Download - Omri's PhD Thesis
A Demographic Framework of Species Diversity
Thesis submitted for the degree of
“Doctor of Philosophy”
By
Omri Allouche
Submitted to the Senate of the Hebrew University of Jerusalem
December 2013
This work was carried out under the supervision of:
Prof. Ronen Kadmon
Acknowledgements
To Ronen, for your dedicated guidance, in every step of the way, and for always pushing me forward. Thanks
for teaching me, in your quiet and assuring way, how to be a better researcher, and more importantly, a
better person.
To all members of the Kadmon lab, throughout the years, for your help and friendship.
To the many friends and colleagues that gave a good advice along the way.
To Masha, my source of power, whose faith encourages me to try to do things my way,
and to mom, dad, Nurit, Hadas, Chen and Bu, for your dedication, care and endless love.
Abstract
Understanding the mechanisms controlling the diversity of ecological communities is one of the oldest and
most challenging questions in ecology. Hubbell's 'unified neutral theory of biodiversity' (Hubbell 2001) is
the most ambitious attempt to date to develop a general theory in ecology, and has been regarded as
"one of the most important contributions to ecology and biogeography of the past half century" (E.O.
Wilson). However, the basic assumptions of Hubbell's theory (strict neutrality and constant community
size) together with the lack of an explicit demographic basis limit the scope of the theory, its applicability,
and its overall explanatory power. Here I present a new, fully analytical framework for studying species
diversity that relaxes the unrealistic assumptions of Hubbell's theory and extends it to non-neutral and
unsaturated communities. The new framework is capable of explaining a surprisingly wide spectrum of
empirically-observed patterns of species diversity including positive, negative, and unimodal relationships
between species diversity and productivity (Waide et al. 1999), linear and curvilinear local-regional
diversity relationships (Ricklefs 1987, Srivastava 1999), gradual and highly delayed responses of species
diversity to habitat loss (Tilman et al. 1994, Ney-Nifle & Mangel 2000), positive and negative responses of
species diversity to habitat heterogeneity (MacArthur 1972, Tews et al. 2004), the increase of species
diversity with area (Arrhenius 1921), and the decrease of species diversity with geographic isolation
(MacArthur & Wilson 1967). For each of these patterns the framework provides novel insights and
testable predictions that cannot be obtained from (and in some cases contrast) current theories of species
diversity. One important result is that all of the above patterns can be obtained without any differences in
overall fitness between the competing species. This finding strongly supports the 'neutral' paradigm
proposed by Hubbell (Hubbell 2001).
Contents
Introduction …………………………………………………………………………………………. 1
Goals of Research …………………………………………………………………………………... 6
Chapter 1 – Paper
Demographic analysis of Hubbell’s neutral theory of biodiversity ……………………………...…..
9
Chapter 2 – Paper
A general framework for neutral models of community dynamics …………………………………..
17
Chapter 3 – Unpublished
Applications of the Demographic Framework and Its Relation to Patch Occupancy Theory ………..
29
Chapter 4 - Paper
Integrating the Effects of Area, Isolation, and Habitat Heterogeneity on Species Diversity: A
Unification of Island Biogeography and Niche Theory …...………………………………………….
45
Chapter 5 – Unpublished
Habitat Heterogeneity, Area, and Species Diversity ………………………………………………. ...
59
Chapter 6 – Paper
Area–heterogeneity tradeoff and the diversity of ecological communities …….……………………..
69
Chapter 7 – Paper
Reply to Hortal - Patterns of bird distribution in Spain support the area–heterogeneity tradeoff ……
77
Chapter 8 – Paper
Reply to Carnicer - Environmental heterogeneity reduces breeding bird richness in Catalonia by
increasing extinction rate …………………………………………………………………………….
81
Discussion and Conclusions ………………………………………………………………………...
Concluding remarks ……….…………………………………………………………………….……
85
97
References …………………………………………………………………………………………… 96
1
Introduction
Explaining the mechanisms regulating the diversity of ecological communities is one of the oldest and most
challenging questions in ecology. One of the most interesting phenomena in ecology is the apparent
contradiction between the complexity of the factors and interactions that determine the abundance of
individual species, and the simplicity of the patterns of species diversity observed at the community level.
The occurrence of similar patterns of species diversity over a wide range of taxa and ecosystems has been
interpreted as a hint that all ecological communities are regulated by a common set of fundamental
mechanisms that are much simpler than could be expected from the immense complexity of such systems
(Bell 2001). In a wider context, Sole and Bascompte (2006) have argued that "at the community level,
different regularities can be observed suggesting the presence of universal principles of community
organization" (Sole & Bascompte 2006, p. 5). Yet, four decades after Robert H. MacArthur and Edward O.
Wilson made the first attempt to formulate a quantitative dynamic theory of species diversity (MacArthur &
Wilson 1967) different patterns of species diversity are usually explained independently and in isolation from
each other. Thus, we have plenty of explanations for the species-area relationship (McGuinness 1984), the
productivity-diversity relationships (Rosenzweig & Abramsky 1993), the local-regional diversity relationship
(Fox & Srivastava 2006), and other empirically-observed patterns of species diversity (Hastings 1980, Hanski
& Gyllenberg 1997, Bell 2001, Gotelli & McCabe 2002, McGill et al. 2007), but so far we lack a general theory,
or even a theoretical framework, that attempts to derive all of these patterns from a unified set of first
principles.
In addition, existing models of ecological communities are highly variable in their assumptions and
underlying mechanisms (Table 1). Typical differences among existing models relate to whether the
environment is treated as homogeneous (most models) or heterogeneous (Tilman 2004, Bell 2005, Chisholm
& Pacala 2011), whether space is treated implicitly (Loreau & Mouquet 1999, Kadmon & Allouche 2007) or
explicitly (Weitz & Rothman 2003, O'Dwyer & Green 2010), whether the model treats all species as
demographically identical (Bell 2000, Dornelas 2010) or incorporates demographic differences among
species (Kondoh 2001, Mouquet & Loreau 2003, Xiao et al. 2009), whether the model incorporates trade-
offs in competitive ability and if so, what kind of trade-offs (Tilman et al. 1994, Sole et al. 2004), and whether
and how it takes into account processes of speciation (Pigolotti & Concini 2009, Rosindell & Phillimore 2011).
Currently it is unclear how these choices affect the conclusions obtained from the models and their
predictions. Answering such questions requires the integration of different factors and mechanisms within a
unified framework and a comparison of the patterns generated by different mechanisms including their
interactions.
2
Table 1: Comparison of selected demographic models of community dynamics with respect to fundamental methodological characteristics. Each model is characterized by a set of eight
properties: whether the results are obtained analytically or by simulations, whether the model is deterministic or stochastic, whether or not the model incorporates differences among
species, whether the model includes speciation and if so, what kind, whether space is treated implicitly or explicitly, whether dispersal is global or local, whether the model incorporates
trade-offs and if so, of what kind, and whether the model incorporates spatial heterogeneity in habitat conditions. Based on Fig. 1, I also note for each model the pattern(s) of diversity it
attempts to explain (a - species-area relationship, b - response to geographical isolation, c - effect of community size, d - response to habitat loss, e - response to habitat heterogeneity, f -
local-regional diversity relationship, g – unimodal response to productivity, h - unimodal response to disturbance, i - interaction between productivity and disturbance.
Results acquisition
Deterministic/ Stochastic
Differences among species
Speciation Space Dispersal Trade-offs*
Habitat heterogeneity
a b c d e f g h i
Bell 2001 Simulations Stochastic No No Explicit Local No No = = = = = = = = =
Pigolotti & Concini 2009 Simulations Stochastic No Instantaneous
Explicit Local No No = = = = = = = = =
O'Dwyer & Green 2010 Analytic Stochastic No No Explicit Local No No = = = = = = = = =
Chave et al. 2002 Simulations Stochastic Yes No Explicit Both C-B, C-D No = = = = = = = = =
Rosindell & Phillimore 2011 Simulations Stochastic No Protracted Implicit Global No No = = = = = = = = =
Loreau & Mouquet 1999 Simulations Deterministic Yes No Implicit Global No No = = = = = = = = =
Allouche & Kadmon 2009 Analytic Stochastic No No Implicit Global No No = = = = = = = = =
Kadmon & Allouche 2007 Analytic. Stochastic Yes No Implicit Global S Yes = = = = = = = = =
Sole et al. 2004 Both Both Yes No Both Both B-D Yes = = = = = = = = =
Tilman et al. 1994 Simulations Deterministic Yes No Implicit Local C-B Yes = = = = = = = = =
Sole et al. 2005 Both Both Yes No Explicit Local No Yes = = = = = = = = =
Mouquet & Loreau 2003 Simulations Stochastic Yes No Implicit Global No Yes = = = = = = = = =
Weitz & Rothman 2003 Simulations Stochastic No No Explicit Local No Yes = = = = = = = = =
Xiao et al. 2009 Simulations Stochastic Yes No Explicit Local C-B No = = = = = = = = =
Cordonier et al. 2006 Both Deterministic Yes No Implicit Global No, C-B No = = = = = = = = =
Dornelas 2010 Simulations Stochastic No No Implicit Global No No = = = = = = = = =
Kondoh 2001 Analytic Deterministic Yes No Implicit Global C-B No = = = = = = = = =
3
Hubbell's "Unified Neutral Theory of Biodiversity" (Hubbell 2001) is the most
ambitious attempt to date to develop a general theory in ecology, and has been regarded as
"one of the most important contributions to ecology and biogeography of the past half
century" (E.O. Wilson). The theory caused a major conceptual shift in ecology by
emphasizing the role of stochasticity and neutral processes in the regulation of ecological
communities (Whitfield 2002, Chave 2004, see also a series of papers in a special feature on
neutral community ecology published in Ecology, issue 87(6)). Holyoak et al. (2005)
considered the neutral theory as one of the four leading paradigms in research of ecological
communities.
Hubbell's Neutral Theory of Biodiversity
Hubbell describes a stochastic, individual-based, 'mainland-island model' (sensu Hanski
1999) where all individuals of all species are demographically equivalent (Hubbell 2001). The
mainland community is regulated by evolutionary processes (speciation and extinction) and
provides immigrants to the island community which is regulated by much faster processes of
local reproduction, mortality, and immigration. For mainland community that is much larger
than the island community, the feedback from the island community to the mainland can be
ignored (Vallade & Houchmandzadeh 2006). The dynamics of the mainland and island
communities are therefore largely uncoupled and we focus on the dynamics of the island
community (as in McKane et al. 2004).
The island is conceptualized as a spatially-implicit landscape that consists of J sites
with each site being able to support at most one individual. The community is 'saturated' in
the sense that all sites are continuously occupied, and the dynamics is modeled as a 'zero-
sum game' in the sense that each time step, a single, randomly drawn individual is killed and
is immediately replaced by a new individual. The replacing individual is either an offspring of
a randomly drawn individual from the local community (with probability 1-m), or a randomly
drawn immigrant from the mainland (with probability m). The likelihood of each species to
replace a death event is determined by its relative abundance in the source community from
which the replacing individual is drawn. Dispersal is assumed to be global so that each site is
accessible from any other site.
Under the assumption of the zero-sum game, the dynamics of a single species in the
local (island) community is given by:
(1) )1(1
)()1()|1( reg
jPJ
Nm
J
NJ
J
NmNNP
4
reg
jj PJ
NJm
J
N
J
NJmNNP
)(
1
)()1()|1(
where )|( xyP is the probability that the number of individuals of the species of interest
will change from x to y during one time-step, m is the probability of replacement by an
immigrant, N is the number of individuals of species j, J is the size of the local community,
and reg
jP is the relative abundance of species j in the mainland community (Hubbell 2001).
The probability that species j will have N individuals in a local community undergoing
Hubbell's (2001) zero-sum game is given by (McKane et al. 2000, 2004, Vallade &
Houchmandzadeh 2003, Volkov et al. 2003):
(2) ),(
),()(
*^
*^
JNPB
NNPNB
N
JNP local
j
where )()()(),( bababaB , reg
jPm
JmP
)1(
)1(^
and ^*
1P
m
mJN
.
Two fundamental concepts of Hubbell's (2001) neutral theory are 'zero-sum'
dynamics and random drift of species abundances. The concept of zero-sum dynamics
implies that each death event in the community is immediately replaced by a new individual,
either from an outer mainland or by birth of an offspring in the local community. Random
drift is present, as (1) the species of the replaced individual, as well as that of the replacing
offspring or immigrant, are randomly drawn according to the relative abundances of species
in the source community (the local community or the outer mainland, respectively), and (2)
the source of the new individual, being an offspring from the local community or an
immigrant from the outer mainland, is also determined randomly.
The analytical solution to Hubbell's model allows derivation of the expected species
abundance distribution given the model's input parameters, and consequently allows a
derivation of the model's best fit to empirical abundance distributions. Albeit its simplicity,
Hubbell's model achieved surprising success in predicting empirical distribution patterns,
and particularly of species abundance distributions (SADs, see Hubbell 2001, He 2005,
Volkov et al. 2005, Volkov et al. 2007). In many cases, Hubbell's theory better fitted
empirical data than the commonly used Log-normal distribution (Hubbell 2001, Volkov et al.
2003, He 2005).
Hubbell's theory is incredibly simple – the whole theory is formulated in terms of
only three parameters. However, this attractive property has two important costs. The first
cost is limited realism: the theory assumes that all species are completely identical ('strict
5
neutrality'); that each death event is immediately followed by a recruitment event ('zero-
sum dynamics'); and that all sites and resources are continuously occupied ('community
saturation'). These assumptions contrast our knowledge about the nature of most ecological
communities. The second cost is that the theory is not derived from the 'first principles' of
population dynamics, namely, the demographic processes of reproduction, mortality, and
migration, which are lumped into a single parameter. This lack of an explicit demographic
basis implies that predictions of the theory cannot be linked to the actual processes that
determine the number of species in a community.
A central requirement in this model is the zero-sum game. Hubbell makes an
unrealistic assumption that communities are saturated and do not change in size, and that
individuals have infinite rates of birth and immigration, so that each available site due to the
death of an individual is immediately taken by a new individual. This assumption contrasts a
fundamental result from models of metapopulation dynamics such as those of Levins (1969
1970) and its derivatives (Hastings 1980, Tilman 1994, Tilman et al. 1997), which show that
species never occupy all available sites. A desired extension of the model is therefore the
ability to infer species abundance distribution and species richness for unsaturated
communities, in which a death is not immediately replaced by birth or immigration, and
there is a temporal variability in community size. Recent papers have relaxed Hubbell's
unrealistic assumptions (Volkov et al. 2003, 2005, He 2005, Etienne et al. 2007), but they are
based on a very problematic assumption, namely, that each species in the community is
totally independent of all other species. This assumption seems as unrealistic as Hubbell's
original assumption of a 'zero-sum dynamics'.
Another major source of criticism against Hubbell's model is its assumption of 'strict'
neutrality, i.e. neutrality in the per-capita probability of death, birth and immigration.
Differences among species in reproduction, mortality, dispersal, and competitive ability are
clearly evident, and trade-offs among them are a major concept in the study of species
coexistence. Although Hubbell limits his model to functionally-equivalent species in a
homogeneous environment, he himself does not ignore the importance of life-history trade-
offs among species, but rather argues that "life history trade-offs equalize the per capita
relative fitness of species in the community" (Hubbell 2001). A useful extension to Hubbell's
model is therefore replacing its assumption of strict neutrality with neutrality in overall
fitness.
The neutral theory of biodiversity has gained considerable popularity as a null model
for community dynamics (Bell 2000, Maurer & McGill 2004, Nee 2005, Alonso et al. 2006,
6
Gotelli & McGill 2006). Nonetheless, in its current formulation, the theory is not applicable
to studies of complex ecological phenomena, such as habitat loss, environmental
heterogeneity, and variation in productivity. All of these factors are known to play an
important role in determining the diversity of ecological communities (Hutchinson 1957,
MacArthur 1972, Ehrlich 1988, Abrams 1995, Fahrig 1997). Thus, there is a need to extend
the framework of Hubbell's model to allow the incorporation of such key determinants of
species diversity.
Goals of Research
In this thesis I present a novel framework for modeling ecological communities, named the
MCD framework (Markovian Community Dynamics). The MCD framework extends Hubbell's
neutral theory of biodiversity, and resolves its main problems: it is formulated in terms of
the fundamental demographic processes of reproduction, mortality, and migration, and
thus, better connects patterns, processes and mechanisms of species diversity, and relaxes
the unrealistic assumptions of Hubbell's theory, thus providing a more realistic framework
for ecological analyses.
The new framework is based on individuals as the basic 'particles' and demographic
processes (reproduction, mortality and migration) as the basic drivers. It tries to cope with
three main challenges: (1) the need for a comprehensive, process-based theory capable of
explaining all the empirically-observed patterns of species diversity, (2) the need for a
theoretical framework capable of bridging gaps and inconsistencies between existing
theories, and (3) the expectation for analytical tractability without sacrificing too much
generality and/or realism. Important features of the new framework are its formulation as a
demographic, individual-based stochastic model, the incorporation of life-history trade-offs;
the explicit derivation of community size from the same stochastic processes that determine
the abundance of individual species, and the ability to incorporate complex ecological
factors such as habitat loss, habitat heterogeneity, variation in productivity and disturbance,
and non-random dispersal. These advantages are gained without sacrificing analytical
tractability. These features make the framework more realistic and more general than
Hubbell's model and previous theories of species diversity, and result in novel and
unexpected insights regarding the mechanisms regulating the diversity of ecological
communities.
The framework presented here extends and unifies leading theories of community
ecology, namely Niche theory (Hutchinson 1957), the theory of Island Biogeography
7
(MacArthur & Wilson 1967), Metapopulation theory (Levins 1969) and the Neutral theory
(Hubbell 2001), into an integrative frame, and allows integration of previously proposed
mechanisms of species diversity such as niche partitioning, competitive trade-offs, and
dispersal limitation within an analytically tractable framework.
As with most theories of species diversity (Hutchinson 1957, MacArthur & Wilson
1967, Tilman 1982, Hubbell 2001, Chase & Leibold 2003), I use the term 'ecological
community' to denote a group of trophically similar species that actually or potentially
compete in a local area for the same or similar resources. Accordingly, the term 'species
diversity' is used for the number of species in a local, trophically defined community. My
focus is on trophically-defined communities, though the conclusions are also applicable to
multitrophic communities.
The following chapters include six published papers (Chapters 1-3, 5-7), two of which
(Chapters 6, 7) as replies to letters criticizing our published work, and one manuscript with
results that were not previously published in the scientific literature (Chapter 4).
Chapters 1, 2 and 3 present the MCD framework and use it to extend Hubbell's
theory and relax its unrealistic assumptions. Chapter 1 provides an explicit derivation of
Hubbell’s local community model from the fundamental processes of reproduction,
mortality, and immigration, and shows that such derivation provides important insights on
the mechanisms regulating species diversity that cannot be obtained from the original
model and its previous extensions. Chapter 2 demonstrates that the MCD framework unifies
existing models of neutral communities and extends the applicability of existing models to a
much wider spectrum of ecological phenomena. We also use the MCD framework to extend
the concept of neutrality to fitness equivalence and explain a wide spectrum of empirical
patterns of species diversity. Chapter 3 presents the MCD framework as an extension of
Patch Occupancy theory, most recognized for the Levins model (1969), into a community of
species that differ in their demographic rates. I also demonstrate the flexibility of the MCD
framework by showing how it can be used to study the effect of complex ecological
mechanisms.
Chapter 4 uses the MCD framework to unify two of the most influential theories in
community ecology, namely, Island Biogeography and Niche Theory. The framework
captures the main elements of both theories and provides new insights about the
mechanisms that regulate the diversity of ecological communities. It also generates
unexpected predictions that could not be attained from any single theory. In 2007, when this
8
work was published, we did not have an analytic solution to the MCD framework. Instead,
we provided a highly-accurate approximation method and relied on numerical simulations to
show its accuracy.
While classical niche theory predicts that species richness should increase with
increasing habitat heterogeneity, the unification of Island Biogeography Theory and Niche
Theory presented in Chapter 3 (Kadmon & Allouche 2007) suggests that habitat
heterogeneity may have positive, negative, or unimodal effects on species richness,
depending on the balance between birth, death, and immigration rates. Hortal et al. (2009)
argued that this prediction contrasts empirical evidence and stems from unrealistic
assumptions of the model. Chapter 5 aims to show that Hortal et al. (2009) misinterpreted
both their data and the assumptions of the model and that a correct analysis of the model
and the data supports rather than contradicts the predictions of Kadmon and Allouche
(2007).
Chapter 6 provides a comprehensive evaluation of the hypothesis that habitat
heterogeneity may have positive, negative, or unimodal effects on species richness. We
analyze an extensive database of breeding bird distribution in Catalonia, perform a meta-
analysis of heterogeneity–diversity relationships in 54 published datasets, and study
simulations in which species may have variable niche widths along a continuous
environmental gradient to show that all support the hypothesis.
The hypothesis that habitat heterogeneity can lead to a decrease in species richness
goes strongly against the intuition of some community ecologists. The manuscript in Chapter
5 was criticized by two groups, which submitted letters to PNAS with their objections to our
analysis and conclusions. Chapters 7 and 8 bring our replies to these letters.
9
Chapter 1 – Paper
Demographic analysis of Hubbell’s neutral theory of biodiversity
Omri Allouche and Ronen Kadmon
Journal of Theoretical Biology (2009) 258:274-280
11
11
12
13
14
15
16
17
Chapter 2 – Paper
A general framework for neutral models of community dynamics
Omri Allouche and Ronen Kadmon
Ecology Letters (2009) 12:1287-1297
18
19
21
21
22
23
24
25
26
27
28
29
Chapter 3
Applications of the Demographic Framework and Its
Relation to Patch Occupancy Theory
.בתת פרק זה מוצגות תוצאות שטרם פורסמו בספרות המדעית
This chapter presents results that were not previously published in the scientific
literature.
31
Patch-occupancy theory (first introduced by Levins (1969, 1970)) is one of the most
influential theories in modern ecology. The theory has caused a paradigm shift in ecology by
emphasizing the crucial role of regional-scale processes in the dynamics of ecological
populations and communities (Hanski & Simberloff 1997, Harding & McNamara 2002). Since
its publication, the theory has been extended to incorporate multispecies interactions and
has been applied to almost any aspect of population and community ecology, including
population persistence (Gotelli 1991, Hanski et al. 1996b), coexistence of competing species
(Levins & Culver 1971, Yu & Wilson 2001), food webs and predator-prey interactions (Holt
1996, Shurin & Allen 2001), habitat fragmentation (Nee & May 1992, Sole et al. 2004), patch
preference (Etienne 2000, Purves & Dushoff 2005), dispersal and rescue effects (Gotelli &
Kelley 1993, Vandermeer & Carvajal 2001), range-abundance relationships (Hanski &
Gyllenberg 1997), disturbance (Hastings 1980), interactions between productivity and
disturbance (Kondoh 2001), source-sink population dynamics (Amarasekare & Nisbet 2001,
Mouquet & Loreau 2003), local-regional relationships of species richness (Mouquet &
Loreau 2003), and succession (Amarasekare & Possingham 2001). Holyoak et al. (2005)
named this modeling framework the 'patch dynamics perspective' and listed it as one of the
four leading paradigms in community ecology.
The basic model of patch-occupancy theory is the classic Levins model (Levins 1969,
1970) - a deterministic, spatially implicit model of metapopulation dynamics, where the
landscape is modeled as an infinite collection of patches, each able to support a local
population. Patches may have only two states – occupied or empty, and within-patch
population dynamics is ignored. Dispersal is assumed to be global, so that any patch is
accessible from any other patch. The change in the proportion of occupied patches is given
by:
(1) dPPbPdt
dP )1(
where P is the proportion of currently occupied patches, b is the rate at which empty
patches are colonized, and d is the rate by which occupied patches go extinct. The
equilibrium proportion of occupied patches, P* = 1 - d/b, is globally stable and requires that
the colonization rate exceeds the extinction rate. Since any metapopulation suffers at least
some level of extinction (i.e., d > 0), any positive equilibrium is associated with at least some
level (d/b) of unoccupied sites.
31
Tilman (1994) extended the Levins metapopulation model into a model of multi-
species dynamics. In this formulation (see also Hastings 1980, Loreau & Mouquet 1999,
Kondoh 2001, Mouquet & Loreau 2003) the 'patches' are rescaled into 'sites' that hold at
most one individual, the extinction parameter, d, is interpreted as mortality rate, and the
colonization parameter, b, is interpreted as reproduction rate. As with the original
metapopulation model, any stable equilibrium is associated with some fraction of empty
sites, implying that ecological communities are never saturated.
Later extensions incorporated immigration from a regional species pool by adding
an immigration term i (Hanski 1999):
(2) (1 ) (1 )dP
bP P dP i Pdt
The patch-occupancy framework has proved extremely useful for studying complex
ecological mechanisms by introducing additional terms into the basic dynamic equations
(see partial list in the first paragraph of this chapter).
The MCD framework provides a Markovian formulation of an individual-based patch
occupancy model with transition rates expressed as functions of reproduction (bk), mortality
(dk), and immigration (ik) of the component species. In this chapter I will show how the MCD
framework can be used to formulate a large number of patch-occupancy models. The MCD
framework thus extends patch-occupancy theory into a general, stochastic theory of
community dynamics with species-specific demographic rates. The incorporation of
stochasticity and a limited number of patches is important for studying the consequences of
habitat loss and fragmentation for the persistence and conservation of rare species (Hanski
et al. 1995, Hanski et al. 1996, Armstrong 2005).
In Chapter 2 (Allouche & Kadmon 2009b), we show that the solution to the MCD
framework is given by:
(3)
1
( ) ( ) ( )MCD
N
P N X N X N
where {1,..., 1}
1
1 0
( )kM
k k
NSk
N m ek m
X N T
;
k
k
k N
kN
N e
gT
r
; ((0,0,...,0)) 1X ; MS is the number of
species in the regional species pool, kN is the abundance of species k in the local
community, 1( ,..., )MSN N N is the local community abundance vector and
{1,..., } 1,..., ,0,...,0n nN N N . We also demonstrated how the transition rates in the MCD
framework can be easily modified to allow the study of a community of independent
32
species, competition for space, habitat heterogeneity, habitat loss, habitat productivity and
more.
In this chapter I exemplify the flexibility of the MCD framework by describing a few
additional extensions of the model. Each application is based on a different expression for
the transition rates k
Ng and
k
Nr , but all applications are based on the analytic solution of the
framework. My aim here is to demonstrate the wide applicability of the new model and to
illustrate its power as a general modeling framework.
The Theory of Island Biogeography
In Chapter 2 (Allouche & Kadmon 2009b) I presented a basic application of the MCD
framework for modeling communities with competition for space. In fact, this extension of
the model is an individual-based formulation of island biogeography theory that attempts to
cope with its main limitations by providing the following advantages: (1) replacement of the
unrealistic concept of constant number of species by the more realistic concept of steady-
state distribution of species numbers, (2) relaxation of the assumption of neutrality, (3)
ability to explicitly incorporate differences among islands in characteristics other than area
and isolation, (4) explicit integration of interspecific competition, (5) complete flexibility in
the distribution of species abundance in the regional species pool, and (6) formulation of the
model in terms of the fundamental demographic processes of birth, death, and migration.
While in the model presented in Chapter 2 (Allouche & Kadmon 2009b) all species
had the same demographic rates b, d and i, the MCD framework can be used to model
species that differ in their rates. In this formulation, each individual of species k dies at rate
dk (which implicitly includes emigration out of the local community) and gives birth to one
offspring at rate bk. A new offspring is immediately dispersed into a random site.
Immigration of species k from the regional species pool to each site in the landscape occurs
at a rate reg
k kP i
I, where reg
kP is the relative abundance of the k'th species in the regional
species pool, ik is the immigration rate of the k'th species, and I is a measure of ‘effective
isolation’ (a modifier of immigration rates) which combines the effects of pure geographic
isolation and the permeability of the medium isolating the local community from the source
of the colonizers. Dispersed offspring and immigrants can only establish in vacant sites
which results in competition for space. The model below translates to the following
transition rates:
33
(4a) reg
k k k kkN
N i Pg b A J
A I
(4b) k
k kNr d N
Equations (4) present a Markovian formulation for the basic equation of the Levins' model
(equation 2), for a finite number of patches. Using the general solution of the demographic
MCD framework, we find that the steady-state distribution of species abundances is given
by equation (3), where:
(4c)
1
1( )
!
kM
k
regNSk k
NJ k
Jk k k
PA J bX N
A d N
, k
k
k
i A
b I ,
and we use the Pochhammer notation 1
0
( )y
y
i
x x i
. This expression also determines
the steady-state distributions of the total number of individuals in the community, the
abundance of each species, and the total number of species (equations 11-14 in Chapter 2).
The presented model provides a powerful platform to study the effects of
fundamental ecological factors on the diversity of ecological communities. These include the
effect of area (through the total number of sites A), geographical isolation (through the
immigration rates ik), local-regional relationships of species diversity (through the properties
of the regional species pool), and the interaction between local and regional determinants of
species diversity (by analyzing the interplay between local competitive ability and regional
abundance of the component species). Furthermore, the fact that the model is completely
flexible in terms of the demographic rates of individual species allows to explicitly
incorporate various forms of demographic trade-offs in the model and to evaluate their
consequences for species diversity. While many previous studies have analyzed conceptually
similar models using numerical simulations, the formulation within the MCD framework
provides for the first time a fully analytical solution that derives the number of species in the
local community, the relative abundance of individual species, and the total number of
individuals, from the (species specific) rates of reproduction, mortality, and immigration.
Figure 1 shows the combined effects of regional species diversity, area, immigration,
and reproduction rates on local species diversity, for a community where species differ in
their per-capita reproduction, mortality, and immigration rates, but are equal in their overall
fitness. In this example fitness equivalence is achieved by incorporating trade-offs between
reproduction and mortality and between immigration and mortality (i.e., k
k
bd
and
34
k
k
id
for all species where λ and are constants). For a given regional species pool,
increasing area enhances and geographical isolation decreases local diversity, in agreement
with the theory of island biogeography (MacArthur & Wilson 1967, Hubbell 2001). Area
increases the number of individuals of each species, which reduces the risk of stochastic
extinction (The ‘More Individuals Hypothesis’, Srivastava & Lawton 1998, Hurlbert 2004).
Increasing immigration enhances species diversity by promoting the likelihood of colonization
by new species (MacArthur & Wilson 1967) and increasing the abundance of rare species,
thus reducing the likelihood of stochastic extinctions (the 'rescue effect', Brown & Kodric-
brown 1977).
The model also explains the demographic mechanisms underlying the local-regional
diversity relationship (Caley & Shluter 1997, Fox & Srivastava 2006). Consistent with
empirical observations, the model predicts that both linear and non-linear ('saturated')
relationships may occur between local and regional diversity (Fig. 1). The functional form of
the relationship depends on the area, the degree of geographic isolation (through the
immigration rates), and the balance between reproduction and mortality (Fig. 1). According
to the model, both area limitation and dispersal limitation increase the curvilinearity of the
local-regional diversity relationship and may turn linear and nearly linear relationships into
'saturated' ones (Fig. 1). This result provides demographic support for recent results based
on colonization-extinction models (He et al. 2005). The model further predicts that saturated
relationships between local and regional diversity may occur even when the community is
far from being saturated and a large portion of the area is accessible for new individuals
(average community size of 962.2 individuals in area of 50,000 and of 38.5 individuals in area
of 2,000, for the green and blue dashed lines in Fig. 1, respectively).
35
0 100 200 300 400 500 600 700 800 900 10000
50
100
150
200
250
300
350
400
Regional Species Richness
Lo
ca
l S
pe
cie
s R
ich
ne
ss
Fig. 1: Combined effects of regional species richness, area, immigration, and reproduction, on local species
richness, for a community where species differ in their demographic rates but are equal in their overall fitness.
Fitness equivalence was introduced by incorporating trade-offs between reproduction and mortality and
between immigration and mortality so that k
k
bd
and k
k
id
for all species, where λ and are
constants. Variation in regional species richness was obtained by generating species pools with log-series
distribution of species abundances and biodiversity numbers (Hubbell 2001) ranging from 1 to 90. For a given
biodiversity number, we used the Poisson distribution (Alonso & McKane 2004) to approximate the expected
number of species with a certain relative abundance in a regional species pool of 10,000,000 individuals. We then
calculated local species diversity for different combinations of area (A), and . Blue lines: A = 2000, =
0.5. Red lines: A = 2000, = 3. Green lines: A = 50000, = 0.5. Black lines: A = 50000, = 3. Dashed lines:
= 0.01. Solid lines: = 0.1.
Studying Communities without Fitness Equivalence
The MCD framework relaxes Hubbell's assumption of strict neutrality. While one can model
species with fitness equivalence, the framework can also be used to study communities
where species differ in their demographic rates and are not equivalent in their fitness, and
one species is better fit than others to the local habitat. While deterministic models predict
that the best competitor would take over the landscape (Tilman 1994, Mouquet & Loreau
2003, Amarasekare et al. 2004), in stochastic models outer immigration is essential for long-
term community viability, as without it stochasticity reduces the abundance of each species
to zero, which becomes an absorbing state. In accordance with this result, a stochastic
model based on the MCD framework predicts that in the absence of immigration even the
best competitor would eventually go extinct, although this may require very long time. On
the other hand, immigration from outside the local community may have profound effects
on the dynamics of the local community and may enable inferior competitors and even
species that cannot maintain viable populations without competitors to coexist with locally
adapted species. I illustrate these predictions by modeling the dynamics of a three-species
community with competitive hierarchy where one species is the best local competitor (i.e.,
36
has the highest reproduction/mortality ratio in the local community), the second species has
a reproduction/mortality ratio greater than 1 but lower than the best competitor, and the
third species has a reproduction/mortality ratio lower than 1 (i.e. it cannot maintain positive
population growth even at the absence of competitors). When immigration is rare, even
very small differences in local competitive ability may lead to large differences in species
abundances, and the best competitor dominates the community (Fig. 2a). These results
support and extend previous simulations indicating that small deviations from strict
neutrality may have profound effects on the distribution of species abundance in ecological
communities (Fuentes 2004).
The model further predicts that frequent immigration of inferior species increases
their local abundance and may compensate for their inferior competitive ability in the local
community ('spatial mass effect', Fig. 2b). Even species with reproduction/mortality ratio
lower than 1 at the absence of competitors may exist (Fig. 2b) and even dominate (Fig. 2c)
the community if their immigration rates are sufficiently higher than those of locally adapted
species ('source-sink dynamics’). Thus, the model may incorporate two different types of life
history trade-offs that may facilitate coexistence – a trade-off between local reproduction
and mortality rates (Fig. 1) and a trade-off between local reproduction and immigration
rates (Fig. 2). Each of these mechanisms has been investigated extensively by previous
models and the model presented here allows for the first time a fully analytical integration
of both mechanisms.
37
Fig. 2: Combined effects of local competitive ability (the ratio between local reproduction (b) and mortality (d)
rates), and immigration rates (i), on relative abundance in the local community. The modelled community has
three species: species 1 is a competitively superior species (b = 2.0, d = 1), species 2 is a competitively inferior
species (b = 1.9, d = 1), and species 3 is an even weaker competitor for which the local community is a sink (b =
0.5, d = 1). Three scenarios are modelled. In (2a) all species have equal immigration rates (i = 0.01 for all species)
and relative abundance is determined by reproduction rates. In (2b) differences among species in immigration
rates (i = 0.01, 0.025, and 0.25 for species 1, 2, and 3, respectively) compensate for the differences in competitive
ability (a trade-off between competitive ability and immigration rates). In (2c) the immigration rate of species 3 is
much higher than those of species 1 and 2 (i = 0.8 vs. 0.01 and 0.025, respectively) and it becomes the most
abundant species in the community. In all graphs A = 300.
Another insight emerging from the model is that the abundance of each species is
determined not solely by its own demographic rates, but also by the demographic rates of
all other species in the community. Figure 2 clearly illustrates this point. While the
demographic rates of the best competitor remain constant under all three scenarios, its
abundance strongly depends on the demographic rates of its competitors (compare Fig. 2a,
38
2b, and 2c). This result can be considered as an ecological counterpart of ‘the red queen
hypothesis’ in evolutionary biology (Van Valen 1973).
Community-level carrying capacity
The classic Levins model is analytically equivalent to the logistic equation (Verhulst 1838).
The logistic equation does not contain an explicit ceiling to population size, but rather
contains a ‘carrying capacity’ beyond which mortality exceeds local reproduction. In a
multispecies formulation, the carrying capacity concept is applied to the entire community
(Haegeman & Etienne 2008). This can be formulated in our model as:
(5a) k reg
k k k kNg b N i P A
(5b) k
k kN
Jr d N
K
The steady-state distribution of species abundances is given by equation (3), where:
(5c)
1
( )! !
kM
k
regNSJ
k kNk
k k k
PbKX N
J d N
, k
k
k
iA
b
Population-level density dependence
Many species suffer a decrease in reproduction and/or survival rates when abundant. Such
population-level density-dependence is often caused by competition for limited resources
and/or the effects of predators and parasites (Begon et al. 1990). Density-dependent
mortality can be incorporated in our model by letting the per-capita rate of mortality
increase with species abundance:
(6a) ( )k reg
k k k kN
A Jg b N i P A
A
(6b) ( )k k
k k k kN
k
Nr d b d N
K
where Kk is the population size of species k above which per-capita mortality exceeds
reproduction. A similar approach can be applied to incorporate density-dependent
reproduction. The steady-state distribution is given by equation (3), where:
(6c)
1
1( )
! 1
kM
k
k
regNSk k
NJ k
Jk k k
k
k N
PA J bX N
A dN
,
kk
k
iA
b , k k
k
k
b d
K
39
Allee effect
In many situations individuals experience reduced reproduction or survival at small
population sizes. This effect, known as the Allee effect (Allee 1931), is of particular
importance for rare species or species-rich communities, where the abundance of each
species is small (Courchamp et al. 2008). One possible way to model the Allee effect is
through a modifier of reproduction rates (Zhou & Zhang 2006):
(7a) ( )k regkk k k kN
k k
N A Jg b N i P A
N A
(7b) k
k kNr d N
where k is a parameter indicating the importance of the Allee effect for species k. In this
formulation the Allee effect influences reproduction rates, but a similar approach can be
applied to affect mortality rates. The steady-state distribution is given by equation (3),
where:
(7c)
12
0
1
1( )
!
k
kM
k
Nreg reg
N k k k k kS
J k m
Jk k k k N
m P m PA J b
X Nd NA
, k
k
k
iA
b .
The Allee effect demonstrates the importance of using a stochastic modeling framework
over deterministic models. While low population size might be sustained in deterministic
models when reproduction and mortality are in equilibrium, stochasticity might reduce
population size below this equilibrium, leading to reduced reproduction due to the Allee
effect, and to population decline to extinction.
Site Selection
Individuals of many species may show preference for vacant sites over occupied sites
(Etienne 2000, Purves & Dushoff 2005). Such site selection can be modeled using a modifier
of reproduction and immigration rates that assigns a larger weight to vacant sites over
occupied sites (Etienne 2000, Purves & Dushoff 2005):
(8a)
( )( )
( )
k reg
k k k kN
v A Jg b N i P A
v A J J
(8b)
k
k kNr d N
The steady-state distribution is given by equation (3), where:
41
(8c)
1
1( )
!
kM
k
regNSk k
NJJ k
k k kJ
PA J bX N
A d N
k
k
k
iA
b ,
1
v
v
For v = 1 this formulation is identical to the demographic formulation of the theory of island
biogeography (equation 2). For infinite v this formulation assumes that all immigrants and
locally-produced offspring are able to establish, given that their number is not larger than
the number of available vacant sites (e.g., Bell 2000, 2001). For 1v equations 6a-c also
describe the following dynamics: a new individual arriving at a new site establishes if the site
is vacant. If the site is occupied, it either dies (with probability 1
v) or moves to a randomly
selected site, and the procedure iterates until establishment or death.
Habitat preference
In real communities immigration and dispersal are rarely random. Instead, species may show
different levels of ‘preference’ to suitable habitats over unsuitable habitats (Purves &
Dushoff 2005). Such ‘habitat preference’ can be caused by active site selection of the
individual (or a dispersal vector), or by the combined effects of environmental
autocorrelation and limited dispersal. The spatially heterogeneous version of our model
presented in Chapter 3 can be extended to incorporate species-specific habitat preference
by adding a weighting parameter kv indicating the preference of species k to its source
habitat over unsuitable habitats. This gives:
(9a) ( )( )
k k k
k k k
H H k Hk reg
k k k kN
H k H H
A J v Ag b N i P A
A v A A A
(9b) k
k kNr d N
The steady-state distribution of species abundances is given by equation (3), where:
(9c)
1 1
( ) 1( 1) !
kkM
k
h
k
N regNSH k kNk k
h h Jh k k H k k
Pb vX N A J
d A v A N
,
kk
k
iA
b
41
Combining multiple mechanisms
For clarity I have separately introduced each ecological mechanism into the model.
However, in principle it is possible to simultaneously include all ecological mechanisms, as
long as equation (3) in Chapter 2 holds. We can thus analyse a very general model of a
community inhabiting a spatially heterogeneous and partially-destructed landscape where
species differ in their demographic rates and different species might be adapted to different
habitats. Individuals reproduce, die and migrate, and may show variable levels of preference
for suitable and/or vacant sites over environmentally-unsuitable or occupied sites. The
dynamics of each species can also be affected by variable levels of Allee effects, as well as
negative density-dependence at the population and/or the community level. Individuals
compete for space via both intraspecific and interspecific competition, and may differ in
their competitive ability within as well as among habitats. Additional mechanisms can be
added by further extensions of the transition rates k
Ng and k
Nr .
Example: Combining Habitat Loss, Site Selection and Habitat Quality
The effects of habitat loss, site selection, and habitat quality can be simultaneously
incorporated into the model using a combination of the relevant transition rates:
(10a) ( )
( )( )
k reg Dk k k kN
D D
v A A Jg b RN i P A
v A A J A J
(10b) k
k kNr d N
With the dimensionality of the model remaining similar. The steady-state distribution is
again given by equation (5), where:
(10c)
1
1( )
!
kM
k
regNS
k kND JJ k
kD k kJ
PA A J bX N R
A A d N
kk
k
iA
b R ,
1
v
v
Analysis of this integrated model enables one to evaluate the combined effects of habitat
quality, habitat loss and site selection on local species diversity. I demonstrate this capability
by analyzing a community where all species are similar in their overall fitness by introducing
appropriate demographic trade-offs (Fig. 3). Habitat loss reduces species diversity by
reducing the size of the community and thus, increasing the likelihood of stochastic
extinction (Fig. 3A-C). As evident from equation (19), habitat loss reduces the size of the
community both directly, by limiting the number of hospitable sites, and indirectly, by
42
reducing effective reproduction rates and increasing mortality of local offspring and
immigrants arriving in inhospitable sites (Casagrandi & Gatto 1999). Both mechanisms
reduce average population size (Fig. 3B) and therefore increase the risk of stochastic
extinction (the ‘More Individuals Hypothesis’, Srivastava & Lawton 1998, Hurlbert 2004).
Increased habitat quality, modelled as an increase in local reproduction rates,
promotes species diversity when habitat loss is large, as it increases community size and
reduces the risk of extinction. However, when habitat loss is small, increased habitat quality
can in fact decrease species diversity (Fig. 3A). In such cases community size is large,
extinction risk is relatively small, and the large number of locally-produced offspring reduces
the number of outer immigrants (which are the only source of new colonizers) that succeed
to establish in the local community (the ‘dilution effect’, Kadmon & Benjamini 2006). When
habitat loss is moderate these contrasting effects can result in unimodal response of species
diversity to habitat quality (Fig. 3A).
Fig. 3: Examples for patterns of species diversity predicted by the new theory. (a) Habitat Heterogeneity. (b)
Habitat loss. (c) Productivity. (d) Regional species diversity. Species were allowed to differ in their demographic
rates but the overall fitness of all species was kept constant by introducing appropriate demographic trade-offs
(i.e., the ratios between birth and death rates (b = k
k
bd
) and between immigration and death (i = k
k
id
) were
kept constant for all species). The sensitivity of each pattern of species diversity to variation in the demographic
rates was evaluated by analyzing the relevant effect under different ratios of birth/death rates (i.e., different levels
of b = k
k
bd
). Analyses of the effects of habitat heterogeneity (Fig. 1a), habitat loss (Fig. 1b), and productivity
(Fig. 1c) were generated using A = 10,000, i = 0.01 and a log-series distribution of species abundances in the
regional species pool, generated using Ewens' sampling method (Ewens 1972) with a biodiversity number of 50
and a mainland community size of 10,000,000 individuals. In the analysis of habitat heterogeneity, the log-series
distribution of species abundance was used for each habitat in the mainland separately. The effect of regional
species diversity (Fig. 1d) was analyzed by varying the total number of species in the regional species pool and
43
their relative abundance. Regional abundance distributions were generated for biodiversity numbers in range 1 to
40. For a given biodiversity number, we used the Poisson distribution (ref 50
) to approximate the expected number
of species with a certain relative species abundance in a regional species pool of 10,000,000 individuals. We then
calculated local species richness for different combinations of birth-to-death ratio (b = 0.5, 1.5, 3, blue, red and
green lines respectively).and immigration-to-death ratio (i = 1, .1, .01, solid, dashed and dotted lines respectively)
while keeping the area constant (A = 10,000).
Site selection, the ability of locally-produced offspring and immigrants to select
vacant sites over occupied sites, leads to increased community size, compensating for the
effect of habitat loss and thus increasing species diversity (Fig. 3A-C, Purves & Dushoff 2005).
This mechanism is particularly important when the community is much below its carrying
capacity, i.e., when habitat quality is relatively low, or when habitat loss is large.
It is interesting to compare the results of our stochastic model with those of an
analogous deterministic model. Under deterministic dynamics we can calculate the steady
state proportion of sites occupied by each species by solving the following set of coupled
differential equations:
(1 )( )
(1 )
regk Dk k k k k k
D D
dP v P Pb P i P d P
dt v P P P P
where Pk is the proportion of sites occupied by species k in steady-state, 1
MS
k
k
P P
is the
proportion of sites occupied by all species, and PD is the proportion of destructed sites.
Deriving expected species diversity from deterministic models is less obvious, as the
probability of absence from the local community is not explicitly given. We estimated
species diversity using the following formula:
*
1
(1 )MS
A
M k
k
SR S P
where *
kP is the steady state relative abundance of species k in the deterministic model.
A comparison of the stochastic model with its deterministic equivalent reveals
considerable similarity in the patterns of mean population size, but qualitative differences in
the patterns of species diversity and distribution of species abundance (Fig. 3). While the
stochastic model predicts unimodal relationship between species diversity and habitat
quality (see also Kadmon & Benjamini 2006), the deterministic model predicts that increased
habitat quality always increases species diversity (Fig. 3D). We explain this difference by the
inability of the deterministic model to account for stochastic extinctions and colonisations,
as species can occupy even infinitesimal fractions of the landscape. This naturally limits the
44
usefulness of deterministic models for studies of habitat loss, where these processes take
prominent role.
In contrast to species diversity, the average population size (i.e. the average number
of individuals of each species in the local community) is highly similar under stochastic and
deterministic dynamics (compare Fig. 3B,E). However, this average value is not sufficient to
determine species richness (Chesson 1978), as even species with large average population
size can be absent from the local community in large portions of the time. Thus, in contrast
to intuition, species richness can be reduced even when the average population size of all
species is increased! (compare Fig. 3A,B).
The stochastic and deterministic versions also disagree in their predictions of the
species abundance distributions. The results in figure 3 were generated using species that
are equal in both their regional abundance and overall fitness and therefore the
deterministic model predicts equal abundance of all species in the local community (Fig. 3F).
In contrast, random drift in the stochastic model generates roughly log-normal species
abundance distributions, as predicted by the neutral theory (Fig. 3C, Hubbell 2001).
Conclusion
The demographic framework can be used to model a wide spectrum of key ecological
mechanisms such as the ‘more individuals hypothesis’ (Srivastava & Lawton 1998, Hurlbert
2004), the 'rescue effect' (Brown & Kodric-Brown 1977), and the ‘dilution effect’ (Kadmon &
Benjamini 2006), and generates predictions consistent with fundamental concepts such as
the principle of competitive exclusion (Gause 1934), source-sink dynamics (Pulliam 1988),
'mass effect' (Shmida & Wilson 1985), and various forms of life-history trade-offs (Kneitel &
Chase 2004). Furthermore, by formulating the model in terms of the fundamental processes
of birth, death, and migration, any prediction of the model can be traced into its underlying
demographic mechanisms. These overall capabilities make the framework a powerful
platform for future research of factors and mechanisms affecting the dynamics, structure,
and diversity of ecological communities.
45
Chapter 4 – Paper
Integrating the Effects of Area, Isolation, and Habitat Heterogeneity
on Species Diversity: A Unification of Island Biogeography and Niche
Theory
Ronen Kadmon and Omri Allouche
American Naturalist (2007) 170(3):443-454
46
47
48
49
51
51
52
53
54
55
56
57
58
59
Chapter 5
Habitat Heterogeneity, Area, and Species Diversity: A (unpublished)
response to Hortal et al. (2009)
.בתת פרק זה מוצגות תוצאות שטרם פורסמו בספרות המדעית
This chapter presents results that were not previously published in the scientific
literature.
61
Hortal et al. (2009) criticized the model proposed by Kadmon and Allouche (2007)
and argued that its predictions are unrealistic and contrast empirical evidence showing that
species richness is positively correlated with the number of habitats. To support their claim
they analyzed the relationship between the number of species and number of habitats
('habitat diversity' in their terminology) in 24 data sets representing a wide spectrum of
insular systems and demonstrated that the relationships were positive in almost all cases.
They further suggested that the contradiction between the predictions of the model
proposed by Kadmon and Allouche (2007) and their empirical results stems from unrealistic
assumptions of the model, and that the "crucial" assumption is that each species is able to
establish and persist in only one type of habitat. Based on these arguments they concluded
that "the model should either be discarded or modified to provide more realistic results".
In this chapter I demonstrate that Hortal et al. (2009) misinterpreted their empirical
results; that a proper analysis of their data does not contradict the predictions of Kadmon
and Allouche (2007); and that relaxing the assumption that each species is able to persist in
only one type of habitat does not change the qualitative predictions of the model. These
new theoretical and empirical findings provide further support to the model proposed by
Kadmon and Allouche (2007).
The model used by Kadmon and Allouche (2007) is a special case of the Markovian
Community Dynamics (MCD) framework (Allouche & Kadmon 2009). The flexibility of the
MCD framework and its analytical tractability make it possible to explore whether and how
relaxing various assumptions of the model used by Kadmon and Allouche (2007) affects its
predictions. For example, using the MCD framework one can relax the assumption that each
species is able to persist in only one type of habitat and introduce complete flexibility in
niche widths of the modeled species. This modification does not change the qualitative
predictions of the model (see the Methodology chapter for analytical solution of the
extended model). As an example I present the effect of habitat heterogeneity on species
richness under two different scenarios: in the first (Fig. 1A) niche width of each species is
randomly selected from a uniform distribution, and in the second (Fig. 1B) niche width of
each species is randomly selected from the empirically observed distribution of niche widths
provided by Hortal et al. (2009, p. 209). Consistent with Kadmon and Allouche (2007),
increasing habitat heterogeneity may have positive, negative, or unimodal effects on species
richness (Fig. 1). It can also be seen that the level of habitat heterogeneity that maximizes
species richness increases with increasing birth rates, as predicted by the original model. In
fact, even simulations performed by Hortal et al. (2009) demonstrate that communities with
61
variable niche widths may show a decrease in species richness with increased habitat
diversity (three out of the four cases in their Fig. 5A). Thus, both our analytical results and
the simulations performed by Hortal et al. (2009) indicate that increasing habitat
heterogeneity may lead to a decrease in species richness even if species are able to exist in
multiple habitats.
The MCD framework can also be used to evaluate the robustness of the results
obtained by Kadmon and Allouche (2007) and Hortal et al. (2009) to their assumptions
concerning the nature of competition and dispersal. Kadmon and Allouche (2007) assumed
that dispersal is completely random and individuals arriving at sites that are already
occupied suffer competitive mortality (preemptive or 'lottery' competition, Amarasekare
2003). In contrast, Hortal et al. (2009) assumed that individuals have perfect ability to select
non-occupied sites, providing that such sites are available. In the Methodology section of
this thesis I show that the MCD framework may account for both situations, as well as for
any intermediate level of site selection ability. As can be expected, increasing the ability to
select vacant sites increases community size, thereby, promoting species richness (Fig. 2A).
The MCD framework can also incorporate a parameter controlling the amount of dispersal
to suitable vs. unsuitable habitats (see appendix for the analytical solution). Such parameter
may represent meta-community structure (Hortal et al. 2009), spatial autocorrelation in
habitat conditions coupled with limited dispersal (Etienne 2000), or habitat selection ability
(Purves & Dushoff 2005). Not surprisingly, reducing dispersal to unsuitable habitats
increases community size, thereby increasing species richness (Fig. 2B). All of these
extensions increase the realism of the model used by Kadmon and Allouche (2007) but do
not change its qualitative predictions.
Hortal et al. (2009) also presented empirical data showing that species richness
almost always increases with increased habitat diversity and argued that this result
contradicts the predictions of Kadmon and Allouche (2007). However, this interpretation is
incorrect because the model proposed by Kadmon and Allouche (2007) predicts the effect of
habitat diversity given an area of a fixed size, while the analysis performed by Hortal et al.
(2009) focused on insular systems representing a wide range of island sizes. A reanalysis of
all data sets that were available to us (all studies that were published in scientific journals, a
total of 20 out of 24 data sets) shows that once the effect of area is controlled for, most
patterns become non-significant (11 data sets), some appear significantly unimodal (three
data sets), and only six show significant positive relationships (Table 1). Clearly, these results
62
cannot be interpreted as evidence against the model proposed by Kadmon and Allouche
(2007).
The prediction that species richness may decrease with increasing habitat
heterogeneity has a simple intuitive explanation: since space is always finite, any increase in
the range of environmental variation while keeping the area constant must lead to a
reduction in the amount of suitable area available for individual species in the community,
thereby increasing the likelihood of stochastic extinctions. This fundamental trade-off
between habitat heterogeneity and area is independent of whether habitat heterogeneity is
quantified by the number of habitats ('habitat diversity' sensu Hortal et al. 2009), by indices
taking into account also the relative abundance of different habitats (e.g. the Shannon
diversity index, Wood et al. 2004), by measures of vegetation diversity (e.g. foliage height
diversity, Ralph 1985), by surrogates for habitat heterogeneity (e.g. elevation diversity,
Debinsky & Brussard 1994), or by the degree of variation in continuous environmental
factors (e.g. temperature range, Ruggiero & Hawkins 2008). Unless all species are fully
generalists with respect to the relevant factor(s), an increase in any of these measures while
keeping the overall area constant reduces the amount of area available for at least some
species in the community. A conceptually similar trade-off is expected to occur between
resource abundance and resource heterogeneity because increasing resource heterogeneity
63
A
B
= 1vh
= 2 = 10
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10
Heterogeneity
Sp
ec
ies
ric
hn
es
s
= 1
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10
Heterogeneity
vs
= 1000= 1012
Sp
ec
ies
ric
hn
es
s
A
B
= 1vh
= 2 = 10
10
20
30
40
50
10
20
30
40
50
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
Heterogeneity
Sp
ec
ies
ric
hn
es
s
= 1
10
20
30
40
50
10
20
30
40
50
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
Heterogeneity
vs
= 1000= 1012
Sp
ec
ies
ric
hn
es
s
Figure 2: Effect of habitat heterogeneity on species
richness for (A) different levels of site selection
ability and (B) different levels of dispersal to
suitable habitats (see appendix for analytical
solutions). For both models, niche width of
individual species is randomly selected from a
uniform distribution between 1-10, area (A) =
10000, reproduction (b) = 5, mortality (d) = 1,
immigration (i) = 0.001, regional species richness
(SM) = 1220 equally abundant species. Results are
based on 1000 realizations of the respective model.
A
Sp
ecie
s r
ich
ness b = 10
= 15= 20
20
0
5
10
15
0 2 4 6 8 10 12 14 16 18 20
Heterogeneity
B
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10
Heterogeneity
Sp
ecie
s r
ich
ness
b = 2= 5= 15
A
Sp
ecie
s r
ich
ness b = 10
= 15= 20
20
0
5
10
15
0
5
10
15
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
Heterogeneity
B
10
20
30
40
50
10
20
30
40
50
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
Heterogeneity
Sp
ecie
s r
ich
ness
b = 2= 5= 15
Figure 1: Effect of habitat heterogeneity on species
richness for different levels of birth rate in a model
where each species can exist in multiple habitats
(see appendix for a description and analytical
solution of the model). (A) Niche width of individual
species is randomly selected from a uniform
distribution between 1 and 10, area (A) = 10000,
mortality (d) = 1, immigration (i) = 0.001, regional
species richness (SM) = 1220 equally abundant
species; (B) Niche width of individual species is
randomly selected from the empirically observed
distribution provided by Hortal et al. (2009, p. 209),
area (A) = 50000, mortality (d) = 1, immigration (i) =
0.001, regional species richness (SM) = 69 equally
abundant species. Results are based on 1000
realizations of the respective model.
64
while keeping the overall amount of resource constant (e.g. increasing variation in prey size
while keeping the overall biomass of prey constant) must reduce the amount of resources
available per individual species, thereby increasing the likelihood of extinction.
The trade-off hypothesis provides a possible explanation for previously unexplained
deviations from the positive heterogeneity-diversity relationship predicted by niche theory.
Kadmon and Allouche (2007) provided two examples for such unexplained patterns. Ralph
(1985) analyzed local-scale patterns of bird diversity using foliage height diversity as a
measure of habitat heterogeneity and in contrast to his expectations found that the
relationship was "paradoxically" unimodal (figure 2 in the original paper). Currie (1991)
analyzed continental-scale patterns of vertebrate richness using tree richness as a measure
of biologically induced habitat heterogeneity and in contrast to his niche-based hypothesis
found that some patterns were strongly unimodal (figure 4 in the original paper). Hortal et
al. (2009) argued that these studies do not support the predictions of Kadmon and Allouche
(2007) because they do not fit the framework of island biogeography and do not use 'habitat
diversity' as the explanatory variable. However, as emphasized above, the trade-off
hypothesis does not assume a particular measure of habitat heterogeneity and is not limited
to insular systems. Tews et al. (2004) provided further examples for negative heterogeneity-
diversity relationships (15% of the studies included in their review) and suggested that such
patterns can be explained by fragmentation effects. This explanation is fully consistent with
the area-heterogeneity trade-off since a major reason for the decrease in species richness
according to this hypothesis is the loss of propagules to unsuitable habitats (Kadmon &
Allouche 2007). However, the novel insight obtained from the model of Kadmon and
Allouche (2007) is that negative heterogeneity-diversity relationships can be obtained even
if all habitats are completely equivalent in the number of species and number of individuals
that they are able to support.
To conclude, an evaluation of the theoretical concerns and empirical data provided
by Hortal et al. (2009) supports the predictions of the model proposed by Kadmon and
Allouche (2009). Moreover, while many studies show positive heterogeneity-diversity
relationships as predicted by classical niche theory, other studies show non-significant,
negative, or unimodal patterns (Table 2). Such variability occurs among different groups of
organisms in the same region (Moreno-Rueda & Pizarro 2009), within the same group of
organisms among different regions (Moreno-Saiz & Lobo 2008), within the same group and
region when the data are analyzed at different spatial scales (Yaacobi et al. 2007), and in the
same group, region, and scale for different measures of habitat heterogeneity (Ruggiero &
65
Hawkins 2008). Currently, no theory is capable of explaining these variable and apparently
conflicting results. The area-heterogeneity trade-off hypothesis provides for the first time a
unified explanation for all empirically observed heterogeneity-diversity relationships and is
also fully consistent with the recognition that both deterministic ('niche based') and
stochastic processes are important in determining the diversity of ecological communities
(Tilman 2004). This fundamental trade-off has been overlooked by all previous theories of
species diversity and currently little is known about its implications for real ecological
communities.
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Nilsson, S. G., J. Bengtsson, and S. As. 1988. Habitat diversity or area per-se: species richness of
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67
Taxon Nislands Linear model Quadratic model Improvement Relationship Source
Adj. R2 AICC Adj. R
2 AICC F
Birds 19 .203 -59.1 .271 -59.4 2.601 Linear Ricklefs & Lovette 1999
Bats 17 -.05 -54.1 -.1 -51.9 .269 NS Ricklefs & Lovette 1999
Butterflies 15 .56 -70.7 .53 -68.4 .149 Linear Ricklefs & Lovette 1999
Terrestrial isopods 13 -.05 -31.5 -.135 -28.9 .156 NS Triantis et al. 2008 - Astipalaia
Terrestrial isopods 12 -.1 -27.4 -.213 -24.6 .067 NS Triantis et al. 2008 - Kalymnos
Plants 42 .108 -185 .257 -193 10.055 Unimodal Kohn & Walsh 1994
Land snails 12 -.097 -27.4 .449 -34 10.904 Unimodal Triantis et al. 2005
Birds 45 .049 -172 .026 -170 .003 NS Davidar et al. 2001
Ground beetles 24 .112 -78 .144 -77.6 1.828 NS Kotze et al. 2000
Bryophytes 32 .476 -131 .477 -129 1.072 Linear Juriado et al. 2006
Mammals 24 .037 -76.1 .042 -74.9 1.107 NS Newmark 1986
All plants 31 .044 -120 .131 -122 4.215 Unimodal Deshaye & Morisset 1988
Halophytic plants 31 .047 -120 .101 -121 2.932 NS Deshaye & Morisset 1988
Birds 73 .25 -333 .275 -335 3.387 Linear Reed 1981
Plants 17 .021 -47.4 -.028 -45.1 .279 NS Nilsson et al. 1988
Carabids 17 -.057 -46.1 -.125 -43.6 .089 NS Nilsson et al. 1988
Snails 17 -.057 -46.1 -.041 -44.9 1.23 NS Nilsson et al. 1988
Birds 44 .223 -177 .218 -175 .751 Linear Haila et al. 1983
Birds 29 .116 -109 .096 -107 .327 Linear Sillén & Solbreck 1977
Plants 37 .024 -133 .035 -133 1.402 NS Rydin & Borgegård 1988
Table 1: Relationships between species richness and habitat heterogeneity after controlling for the effect of area on both variables in 20 previously published data sets. Note – Analyses were performed for all data sets analyzed by Hortal et al. (2009) that were published in scientific journals (20 out of 24 data sets). Each data set was analyzed in three steps.
First, habitat heterogeneity (number of habitats) was regressed against area using ordinary least squares regression and the residuals of the model were determined. Second, log species
richness was regressed against log area and the predicted values of log species richness were back transformed into units of species richness in order to determine the residuals of species
richness over area. In the third step, the residuals of species richness over area were regressed against the residuals of habitat heterogeneity over area using both linear and quadratic
regression models and the improvement of the quadratic model over the linear model was determined using F test. We also determined the small sample size–corrected Akaike Information
Criterion (AICC) for both types of models to verify that statistically significant improvements were associated with lower AICC values and confirmed that the coefficient of the quadratic term
was negative with the inflation point occurring within the range of the data. Statistically significant values of R2 and F are marked in bold.
68
Source Taxon Region Scale Measure Pattern
Moreno-Rueda & Pizaro 2009 Amphibians Iberian peninsula 10 × 10 km Habitat diversity NS
Reptiles Iberian peninsula 10 × 10 km Habitat diversity NS
Birds Iberian peninsula 10 × 10 km Habitat diversity Positive
Mammals Iberian peninsula 10 × 10 km Habitat diversity NS
Debinski & Brussard 1994 Butterflies Glacier National Park 1 × 1 km Elevation diversity Negative
Birds Glacier National Park 1 × 1 km Elevation diversity NS
Moreno-Saiz & Lobo 2008 Ferns Iberian peninsula Region 2 50 × 50 km Altitude range Unimodal
Ferns Iberian peninsula Region 3 50 × 50 km Altitude range Positive
Ferns Iberian peninsula Region 4 50 × 50 km Altitude range Unimodal
Ferns Iberian peninsula Region 5 50 × 50 km Altitude range Positive
Ferns Iberian peninsula Region 7 50 × 50 km Altitude range NS
Ferns Iberian peninsula Region 8 50 × 50 km Altitude range Unimodal
Ferns Iberian peninsula Region 9 50 × 50 km Altitude range NS
Yaacobi et al. 2007 Beetles (Carabidae) Beit Guvrin, Israel <2000 m2 Patch spatial heterogeneity Unimodal
Beetles (Carabidae) Beit Guvrin, Israel >2000 m2) Patch spatial heterogeneity NS
Beetles (Tenebrionidae) Beit Guvrin, Israel <2000 m2 Patch spatial heterogeneity NS
Beetles (Tenebrionidae) Beit Guvrin, Israel >2000 m2) Patch spatial heterogeneity Unimodal
Ruggiero & Hawkins 2008 Birds Western mountains of the New World 50 × 50 km Temperature range Negative
Birds Western mountains of the New World 50 × 50 km Altitude SD Positive
Table 2. Examples for documented variability in heterogeneity-diversity relationships among taxa, regions, scales, and measures of heterogeneity.
69
Chapter 6 – Paper
Area–heterogeneity tradeoff and the diversity of ecological
communities
Omri Allouche, Michael Kalyuzhni, Gregorio Moreno-Rueda, Manuel Pizarro and
Ronen Kadmon
PNAS (2012) 109(43):17495-17500
71
71
72
73
74
75
76
77
Chapter 7 – Paper
Reply to Hortal et al.:
Patterns of bird distribution in Spain support the area–heterogeneity
tradeoff
Omri Allouche, Michael Kalyuzhni, Gregorio Moreno-Rueda, Manuel Pizarro and
Ronen Kadmon
PNAS (2013) 110 (24) : E2151-E2152
78
79
81
81
Chapter 8 – Paper
Reply to Carnicer et al.:
Environmental heterogeneity reduces breeding bird richness in
Catalonia by increasing extinction rates
Omri Allouche, Michael Kalyuzhni, Gregorio Moreno-Rueda, Manuel Pizarro and
Ronen Kadmon
PNAS (2013) 110 (31) : E2861-E2862
82
83
84
85
Discussion and Conclusions
Hubbell's "Unified Neutral Theory of Biodiversity" (Hubbell 2001) is one of the most
ambitious attempts to date to develop a general theory in ecology. The theory caused a
major conceptual shift in ecology by emphasizing the role of stochasticity and neutral
processes in the regulation of ecological communities (Whitfield 2002, Chave 2004, see also
a series of papers in a special feature on neutral community ecology published in Ecology,
issue 87(6)). The neutral theory extends the scope of one of the most influential theories in
community ecology, the theory of island biogeography (MacArthur & Wilson 1967), by
formulating it as a stochastic individual-based model. This formulation allowed Hubbell to
provide for the first time a unified explanation for the number of species in a community
and the distribution of species abundances. An important contribution of Hubbell’s (2001)
formulation was the recognition of demographic stochasticity and dispersal limitation as
fundamental drivers of species diversity (Alonso et al. 2006). Yet, Hubbell's model makes
several unrealistic assumptions – it assumes that (1) individuals are neutral, that (2) the
environment is spatially homogeneous, that (3) the total number of individuals is
continuously constant, and that (4) each death event is immediately followed by birth or
immigration of a new individual (‘zero-sum dynamics’). These assumptions are only made for
analytical tractability and contrast our knowledge concerning the dynamics of most
ecological communities.
In this thesis, I present a new framework for studying species diversity, the MCD
(Markovian Community Dynamics) framework, that is based on individuals as the basic
particles of ecological communities and on the demographic processes that they undergo –
birth, death and migration. The MCD framework explicitly accounts for the demographic
processes (reproduction, mortality and migration) that drive change in the number of
individuals and species composition of ecological communities. I provide a mathematical
formulation and an analytic solution to the Markovian framework (Allouche & Kadmon
2009b, Chapter 2) that includes Hubbell's (2001) island-mainland model as a special case, as
well as other suggested neutral models (Allouche & Kadmon 2009b, Chapter 2). The
framework relaxes the unrealistic assumptions of Hubbell’s (2001) theory: (1) species may
differ in their demographic rates, (2) islands may differ in properties other than area and
isolation and may be internally heterogeneous, (3) the total number of individuals in the
community fluctuates according to the stochasticity in the demographic rates of individual
species, and (4) there is no coupling between mortality and reproduction or immigration.
86
These advantages imply an improvement in generality and realism, without sacrificing
analytical tractability.
Several previous modifications of Hubbell’s model relaxed the assumption of zero-
sum dynamics but, alternatively, assumed that the dynamics of each species is independent
of all other species in the community (Volkov et al. 2003, He 2005, Etienne et al. 2007). This
assumption is also unrealistic, and implies that the size of the community is unlimited. To
solve this problem, the demographic parameters need to be tuned to appropriate values,
forcing birth rates to be lower than death rates. In the MCD framework presented here the
size of the community has an upper limit (consistent with the concept of finite resource
availability) but the actual size of the community is determined by the balance between the
(species specific) rates of reproduction, mortality, and immigration. Other forms of resource
competition (e.g. a community level carrying capacity above which mortality rates exceed
reproduction) can also be easily incorporated through appropriate formulation of the
transition rates (Allouche & Kadmon 2009b, brought in Chapter 2).
Another important advantage of the MCD framework over Hubbell’s (2001) theory is
its explicit derivation from the fundamental demographic processes. Hubbell’s model is
often being considered as a demographic model, but it actually combines the demographic
processes of reproduction, mortality, and immigration into a single parameter (m)
representing the probability that a dying individual would be replaced by an immigrant from
the regional species pool. While such reduction in the number of parameters has the
advantage of reducing model complexity, we showed (Allouche & Kadmon 2009a, brought in
Chapter 1) that it may hide important mechanisms and may lead to incomplete and even
misleading conclusions concerning fundamental mechanisms of species diversity.
Explaining the Success of Hubbell's Theory in Fitting Empirical
Patterns
Hubbell's model was shown to provide remarkable fit to empirical species
abundance distributions, that outperformed the leading alternative, the log-normal
distribution (Hubbell 2001, Volkov et al. 2003, He 2005, though see McGill et al. (2007) for
criticism). This result is often interpreted as support for the relative importance of
stochasticity and dispersal limitation over ecological niche in structuring ecological
communities, but is very surprising, given the unrealistic assumptions of Hubbell’s model.
Hubbell himself was well aware of the paradox between the simplicity and
unrealistic assumptions of his theory and its success in explaining empirically observed
87
patterns, and argues that while his assumptions are uncorroborated, they provide
reasonable simplifications of reality. According to Hubbell in reality “life history trade-offs
equalize the per capita relative fitness of species in the community" (Hubbell 2001, page
346). However, Hubbell chose to model the equal fitness of species in the community by
assuming strict neutrality, which disallows the very same trade-offs that create fitness
equivalence. Another simplifying assumption Hubbell makes is of constant community size.
Observing the linear increase of community size (the number of individuals) with area, an
almost universal ecological pattern, Hubbell deduces that ecological communities are always
saturated, and that their dynamics follow a zero-sum game (Hubbell 2001, page 53). While
Hubbell sees this as a “first approximation” (Hubbell 2001, page 53) and does not deny that
the total number of individuals fluctuates in time, his model does not account for this
circumstance, and alternatively assumes infinite reproduction, so that each dying individual
is immediately replaced by a new offspring. This assumption seems reasonable for highly-
productive communities, such as tropical forests, but does not seem to hold for low-
productive communities, such as deserts and boreal forests.
The above-mentioned paradox led some to argue that Hubbell’s model is robust to
its assumptions, and that these assumptions can be relaxed without largely affecting
predictions. Volkov et al. (2003, 2005), He (2005) and Etienne et al. (2007) all relax the
assumption of constant community size and zero-sum dynamics, without affecting predicted
patterns. However, these models alternatively assume an unrealistic assumption, namely
that each species is totally independent of other all species, and thus ignore interspecies
competition. Volkov et al. (2005) also argue that species abundance distributions obtained
from the neutral theory are practically indistinguishable from those obtained from a theory
which incorporates density-dependence (though see Chave et al. (2006) for criticism). Purves
and Pacala (2005) show that neutral theory is robust to the introduction of deterministic
processes, such as niche structuring, when diversity is large. Several papers (Volkov et al.
2003, 2005, He 2005, Etienne et al. 2007) have relaxed the assumption of per-capita equality
in the demographic rates, provided that the overall fitness is constant among species,
thereby allowing trade-offs, but made an the unrealistic assumption of independent species.
The MCD framework presented in this thesis relaxes the unrealistic assumptions of
Hubbell's theory, including its ignorance of the temporal variability in community size and its
assumption of full neutrality in the life-history traits of all species. In Chapter 2 (Allouche and
Kadmon 2009b) we demonstrate that Hubbell's model provides highly accurate estimates of
species richness in an extended model that is based on the MCD framework. This
88
remarkable result indicates that even though the fundamental assumptions of Hubbell's
neutral theory are unrealistic and contrast our knowledge about ecological communities, the
results of the theory are still valid when these assumptions are relaxed.
Specifically, chapter 2 (Allouche & Kadmon 2009b) shows that given a community size,
the abundance distribution of Hubbell's model, termed the DLM (Dispersal Limited
Multinomial, Etienne et al. 2007, Allouche & Kadmon 2009a), is equal to the abundance
distribution in all of its extensions. This surprising result demonstrates the robustness of the
neutral theory to relaxation of its fundamental assumptions, and may help explaining the
paradox between its unrealistic assumptions and its ability to accurately estimate observed
patterns of species diversity (Hubbell 2001, Volkov et al 2003, Volkov et al 2005, Chave et al
2006).
Relation of the MCD Framework to Theories of Species Diversity
Island Biogeography Theory
The essence of island biogeography theory (MacArthur & Wilson 1967) is that species
diversity in a local community reflects a dynamic balance between colonization (arrival of
new species) and extinction of species already present in the community. According to the
theory, species composition is constantly changing over time, as new species replace those
that go extinct. This concept of a dynamic equilibrium was in sharp contrast to the
deterministic view of community ecology that dominated ecological theory when the theory
was published.
The theory further emphasizes the role of area and geographical isolation as the
main determinants of species diversity. Based on the assumption that colonization rates are
determined by the degree of geographical isolation and extinction rates are determined by
the size (area) of the island, the theory predicts that species diversity should be positively
correlated with island size and negatively correlated with the degree of isolation. Later
developments of the theory recognized that isolation may also influence extinction rates
because islands close to the mainland are characterized by higher immigration rates than
remote islands, which reduces the likelihood of stochastic extinctions (the 'rescue effect',
Brown & Kodrik-Brown 1977). It has further been suggested that the area of an island may
influence the rate of colonization because large areas receive more colonizers than small
areas (the 'passive sampling hypothesis', Connor & McCoy 1979). These extensions of the
89
original theory still consider area and isolation as the primary determinants of species
richness.
In Chapter 3 I demonstrate a basic application of the MCD framework that can be
seen as an individual-based formulation of island biogeography theory. This basic model
attempts to cope with the main limitations of the original theory while capturing its main
elements – species diversity is dynamically determined as the balance between extinction
and colonization events, which in turn are affected by the area of the local community and
the degree of isolation. However, the MCD framework moves the focus from the species
level to the level of the individuals. The species-level processes of extinction and
colonization therefore directly stem from the individual-level processes of birth, death and
immigration. The MCD framework is thus formulated in terms of the ‘first principles’ of
population dynamics, and can improve understanding of how mechanisms operating at the
level of the individuals affect patterns and processes at the higher levels of the populations
and the whole community.
In Chapter 3 (Kadmon & Allouche 2007) we use this application to analyze the
combined effects of area, isolation, and regional species diversity on local species diversity.
To control for differences among species in local competitive ability, we allow species to
differ in their per-capita demographic rates, but keep the overall fitness of all species equal
by incorporating trade-offs between reproduction and mortality and between immigration
and mortality (i.e., k
k
bd
and k
k
id
for all species, where λ and are constants).
The MCD framework produces patterns consistent with the theory of island biogeography
(MacArthur & Wilson 1967, He et al. 2005), regarding the effect of area and isolation on
local diversity (Fig. 1 in Chapter 3). In addition, our model also accounts for mechanisms
known to affect ecological communities - the ‘More Individuals Hypothesis’ (Srivastava &
Lawton 1998) and the ’rescue effect’ (Brown & Kodric-brown 1977), and allows the study of
their combined effects on species diversity of ecological communities.
Levins' Model and the Patch Occupancy Theory
Levins' model (Levins 1969) has caused a paradigm shift in ecology by emphasizing the
crucial role of regional-scale processes in the dynamics of populations and communities
(Hanski & Simberloff 1997, Harding & McNamara 2002). Since its publication, the model has
been extended to include multi-species interactions, and has been applied to almost any
aspect of population and community ecology, including population persistence (Gotelli
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1991, Hanski et al. 1996), coexistence of competing species (Levins & Culver 1971, Yu &
Wilson 2001), food webs and predator-prey interactions (Holt 1996, Shurin & Allen 2001),
habitat fragmentation (Nee & May 1992, Sole et al 2004), patch preference (Etiene 2000,
Purves & Dushoff 2005), rescue effect (Gotelli & Kelley 1993), the quality of the matrix
habitat (Vandermeer & Carvajal 2001), range-abundance relationship (Hanski & Gyllenberg
1997), disturbance (Hastings 1980), interactions between productivity and disturbance
(Kondoh 2001), source-sink dynamics (Amarasekare & Nisbet 2001, Mouquet & Loreau
2003), local-regional relationships of species richness (Mouquet & Loreau 2003), and
succession (Amarasekare & Possingham 2001). Holyoak et al. (2005) named this modeling
framework the 'patch dynamics perspective' and listed it as one of the four leading
paradigms in community ecology.
In Chapter 3 I show that the MCD framework can be seen as an extension of the
Levins' model into an individual-based, multi-species community where species may differ in
their demographic rates. In Chapter 3 (Kadmon & Allouche 2007) we show that the MCD
framework can integrate Levins' model and Hubbell's model. This integration reconciles an
apparent contradiction between the two models: while Hubbell's model explicitly assumes
that ecological communities are continuously saturated and vacant sites never exist, Levin's
model predicts exactly the opposite, i.e., that ecological communities are never saturated
and empty sites always exist in the community. For example, Tilman (1994) developed a
multispecies version of Levin's models and noted that "The most interesting feature [of the
model] is that a sessile species can never fill a habitat. For a species to completely fill a
spatially structured habitat, the species would have to be immortal or have infinite dispersal
abilities, both of which are biologically unrealistic". This conclusion is a key element of many
current theories in population and community ecology (Levin 1974, Hastings 1980, Tilman
1994, Holt 1996, Tilman & Lehman 1997).
Stochastic Niche Theory
Applications of the MCD framework that incorporate habitat heterogeneity (Kadmon &
Allouche 2007, Allouche & Kadmon 2009b) bear some resemblance to Tilman’s (2004)
'stochastic niche theory'. Similar to the MCD framework, Tilman’s theory integrates the main
elements of niche theory (habitat heterogeneity and niche partitioning) and neutral theory
(dispersal limitation and demographic stochasticity) within a unified framework. In Tilman’s
model, individuals compete for limiting resources along an environmental gradient that
provides opportunities for niche partitioning, but new immigrants become established only
91
if they survive stochastic mortality while growing on the resources left unutilized by existing
species (Tilman 2004). The MCD framework defines a conceptually similar model where the
limiting resource is space and niche partitioning is incorporated by letting each species be
adapted to a particular habitat, rather than having a unique optimum along a continuous
niche axis (Kadmon & Allouche 2007). As in Tilman’s (2004) model, individuals compete for
limited resources in a spatially heterogeneous environment, and new immigrants become
established only if they survive stochastic mortality while growing on sites left unutilized by
existing species.
Limitations of the MCD framework
Some limitations of the MCD framework should also be taken into account. First, similarly to
theory of island biogeography, the MCD framework does not explicitly incorporate
speciation and therefore its applications are limited to ecological time scales. While the
MCD framework can be extended to incorporate speciation by appropriate modifications of
the transition rates, I focus on processes operating at ecological, rather than evolutionary
time scales. Ecological time scales are more important for purposes of conservation planning
and management. It should still be appreciated that in contrast to other recent extensions
of theory of island biogeography (e.g. Whittaker et al. 2008), the MCD framework cannot
explain patterns of species diversity at the spatial and temporal scales where speciation
plays an important role.
Another limitation is the formulation of the MCD framework as a spatially-implicit,
rather than spatially-explicit model. This choice, made for analytical tractability, prevents
direct consideration of the effect of dispersal distance on the spatial distribution of
individuals in the community and therefore reduces the predictive power of the framework.
While any application of the MCD framework can be analyzed using a spatially-explicit
model with limited dispersal distance, at this stage such applications cannot be analyzed
using our general analytical solution. Another consequence of this limitation is the inability
of the framework to deal with scale-dependent processes which are extremely important
for biodiversity conservation (Whittaker et al. 2001).
Finally, like most previous theories of species diversity, the MCD framework focuses
on trophically-defined communities and cannot deal with the mutual dynamics of predator-
prey and herbivore-plant interactions. Trophic interactions play an important role in
determining the diversity of ecological communities (Paine 1966, Sole & Monyoya 2001,
Worm & Duffy 2003) and empirical studies suggest that herbivory and predation might be
92
important in determining observed patterns of decline in native species diversity (Crooks &
Soule 1999, McIntyre et al. 2003, McKenzie et al. 2007, Pelicice & Agostinho 2009). The MCD
framework is capable of incorporating species-specific mortality due to predation or
herbivory, but is unable to deal with feedback processes between species in different
trophic levels.
A major limitation of the framework, shared by all demographic models, is the
immense difficulty of calibrating the model with empirical data. A demographic model
requires explicit characterization of reproduction, mortality and migration ability of each
species, as well as data on factors affecting these processes. While mortality rates can be
measured relatively easily in some systems (Condit et al. 1995, HilleRisLambers et al. 2002,
Metz et al. 2010), it is much more difficult to quantify reproduction, and even more so
migration. Indeed, at present obtaining these values for each species in a multi-species
community seems impractical.
Community-level demographic models are extremely limited in their mathematical
complexity. Even models of moderate complexity need to characterize each species by
ecological properties such as demographic rates, competitive ability, environmental
tolerance, niche position, etc., and to specify the mechanisms by which species interact with
each other and with their environment. The resulting complexity, particularly when
combined with demographic stochasticity, makes the mathematic analysis of such models
extremely challenging.
Ecologists have traditionally coped with this complexity by limiting the analysis to a
small number of species (MacArthur & Levins 1967), assuming deterministic dynamics
(Tilman et al. 1994), or focusing on highly simplified, mostly neutral communities (Hubbell
2001). However, the palette of mathematical tools available to ecologists nowadays is
rapidly increasing, allowing the analytic solution and approximation of highly complex
models (e.g. Ovaskainen & Cornell 2006, Govindarajan et al. 2007, Allouche & Kadmon
2009b, Banavar et al. 2010, O'Dwyer & Green 2010). Moreover, the phenomenal
exponential increase in computation power over the past decades (the so-called Moore's
law, Moore 1965) now allows ecologists to perform highly complex simulations over large
scales and obtain accurate results which do not fall much shorter of an analytic analysis. I
therefore believe that in the future mathematical complexity will play a lesser role in
limiting the analysis of community-level demographic models. The MCD framework provides
a simple requirement for the model's parameters to obtain analytical tractability. Still,
computing a specific value for a model might require considerable computation power.
93
Computational methods such as the Metropolis-Hastings algorithm (Metropolis et al. 1953,
Hastings 1970) and the Gibbs sampler (Geman & Geman 1984) can be used to reduce the
required computation power.
Inspired by the importance of ecological insights that can be obtained from
community-level demographic analyses, several large-scale projects have invested huge
efforts in detailed measurements of species demographic characteristics (e.g. Goldberg &
Turner 1986, Pacala et al. 1996, Condit et al. 1999, Lauenroth & Adler 2008, Adler &
HilleRisLambers 2008). Recent advances in remote sensing (Lamar & McGraw 2005, Dietze &
Moorcroft 2011), GPS technology (Sims et al. 2009, Hebblewhite & Haidon 2010, Morales et
al. 2010) and molecular tools (Jordano et al. 2007, Maruvka & Shnerb 2009) open new
opportunities for measuring demographic processes and give hope that in the future such
data will be available for larger spatial and temporal scales.
While these recent advances are promising, I believe the MCD framework, and
demographic models of species diversity in general, should be used for gaining general
insights about the factors shaping ecological communities, rather than for modeling the
dynamics of specific communities. In my view, general theories of complex systems should
provide insights on mechanisms underlying general phenomena, rather than quantitative
predictions for particular systems. For this purpose, crude estimates of demographic
parameters with subsequent sensitivity analyses might be sufficient, and there is no need to
accurately estimate all demographic parameters of all species in a concrete community. As
Hansson (2003) stated, "very general theory… seem[s] therefore hardly ever able to explain
the precise working or outcome of local ecological interactions". In fact, most leading
theories of species diversity do not produce exact quantitative predictions, and their input
parameters do not translate into clear quantifiable measures. Notable examples are the
theory of island biogeography (MacArthur & Wilson 1967), and niche theory (Hutchinson
1957). Yet, no ecologist questions the contribution of such theories to community ecology.
Towards a General Theory of Species Diversity
In a seminal paper entitled "Are there general laws in ecology", Lawton (1999) argued that
"community ecology is a mess, with so much contingency that useful generalizations are
hard to find". He further proposed that "the great majority, probably all of our actual laws
(in the sense of widely observable tendencies) cannot be derived from first principles". The
essence of this view is that the mechanisms controlling the diversity of ecological
94
communities are so complex, and the laws are so contingent, such that no single theory can
fit a wide range of systems (see also Hansson 2003, Simberloff 2004).
In this PhD thesis I present a novel modeling framework that is formulated in terms
of the 'first principles' of population dynamics - the fundamental demographic processes of
birth, death, and migration of individuals, and is capable of explaining the major patterns of
species diversity. Personally, I see this framework as an attempt to lay the ground for a
general, demographic theory of species diversity. It is therefore beneficial to consider our
expectations from some a theory first.
Developing a 'General' Theory of Species Diversity
The greatest challenge in the development of a general theory of species diversity is finding
an optimal balance between generality, realism, and simplicity (Levins 1966). While a model
focusing on a particular question might be unrealistic in many respects and still capture the
realism required for dealing with the particular problem at hand, a general theory
attempting to explain a wide spectrum of phenomena needs to be realistic in multiple
respects, and therefore requires a higher degree of overall realism.
Like other theories of complex systems (e.g. statistical mechanics), a general theory
should provide insights on mechanisms underlying general phenomena, and should
therefore be evaluated based on its capability to explain and predict qualitative patterns
rather than the state of a particular system in a particular space or time (Hansson 2003).
This expectation may sound a rather weak test, taking into account that 'qualitative
patterns' usually refer to a limited range of patterns (e.g., positive, negative, unimodal, or
cyclic patterns). However, even simple patterns often emerge from complex underlying
interactions and mechanisms that are very difficult to understand without explicit
quantitative formulation of the underlying processes (Chave et al. 2002, Mouquet & Loreau
2003, Bell 2005, Sole & Bascompte 2006, Chisholm & Pacala 2011). The requirement to
simultaneously produce all major patterns of species diversity thus presents a substantial,
non-trivial challenge (McGill et al. 2007).
A general theory of species diversity is expected to provide mechanistic
explanations to the major empirically observed patterns of species diversity (Figure 1) and
be testable using empirical data. Finally, as with other fields of the natural sciences, the
ultimate expectation from a general theory of species diversity is to provide novel insights
and new predictions that cannot be obtained from existing theories.
95
The Importance of Demography
It is now widely recognized that macroscopic patterns of complex systems such as ecological
communities could best be understood as emergent properties of the collective behavior of
many units at smaller scales (Levin 1992, Maurer 1999). The most basic elements of
ecological communities are individuals, and changes in the number of species in ecological
communities can only result from changes in the number of individuals of the component
species. Demography is the study of variation in the number of individuals over space and
time and therefore a fully mechanistic theory of species diversity must be formulated with
individuals as the basic 'particles' and demographic processes (i.e., reproduction, mortality,
and migration) as the main drivers. Thus, the essence of an individual-based demographic
approach for the study of species diversity is the focus on fundamental processes as a
means for understanding mechanisms of empirically-observed patterns. This approach is
very different from the statistical 'pattern based' approach that dominates the field of
macroecology (McGill 2003).
Although no ecologist would deny that demographic processes are the fundamental
drivers of species diversity, most leading theories of species diversity lack an explicit
demographic basis. For example, MacArthur and Wilson's (1967) theory of island
biogeography is formulated in terms of species as the basic 'particles' and colonization and
extinction processes as the basic drivers of species diversity. Hubbell's (2001) recent
extension of island biogeography theory into a 'unified neutral theory of biodiversity and
biogeography' adopts an individual-based modeling approach, but in his formulation all
demographic processes are summarized by a single parameter (m, the probability that a
dying individual will be replaced by an immigrant) representing the combined effects of
reproduction, mortality, and immigration. This formulation limits the scope of the theory
and hides important interactions between the fundamental demographic processes
(Allouche & Kadmon 2009). Other leading theories of species diversity (e.g. Hutchinson's
(1957) hypervolume model of the ecological niche, Levins' (1968) community matrix,
Grime's (1977) C-S-R theory, Tilman's (1982) resource-ratio theory) are even less connected
to the fundamental demographic processes of reproduction, mortality and migration.
Primary versus Secondary Parameters
The colonization and extinction rates in the theory of Island Biogeography (MacArthur &
Wilson's 1967) can be considered as 'secondary parameters', as they can be derived from
the underlying demographic processes of reproduction, mortality, migration and speciation.
96
In contrast, these demographic processes represent the most direct processes that affect
the size of populations, the relative abundance of individual species, and the total number
of species in a community, and can thus be considered 'primary parameters'. This distinction
holds also for population dynamics models. For example, the intrinsic growth rate in the
logistic model is a secondary parameter lumping reproduction and mortality rates, and so
are the colonization and extinction parameters in classical metapopulation dynamic models
(Levins 1969).
While secondary parameters can be derived from underlying primary parameters,
most theories do not explicitly define the linkage between secondary parameters and the
underlying demographic processes, which may hide important interactions between the
relevant parameters and thus lead to incomplete and even misleading conclusions
concerning their role in the dynamics of the relevant system. For example, the theory of
island biogeography (MacArthur & Wilson 1967, Hubbell 2001) argues that species diversity
is determined by the balance between the rates of colonization, set by the degree of
geographical isolation, and extinction, set by the size of the island. A demographic
formulations of the theory demonstrate that colonization rates can also be influenced by
the size of the area (through the 'passive sampling' effect, Connor & McCoy 1979), and that
extinction rates can be influenced by the degree of geographical isolation (through the
'rescue effect', Brown & Kodric-Brown 1977).
Hubbell's (2001) neutral theory argues that species diversity is determined by the
size of the local community (J, the total number of individuals of all species) and the
probability that a new individual is replaced by an immigrant from the mainland (m).
Hubbell recognizes that these parameters, given as free, independent parameters, are in
fact secondary parameters derived from the primary demographic processes, but his
formulation does not explicitly provide mathematical account of this derivation. A
demographic formulation of Hubbell's (2001) model (Allouche & Kadmon 2009a, brought in
Chapter 1) elucidates important insights not recognized by the non-demographic model, for
example that the fundamental parameters of the model (J and m) are not independent, as a
larger community (larger J) produces more offspring that compete with immigrants over
vacant sites, thereby reducing the probability of replacement by an immigrant (smaller m).
This mechanism leads to a counterintuitive trade-off between community size and species
diversity.
97
Concluding Remarks
Ecological research over the years has pointed to the existence of a wide spectrum of
general patterns of species diversity that can be considered as 'semi-universal' in the sense
that they are found over very different life forms and ecosystems. These patterns include
the increase of species diversity with area (Arrhenius 1921), the decrease of species diversity
with geographical isolation (MacArthur & Wilson 1967), the unimodal response of species
diversity to productivity (Waide et al. 1999), the increase of species diversity with habitat
heterogeneity (MacArthur 1972, Tews et al. 2004), the decrease of species diversity with
habitat loss (Tilman et al. 1994, Ney-Nifle & Mangel 2000), and the increase of local species
diversity with regional species diversity (Ricklefs 1987, Srivastava 1999).
Hubbell's neutral theory of biodiversity (2001) caused a conceptual shift in
community ecology, by emphasizing the importance of demographic stochasticity and
random drift in shaping ecological communities. A highly attractive property of Hubbell's
theory is its simplicity – the whole theory is formulated in terms of only three parameters.
However, this advantage has a cost – the theory is not derived from the 'first principles' of
population dynamics, namely, the demographic processes of reproduction, mortality, and
migration. This lack of explicit demographic basis strongly limits the scope of the theory, its
applicability, and its overall explanatory power.
In my PhD work I attempted to lay the ground for a new, fully analytical theory of
species diversity that will be formulated in terms of the fundamental demographic
processes of reproduction, mortality, and migration, and thus, will better connect patterns,
processes and mechanisms of species diversity. We developed the MCD framework, a highly
flexible framework for modeling ecological communities that is based on the theory of
Markov processes, and provided an analytic solution for it. The MCD framework is able to
simultaneously produce all major patterns of species diversity (Fig. 1) as emergent from the
underlying demographic processes of reproduction, mortality and migration of individuals,
including positive, negative, and unimodal relationships between species diversity and
productivity (Waide et al. 1999), linear and curvilinear local-regional diversity relationships
(Ricklefs 1987, Srivastava 1999), gradual and highly delayed responses of species diversity to
habitat loss (Tilman et al. 1994, Ney-Nifle & Mangel 2000), positive and negative responses
of species diversity to habitat heterogeneity (MacArthur 1972, Tews et al. 2004), the
increase of species diversity with area (Arrhenius 1921), and the decrease of species
diversity with geographic isolation (MacArthur & Wilson 1967). As presented in the various
chapters of this work, the flexibility of the MCD framework enable to obtain novel insights
98
and testable predictions that cannot be obtained from (and in some cases contrast) current
theories of species diversity. A particularly interesting finding is that all of the above
patterns can be obtained without any differences in overall fitness between the competing
species. This finding demonstrates that the fundamental elements of the 'neutral' theory are
sufficient to explain a wider range of phenomena than has originally been proposed by
Hubbell (2001).
Fig. 1: Left: Empirically observed patterns of species diversity. Right: Patterns obtained using the MCD
framework. (a) the species-area relationship, (b) decrease of species diversity with geographic isolation, (c)
increase of species diversity with the number of individuals in the community, (d) increase of species diversity
with habitat heterogeneity, (e) decrease of species diversity with habitat loss, (f) increase of local species
diversity with regional diversity, (g) unimodal response of species diversity to productivity, (h) unimodal
response of species diversity to disturbance, (i) positive interaction between the effects of productivity and
disturbance on species diversity.
While Hubbell's theory received much attention, it seems that many ecologists see it
as a toy model for making null hypotheses, and refuse to accept the theory as a valid tool for
obtaining insights and making informed predictions about real ecological communities. On a
personal note, during my PhD research we have experienced great resistance to the
conclusions derived from the MCD framework. While the framework was appreciated for
the analytic advances it carries, attempts to use it to understand patterns of empirical
communities received much criticism – some of it resulted in Chapters 6 and 7.
For many ecologists, the complexity of the mechanisms controlling the diversity of
ecological communities and the contingency of ecological laws lead to a conclusion that no
single theory can fit a wide range of systems (Lawton 1999, Hansson 2003, Simberloff 2004).
As an example, my own work on this subject was cited (December 2013) a total of 85 times
(combining Kadmon & Allouche 2007, Allouche & Kadmon 2009a, Allouche & Kadmon 2009b
and Allouche et al. 2012) while a single paper from my M.Sc. studies focusing on modeling of
single species distribution was cited 258 times (Allouche & Kadmon 2006) and two related
99
papers from this work (Tsoar et al. 2007, Allouche et al. 2008) were cited together 198
times. My interpretation of this difference is that ecologists still believe that a general
theory of species diversity is less practical as a tool for understanding or predicting
ecological patterns than models focusing on simpler phenomena such as the distribution
range of a single species.
Still, I believe that developing a general theory of species diversity is both a feasible
and timely challenge. Alonso et al. (2006) argue that “neutral theory in ecology is a first
approximation to reality. Ideal gases do not exist, neither do neutral communities. Similar to
the kinetic theory of ideal gases in physics, neutral theory is a basic theory that provides the
essential ingredients to further explore theories that involve more complex assumptions”. I
believe that the MCD framework presented here, together with other demographic models
of ecological communities (Chave et al. 2002, Tilman 2004, Gravel et al.2006, Ovaskainen &
Cornell 2006), are a step in this direction, and hope they will lay the ground to a new
generation of analytically tractable theories of species diversity that will be much more
general and realistic than existing theories and consequently, will serve as a more solid basis
for biodiversity protection and conservation planning.
111
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את מציגים 8-ו 7 פרקים. ממנו הנובעות והמסקנות שערכנו הניתוח על ביקורת, 5 בפרק המוצג
.אלה לביקורות תגובותינו
MCD-ה מסגרת). 1 איור( ביולוגי מגוון של יליםמוב דגמים כמה מוכרים האקולוגית בספרות
, והגירה תמותה, ילודה של הדמוגרפיים התהליכים מתוך הללו הדגמים כל את זמנית בו לייצר מסוגלת
בשטח עליה עם המינים בעושר עליה כוללים אלו דגמים. אקולוגית חברה כל בבסיס העומדים
)Arrhenius 1921( ,גיאוגרפי דודבי בשל המינים בעושר ירידה )MacArthur & Wilson 1967( ,השפעה
, )Waide et al. 1999( המינים עושר על סביבתית פרודוקטיביות של אונימודלית או שלילית, חיובית
ירידה, (Ricklefs 1987, Srivastava 1999) המקומי לזה האזורי המינים עשר בין קעור או ליניארי קשר
Tilman et al. 1994, Ney-Nifle( גידול בית להרס אקולוגית חברה בתבתגו שבירה נקודת וקיום מתונה
and Mangel 2000 (המינים עושר על סביבתית הטרוגניות של שלילית או חיובית והשפעה
)MacArthur 1972, Tews et al. 2004 .(ה מסגרת של הגדולה הגמישות-MCD תובנות לייצר מאפשרת
המקובלות לאלו המנוגדות או, קיימות מתיאוריות לייצר ניתן שלא, לבדיקה הניתנות והיפותזות חדשות
במודל גם להתקבל יכולים המתוארים הדגמים שכל הוא מעניין ממצא. האקולוגית בספרות היום
שבתיאוריה המרכזיים שהאלמנטים מראה זה ממצא. השונים המינים בין הבדל אין בו, נייטראלי
. שלו התיאוריה במסגרת האבל שהציע מזה אף הגדול תופעות ןמגוו להסביר מסוגלים הנייטראלית
לנסיונות הזמן שבשל מאמין אני, מאחדת תיאוריה חסר עדיין האקולוגי שהמחקר אף אל
התיאוריה" כי טענו) 2006( וחובריו Alonse. החברות ברמת ביולוגי למגוון כוללת תיאוריה לפיתוח
חברות גם וכך, קיימים אינם אידיאליים גזים. למציאות ראשון קירוב היא באקולוגיה הניטראלית
היא הנייטראלית התיאוריה, בפיזיקה אידיאליים גזים של הקינטית לתיאוריה בדומה. אקולוגיות
מורכבות הנחות המשלבות תיאוריות לפיתוח ההכרחיים הרכיבים את המספקת בסיסית תיאוריה
חברות של נוספים דמוגרפיים מודלים עם ביחד ,כאן שהוצגה MCD-ה שמסגרת מקווה אני ".יותר
תיאוריות של חדש לדואר הקרקע את יניחו אלו שמודלים ומקווה, זה בכיוון צעד הם, אקולוגיות
, מכך וכתוצאה, הקיימות מהתיאוריות יותר ומקיפות כלליות שתהיינה מינים מגוון של אנליטיות
.טבע שימור לפרויקטי יותר איתן כבסיס תשמשנה
ובפרט, חברות של באקולוגיה מובילות תיאוריות ומאחדת מרחיבה כאן המוצגת המסגרת
, )MacArthur & Wilson 1967( איים של הביוגיאוגרפיה תורת, )Hutchinson 1957( הנישה תורת
Patch, הכתמים כתורת גם מוכרת, Metapopulation theory, Levins 1969( העל-אוכלוסית תורת
Occupancy Theory (הנייטראלית והתיאוריה )Hubbell 2001( ,אינטגרטיבית מסגרת לכדי ,
המרה יחסי, נישות חלוקת כגול, ביולוגי מגוון על במשפיעים שונים מנגנונים איחוד המאפשרת
.חישובית פתירה תיאורטית מסגרת לכדי, הפצה ומגבלת תחרותיים
של הנייטראלית התיאוריה בתלהרח בה ומשתמשים MCD-ה מסגרת את מציגים 1-3 פרקים
לבין האבל של המודל בין מתמטית מקשר 1 פרק. שלה ריאליסטיות-הלא בהנחות הצורך וביטול האבל
על תובנות מספק כזה שניתוח ומראה, והגירה תמותה, ילודה של הבסיסיים הדמוגרפיים התהליכים
. שלו קודמות והרחבות האבל של ריהמקו המודל מתוך ברורות שלא, ביולוגי מגוון המבקרים המנגנונים
לאפשר ובכך בספרות המוכרים נייטראלים מודלים לאחד MCD-ה מסגרת יכולת את מדגים 2 פרק
הגדרת את מרחיבים גם אנחנו זה במאמר. אקולוגיות תופעות של בהרבה רחב למגוון שלהם יישום
דגמים של גדול מספר יםומסביר פרט כל בכשירות לנייטראליות האבל של הנוקשה הנייטראליות
Patch( הכתמים תיאורית של כהרחבה MCD-ה מסגרת את מציג 3 פרק. מינים מגוון של אמפיריים
Occupancy theory, Levins 1969( ,המינים בה חברה למידול הכתמים בתיאורית שימוש המאפשרת
.שלהם הדמוגרפיים בקצבים נבדלים
באקולוגיה ביותר המשפיעות מהתיאוריות תייםש בין MCD-ה מסגרת באמצעות מאחד 4 פרק
האלמנטים בין משלבת המסגרת. הנישה ותורת איים של הביוגיאוגרפיה תיאורית – חברות של
ביולוגי מגוון המבקרים המנגנונים לגבי חדשות תובנות מספקת, התיאוריות שתי של הבסיסיים
. בנפרד מהתיאוריות אחת מאף לקבל ניתן שלא, חדשניות היפותזות ומייצרת, אמפיריות בחברות
האיחוד, הגידול בית בהטרוגניות העליה עם גדל תמיד המינים עושר כי צופה הנישה תורת בעוד, בפרט
טוען) Kadmon & Allouche 2007( בפרק המוצג איים של הביוגיאוגרפיה לתיאורית הנישה תורת בין
עושר על) ∩ בצורת( אונימודלי אפקט או ירידה, לעליה להוביל יכולה הגידול בית בהטרוגניות שעליה
מנוגדת זו שפרדיקציה טענו) 2009( וחובריו Hortal. והגירה תמותה, ילודה שבין באיזון כתלות, המינים
Hortal כי להראות מנסה 5 פרק. המודל של ריאליסטיות לא מהנחות ונובעת אמפיריות לתוצאות
ניתוח. המודל של ההנחות ואת שברשותם האמפיריים יםהנתונ את נכון לא פירשו) 2009( וחובריו
& Kadmon( 3 בפרק המוצגות הפרדיקציות של, הפרכה ולא, תמיכה מספק והנתונים המודל של מתוקן
Allouche 2007.(
הגידול בבית הטרוגניות לפיה 3 בפרק שהוצגה ההיפותזה של מקיף ניתוח מספק 6 פרק
נתונים מאגר ניתחנו כך לצורך. המינים עושר על אונימודלית עההשפ או ירידה, לעליה לגרום עשויה
54-ב מינים עושר-הטרוגניות דגמי של על-ניתוח ביצענו, בקטלוניה ציפורים מיני תפוצת של מקיף
הנישה ברחבי נבדלים מינים בהן סימולציות ובחנו, האקולוגית בספרות שהתפרסמו נתונים מאגרי
.3 בפרק שהוצגה בהיפותזה תומכים שביצענו הניתוחים כל. רציף סביבתי גרדיאנט לאורך שלהם
לאינטואיציה מנוגדת המינים בעושר לירידה להוביל עשויה מרחבית שהטרוגניות ההיפותזה
המאמר פורסם בו, PNAS העת לכתב הגישו חוקרים קבוצות שתי. חברות של אקולוגיה חוקרי כמה של
הם פרטים) 1(-ש מניחה היא. ריאליסטיות לא הנחות הכמ מניחה האבל של התיאוריה, עדיין
דינמיקת) 4(, קבוע בחברה הכולל הפרטים מספר) 3(, מרחבית הומוגנית הסביבה) 2(, נייטראליים
קיים פרט של צאצא או מהגר( חדש פרט י"ע בהחלפה מיידית מלווה מוות אירוע כל -' אפס-סכום'
לידע ומנוגדת, האבל של התיאוריה שבבסיס המודל תא מתמטית לפשט נועדו אלה הנחות). בחברה
.אמפיריות אקולוגיות חברות על שלנו
של הבסיסיים התהליכים מתוך ישירות כנובעת מנוסחת לא האבל של התיאוריה, בנוסף
. והגירה תמותה, ילודה של הדמוגרפיים התהליכים –) Population Dynamics( חברה דינמיקת
לקשר קושי קיים, כך עקב. יחיד פרמטר לכדי מתומצתים הדמוגרפיים טריםהפרמ האבל של בתיאוריה
. המינים עושר את הקובעים הבסיסיים הדמוגרפיים לתהליכים התיאוריה של הפרדיקציות את
-ה מסגרת הנקראת, אקולוגיות חברות למידול חדשה תיאורתית מסגרת מציג אני זו בתזה
MCD )MCD framework - Markovian Community Dynamics .(ה מסגרת-MCD את מרחיבה
באמצעות מנוסחת המסגרת – שבה העיקריות הבעיות את ופותרת האבל של הנייטראלית התיאוריה
בין יותר טוב חיבור מאפשרת ולכן, והגירה תמותה, ילודה של הבסיסיים הדמוגרפיים התהליכים
של ריאליסטיות הלא ההנחות את מרככת המסגרת. ביולוגי מגוון של ומנגנונים תהליכים, דגמים
. אקולוגיות תופעות לחקר יותר ריאליסטית מסגרת מספקת ובכך, האבל של התיאוריה
כעל דמוגרפיים תהליכים ועל הבסיסיים' חלקיקים'כ פרטים על מבוססת החדה המסגרת
בתיאוריה הצורך) 1: (עיקריים אתגרים שלושה עם להתמודד מנסה המסגרת. הבסיסיים התהליכים
של הנצפים הדגמים כל את להסביר מסוגלת שתהיה, Bottom-up בגישת תהליכים המבוססת, כוללת
בפרדיקציות התאמות אי בין לגשר המסוגלת תיאורתית במסגרת הצורך) 2(, ביולוגי מינים מגוון
בכלליות גדולה התפשרות ללא, אנליטית פתירה למסגרת השאיפה) 3(, קיימות תיאוריות בין העולות
כמודל שלה הניסוח הן כאן המוצגת MCD-ה מסגרת של חשובות תכונות. התיאוריה של ריאליזם או
, )Life-history trafe-offs( חיים באסטרטגיות המרה-יחסי שילוב, פרטים-מבוסס סטוכסטי דמוגרפי
, השונים םהמיני שפעת את הקובעים סטוכסטיים תהליכים אותם מתוך החברה גודל של מפורש חילוץ
שונות, מרחבית הטרוגניות, גידול בית הרס דוגמת מורכבים אקולוגיים מנגנונים בקלות לשלב והיכולת
.אקראית-לא והפצה, והפרעה בפרודוקטיביות
MCD )Allouche & Kadmon-ה למסגרת אנליטי ופתרון מתמטי ניסוח מספק אני זו בתזה
2009b ,נייטראליים מודלים גם כמו, מיוחד כמקרה) 2001( בלהא של הבסיסי המודל את שכולל) 2 פרק
, האבל של התיאוריה של ריאליסטיות הלא ההנחות את מרככת המסגרת. בספרות שהוצגו אחרים
בגורמים להיבדל יכולים איים) 2(, שלהם הדמוגרפיים בקצבים להיבדל יכולים מיינים) 1: (ובפרט
על משתנה בחברה הפרטים מספר) 3(, מרחבית רוגנייםהט להיות ויכולים, ובידוד לשטח מעבר נוספים
אין) 4(-ו, החברה את המרכיבים מהמינים אחד כל שפעת על המשפיעים דמוגרפיים תהליכים אותם פי
שיפור מביאים אלה יתרונות. האבל של מהמודל בשונה, ילודה או הגירה לבין תמותה בין צימוד
. אנליטי לפתרון יתנתנ שעדיין, התיאוריה של וריאליזם בכלליות
תקציר
הותיקות השאלות אחת היא אקולוגיות בחברות הביולוגי המגוון את המווסתים המנגנונים הסברת
של הגדולה המורכבות בין הבולטת הסתירה היא מעניינת תופעה. באקולוגיה ביותר והמאתגרות
ברמת שנצפים הדגמים טותפש לבין, ביניהם והאינטרקציות השונים המינים על המשפיעים הגורמים
שכל מרמזים, ואקוסיסטמות טקסות של גדול טווח לאורך המתקיימים, הדומים הדפוסים. החברה כלל
הגדולה מהמורכבות מהצפוי בהרבה הפשוט, בסיסיים מנגנונים סט י"ע נשלטות האקולוגיות החברות
). Bell 2001( כאלה מערכות של
למשל מוכרים האקולוגית בספרות. מזה זה תלוי בלתי פןבאו לרב מוסברים אלה דגמים, עדיין
& Rosenzweig( מגוון-פרודוקטיביות, )McGuinness 1984( מינים מספר-השטח לדגם רבים הסברים
Abramsky 1993( ,מקומי-אזורי מינים עושר )Fox & Srivastava 2006 (מגוון של אחרים נצפים ודגמים
Hastings 1980, Hanski & Gyllenberg 1997, Bell 2001, Gotelli & McCabe 2002, McGill( ביולוגי
et al. 2007( ,את באחת להסביר המנסה, תיאורטית מסגרת ואפילו, כוללת תיאוריה חסרה עדיין איך
").אקסיומות(" יסוד עקרונות של מוגדר סט מתוך הללו הדגמים כל
הנסיונות אחד היא) Hubbell 2001" (ביולוגי מגוון של המאוחדת הנייטראלית התיאוריה"
תפקיד את בהדגשתה ער דיון עוררה התיאוריה. באקולוגיה כוללת תיאוריה לפיתוח כה עד השאפתניים
סדרת וכן, Whitfield 2002, Chave 2004( אקולוגיות חברות בעיצוב נייטראליים ותהליכים המקריות
ראו) 2005( וחובריו Holyoak). Ecology העת כתב של) 6(87 בגליון נייטראלית אקולוגיה על מאמרים
.אקולוגיות חברות במחקר המובילות הפרדיגמות מארבע כאחת הנייטראלית בתיאוריה
מתקיימת הנבחנת החברה בו, פרטים-מבוסס סטוכסטי מודל מציגה הנייטראלית התיאוריה
הנייטראלית התפיסה. )החיצונית הסביבה" (יבשת" עם באינטרקציה הנמצא) הנבחן האיזור" (אי"ב
י"ע נשלטת באי הדינמיקה. דמוגרפית כזהים, שלהם המין בזהות תלות ללא, הפרטים כל את ממדלת
של אבולוציוניים תהליכים י"ע נשלטת היבשת בעוד, והגירה תמותה, ילודה של דמוגרפיים תהליכים
השינויים את מזניח מודלה, בהרבה איטיים אלה אבולוציוניים שתהליכים כיוון. והכחדה ספציאציה
לנתח להאבל אפשר המתואר המודל ניסוח. בהרכבה קבועה כאל ליבשת ומתייחס בהם האיטיים
מאוחד הסבר לספק ולראשונה, האקולוגית בחברה המשקל שיווי והתפלגות הדינמיקה את מתמטית
. השונים המינים של השפעה ולהתפלגות בחברה המינים לעושר
והסחיפה' אפס-סכום' דינמיקת הם האבל של הנייטראלית בתיאוריה םמרכזיי עקרונות שני
מיידית מלווה בחברה מוות אירוע שכל קובע' אפס-סכום' דינמיקת עקרון. המינים שפעת של האקראית
הגירה עקב או, בחברה הקיימים הפרטים מאחד והפצה ילודה עקב, חדש פרט י"ע החלפה באירוע
נבחרים, המחליף המין גם כמו, המוחלף המין) 1(-ש בכך אתמתבט האקראית הסחיפה. מבחוץ
הנולד חדש פרט או מבחוץ מהגר – חדש לפרט המקור) 2(-ו, שלהם היחסית השפעה פ"ע באקראי
. באקראי נקבע – האי בתוך ומופץ
דמוגרפית בסטוכסטיות ההכרה היא) 2001( האבל של הנייטראלית התיאוריה של משמעותית תרומה
).Alonso et al. 2006( אקולוגיות חברות בעיצוב משמעותיים גורמים כשני הפצה ובמגבלת
:של בהנחיתו שתהנע זו עבודה
קדמון רונן פרופסור
מסגרת תיאורטית חדשה למגוון מינים ביולוגי
לפילוסופיה דוקטור תואר קבלת לשם חיבור
מאת
אלוש עמרי
בירושלים בריתהע האוניברסיטה לסנט הוגש
3102 דצמבר
