A Probabilistic Test of the Neutral
Model
C. M. Mutshinda1, R.B. O’Hara1, I.P. Woiwod2
1University of Helsinki, and 2Rothamsted
Research, UK.
Plan of the talk
Introduction
Model
Results
Conclusion
Suggestions
INTRODUCTION
•There is a long-standing interest in identifying the
mechanisms underlying the dynamics of ecological
communities
•The list of presumed mechanisms is still growing
•Existing theories can be subdivised in two categories:
neutral and non-neutral models
•The debate between the two sides is still very much
alive
An ecological community is a group of trophically
similar species that actually or potentially compete in a
local area for the same or similar resources.
•Neutral models assume Ecological Equivalence of
species, i.e. same demographic properties (birth
death immigration speciation rates) for all individuals
irrespective of species.
Consequence: Species richness and relative
species abundance distributions (SAD) are
assumed to be generated entirely by drift
between species
•Non-neutral models consider that species may
differ in their demographic properties, their
competitive abilities or their responses to
environmental fluctuations
The most documented version of neutral models is
the Unified Neutral Theory of Biodiversity and
Biogeography (UNTBB) developed by Hubbell in
2001.
From now on, neutral theory refers to Hubbell's
model
• The UNTBB considers communities on two scales of
communities:
Local Community
Governed by birth, death, immigration (from a
metacommunity)
Dynamics taking place an ecological time scale.
Metacommunity
Include an additional mechanism of speciation taking
place on an evolutionary time scale.
•Main Assumptions of the UNTBB:
 Ecological Equivalence
 Zero-Sum (ZS) assumption : constant
community size (saturated communities)
Consequences of the assumptions
Relative Species Abundance entirely
genarated by random Drift
A typical SAD, the zero – sum multinomial
(ZSM).
•Criticisms of the UNTBB: have concerned both
assumptions
Ecological Equivalence (e.g. Mauer &Mc Gill
2004; Poulin 2004; Chase2005)
Zero-Sum assumption (e.g. Alder 2003; McGill
2003; Williamson & Gaston 2005 )
The critics of the ZSM have generally assumed
equilibrium and have proceeded by comparing the fit
of the ZSM to a theoretical distribution mainly the
Lognormal
However, over the last 30 years, ecologists have been
moving away from equilibrium ideas (e.g. Wallington et
al. 2005), but Hubbell leaps straight back in.
A dynamical model such as the UNTBB can be
examined without assuming equilibrium.
A sensible way of examining the neutral model would
would consist of fitting the model to the data and
assessing:
how realistic the parameter estimates are
if the changes in the abundance of the species
can be explained by the model with a realistic
community size
We Develop and fit a discrete-time neutral model
identical to Hubbell's in all other aspects except that
We relax the assumption of constant
community size
Data
3 macro-moth (Lepidoptera) time series from the
Rothamsted Insect Survey light-traps network in
the UK: Geescroft I & II (from the Rothamsted farm
in Hertfordshire) and Tregaron (from a Nature
reserve in mid-Wales)
Number of species and years:
Geescroft I (352, 40); Geescroft II (319, 26); Tregaron
(371, 28).
THE MODEL
•Process Model
Nber of ind. of species i at time t
N i , t ~ Pois ( i , t )




J
1

m
*
Cm

*
P




i
,
t
t

1
t
i
,
t

1
t
i
,
t
Relative abundance of sp. i at t-1
Immigration rate at time t
N i , 1 ~ Pois (  i )
J t  i Ni , t :community
size at time t
Pi , t  Pi
(time-scale separation)
i , t  1  mt  * Ni , t 1  mt * JPi , t 
JPi , t  J t 1 * Pi
•Sampling Model
yi , t ~ Pois  N i , t * qt 
Sampling rate (observed proportion) at time t
The same analyses were carried out on the geometrid
(Geometridae) species alone which are known to
respond in a similar way to light (Taylor and French
1974).
Nber of geometrid species in the 3 datasets: 135, 127
& 135 respectively.
Model Fitting
 Bayesian approach
Noninformative priors
mt : Beta(1,1) JPi , t : U (5,100)
qt :   0.1, 0.2
i ~   0.01, 0.01
We used MCMC via OpenBUGS to fit the model
RESULTS
Fig. 1: Unrealistic Community sizes
Greescroft II
2.5
2.0
2.0
2.0
2000
1990
1980
2000
2.5
1995
2.5
1995
3.0
Expected
1990
3.0
3.0
1985
3.5
1980
3.5
3.5
1975
4.0
1970
Log10(Community size)
4.0
1990
Observed
4.0
C
1985
B
1980
A
Tregaron
1975
Greescroft I
Fig. 2: Unrealistic Sampling Rates
1970
1980
1990
2000
15
20
Tregaron
0
5
10
15
10
0
5
10
5
0
Sampling Rate
15
20
Greescroft II
20
Greescroft I
1975
1985
1995
1980
The horizontal dashed line is drawn at height 1!
1990
2000
CONCLUSION
The neutral model does not fit the data well as it
would need parameter values that are impossible
Thus, random drift alone cannot explain the
variation in species abundances
Possible reasons for the excess of temporal variation:
A number of important mechanisms are simply
ignored. These include:
environmental stochasticity
Density-dependence
Species heterogeneity
Effects of species interactions
SUGGESTIONS
The model can be extended to include the missing
components, this will result in a complex model
Complex models can be developed and fitted
under the hierarchical Bayesian framework
Ecological hypotheses such as neutral community
structure can be examined from the results
We examined if parameters of such a model may be
identifiable, we developed a dynamical model including
environmental stochasticity and interaction coefficients
The model was fitted to a dataset comprising 10 among
the most abundant species at Geescroft I
All the parameters turned out to be identifiable
Scientific and common names of the 10 species
Nber
Scientific name
Common name
1
Selenia dentaria
Early Thorn
2
Selenia tetralunaria
Purple Thorn
3
Apeira syringaria
Lilac beauty
4
Odontopera bidentata
Scalloped Hazel
5
Colotois pennaria
Feathered Thorn
6
Crocallis elinguaria
Scalloped oak
7
Opistograptis luteolata
Brimstone moth
8
Ourapteryx sambucaria
Swallow-tail
9
Opocheima pilosaria
Pale brinbley beauty
10
Lycia hispidaria
Brindley beauty
Process model
S
 



N

j 1 i , j j , t 


E  Ni , t 1   i , t  Ni , t exp ri , t 1 
 

Ki

 
ri , t
: density-independent per capita growth rate of species i at time t,
 i, j
:per capita effect of species j on the growth of species i,
Ki
S
:carrying capacity for species i,
: number of species in the community
N i , t ~ Pois  i , t 1  , t  2
Ni , 1 ~ Pois  i 
Sampling model
yi , t ~ Pois  Ni , t * qi , t 
Parameter model
r i , t ~ N  i ,  i
1

 i , j ~ N  0,  1 
Priors
i ~ N (0,0.1)
 i ~ (0.001, 0.001)
qi , t ~ Beta(1,1)
Ki ~ Exp(0.0001)
i ~   0.01, 0.01
Model fitting by MCMC via OpenBUGS
Results
• Significant differences in species-specific
environmental variances
•The posterior estimates of the interaction coefficients
,
reveal a significant negative effect of the Opistograptis
luteolata (species #7) on the reminder as illustrated in
the following table
The results suggest a non-neutral community structure
posterior means of the interaction coefficients
Species
1
2
3
4
5
6
7
8
9
10
1
-0.32
0.12
-0.02
0.13
0.13
0.08
0.69
0.05
0.08
0.00
2
0.31
-1.07
-0.05
0.01
0.00
-0.11
0.58
-0.02
-0.11
0.07
3
0.27
0.05
0.00
0.09
0.11
0.13
0.52
0.05
0.05
0.01
4
0.13
0.01
0.00
-0.10
0.03
0.08
0.32
0.04
0.13
0.01
5
0.15
-0.02
0.00
0.08
0.19
0.03
0.53
0.03
0.13
-0.01
6
0.01
-0.09
0.01
0.00
-0.04
0.21
0.51
0.10
0.14
-0.03
7
0.3
0.26
-0.02
0.21
0.02
-0.13
0.94
-0.01
0.12
-0.05
8
0.29
0.05
0.02
0.07
0.10
0.10
0.65
0.03
0.10
0.01
9
0.20
0.04
0.02
0.06
0.07
0.07
0.06
0.04
0.04
0.00
10
0.2
0.06
0.02
0.08
0.09
0.08
0.58
0.05
0.06
-0.02
posterior means of the interaction coefficients
Remarks
•Real communities are typically much larger than 10
species. Hence, The dimensionality of the model
may be too large
•Some interaction coefficients are almost zero or
insignificant, it might be worth not estimating them
•Sensible ways of pulling the model's dimensionality
down to a tractable level are needed, and this is
where variable selection comes into play.
Work in Progress
We are now working on Bayesian variable selection
methods such as Gibbs Variable Selection, Stochastic
Search Variable Selection or Reversible Jump MCMC
to extend the applicability of the model to large
community datasets.
THANK YOU
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