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A Demographic Framework of Species Diversity

Thesis submitted for the degree of
“Doctor of Philosophy”
By

Omri Allouche
...
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. Thank...
Abstract
Understanding the mechanisms controlling the diversity of ecological communities is one of the oldest and
most ch...
Contents
Introduction ………………………………………………………………………………………….

1

Goals of Research …………………………………………………………………………………...

6

Cha...
Introduction
Explaining the mechanisms regulating the diversity of ecological communities is one of the oldest and most
ch...
Results
acquisition

Deterministic
/ Stochastic

Bell 2001
Pigolotti & Concini 2009

Simulations
Simulations

Stochastic
S...
Hubbell's "Unified Neutral Theory of Biodiversity" (Hubbell 2001) is the most
ambitious attempt to date to develop a gener...
P( N  1 | N j )  (1  m)

(J  N ) N
( J  N ) reg
m
Pj
J
J 1
J

where P( y | x) is the probability that the number of...
neutrality'); that each death event is immediately followed by a recruitment event ('zerosum dynamics'); and that all site...
Gotelli & McGill 2006). Nonetheless, in its current formulation, the theory is not applicable
to studies of complex ecolog...
(MacArthur & Wilson 1967), Metapopulation theory (Levins 1969) and the Neutral theory
(Hubbell 2001), into an integrative ...
work was published, we did not have an analytic solution to the MCD framework. Instead,
we provided a highly-accurate appr...
Chapter 1 – Paper
Demographic analysis of Hubbell’s neutral theory of biodiversity

Omri Allouche and Ronen Kadmon
Journal...
11
11
12
13
14
15
16
Chapter 2 – Paper
A general framework for neutral models of community dynamics

Omri Allouche and Ronen Kadmon
Ecology Let...
18
19
21
21
22
23
24
25
26
27
28
Chapter 3
Applications of the Demographic Framework and Its
Relation to Patch Occupancy Theory

.‫בתת פרק זה מוצגות תוצאות...
Patch-occupancy theory (first introduced by Levins (1969, 1970)) is one of the most
influential theories in modern ecology...
Tilman (1994) extended the Levins metapopulation model into a model of multispecies dynamics. In this formulation (see als...
species, competition for space, habitat heterogeneity, habitat loss, habitat productivity and
more.
In this chapter I exem...
(4a)

 N i P reg 
k
g N   bk k  k k   A  J 
I 
 A

(4b)

k
rN  dk N k

Equations (4) present a Markovian formu...
ik

dk

  for all species where λ and  are constants). For a given regional species pool,

increasing area enhances and...
400

Local Species Richness

350
300
250
200
150
100
50
0

0

100

200

300

400

500

600

700

800

900

1000

Regional ...
has the highest reproduction/mortality ratio in the local community), the second species has
a reproduction/mortality rati...
Fig. 2: Combined effects of local competitive ability (the ratio between local reproduction (b) and mortality (d)
rates), ...
2b, and 2c). This result can be considered as an ecological counterpart of ‘the red queen
hypothesis’ in evolutionary biol...
Allee effect
In many situations individuals experience reduced reproduction or survival at small
population sizes. This ef...
 A  J  1 J J S  bk 
X (N ) 
  
 A J
k 1  d k 
M

(8c)

k 

Nk

 P 

reg
k k
Nk

Nk !

ik
v
A,  
bk...
Combining multiple mechanisms
For clarity I have separately introduced each ecological mechanism into the model.
However, ...
reducing effective reproduction rates and increasing mortality of local offspring and
immigrants arriving in inhospitable ...
their relative abundance. Regional abundance distributions were generated for biodiversity numbers in range 1 to
50
40. Fo...
usefulness of deterministic models for studies of habitat loss, where these processes take
prominent role.
In contrast to ...
Chapter 4 – Paper
Integrating the Effects of Area, Isolation, and Habitat Heterogeneity
on Species Diversity: A Unificatio...
46
47
48
49
51
51
52
53
54
55
56
57
58
Chapter 5
Habitat Heterogeneity, Area, and Species Diversity: A (unpublished)
response to Hortal et al. (2009)

.‫בתת פרק ...
Hortal et al. (2009) criticized the model proposed by Kadmon and Allouche (2007)
and argued that its predictions are unrea...
variable niche widths may show a decrease in species richness with increased habitat
diversity (three out of the four case...
cannot be interpreted as evidence against the model proposed by Kadmon and Allouche
(2007).
The prediction that species ri...
A

A

50
b=2
=5
= 15

40

Species richness

Species richness

50

30
20
10

1

2

3

4

5

6

7

8

vs = 1

30
20
10

9 10...
while keeping the overall amount of resource constant (e.g. increasing variation in prey size
while keeping the overall bi...
Hawkins 2008). Currently, no theory is capable of explaining these variable and apparently
conflicting results. The area-h...
Moreno-Rueda, G., and M. Pizarro. 2009. Relative influence of habitat heterogeneity, climate, human
disturbance, and spati...
Taxon

Nislands

Birds
Bats
Butterflies
Terrestrial isopods
Terrestrial isopods
Plants
Land snails
Birds
Ground beetles
Br...
Source
Moreno-Rueda & Pizaro 2009

Debinski & Brussard 1994
Moreno-Saiz & Lobo 2008

Yaacobi et al. 2007

Ruggiero & Hawki...
Chapter 6 – Paper
Area–heterogeneity tradeoff and the diversity of ecological
communities

Omri Allouche, Michael Kalyuzhn...
71
71
72
73
74
75
76
Chapter 7 – Paper
Reply to Hortal et al.:
Patterns of bird distribution in Spain support the area–heterogeneity
tradeoff

...
78
79
81
Chapter 8 – Paper
Reply to Carnicer et al.:
Environmental heterogeneity reduces breeding bird richness in
Catalonia by inc...
82
83
84
Discussion and Conclusions
Hubbell's "Unified Neutral Theory of Biodiversity" (Hubbell 2001) is one of the most
ambitious ...
These advantages imply an improvement in generality and realism, without sacrificing
analytical tractability.
Several prev...
patterns, and argues that while his assumptions are uncorroborated, they provide
reasonable simplifications of reality. Ac...
remarkable result indicates that even though the fundamental assumptions of Hubbell's
neutral theory are unrealistic and c...
original theory still consider area and isolation as the primary determinants of species
richness.
In Chapter 3 I demonstr...
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
Omri's PhD Thesis
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Omri's PhD Thesis

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My thesis in Computational Ecology was carried under the supervision of Prof. Ronen Kadmon, and submitted on Dec 2013 to the senate of the Hebrew University in Jerusalem.

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  1. 1. 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
  2. 2. This work was carried out under the supervision of: Prof. Ronen Kadmon
  3. 3. 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.
  4. 4. 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).
  5. 5. 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
  6. 6. 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 tradeoffs 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. 1
  7. 7. Results acquisition Deterministic / Stochastic Bell 2001 Pigolotti & Concini 2009 Simulations Simulations Stochastic Stochastic Differences among species No No O'Dwyer & Green 2010 Chave et al. 2002 Rosindell & Phillimore 2011 Loreau & Mouquet 1999 Allouche & Kadmon 2009 Kadmon & Allouche 2007 Sole et al. 2004 Tilman et al. 1994 Sole et al. 2005 Mouquet & Loreau 2003 Weitz & Rothman 2003 Xiao et al. 2009 Cordonier et al. 2006 Dornelas 2010 Kondoh 2001 Analytic Simulations Simulations Simulations Analytic Analytic. Both Simulations Both Simulations Simulations Simulations Both Simulations Analytic Stochastic Stochastic Stochastic Deterministic Stochastic Stochastic Both Deterministic Both Stochastic Stochastic Stochastic Deterministic Stochastic Deterministic No Yes No Yes No Yes Yes Yes Yes Yes No Yes Yes No Yes Speciation Space Dispersal Tradeoffs* No Instantaneou s No No Protracted No No No No No No No No No No No No Explicit Explicit Local Local No No Explicit Explicit Implicit Implicit Implicit Implicit Both Implicit Explicit Implicit Explicit Explicit Implicit Implicit Implicit Local Both Global Global Global Global Both Local Local Global Local Local Global Global Global No C-B, C-D No No No S B-D C-B No No No C-B No, C-B No C-B Habitat heteroge neity No No a b c d e f g h i No No No No No Yes Yes Yes Yes Yes Yes No No No No = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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. 2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  8. 8. 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 'zerosum 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) P( N  1 | N )  (1  m) N (J  N ) N  m (1  Pjreg ) J J 1 J 3
  9. 9. P( N  1 | N j )  (1  m) (J  N ) N ( J  N ) reg m Pj J J 1 J where P( y | x) 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, reg and Pj 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) local j P ^ *  J  B( N  P , N  N ) (N )    ^ *  N  B( P , N  J ) where B(a, b)  (a)(b) (a  b) , P ^  J m m( J  1) reg Pj and N *   P^ . (1  m) 1 m 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 4
  10. 10. neutrality'); that each death event is immediately followed by a recruitment event ('zerosum 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 tradeoffs 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, 5
  11. 11. 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 6
  12. 12. (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 7
  13. 13. 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 metaanalysis 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. 8
  14. 14. 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 9
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  22. 22. Chapter 2 – Paper A general framework for neutral models of community dynamics Omri Allouche and Ronen Kadmon Ecology Letters (2009) 12:1287-1297 17
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  34. 34. 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. 29
  35. 35. 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) dP  bP(1  P)  dP dt 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
  36. 36. Tilman (1994) extended the Levins metapopulation model into a model of multispecies 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) dP  bP(1  P)  dP  i(1  P) dt 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: 1 (3)   PMCD ( N )    X ( N )  X ( N )  N  where X ( N )  SM Nk 1  T k 1 m0 k N{1,..., k 1}  mek ; T  k N k gN X ((0,0,...,0))  1 ; S M is the number of k rN ek ; species in the regional species pool, N k is the abundance of species k in the local community, N  ( N1 ,..., N SM ) is the local community abundance vector and N{1,...,n}   N1 ,..., Nn , 0,..., 0  . We also demonstrated how the transition rates in the MCD framework can be easily modified to allow the study of a community of independent 31
  37. 37. 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 k k the transition rates g N and rN , 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 steadystate 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 Pkreg ik , where Pkreg is the relative abundance of the k'th species in the regional I 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: 32
  38. 38. (4a)  N i P reg  k g N   bk k  k k   A  J  I   A (4b) k rN  dk N k 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) X (N )   A  J  1 J A J b    dk  k 1  k  SM Nk  P  reg k k Nk Nk ! and we use the Pochhammer notation ( x) y  , k  ik A , bk I y 1   x  i  . This expression also determines i 0 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., 33 bk dk   and
  39. 39. ik dk   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 & Kodricbrown 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). 34
  40. 40. 400 Local Species Richness 350 300 250 200 150 100 50 0 0 100 200 300 400 500 600 700 800 900 1000 Regional Species Richness 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 bk dk  and ik dk  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  . Blue lines: A = 2000,  =  = 0.5. Black lines: A = 50000,  = 3. Dashed lines:  calculated local species diversity for different combinations of area (A), 0.5. Red lines: A = 2000, = 0.01. Solid lines:   = 3. Green lines: A = 50000,  and = 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 longterm 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., 35
  41. 41. 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. 36
  42. 42. 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, 37
  43. 43. 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 g N  bk Nk  ik Pkreg A (5b) k rN  d k N k J K The steady-state distribution of species abundances is given by equation (3), where: (5c) K J SM  bk  X (N )    J ! k 1  d k  Nk  P  reg k k Nk ! Nk , k  ik A bk 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: A J A (6a) k g N  (bk N k  ik Pkreg A) (6b)  N  k rN   d k  (bk  d k ) k  N k Kk   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) X (N )  k   A  J  1 J AJ b    k  k 1  k  SM Nk  P  reg k k Nk d  N k ! k  1  k  Nk ik b  dk A, k  k Kk bk 38 ,
  44. 44. 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 g N  (bk N k (7b) Nk A J  ik Pkreg A) N k  k A k rN  d k N k 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: N k 1 (7c) X (N )   A  J  1 J A J b    dk  k 1  k  SM Nk  m m 0 2   k Pkreg m   k Pkreg k N k ! k  N k  , k  ik A. bk 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 g N  (bk N k  ik Pkreg A) v( A  J ) v( A  J )  J k rN  d k N k (8b) The steady-state distribution is given by equation (3), where: 39
  45. 45.  A  J  1 J J S  bk  X (N )      A J k 1  d k  M (8c) k  Nk  P  reg k k Nk Nk ! ik v A,   bk 1 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 v  1 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 ) or moves to a randomly v 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 vk indicating the preference of species k to its source habitat over unsuitable habitats. This gives: (9a) k g N  (bk N k  ik Pkreg A) (9b) AH k  J H k vk AH k AH k vk AH k  ( A  AH k ) k rN  dk N k The steady-state distribution of species abundances is given by equation (3), where: k   bk   vk X ( N )    Ah  J h  1 J     h  AH (vk  1)  A   h 1 k 1  d k   k  H (9c) k  SM N ik A bk 41 Nk  P  reg k k Nk Nk ! ,
  46. 46. 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 k k added by further extensions of the transition rates g N and rN . 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 g N  (bk RN k  ik Pkreg A) (10b) v( A  AD  J ) v( A  AD  J )  AD  J k rN  d k N k With the dimensionality of the model remaining similar. The steady-state distribution is again given by equation (5), where:  A  AD  J  1 J J S  bk X (N )     A  AD  J k 1  d k M (10c) k   R  Nk  P  k reg k Nk Nk ! v ik A,   bk R 1 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 41
  47. 47. 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 = bk dk ) and between immigration and death (i = ik dk ) 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 = bk dk ). 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 42
  48. 48. their relative abundance. Regional abundance distributions were generated for biodiversity numbers in range 1 to 50 40. For a given biodiversity number, we used the Poisson distribution (ref ) 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: dPk v(1  PD  P)  (bk Pk  ik Pkreg )  d k Pk dt v(1  PD  P)  PD  P where Pk is the proportion of sites occupied by species k in steady-state, P  SM P k 1 k 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: SM SR  SM   (1  Pk* ) A k 1 where Pk* 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 43
  49. 49. 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. 44
  50. 50. 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 45
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  64. 64. 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. 59
  65. 65. 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
  66. 66. 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 61
  67. 67. 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 62
  68. 68. A A 50 b=2 =5 = 15 40 Species richness Species richness 50 30 20 10 1 2 3 4 5 6 7 8 vs = 1 30 20 10 9 10 = 1000 = 1012 40 1 2 Heterogeneity B B 15 Species richness Species richness b = 10 = 15 = 20 10 5 5 6 7 8 9 10 8 9 10 vh = 1 2 4 6 8 10 12 14 16 18 20 Heterogeneity =2 = 10 40 30 20 10 0 4 50 20 0 3 Heterogeneity 1 2 3 4 5 6 7 Heterogeneity Figure 1: Effect of habitat heterogeneity on species Figure 2: Effect of habitat heterogeneity on species richness for different levels of birth rate in a model richness for (A) different levels of site selection where each species can exist in multiple habitats ability and (B) different levels of dispersal to (see appendix for a description and analytical suitable habitats (see appendix for analytical solution of the model). (A) Niche width of individual solutions). For both models, niche width of species is randomly selected from a uniform individual species is randomly selected from a distribution between 1 and 10, area (A) = 10000, uniform distribution between 1-10, area (A) = mortality (d) = 1, immigration (i) = 0.001, regional 10000, reproduction (b) = 5, mortality (d) = 1, species richness (SM) = 1220 equally abundant immigration (i) = 0.001, regional species richness species; (B) Niche width of individual species is (SM) = 1220 equally abundant species. Results are randomly selected from the empirically observed based on 1000 realizations of the respective model. 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. 63
  69. 69. 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 heterogeneitydiversity 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 & 64
  70. 70. 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. Literature Cited Allouche, O., and R. Kadmon. 2009. A general framework for neutral models of community dynamics. Ecology Letters 12:1287-1297. Amarasekare, P. 2003. Competitive coexistence in spatially structured environments: a synthesis. Ecology Letters 6:1109-1122. Currie, D. J. 1991. Energy and Large-Scale Patterns of Animal-Species and Plant-Species Richness. American Naturalist 137:27-49. Davidar, P., K. Yoganand, and T. Ganesh. 2001. Distribution of forest birds in the Andaman islands: importance of key habitats. Journal of Biogeography 28:663-671. Debinski, D. M., and P. F. Brussard. 1994. Using Biodiversity Data to Assess Species-Habitat Relationships in Glacier National-Park, Montana. Ecological Applications 4:833-843. Deshaye, J., and P. Morisset. 1988. Floristic richness, area, and habitat diversity in a hemiarctic archipelago. Journal of Biogeography 15:747-757. Etienne, R. S. 2000. Local populations of different sizes, mechanistic rescue effect and patch preference in the Levins metapopulation model. Bulletin of Mathematical Biology 62:943-958. Haila, Y., O. Jarvinen, and S. Kuusela. 1983. Colonization of islands by land birds - prevalence functions in a finnish archipelago. Journal of Biogeography 10:499-531. Hortal, J., K. A. Triantis, S. Meiri, E. Thebault, and S. Sfenthourakis. 2009. Island Species Richness Increases with Habitat Diversity. American Naturalist 174. Juriado, I., A. Suija, and J. Liira. 2006. Biogeographical determinants of lichen species diversity on islets in the West-Estonian Archipelago. Journal of Vegetation Science 17:125-+. Kadmon, R., and O. Allouche. 2007. Integrating the effects of area, isolation, and habitat heterogeneity on species diversity: A unification of island biogeography and niche theory. American Naturalist 170:443-454. Kohn, D. D., and D. M. Walsh. 1994. Plant species richness – the effect of island size and habitat diversity. Journal of Ecology 82:367-377. Kotze, D. J., J. Niemela, and M. Nieminen. 2000. Colonization success of carabid beetles on Baltic islands. Journal of Biogeography 27:807-819. Moreno Saiz, J. C., and J. M. Lobo. 2008. Iberian-Balearic fern regions and their explanatory variables. Plant Ecology 198:149-167. 65
  71. 71. Moreno-Rueda, G., and M. Pizarro. 2009. Relative influence of habitat heterogeneity, climate, human disturbance, and spatial structure on vertebrate species richness in Spain. Ecological Research 24:335-344. Newmark, W. D. 1986. Species area relationship and its determinants for mammals in western northamerican national-parks. Biological Journal of the Linnean Society 28:83-98. Nilsson, S. G., J. Bengtsson, and S. As. 1988. Habitat diversity or area per-se: species richness of woody-plants, carabid beetles and land snails on islands. Journal of Animal Ecology 57:685-704. Purves, D. W., and J. Dushoff. 2005. Directed seed dispersal and metapopulation response to habitat loss and disturbance: application to Eichhornia paniculata. Journal of Ecology 93:658-669. Ralph, C. J. 1985. Habitat association patterns of forest and steppe birds of northern patagonia, argentina. Condor 87:471-483. Reed, T. 1981. The number of breeding landbird species on British Islands. Journal of Animal Ecology 50:613-624. Ricklefs, R. E., and I. J. Lovette. 1999. The roles of island area per se and habitat diversity in the species-area relationships of four Lesser Antillean faunal groups. Journal of Animal Ecology 68:1142-1160. Ruggiero, A., and B. A. Hawkins. 2008. Why do mountains support so many species of birds? Ecography 31:306-315. Rydin, H., and S. O. Borgegard. 1988. Plant-species richness on islands over a century of primary succession - lake hjalmaren. Ecology 69:916-927. Sillen, B., and C. Solbreck. 1977. Effects of area and habitat diversity on bird species richness in lakes. Ornis Scandinavica 8:185-192. Tews, J., U. Brose, V. Grimm, K. Tielborger, M. C. Wichmann, M. Schwager, and F. Jeltsch. 2004. Animal species diversity driven by habitat heterogeneity/diversity: the importance of keystone structures. Journal of Biogeography 31:79-92. Tilman, D. 2004. Niche tradeoffs, neutrality, and community structure: A stochastic theory of resource competition, invasion, and community assembly. Proceedings of the National Academy of Sciences of the United States of America 101:10854-10861. Triantis, K. A., M. Mylonas, M. D. Weiser, K. Lika, and K. Vardinoyannis. 2005. Species richness, environmental heterogeneity and area: a case study based on land snails in Skyros archipelago (Aegean Sea, Greece). Journal of Biogeography 32:1727-1735. Triantis, K. A., S. Sfenthourakis, and M. Mylonas. 2008. Biodiversity patterns of terrestrial isopods from two island groups in the Aegean Sea (Greece): Species-area relationship, small island effect, and nestedness. Ecoscience 15:169-181. Wood, D. R., L. W. Burger, J. L. Bowman, and C. L. Hardy. 2004. Avian community response to pinegrassland restoration. Wildlife Society Bulletin 32:819-828. Yaacobi, G., Y. Ziv, and M. L. Rosenzweig. 2007. Effects of interactive scale-dependent variables on beetle diversity patterns in a semi-arid agricultural landscape. Landscape Ecology 22:687-703. 66
  72. 72. Taxon Nislands Birds Bats Butterflies Terrestrial isopods Terrestrial isopods Plants Land snails Birds Ground beetles Bryophytes Mammals All plants Halophytic plants Birds Plants Carabids Snails Birds Birds Plants 19 17 15 13 12 42 12 45 24 32 24 31 31 73 17 17 17 44 29 37 Linear model 2 Adj. R AICC .203 -59.1 -.05 -54.1 .56 -70.7 -.05 -31.5 -.1 -27.4 .108 -185 -.097 -27.4 .049 -172 .112 -78 .476 -131 .037 -76.1 .044 -120 .047 -120 .25 -333 .021 -47.4 -.057 -46.1 -.057 -46.1 .223 -177 .116 -109 .024 -133 Quadratic model 2 Adj. R AICC .271 -59.4 -.1 -51.9 .53 -68.4 -.135 -28.9 -.213 -24.6 .257 -193 .449 -34 .026 -170 .144 -77.6 .477 -129 .042 -74.9 .131 -122 .101 -121 .275 -335 -.028 -45.1 -.125 -43.6 -.041 -44.9 .218 -175 .096 -107 .035 -133 Improvement F 2.601 .269 .149 .156 .067 10.055 10.904 .003 1.828 1.072 1.107 4.215 2.932 3.387 .279 .089 1.23 .751 .327 1.402 Relationship Source Linear NS Linear NS NS Unimodal Unimodal NS NS Linear NS Unimodal NS Linear NS NS NS Linear Linear NS Ricklefs & Lovette 1999 Ricklefs & Lovette 1999 Ricklefs & Lovette 1999 Triantis et al. 2008 - Astipalaia Triantis et al. 2008 - Kalymnos Kohn & Walsh 1994 Triantis et al. 2005 Davidar et al. 2001 Kotze et al. 2000 Juriado et al. 2006 Newmark 1986 Deshaye & Morisset 1988 Deshaye & Morisset 1988 Reed 1981 Nilsson et al. 1988 Nilsson et al. 1988 Nilsson et al. 1988 Haila et al. 1983 Sillén & Solbreck 1977 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 2 was negative with the inflation point occurring within the range of the data. Statistically significant values of R and F are marked in bold. 67
  73. 73. Source Moreno-Rueda & Pizaro 2009 Debinski & Brussard 1994 Moreno-Saiz & Lobo 2008 Yaacobi et al. 2007 Ruggiero & Hawkins 2008 Taxon Amphibians Reptiles Birds Mammals Butterflies Birds Ferns Ferns Ferns Ferns Ferns Ferns Ferns Beetles (Carabidae) Beetles (Carabidae) Beetles (Tenebrionidae) Beetles (Tenebrionidae) Birds Birds Region Iberian peninsula Iberian peninsula Iberian peninsula Iberian peninsula Glacier National Park Glacier National Park Iberian peninsula Region 2 Iberian peninsula Region 3 Iberian peninsula Region 4 Iberian peninsula Region 5 Iberian peninsula Region 7 Iberian peninsula Region 8 Iberian peninsula Region 9 Beit Guvrin, Israel Beit Guvrin, Israel Beit Guvrin, Israel Beit Guvrin, Israel Western mountains of the New World Western mountains of the New World Scale 10 × 10 km 10 × 10 km 10 × 10 km 10 × 10 km 1 × 1 km 1 × 1 km 50 × 50 km 50 × 50 km 50 × 50 km 50 × 50 km 50 × 50 km 50 × 50 km 50 × 50 km <2000 m2 >2000 m2) <2000 m2 >2000 m2) 50 × 50 km 50 × 50 km Measure Habitat diversity Habitat diversity Habitat diversity Habitat diversity Elevation diversity Elevation diversity Altitude range Altitude range Altitude range Altitude range Altitude range Altitude range Altitude range Patch spatial heterogeneity Patch spatial heterogeneity Patch spatial heterogeneity Patch spatial heterogeneity Temperature range Altitude SD Table 2. Examples for documented variability in heterogeneity-diversity relationships among taxa, regions, scales, and measures of heterogeneity. 68 Pattern NS NS Positive NS Negative NS Unimodal Positive Unimodal Positive NS Unimodal NS Unimodal NS NS Unimodal Negative Positive
  74. 74. 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 69
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  82. 82. 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 77
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  86. 86. 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 81
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  90. 90. 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. 85
  91. 91. These advantages imply an improvement in generality and realism, without sacrificing analytical tractability. Several previous modifications of Hubbell’s model relaxed the assumption of zerosum 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 86
  92. 92. 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 highlyproductive communities, such as tropical forests, but does not seem to hold for lowproductive 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 87
  93. 93. 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 88
  94. 94. 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., bk dk   and ik dk   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 89

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