הרצאת סיום הדוקטורט שלי במכון למדעי החיים באוניברסיטה העברית.

1. Extending Hubbell's neutral
theory of biodiversity using a
demographic model
Omri Allouche
Prof. Ronen Kadmon
EEB, HUJI December 2012

2. ?
The Big Question:
Can there be
one, unifying theory of
species diversity?

3. Ecological Communities show Semi-
universal Patterns
a b c
Species diversity
Area Isolation Habitat diversity
Species diversity
d e
ar
ne
Li
Saturated Disturbance level:
Low
Habitat loss Regional diversity Medium
High
f g h
Species diversity
Disturbance
Productivity Disturbance Productivity

4. 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

5. 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.

6. Hubbell’s Neutral Theory
of Biodiversity
“Everything should be made as
simple as possible, but not any
simpler”
A. Einstein

7. 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

8. Hubbell’s Neutral Theory of
Biodiversity
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

9. Analytic solution to Hubbell’s model
J B( n P * , n* n)
J Community size P(n)
n B( P* , n* J)
m Probability that a B(a, b) (a) (b) (a b)
replacing individual is m( J 1) j J m
P* Preg N* P*
an immigrant (1 m) 1 m
• Abundance distribution of each species
= The probability of each species to have 0,1,2,…
individuals in the local community
• Species richness

10. Determinants of Species Richness
in Hubbell’s model
J Community size
m
m Probability that
a replacing
individual is an
SR
immigrant J

11. Neutral models fit empirical data
Coral reef – various scales
Volkov et al. 2007 Nature

12. Trees
Neutral models fit
empirical data
Tropical forests
He 2005 Func. Ecol.
Volkov et al. 2005 Nature

13. 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

14. CRITICISM AGAINST THE NEUTRAL THEORY
Assumptions:
Constant community size
Confined demography
Strict neutrality
Applicability:
Limited scope

15. Individual-based models and the
importance of Demography
ΔN = B – D + I - E

16. Is Hubbell’s model demographic?
Includes processes of
birth, death and
immigration.
J community size
m probability that a replacing
individual is an immigrant
The demographic equation: ΔN = B – D + I - E

17. The Aim:
Develop a general demographic
framework for modeling ecological
communities
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

18. 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

19. Possible events:
Increase by 1 in the number of
Local reproduction
individuals of species k
Immigration
k
g 
N
Decrease by 1 in the number of
individuals of species k Mortality
k
rN
 Emigration

20. The MCD Framework
Possible transitions:
from a state of
5 individuals of species 1 and
3 individuals of species 2:

21. 
dP( N ) SM    k
  k  k k
P N ek rN 
ek
P N ek g N
 
ek
P N gN
 rN

dt k 1
 1 if k = l
ek l
0 otherwise
Possible events:
Increase by 1 in the number of k
individuals of species k
gN

Decrease by 1 in the number of k
individuals of species k
r 
N

22. Analytic Solution
k
  1
  SM N k 1 g N{1,...,k 1}
 
m ek
PMCD ( N ) X (N ) X (N ) X (N ) k

N k 1 m 0 rN{1,...,k 1}
 
( m 1) ek

Community size p( J ) 
PMCD ( N )
N: Nk J
k
local

Abundance Pk ( n) 
PMCD ( N )
N : Nk n
SM
Species Richness SR (1 Pklocal (0))
k 1

23. 1st application:
Single Species
Population with
Competition for Space

24. 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

25. Mortality:
Local reproduction:
Immigration from the regional
pool:
This is the stochastic formulation of
the Levins’ model!

26. Levins’ Model of Metapopulation Dynamics
(1969)
A deterministic model of reproduction and mortality
* e
P 1
c
Levins’ model is a special case of the MCD framework.
Communities are never saturated
The community never occupies all available sites

27. 2nd application:
Multispecies
Community with
Competition for Space

28. 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

29. Mortality:
k
rN
 dk Nk
dk Nk dt
Local reproduction:
A J
bk N k dt k
gN

A
A J
reg
Immigration from the regional bk Nk iP A
k k
pool: A
ik Pkreg ( A J )dt
The solution:
ik
A
 A J 1 SM
bk
Nk
k Pkreg k
bk
J Nk
X (N ) J
y 1
A k 1 dk Nk ! ( x) y x i
i 0

30. Multispecies Community with Competition for Space
• Area A
• Geographic isolation i
• Habitat quality b, d
• Habitat adaptation bk, dk
• Life-history trade-offs bk / d k
• Local-regional relationship
• Mechanisms:
• More Individuals Hypothesis
• Rescue Effect
• Dilution effect

31. CRITICISM AGAINST THE NEUTRAL THEORY
Assumptions:
Constant community size
Confined demography
Strict neutrality
Applicability:
Limited scope

32. The MCD Framework
• Individual-based
• Explicit consideration of demographic processes:
Reproduction, mortality and migration
• Demographic differences among species
• Analytically tractable
• Highly flexible

33. Connecting Hubbell’s
Model to the
MCD Framework

34. Connecting Hubbell’s model
to the MCD framework
i Immigration
b Birth i
d Death m
b
A Area
SR
d
J Community size J
m Probability of replacement A
by immigrant positive influence
negative influence

35. 1000
J 100
10
1
0.1
m
0.01
0.001
100
b
SR
10
0.1 1 10
Basic Reproductive Rate

36. 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

 PMCD ( N ) 
PMCD ( N | J ) PDLM ( N | J )
p( J )

37. The MCD Framework as
a General Framework
for Neutral Models

38. General framework for Neutral Models
Hubbell’s zero-sum model
(Hubbell 2001)
k reg A J k
g 
N
(bNk iP k A) rN
 dk J k
A
iA
where: b dA i dA m
iA b( A 1)
Independent species
(Volkov et al. 2003, 2005, He 2005, Etienne et al. 2007)
k
gN
 bN k iPkreg k
rN
 dk Nk
Community-level density-dependence
(Haegeman & Etienne 2008)
k
g k
 b( J ) Nk i( J ) P k
reg r 
N
d ( J ) Nk
N

39. Relaxing Hubbell’s
assumption of
Strict Neutrality
Assumptions:
Constant community size
Confined demography
Strict neutrality
Applicability:
Limited scope

40. Are species “Life history trade-offs
neutral? (?!) equalize the per capita relative
fitness of species in the
community, which set the
stage for ecological drift”
Annual Survival
S. Hubbell
(2001)
Median Annual Growth

41. Are species neutral?
ik
A
 A J 1 SM
bk
Nk
k Pkreg k
bk
J Nk
X (N ) J
y 1
A k 1 dk Nk ! ( x) y x i
i 0
reproducti on immigratio n
Fitness = mortality
,
mortality
• 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!

42. CRITICISM AGAINST THE NEUTRAL THEORY
Assumptions:
Constant community size
Confined demography
Strict neutrality
Applicability:
Limited scope

43. Examples of More Complex Demography
k reg
AHk J Hk vk AHk
Habitat preference g 
N
(bk Nk i P A)
k k
k
rN
 dk Nk
AHk vk AHk ( A AHk )
Site selection k v( A J )
gN
 (bk N k ik Pkreg A) k
rN
 dk Nk
v( A J ) J
Allee effect k Nk A J k
gN
 (bk N k ik Pkreg A) rN
 dk Nk
Nk k A
A J k Nk
Density dependence g k
 (bk Nk reg
i P A) rN
 dk (bk dk ) Nk
N k k
A Kk
k k J
Carrying capacity gN
 bk Nk ik Pkreg A rN
 dk Nk
K

44. Sample application:
Habitat Heterogeneity
b c
a b c
Species diversity
Isolation Area
Habitat diversity Isolation Habitat
ecies diversity
e d e
ar
ar ine
Line L
D

45. Habitat Heterogeneity
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

46. k reg
AH k J Hk
g 
N
(bk N k ik P k A)
A
k
rN
 dk Nk
 1 H SM
bk
Nk
k Pkreg
Nk
X (N ) Ah Jh 1
AJ h 1
Jh
k 1 dk Nk !

47. 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”

48. Empirical Test of the AHTO
Predictions

49. 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

51. Increasing Niche Width shifts the inflection point
to Higher Heterogeneity values

52. 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

53. Sample application:
Habitat Loss

54. HABITAT LOSS
• Classically studied using
Species-Area curves
• No mechanistic view
• Mostly deterministic
models

55. Habitat Loss
“The greatest existing
threat to biodiversity “
100 Reproduction
1.5
80 3
Species richness
5
k A AD J 60 10
gN
 (bk Nk ik Pkreg A)
A
k 40
rN
 dk Nk
20
AD = The number of
0
destroyed sites 0 20 40 60 80 100
Habitat loss (%)
 A AD J 1 SM
bk
Nk
k Pkreg ik
J Nk
X (N ) , k A
AJ k 1 dk Nk ! bk

56. The MCD Framework
• General framework for modeling ecological communities
• Individual-based
• Stochastic
• Basic demographic processes
• Demographic differences among species
• Analytically tractable
• Relaxes the unrealistic assumptions of Hubbell’s model

57. The MCD Framework
• Provides a stochastic, multispecies version of Levins’
model
• A general framework for neutral models
• Highly flexible
• Useful for the study of complex ecological phenomena
• Provides novel predications

58. Semi-universal patterns
The model Reality
a b c a b c
Species diversity
500
Species diversity
100
Species Diversity
120
400
80
300 80
60
200
40 1.5
100 40 3
5
0 10
20 0
0 2 4 6 8 10 0 200 400 600 800 1000 2 4 6 8 10
Area
Area x 10 4 Isolation
Isolation Habitat diversity Area Isolation Habitat diversity
Heterogeneity
400
Species diversity
Species diversity
100
Species Diversity
d 1.5
e d e
80 3 300
5
60 10
200 ar ar
ne ne
40 Li Li
100
20 Disturbance level: Disturbance level:
Saturated Saturated
0 Low Low
0 20 40 60 80 100 00 200 400 600 800 1000
Habitat loss Regional diversity Medium Habitat loss Regional diversity Medium
Habitat loss (%) Regional Diversity High High
f g h f g h
Species diversity
Species diversity
100 100 500
Species Diversity
Low
Medium
High
Disturbance
Disturbance
80
50 300
60
b=1.5
5
40 10
20
0 100
0.2 0.4 0.6 0.8 1 0 1 2 0 2 4 6 8 10
Productivity Disturbance
Disturbance Productivity
Productivity Productivity Disturbance Productivity
Productivity

59. Limitations of The MCD Framework
• Solution applies only in some cases
• Single trophic level
• No competitive takeover
• Implicit space
• No sex
• No evolutionary processes

60. CRITICISM AGAINST THE NEUTRAL THEORY
Assumptions:
Constant community size
Confined demography
Strict neutrality
Applicability:
Limited scope

62. “… neutral theory in ecology is a first approximation to
reality. Ideal gases do not exist, neither do neutral
communities. Similar to the kinetic theory
of ideal gases in physics,
neutral theory is a basic theory that provides
the essential ingredients to further explore
theories that involve more complex
assumptions”

63. Special thanks to Ronen Kadmon, our lab members – past and
present, Uzi Motro, Lewi Stone, Guy Sella, Gur Yaari, Nadav
Shnerb, Sarit Levy, Jonathan Rubin, Liat Segal and many many
others
AND TO YOU FOR LISTENING :)