Null models in Ecology
Diane Srivastava
Sept 2010
The big questions
• What constitutes a null model?
• What biological assumptions are behind the
deterministic constraints in null models?
• How do these constraints affect our ability to
detect “interesting” patterns?
• Is a process or a pattern assumed to be
stochastic in null models?
• Are neutral models null models?
What is a null model?
Gotelli and Graves (1996):
“A null model is a pattern-generating model that
is based on randomization of ecological
data…Certain elements of the data are held
constant and others are allowed to vary
stochastically…The randomization is designed
to produce a pattern that would be expected
in the absence of a particular ecological
mechanism”
Two views of null models:
• Statistical descriptions of randomized data
(Simberloff 1983)
• Simulations of random assembly processes
(Colwell and Winkler 1984, Gotelli and Graves
1996)
Example: A null model for the trophicrank hypothesis
Populate each bromeliad with individuals by randomly
sampling regional pool
Stop populating a bromeliad when it reaches its
individual capacity (related to size)
Frequency
From data, construct regional pool of individuals
100
90
80
70
60
50
40
30
20
10
Repeat 1000s of time
Compare observed trophic-rank effect
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
Calculate difference in species-area curves between
predators and prey
-0.10
0
Predator – Prey Z values
Example: A null model for the trophicrank hypothesis
From data, construct regional pool of individuals
Populate each bromeliad with individuals by sampling
regional pool
Stop populating a bromeliad when it reaches its
individual capacity (related to size)
Calculate difference in species-area curves between
predators and prey
Repeat 1000s of time
Compare observed trophic-rank effect
What aspects of the data were
held constant?
Example: A null model for the trophicrank hypothesis
From data, construct regional pool of individuals
Populate each bromeliad with individuals by sampling
regional pool
Stop populating a bromeliad when it reaches its
individual capacity (related to size)
Calculate difference in species-area curves between
predators and prey
Repeat 1000s of time
Compare observed trophic-rank effect
What aspects of the data were
held constant?
What was randomized?
Example: A null model for the trophicrank hypothesis
From data, construct regional pool of individuals
Populate each bromeliad with individuals by sampling
regional pool
Stop populating a bromeliad when it reaches its
individual capacity (related to size)
Calculate difference in species-area curves between
predators and prey
Repeat 1000s of time
Compare observed trophic-rank effect
What aspects of the data were
held constant?
What was randomized?
What ecological process(es) were
removed by the null model?
Example: A null model for the trophicrank hypothesis
From data, construct regional pool of individuals
Populate each bromeliad with individuals by sampling
regional pool
Stop populating a bromeliad when it reaches its
individual capacity (related to size)
Calculate difference in species-area curves between
predators and prey
Repeat 1000s of time
Compare observed trophic-rank effect
What aspects of the data were
held constant?
What was randomized?
What ecological process(es) were
removed by the null model?
What biological assumptions did I
make?
Null models assume:
• Ecological hypotheses are falsifiable (sensu
Popper)
• The simplest hypothesis is the best (Occam’s
razor)
• Ecological processes can be removed from the
data by simulation (now the fun begins…)
A brief history of null models
• First null models tested S/G ratios in communities
(Maillefer 1929, William 1947)
• Rarefaction techniques for species richness estimates
developed in 1960s (Sanders 1968)
• Null models developed for niche overlap and body size
limits in 1970s (Brown 1973, Sale 1974)
• First neutral model for relative abundance: Calwell 1976
• Passive sampling models for species-area relationships in
1980s (Colwell 1981)
Null models developed for:
-
Species-area relationship
Diversity constancy through time
Food-web structure
Niche overlap
Limiting similarity to body sizes
Species: genus ratios in communities
Phylogenetic diversity within communities
Nestedness
Species co-occurrence
Null models developed for:
-
What process might your
null model remove?
Species-area relationship
Diversity constancy through time
What large dataset might
Food-web structure
you randomly sample to
generate a community Niche overlap
level null pattern?
Limiting similarity to body sizes
Species: genus ratios in communities
Phylogenetic diversity within communities
Nestedness
Species co-occurrence
Species co-occurence
Do species distributions reflect negative effects of competition?
If so, species should not be distributed randomly – some species
combinations should be missing or rare
Checkerboard distribution:
siteA siteB
Species A
1
0
Species B
0
1
(Unfortunately this pattern can occur for other reasons like
habitat filtering!)
Species co-occurence
1960s/70s: Chris Pielou developed null model for co-occurrence that involved
randomizing a species list amongst sites, and comparing to observed.
Also developed variance test for co-occurrence.
1975: Jared Diamond published 7 “assembly rules” based on observations of
birds on islands.
1979: Connor and Simberloff countered Diamond with randomization tests of
co-occurrence
1980s: Gilpin and Diamond argue C&S’s constraints bias randomization test to
no finding patterns; Schluter’s variance ratio test
1990s: Gotelli and Entsminger release ECOSIM
Species co-occurrence: constraints
Pielou and Pielou (1968): Rows and columns both equiprobable
Connor and Simberloff (1979): Rows and column totals both
fixed + restricted each species to sites with a species richness
in its observed range.
Gilpin and Diamond (1982): Cell probability proportional to
observed row and column totals.
Species co-occurrence: constraints
Pielou and Pielou (1968): Rows and columns both equiprobable
Species equally likely to provide next colonist, sites equally likely
to receive next colonist
Connor and Simberloff (1979): Rows and column totals both
fixed + restricted each species to sites whose species richness
was in its observed range.
Gilpin and Diamond (1982): Cell probability proportional to
observed row and column totals.
Species co-occurrence: constraints
Pielou and Pielou (1968): Rows and columns both equiprobable
Species equally likely to provide next colonist, sites equally likely
to receive next colonist
Connor and Simberloff (1979): Rows and column totals both
fixed + restricted each species to sites with a species richness
in its observed range. Regionally rare species remain rare,
species rich islands remain species rich, incidence functions of
species are preserved.
Gilpin and Diamond (1982): Cell probability proportional to
observed row and column totals.
Species co-occurrence: constraints
Pielou and Pielou (1968): Rows and columns both equiprobable
Species equally likely to provide next colonist, sites equally likely
to receive next colonist
Connor and Simberloff (1979): Rows and column totals both
fixed + restricted each species to sites with a species richness
in its observed range. Regionally rare species remain rare,
species rich islands remain species rich, incidence functions of
species are preserved.
Gilpin and Diamond (1982): Cell probability proportional to
observed row and column totals. Sites are targets that differ
in likelihood of being hit, some species have traits that
predispose them to be next colonist
Species co-occurrence: constraints
Site saturation
Species
regional
abundance
Site accessibility
Species
colonization
ability
Species co-occurrence: constraints
Site accessibility
Site saturation
Species
regional
abundance
Deterministic constraint
Species
colonization
ability
?
Modeled as stochastic
Species co-occurrence: constraints
Site
accessibility
Site saturation
Paradox:
To incorporate these constraints, need to assume that
observed regional occurrence of species or occupancy
Species– in order to test
of sites is independent of competition
colonization
for effects of competition on
species occurrence and
Species
ability
site occupancy!
regional
abundance
Deterministic constraint
?
Modeled as stochastic
Species co-occurrence: the pairs?
• Randomization tests look at overall patterns in cooccurrence, little meaning to pairs with high C-scores
(Gotelli and Graves 1996)
• C-scores of individual pairs influenced by abundance
of species
• Need to conduct separate randomization tests of just
the pair to determine significance.
Species co-occurrence in the age of
meta-analyses
• Meta-analysis of 96 datasets suggests that most have nonrandom associations (fixed row, fixed column method; Gotelli
and McCabe 2002)
What is a null model? (Revisited)
Graham Bell (2000) argued that there are two
distinct types of null model: Statistical and
dynamic.
Statistical: “output varies stochastically” (e.g.
randomization tests of co-occurrence)
Dynamic: “input to the system varies stochastically”
(e.g. neutral model: mutation and demography
are stochastic)
Statistical null vs. neutral
(Statistical) null
Neutral
What
pattern
process
Species
Interactions?
No
No
Species
Equivalence?
No
Yes
Random processes do not cause
random patterns!
Neutral models show correlation of
species between sites (top) or
though time (bottom) simply
because of dispersal limitation.
Spatial covariance
Such non-random patterns are not
expected under a statistical null
model.
Temporal covariance
Simulated neutral communities
have significant C-scores (Ulrich
2004, Bell et al. 2005)
(Bell et al 2006, Bell 2000)
Can neutral processes explain nonrandom species associations?
Perhaps only in
part!
C-scores in real
data are much
higher than
expected from
neutral models
Neutral model:
mean = 0.5
100 datasets:
mean = 2.67
(Ulrich 2004)
(Gotelli and McCabe 2002)
ECOSIM Lab
West Indies Finches
West Indies finches,
From Gotelli and Abele 1982
Your task:
Experiment with trying all
possible combinations of
equiprobable, proportional
and fixed row/column totals!
What changes in constraints
have the most effect?
Gotelli 2000: Fixed row methods have
lowest type 1 (&2) errors
Truly random
SIM2: Column totals equiprobable
SIM4: Columns totals proportional
SIM9: Column totals fixed (Connor
& Simberloff 1979)
Finches