T

and −µ/T , respectively (T > 0 and µ 6 0). Eq.(A.10) yields

Emax = T +∑

k

εkf ′(

εk − µ

T

)(A.12)

and

Nmax = −T

µ+∑

k

f ′(

εk − µ

T

). (A.13)

The free energy, F , is given by

F ≡ −T ln Q = −Emax + µNmax + F0 + F1, (A.14)

where

F0 = T∑

k

f

(εk − µ

T

)(A.15)

91

and

F1 =1

2T ln

1 −

∑

k

ε2k

+ µ2

T 2f ′′(

εk − µ

T

) . (A.16)

The last term, F1, can be neglected for large enough systems because F1/F0 ∝ ln(V )/V ≪

1, where V is the characteristic size of the system.

Using Eq.(A.14), one can find the entropy S:

S = −∂F

∂T= −

∑

k,n

Pn,k(T ) lnPn,k(T )

Pn, (A.17)

where

Pn,k(T ) =Pne−εkn/T

∑m Pme−εkm/T

. (A.18)

The average population of the k-th level is given by

〈nk〉 =∂F

∂εk= f ′

(εk − µ

T

)(A.19)

and the average population and the total energy of the system are defined as 〈N〉 =

∑k〈nk〉 and 〈E〉 =

∑k εk〈nk〉, respectively. Thus one can rewrite Eqs.(A.12) and

(A.13) as

Emax = T + 〈E〉 (A.20)

92

and

Nmax = − 1

ln(α)+ 〈N〉, (A.21)

where α ≡ exp(µ/T ).

The above equations demonstrate the dual role of the temperature and the chem-

ical potential, which was discussed earlier and represent the central result of our deriva-

tion.

Let us postulate that εk = k1/(d+1) with d > −1, i.e. the number of energy levels

per unit energy interval (the density of states) scales as V εd, where V is the size of the

system. In what follows we will work in units in which V is set equal to 1.

In a continuum formulation, one obtains the following expressions for the average

energy 〈E〉 and population 〈N〉 of the ecosystem:

〈E〉 = Td+2I1(α) (A.22)

and

〈N〉 = Td+1I0(α), (A.23)

where

Im(α) = (d + 1)

∫ ∞

0f′[t − ln(α)]t

d+mdt. (A.24)

93

In the next section we will show, by explicitly calculating the integrals in Eq.(A.24),

that for systems with Bose-Einstein, Fermi-Dirac and Boltzmann statistics, I0,1(α) are

0 when α = 0 and monotonically increase as α approaches 1.

From Eqs.(A.20)-(A.23) one can see that, for a given Emax, the temperature T

cannot become lower than some value Tmin, which occurs when there is no constraint

on the total population, i.e Nmax = ∞. Similarly, the system reaches the maximum

T = Emax when Nmax = 0 and the system is empty.

In order to analyze whether the system can undergo a phase transition, we will

use a scheme similar to the familiar one in Bose-Einstein condensation[34]. Let us fix

the value of Nmax in Eq.(A.21) and vary the temperature (this can be done by varying

Emax). At very high temperatures, the values of α are very small and thus one can

neglect the first term (imbalance) in the rhs of Eq.(A.21). This means that the system

is populated to its full capacity Nmax. As we decrease the temperature, α approaches 1

and the imbalance can no longer be neglected (the system undergoes depletion). If I0(1)

is finite then one can introduce a critical temperature Tc

Tc =

(NmaxI0(1)

) 1d+1

(A.25)

above which 〈N〉 ≈ Nmax and below which the system undergoes a rapid depletion.

Note that in our ensemble the imbalance −1/ ln(α) acts as a zero groundstate level in

the conventional grand canonical ensemble: the macroscopic depletion of the population

of the former is analogous to the macroscopic occupation of the groundstate of the latter.

94

Finally, let us note that α = 1 is related to two cases: first, it enters the expression

for Tc and, second, α = 1 when the system has just the Emax constraint (Nmax = ∞).

This suggests that Tmin = Tc provided that Tc exists (i.e. I0(1) is finite). Indeed, let

us consider the following scenario: when Nmax = ∞ the system organizes at Tmin with

some average population NM and from Eq.(A.23) it follows that Tmin =

(NMI0(1)

) 1d+1

.

But this is also the critical temperature for the system with Nmax = NM . Note that

in the simulations the actual transition temperature T ′c

slightly differs from Tc since for

finite Nmax the value of α cannot reach 1.

A.2 Theoretical and numerical results for systems with different statis-

tics

We now consider four distinct cases and demonstrate that the behavior observed

in simulations is in excellent accord with the theoretical predictions.

A.2.1 Boltzmann Statistics (Figures A.1, A.2 and A.3)

A.2.1.1 Theory

For Boltzmann statistics (bn = b and dn = dn) the average population of the k-th

level and Eqs.(A.20) and (A.21) give

〈nk〉 = αre−εk/T , (A.26)

Emax = T + (d + 1)Γ(d + 2)αrTd+2 (A.27)

95

and

Nmax = − 1

lnα+ Γ(d + 2)αrT

d+1, (A.28)

where r = b/d and Γ(x) =∫∞0

tz−1e−tdt is the gamma function.

One can see that I0(1) is finite for any d. Thus the system with Boltzmann

statistics can undergo a first order phase transition (see Figure A.3).

From Eq.(A.26) it follows that the average population of any level cannot exceed

r, hence r should be large (we used r = 100 in simulations).

A.2.1.2 Simulations

At any given time step, a level is randomly picked and a random number R in the

interval [0, 1) is generated. If R < b/(n + 1) and there is sufficient energy and available

particles in the reservoir a birth event occurs (here n is the occupancy of the level). If

R > 1− d and the level is occupied, a death event occurs. Otherwise no action is taken.

Note that in this scheme the birth and death rates are chosen to be bn = b/(n + 1) and

dn = d (this is equivalent to bn = b and dn = dn since only the ratio bn/dn+1 enters the

expression for the probability). This choice is helpful because both bn and dn are finite

for arbitrary n.

The absolute value of b (recall that d = b/r) is not important provided that it is

less than 0.5.

96

A.2.2 Fermi-Dirac Statistics (Figures A.4, A.5, A.6 and A.7)

A.2.2.1 Theory

For Fermi-Dirac statistics (b0 = b, d1 = d) the average population of the k-th

level and Eqs.(A.20) and (A.21) give

〈nk〉 =αre−εk/T

1 + αre−εk/T, (A.29)

Emax = T − (d + 1)Γ(d + 2)Lid+2(−αr)Td+2 (A.30)

and

Nmax = − 1

ln α− Γ(d + 2)Lid+1(−αr)Td+1, (A.31)

where Lin(z) =∑∞

k=1zk/kn is a polylogarithm function.

As in the case of Boltzmann statistics, the system can undergo a phase transition

for any d.

Note that Eq.(A.29) can be represented as

〈nk〉 =e−(εk−µ∗)/T

1 + e−(εk−µ∗)/T, (A.32)

where µ∗ = µ + T ln r is an ‘effective’ chemical potential and when positive can be

associated with the Fermi energy εF . If a system has a constraint on Emax only, then

µ = 0 and εF = T ln r which means that at small temperatures only the levels with

energies less than εF are filled (this effect is shown in Fig. A.6: if one increases r and

decreases T so that T ln r remains finite, one would observe a sharper effect).

97

A.2.2.2 Simulations

The simulation algorithm is analogous to the case of Boltzmann statistics:

At any given time step, a level is randomly picked and a random number R in the interval

[0, 1) is generated. If the level is empty and R < b and there are sufficient energy and

particles in the reservoir a birth event occurs. If R > 1 − d and the level is occupied a

death event occurs. Otherwise no action is taken.

A.2.3 Bose-Einstein Statistics (Figures A.8 - A.14)

A.2.3.1 Theory

For Bose-Einstein statistics (bn = b(n + 1), dn = dn) the average population of

the k-th level and Eqs.(A.20) and (A.21) give

〈nk〉 =αre−εk/T

1 − αre−εk/T, (A.33)

Emax = T + (d + 1)Γ(d + 2)Lid+2(αr)Td+2 (A.34)

and

Nmax = − 1

lnα+ Γ(d + 2)Lid+1(αr)T

d+1. (A.35)

Let us consider the behavior of the system with different values of r.

When r < 1, the system is underpopulated, i.e. 〈nk〉 is finite for any value of T .

Since Lid+2(r) is finite for any value of d > −1, the system undergoes a phase transition.

98

When r = 1, Lid+1(1) diverges when d 6 0 and is finite otherwise. Thus the

system can undergo a continuous phase transition only for a class of density of states

with d > 0 (see Figure A.14).

Finally, when r > 1, one finds, generally, for sufficiently large Emax (and Nmax =

∞) that the occupancy of the excited levels is small and independent of Emax with

the population of the ground state increasing proportional to Emax. This arises from

Eq. (A.33): 〈n1〉 → ∞ and 〈n2〉, 〈n3〉, ... are finite as T → ε1/ ln(r) from below. The

behavior of the system is qualitatively independent of d.

A.2.3.2 Simulations

The simulation algorithm is the following:

At any given time step, a level is randomly picked and a random number R in the interval

[0, 1) is generated. If R < b and there are sufficient energy and particles in the reservoir a

birth event occurs. If R > 1−d and the level is occupied a death event occurs. Otherwise

no action is taken. Note that the birth and death rates are chosen as bn = b and dn = d

(these rates lead to the same ratio bn/dn+1 as bn = b(n + 1) and dn = dn).

A.2.4 Ecological case (Figures A.15 and A.16)

A.2.4.1 Theory

For the ecosystem obeying logarithmic distribution (bn = b(n + δn,0), dn = dn)

the average population of the k-th level is[84]

〈nk〉 =αre−εk/T

[1 − αre−εk/T ][1 − ln(1 − αre−εk/T )]. (A.36)

99

Here, without loss of generality, we choose the speciation rate b0 to be equal to b.

One can see that, apart from the factor 1 − ln(1 − αre−εk/T ), the expression for

〈nk〉 is similar to that of Bose-Einstein statistics. Indeed, a comparison of Figures A.15

and A.16 with Figures A.8 and A.9 demonstrates that systems in both cases have very

similar behavior. As in the Bose-Einstein case, the system is able to undergo a phase

transition only when d > 0 when r = 1.

In ecological systems, one would expect, in the simplest scenario, that there ought

to be a co-existence of all species in our model with an infinite population of each when

there are no constraints whatsoever or equivalently when Emax = Nmax = ∞. This

ecologically meaningful case corresponds to r = 1.

A.2.4.2 Simulations

The simulation algorithm is the following:

At any given time step, a level is randomly picked and a random number R in the interval

[0, 1) is generated. If the level is empty (occupied) and R < b (R < bn/(n+1)) and there

are sufficient energy and particles in the reservoir a birth event occurs. If R > 1− d and

the level is occupied a death event occurs. Otherwise no action is taken. Note that the

birth and death rates are chosen as bn = b[n/(n + 1) + δn,0] and dn = d.

A.3 Conclusions

We conclude with a brief discussion on the relationship of the novel ensemble that

we have studied here with standard ensembles in physics. The familiar microcanonical

ensemble is obtained on replacing both Θ functions in Eq. (A.4) by Dirac delta functions.

100

Indeed, by implementing the steepest descent method on Emax term, one obtains the

canonical ensemble whereas on using the steepest descent method on both the Emax

and Nmax terms, one obtains the grand canonical ensemble.

Our numerical scheme is readily modified for the study of the canonical ensemble.

At each timestep, two events occur (provided that the energy constraint is satisfied): two

levels are chosen randomly and a death of a particle in one level is followed by a birth

of the particle in another, provided that the energy constraint is satisfied (otherwise no

action is taken). No action is taken if the first level chosen has no particles in it. We have

implemented this scheme and confirmed that it is in excellent accord with theoretical

expectations.

As a possible application of our model, one can consider the effect of photoex-

citation (and/or photoionization), which occurs when the radiation produced when an

external source interacts with the surrounding atomic gas (e.g., planetary nebulae or OB

star associations embedded in gas clouds[29]). In this case, the processes of birth/death

are represented by excitation/deexcitation (ionization/recombination). The maximum

number of electrons that can possibly go into excited states (or, in the case of ionization,

leave the atom) corresponds to Nmax and the radiation flux can be associated with

Emax. One would expect then that the stimulated emission from the gas will follow the

phase transition scenario described here, i. e. on decreasing Emax below some critical

value one would observe a rapid decline in the flux of stimulated emission.

A more direct example of our ensemble is a shopping game. Consider a con-

sumer shopping in a supermarket. The energy levels correspond to the different types

of products (the products are distinguished from each other by their price only). The

101

total amount of money that the consumer has corresponds to Emax. The analog of

Nmax is the limit on the maximum number of items that the consumer could buy and

is determined, say, by the size of the consumer’s shopping cart. The dynamics of the

game consists of the following rules. The analog of birth is selection of an item from the

store shelf and adding to the cart provided that the number of items in the cart does

not exceed the threshold and provided that the shopper has sufficient money to buy all

the merchandise in the cart. The removal of an item from the cart and returning it to

the shelves corresponds to a death event.

Let us reformulate the rules discussed earlier in the language of this shopping

game. For Boltzmann statistics, the addition event corresponds to the placement of an

item of a randomly picked product in the cart and the death event is the removal of a

random item from the cart. For Bose-Einstein statistics, the addition event is the same

as for Boltzmann statistics and the death event corresponds to the removal of an item of

a randomly picked product already in the cart. For this case, r is a measure of the ratio

of addition to removal attempts. Fermi statistics has the same rules as Bose-Einstein

statistics with the constraint that at most there is just one item in the cart of any given

product. The ecology case consists of addition of an item of a product already contained

in the cart with a probability proportional to the number of such items, a non-zero

probability of the addition of an item of a product not already represented in the cart

and the removal of a randomly picked item present in the cart. Note that the above

rules are not unique and there are many ways to obtain any desired statistics.

Our two key results can be stated as follows. First, the average quantity of money

remaining in the shopper’s wallet (the imbalance) is non-zero and determines the relative

102

numbers of items of different products represented in the cart. The imbalance magnitude

plays the role of temperature in the system. For a given Emax = EM , with no Nmax

constraint, let the average number of items in the cart be denoted by NM . The novel

transition that we observe corresponds to the case in which Nmax = NM and occurs on

varying Emax or the total money in the wallet. There is a sharp depletion in the number

of items in the cart as Emax drops below EM . Interestingly this phase transition occurs

for any density of states for the Boltzmann and Fermi-Dirac cases and is a first order

but only for the ‘right’ density of states, when r = 1, for the ecology and Bose-Einstein

cases, where it becomes a continuous transition.

0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Emax

Tm

in

Fig. A.1. The results of the simulations of the novel ensemble with Boltzmann statistics.

r = 100, Nmax = ∞, εk = k2/3, k = 1..1000. The solid line denotes the theoreticalprediction.

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

25

30

35

40

T

⟨N⟩

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

10

20

30

40

50

60

70

80

90

100

T

⟨E⟩

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−20

0

20

40

60

80

100

120

140

160

T

c v

Fig. A.2. The results of the simulations of the novel ensemble with Boltzmann statistics.

r = 100. εk = k2/3, k = 1..1000, Nmax = 35, Tmin ≈ 0.62. Here Cv = ∂〈E〉/∂T is thespecific heat of a system. The peak in the specific heat occurs at the phase transition.The solid line denotes the theoretical prediction.

104

100 200 300 400 500 600 700 800 900 1000

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

T

µ

100 200 300 400 500 600 700 800 900 1000−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1x 10

−7

T

dµ/

dT

100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6x 10

6

T

Cv

100 200 300 400 500 600 700 800 900 1000

2

2.5

3

3.5

4

4.5

5

5.5

6

T

Cv/⟨N

⟩

Fig. A.3. Boltzmann Statistics. r = 1, d = 1, Nmax = 106.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

20

40

60

80

100

120

140

160

180

Emax

Tm

in

Fig. A.4. The results of the simulations of the novel ensemble with Fermi-Dirac statis-

tics. r = 100, Nmax = ∞, εk = k3/2, k = 1..1000. The solid line denotes the theoreticalprediction.

105

0 50 100 150 200 250 300 350 400 450 50015

20

25

30

35

40

45

50

55

60

65

70

T

⟨N⟩

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5x 10

4

T⟨E

⟩

0 50 100 150 200 250 300 350 400 450 50040

60

80

100

120

140

160

180

T

c v

Fig. A.5. The results of the simulations of the novel ensemble with Fermi-Dirac statis-

tics. r = 100, εk = k3/2, k = 1..1000, Nmax = 65, Tmin ≈ 115. The peak in the specificheat occurs at the phase transition. The solid line denotes the theoretical prediction.

106

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

εk

⟨nk⟩

Fig. A.6. Plot of 〈nk〉 versus εk for the system with Fermi-Dirac statistics. r = 100,

Nmax = ∞, Tmin = 3.1, εF = 14.2, εk = k2/3. The solid line denotes the theoreticalprediction.

400 500 600 700 800 900 1000

−0.5

−0.4

−0.3

−0.2

−0.1

0

T

µ

400 500 600 700 800 900 1000−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−3

T

dµ/

dT

400 500 600 700 800 900 10001.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5x 10

6

T

Cv

400 500 600 700 800 900 10001

2

3

4

5

6

7

T

Cv/⟨N

⟩

Fig. A.7. Fermi Statistics. r = 1, d = 1, Nmax = 106.

107

0 200 400 600 800 1000 12000

1

2

3

4

5

6

Emax

Tm

in

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

50

100

150

200

250

300

350

400

450

Emax

Tm

in

Fig. A.8. The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 1, Nmax = ∞, εk = k0.5, k = 1..100, 000 (left), εk = k3/2, k =1..1000(right). The solid line denotes the theoretical prediction.

108

1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

T

⟨N⟩

0 100 200 300 400 500 6000

20

40

60

80

100

120

140

160

180

200

T

⟨N⟩

1 2 3 4 5 6 7 8 90

200

400

600

800

1000

1200

T

⟨E⟩

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5x 10

4

T

⟨E⟩

1 2 3 4 5 6 7 8 90

50

100

150

200

250

T

c v

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90

T

c v

Fig. A.9. The results of the simulations of the novel ensemble with Bose-Einsteinstatistics. r = 1.Left panel: εk = k0.5, k = 1..100, 000, Nmax = 70, Tmin ≈ 4.87.

Right panel: εk = k3/2, k = 1..1000, Nmax = 275, Tmin ≈ 115. The peak in the specificheat occurs at the phase transition. Note the absence of a phase transition when d isnegative. The solid line denotes the theoretical prediction.

109

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

100

200

300

400

500

600

700

800

900

1000

Emax

Tm

in

0 100 200 300 400 500 6000

2

4

6

8

10

12

Emax

Tm

in

Fig. A.10. The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.33/0.67. Nmax = ∞. Left panel: εk = k3/2, k = 1..1000. Right panel :

εk = k2/3, k = 1..10000. The solid line denotes the theoretical prediction.

110

0 200 400 600 800 1000 1200 14000

5

10

15

20

25

30

35

40

45

T

⟨N⟩

0 5 10 15 20 250

2

4

6

8

10

12

14

16

T

⟨N⟩

0 200 400 600 800 1000 1200 14000

0.5

1

1.5

2

2.5

3

3.5x 10

4

T

⟨E⟩

0 5 10 15 20 250

50

100

150

200

250

300

350

400

450

500

T

⟨E⟩

0 200 400 600 800 1000 1200 14000

5

10

15

20

25

30

T

c v

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

T

c v

Fig. A.11. The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.33/0.67. Left panel: Nmax = 45, εk = k3/2, k = 1..1000, Tmin ≈ 545.

Right panel: Nmax = 16, εk = k2/3, k = 1..10000, Tmin ≈ 7.5.

111

0 10 20 30 40 50 60 70 80 90 1000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Emax

T

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

10

20

30

40

50

60

70

80

90

100

T

⟨ N⟩

Fig. A.12. The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.67/0.33, Nmax = ∞, εk = k2/3, k = 1..1000. Solid lines and circlesrepresent theory and simulations, respectively. Note the deviation from the theory asthe system approaches the “infinite” temperature ε1/ ln(r). The solid line denotes thetheoretical prediction.

1 1.5 2 2.5 3 3.50

10

20

30

40

50

60

70

80

90

100

T

<E><N>

Fig. A.13. The results of the simulations of the novel ensemble with Bose-Einstein

statistics. r = 0.67/0.33, Nmax = 50, εk = k2/3, k = 1..1000

112

100 200 300 400 500 600 700 800 900

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

T

µ

100 200 300 400 500 600 700 800 900−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

x 10−4

T

dµ/

dT

100 200 300 400 500 600 700 800 9000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

6

T

Cv

100 200 300 400 500 600 700 800 9002

2.5

3

3.5

4

4.5

T

Cv/⟨N

⟩

Fig. A.14. Bose Statistics. r = 1, d = 1, Nmax = 106.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

100

200

300

400

500

600

Emax

T

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

4

4.5

5

Emax

T

Fig. A.15. The results of the simulations of the model of an ecosystem with r = 1,

Nmax = ∞. Left panel: εk = k3/2, k = 1..1000. Right panel : εk = k2/3, k = 1..1000.The solid line denotes the theoretical prediction.

113

0 100 200 300 400 500 600 70040

60

80

100

120

140

160

T

⟨N⟩

1 2 3 4 5 6 71

2

3

4

5

6

7

8

9

10

T

⟨N⟩

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3

3.5x 10

4

T

⟨E⟩

1 2 3 4 5 6 70

10

20

30

40

50

60

70

80

90

100

T

⟨E⟩

0 100 200 300 400 500 600 70020

30

40

50

60

70

80

90

T

Cv

1 2 3 4 5 6 78

10

12

14

16

18

20

22

24

T

Cv

Fig. A.16. The results of the simulations of the model of an ecosystem with r = 1. Left

panel: Nmax = 200, εk = k3/2, k = 1..1000, Tmin ≈ 400. Right panel: Nmax = 10,

εk = k2/3, k = 1..1000, Tmin ≈ 3.6. The peak in the specific heat occurs at the phasetransition. Note that the similarity between this figure and Fig. A.9 for Bose-Einsteinstatistics.

114

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Vita

Igor Volkov

Department of Physics

104 Davey Lab

The Pennsylvania State University

University Park, PA 16802-6300

volkov@psu.edu

(814) 880-7930 (Phone)

(814) 865-3604 (Fax)

Education

1998 — : Graduate Student at The Pennsylvania State University.

Thesis advisor: Jayanth Banavar

Diploma 1996 : Physics, Belarusian State University, Belarus.

Advisor: Serguei Zavtrak