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Extending Hubbell's neutral theory of biodiversity using ademographic model

Omri Allouche

Prof. Ronen Kadmon

EEB, HUJI December 2012

? Can there be one, unifying theory of species diversity?

The Big Question:

Ecological Communities show Semi-universal Patterns

Disturbance level:

Medium

Habitat diversity

Habitat loss

ProductivityDisturbanceProductivity

Isolation

Regional diversity

f g hD

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

f g hD

Saturated

Linear

Disturbance level:

Medium

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

f g hD

Saturated

Linear

Habitat diversity

Habitat loss

Isolation

Regional diversity

f g hD

Habitat diversityHabitat diversity

Habitat lossHabitat loss

ProductivityProductivityDisturbanceDisturbanceProductivityProductivity

IsolationIsolation

Regional diversityRegional diversity

AreaArea

f g hD

Saturated

Linear

Theories of community ecology

• Explain the distribution, abundance and interactions

of species

• Different assumptions

• Emphasize different factors as structuring ecological

communities

• Lack of a unified theory

The “godfathers” of modern community ecology

Niche theory (Hutchinson 1957)

• Communities are mainly deterministic assemblages of species, which have different niche characteristics

The theory of island biogeography (MacArthur and

Wilson 1967)

• Community structure is constantly changing. Species richness is determined by the balance between processes of extinction and colonization.

“Everything should be made as

simple as possible, but not any

simpler”

A. Einstein

Hubbell’s Neutral Theory of Biodiversity

Designing the Simplest Model of a Community

1. Individual-based

2. Individuals die and give birth

3. Functionally equivalent species

4. Single trophic level

1. No niches

2. No competitive hierarchy

3. No predation

4. No heterogeneity

5. No temporal variation

6. No dispersal limitation - Global dispersal

7. No differences among species

1. Island receiving immigrants from an

outer mainland

2. Constant community size – all sites are

occupied

3. In each time-step, one individual dies

and is immediately replaced by:

Immigrant m

Local offspring (1-m)

4. Neutral species – individuals are equal

in probability of death and

replacement, regardless of their

species identity

Hubbell’s Neutral Theory of Biodiversity

)()()(),( bababaB

)1(* **

Analytic solution to Hubbell’s model

• Abundance distribution of each species

= The probability of each species to have 0,1,2,… individuals in the local community

• Species richness

J Community size

m Probability that a replacing individual is an immigrant

J Community size

m Probability that a replacing individual is an immigrant

Determinants of Species Richness in Hubbell’s model

Neutral models fit empirical data

Volkov et al. 2007 Nature

Coral reef – various scales

Neutral models fit empirical data

He 2005 Func. Ecol.Volkov et al. 2005 Nature

Tropical forests

Hubbell’s model - Features

• Individual-based

• Analytically-tractable

• Fits species abundance distributions well

• Stochastic

• Emphasizes chance as structuring factor

• Non-equilibrium view of community structure

• Questions the importance of niches and differences among functionally-equivalent species

CRITICISM AGAINST THE NEUTRAL THEORY

Assumptions:

Constant community size

Confined demography

Strict neutrality

Applicability:

Limited scope

ΔN = B – D + I - E

Individual-based models and the importance of Demography

Is Hubbell’s model demographic?

J community size

m probability that a replacing individual is an immigrant

The demographic equation: ΔN = B – D + I - E

Includes processes of birth, death and immigration.

Demographic formulation of community dynamics

Relaxes the unrealistic assumptions of Hubbell’s model

Analytic solution

Able to qualitatively produce known patterns of species-diversity

Useful for the study of complex ecological phenomena

Develop a general demographic framework for modeling ecological

communities

The Aim:

The MCD Framework

The state of the community is described by the number of individuals of each species:

Community size: 5+3+2+7+3 = 20

Abundance of species 4: 2

Species richness: 5

Mortality

Emigrationk

Decrease by 1 in the number of individuals of species k

Possible events:

Local reproduction

Immigrationk

Increase by 1 in the number of individuals of species k

Possible transitions:from a state of

5 individuals of species 1 and

3 individuals of species 2:

The MCD Framework

Nr Decrease by 1 in the number of

individuals of species k

Possible events:

Ng Increase by 1 in the number of

individuals of species k

Sk k k k

k kN e N e N Nk

dP NP N e r P N e g P N g r

1 if =

0 otherwisek l

Abundance

Species Richness

{1,..., 1}

1 0 ( 1)

kk m N m e

Community size

Analytic Solution

( ) ( ) ( )MCD

P N X N X N

( ) ( )kk

p J P N

( ) ( )k

P n P N

(1 (0))MS

1st application:

Single Species Population with

Competition for Space

• Island of area A

• Single species

• Rates of:

Birth b

Death d

Immigration i

• Individuals only establish in vacant sites

SINGLE SPECIES POPULATION WITH

COMPETITION FOR SPACE

Local reproduction:

Mortality:

Immigration from the regional

This is the stochastic formulation of the Levins’ model!

The community never occupies all available sites

Communities are never saturated

A deterministic model of reproduction and mortality* 1

Levins’ Model of Metapopulation Dynamics (1969)

Levins’ model is a special case of the MCD framework.

2nd application:

Multispecies Community with

Competition for Space

• Island of area A

• Multiple species

• Rates of:

Birth bk

Death dk

Immigration ik

• Relative regional abundance

• Individuals only establish in vacant sites

MULTISPECIES COMMUNITY WITH COMPETITION FOR SPACE

A Jb N dt

k kd N dt

( )reg

k ki P A J dt

Local reproduction:

Mortality:

Immigration from the regional

k kNr d N

k k k k

A Jb N i P A

k kNJ k

Jk k k

PA J bX N

The solution:

• Area A

• Geographic isolation i

• Habitat quality b, d

• Habitat adaptation bk, dk

• Life-history trade-offs bk / dk

• Local-regional relationship

• Mechanisms:

• More Individuals Hypothesis

• Rescue Effect

• Dilution effect

Multispecies Community with Competition for Space

Assumptions:

Confined demography

Strict neutrality

Applicability:

Limited scope

• Explicit consideration of demographic processes:

Reproduction, mortality and migration

• Demographic differences among species

• Analytically tractable

• Highly flexible

The MCD Framework

Connecting Hubbell’s Model to the

MCD Framework

i Immigration

b Birth

d Death

A Area

J Community size

m Probability of replacement by immigrant

Connecting Hubbell’s model to the MCD framework

positive influence

negative influence

0.1 1 10

Basic Reproductive Rate

Connecting Hubbell’s model to the MCD framework

• Hubbell’s model forces a constant community size

• In our model community size changes according to the demographic processes that affect species abundance

• Still, given a community size J, the abundance distribution in both models is equal

( )( | ) ( | )

MCD DLM

P NP N J P N J

The MCD Framework as

a General Framework for Neutral Models

General framework for Neutral Models

k kNr d N

k kNr d J

Hubbell’s zero-sum model

( )k reg

A Jg bN iP A

k kNg bN iP

( ) ( )k reg

k kNbg N PJ i J ( )k

kNr d J N

(Hubbell 2001)

(Volkov et al. 2003, 2005, He 2005, Etienne et al. 2007)

(Haegeman & Etienne 2008)

iA b Awhere: b dA i dA

Independent species

Community-level density-dependence

Relaxing Hubbell’s assumption of

Strict Neutrality

Assumptions:

Confined demography

Strict neutrality

Applicability:

Limited scope

Median Annual Growth

Are species neutral? (?!)

“Life history trade-offsequalize the per capita relative fitness of species in the community, which set the stage for ecological drift”

S. Hubbell (2001)

k kNJ k

Jk k k

PA J bX N

Are species neutral?

Fitness = mortality

nimmigratio,

mortality

onreproducti

• Each species has specific demographic rates

• Replacing strict neutrality with trade-offs

• Species are equal in their fitness

• Analytic solution still applies

• No effect on species abundance and species richness!

Assumptions:

Confined demography

Strict neutrality

Applicability:

Limited scope

Examples of More Complex Demography

Habitat preference

Site selection

Allee effect

Density dependence

Carrying capacity

H Hk reg

k k k kN

A Jg b N i

v A AP

A AA k

k kNr d N

( )( )k reg

k k k kN

v A J Jg b N i P A k

k kNr d N

( )k reg

A Jg b N i P A

( )k reg

k k k kN

A Jg b N i P A

A ( ) k

Kr bd N

k k k kNg b N i P A

k kNr d

k kNr d N

Sample application:

Habitat Heterogeneity

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Habitat diversity

Habitat loss

Isolation

Regional diversity

IsolationIsolation

AreaArea

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Habitat diversity

Habitat loss

Isolation

Regional diversity

IsolationIsolation

AreaArea

Saturated

Linear

1. An island consisting of A sites, divided among H habitats

2. Each species is able to establish and persist in only one habitat

3. Individuals disperse and immigrate to random sites

Habitat Heterogeneity

( ) k kk reg

k k k k

Ag b N

k kNr d N

1( ) 1

regNSH k k

Nkh hJ J

h k k k

PbX N A J

The Area-Heterogeneity Tradeoff (AHTO)

“Unless niche width of all species is

unlimited, any increase in

environmental heterogeneity within

a fixed space must lead to a

reduction in the average amount of

effective area available for individual

species”

Empirical Test of the AHTO

Predictions

Study System

• Breeding bird distributions in Catalonia (NE Spain)

• 372 UTM cells of 10x10 km

• Measure Elevation Range in a radius around each cell

• Two distinct surveys

– 1975-1982 with most data collected in 1980-82

– 1999-2002

• Interval between the two surveys (two decades) was larger than the typical lifespan of most bird species

• 10x10km is small enough to detect extinction events but large enough to show within-cell heterogeneity in conditions

• Data include estimates of species abundance

Increasing Niche Width shifts the inflection point to Higher Heterogeneity values

AHTO – Meta-analysis

• 54 datasets with data on

area, elevation range and species

richness

• 43 show positive relationship

• After correcting for the effect of area:

• 6 positive

• 14 unimodal

• 30 non-significant

• Similar results for habitat diversity

Sample application:

Habitat Loss

HABITAT LOSS

• Classically studied using Species-Area curves

• No mechanistic view

• Mostly deterministicmodels

0 20 40 60 80 1000

Habitat loss (%)

Specie

ichness

Reproduction

AD = The number of destroyed sites

Habitat Loss“The greatest existing threat to biodiversity “

( )k reg

k k kD

A Jg b N i P A

k kNr d N

1( ) ,

k kND J k

Jk k k

PA A J bX N

The MCD Framework

• General framework for modeling ecological communities

• Stochastic

• Basic demographic processes

• Demographic differences among species

• Analytically tractable

• Relaxes the unrealistic assumptions of Hubbell’s model

The MCD Framework

• Provides a stochastic, multispecies version of Levins’

• A general framework for neutral models

• Highly flexible

• Useful for the study of complex ecological phenomena

• Provides novel predications

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Habitat diversity

Habitat loss

Isolation

Regional diversity

IsolationIsolation

AreaArea

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Disturbance level:

Medium

Disturbance level:

Medium

Habitat diversity

Habitat loss

Isolation

Regional diversity

Saturated

Linear

Habitat diversity

Habitat loss

Isolation

Regional diversity

IsolationIsolation

AreaArea

Saturated

Linear

2 4 6 8 100

Heterogeneity

0 200 400 600 800 10000

Regional Diversity

0 200 400 600 800 100020

Isolation

0 2 4 6 8 10100

Productivity

Medium

RealityThe model

0 2 4 6 8 10

0 20 40 60 80 1000

Habitat loss (%)

0.2 0.4 0.6 0.8 1

Productivity

b=1.55

Disturbance

Semi-universal patterns

Limitations of The MCD Framework

• Solution applies only in some cases

• Single trophic level

• No competitive takeover

• Implicit space

• No sex

• No evolutionary processes

Assumptions:

Confined demography

Strict neutrality

Applicability:

Limited scope

“… neutral theory in ecology is a first approximation to reality. Ideal gases do not exist, neither do neutral

communities. Similar to the kinetic theory of ideal gases in physics,

neutral theory is a basic theory that provides the essential ingredients to further explore

theories that involve more complexassumptions”

Special thanks to Ronen Kadmon, our lab members – past and present, Uzi Motro, Lewi Stone, Guy Sella, Gur Yaari, NadavShnerb, Sarit Levy, Jonathan Rubin, Liat Segal and many manyothers

AND TO YOU FOR LISTENING :)

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