Modelling the joint abundance of more
plant species - pin-point cover data
Christian Damgaard
Department of Bioscience
Aarhus University
Bioscience – Aarhus University
Why model the joint distribution of
spatial plant abundance data?
If we know a species is present at a position then the probability that
another species is present at the same position is reduced
More used information → More statistical power
Possible to fit ecological models to multivariate abundance data
instead of summarizing the variation by ad hoc distance methods
e.g. Bray-Curtis distances in ordination techniques
Knowledge on spatial covariance patterns
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Plant species are patchy distributed
Large-scale ecological processes (among-sites):
environmental drivers
extinction / colonization of sites
Small-scale ecological process (within-sites):
size of individuals
limited dispersal
density-dependent population growth
inter-specific competition
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Cover at a site is beta distributed
Plant species are patchy distributed ⇒
Single species cover data are L or U - shaped
𝜈 ~ 𝐵𝑒𝑡𝑎
𝑞
𝛿
− 𝑞,
1−𝑞 1−𝛿
𝛿
𝐸 𝜈 =𝑞
Var 𝜈 = 𝛿 𝑞 (1 − 𝑞)
q : mean cover at the site
d : intra-plot correlation
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𝑞 = 0.3, 𝛿 = 0.3
Dry heathlands
The cover of the two dominating species on dry heathlands, Calluna
vulgaris and Deschampsia flexuosa, and all other higher plant
species covary
179 Danish dry heathland sites with 20 – 40 plots
Mean cover was 0.257,0.208,and 0.535, respectively
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Spatial aggregation – single species
The beta-binomial distribution adequately model the
distribution of single species pin-point cover data
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Joint pin-point cover distribution
The beta-binomial distribution may be generalized to n species using
the Dirichlet-multinomial distribution
𝑦𝑖 pin-point hits of species i;
𝑝𝑖 relative cover of species i;
𝑌~𝑀𝑛
𝑛 𝑦𝑖
𝑛 𝑦𝑖
≥ # of grid points
𝑛 𝑝𝑖 = 1
, 𝑝1 , … , 𝑝𝑛−1 , 1 − 𝑝1 − ⋯ − 𝑝𝑛−1
Λ 𝑝1 , … , 𝑝𝑛−1 ~𝐷𝑖𝑟
𝑞1 −𝑞1 𝛿
𝑞𝑛−1 − 𝑞𝑛−1 𝛿 1 − 𝛿 𝑞1 −𝑞1 𝛿
𝑞𝑛−1 − 𝑞𝑛−1 𝛿
,…,
,
−
− ⋯−
𝛿
𝛿
𝛿
𝛿
𝛿
𝐸 𝑝1 , … , 𝑝𝑛−1 = 𝑞1 , … , 𝑞𝑛−1
𝛿 intra-plot correlation due to spatial aggregation
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Environmental gradient
The possible relationships between an abiotic driver, 𝑧𝑗 , and the
spatial distribution of n species may be modelled as,
(𝑞1,𝑗 , … , 𝑞𝑖,𝑗 , … , 𝑞𝑛−1,𝑗 )~ 𝑁(𝛼𝑖 + 𝛽𝑖 𝑧𝑗 , Σ)
Σ is the covariance matrix in the multinormal distribution
with parameters 𝜎𝑑,𝑖 and 𝜌 in the three species case
Precipitation
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Estimation
Hierarchical Bayesian modelling approach
Mean site cover was modelled by latent variables
MCMC - Metropolis-Hastings algorithm
Statistical inferences on the parameters of interest were based on
the marginal posterior distribution of the parameters
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Spatial aggregation within sites
𝜹 measures the intra-plot correlation due to spatial aggregation
Parameter
2.50%
50%
𝛅
0.197
0.206
0.215
-
𝛼𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
-0.194
0.025
0.190
0.59
𝛼𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
0.186
0.345
0.514
1
𝛽𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
-4.12E-06
0.000193
0.000459
0.9725
𝛽𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
-0.00041
-0.00022
-2.60E-05
0.0113
𝜎𝑑,𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
0.122
0.139
0.159
-
𝜎𝑑,𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
0.107
0.120
0.136
-
𝜌
-0.299
-0.142
0.0317
0.0544
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97.50% P(X > 0)
Spatial aggregation parameter (𝛿)
If we do not take the spatial aggregation into account then statistical
inference is biased (Damgaard 2013)
Like pseudo-replication
A serious and often made error in the analysis of plant cover
The spatial aggregation parameter may be used to test different
hypotheses on the role of spatial aggregation at the level of the
community
Test of neutrality (Damgaard and Ejrnæs 2009)
Does species aggregation reduce the importance of competitive
interactions? (Damgaard 2011)
Does climate change or nitrogen deposition change the spatial structure
or increase size of plants? (Damgaard et al. 2012)
Bioscience – Aarhus University
Effects of precipitation on dry heaths
(𝑞1,𝑗 , … , 𝑞𝑖,𝑗 , … , 𝑞𝑛−1,𝑗 )~ 𝑁(𝛼𝑖 + 𝛽𝑖 𝑧𝑗 , Σ)
Parameter
Σ = 𝑓(𝜎𝑑,𝑖 , 𝜌)
2.50%
50%
97.50% P(X > 0)
𝛿
0.197
0.206
0.215
-
𝛼𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
-0.194
0.025
0.190
0.59
𝛼𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
0.186
0.345
0.514
1
𝛽𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
-4.12E-06
0.000193
0.000459
0.9725
𝛽𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
-0.00041
-0.00022
-2.60E-05
0.0113
𝜎𝑑,𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
0.122
0.139
0.159
-
𝜎𝑑,𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
0.107
0.120
0.136
-
𝜌
-0.299
-0.142
0.0317
0.0544
Relatively high precipitation is correlated with low cover of D. flexuosa and
high cover of C. vulgaris
Bioscience – Aarhus University
Spatial covariation among sites
(𝑞1,𝑗 , … , 𝑞𝑖,𝑗 , … , 𝑞𝑛−1,𝑗 )~ 𝑁(𝛼𝑖 + 𝛽𝑖 𝑧𝑗 , Σ)
Parameter
Σ = 𝑓(𝜎𝑑,𝑖 , 𝜌)
2.50%
50%
97.50% P(X > 0)
𝛿
0.197
0.206
0.215
-
𝛼𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
-0.194
0.025
0.190
0.59
𝛼𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
0.186
0.345
0.514
1
𝛽𝐶. 𝑣𝑢𝑙𝑔𝑎𝑟𝑖𝑠
-4.12E-06
0.000193
0.000459
0.9725
𝛽𝐷.𝑓𝑙𝑒𝑥𝑢𝑜𝑠𝑎
-0.00041
-0.00022
-2.60E-05
0.0113
𝝈𝐝,𝐂. 𝒗𝒖𝒍𝒈𝒂𝒓𝒊𝒔
0.122
0.139
0.159
-
𝝈𝐝,𝐃.𝒇𝒍𝒆𝒙𝒖𝒐𝒔𝒂
0.107
0.120
0.136
-
𝛒
-0.299
-0.142
0.0317
0.0544
Almost significant negative large-scale spatial covariation indicates that the
oceanic–continental climatic gradient determine the abundance of the two
species
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Conclusions
The framework allows modelling of the spatial covariation in cover at
two levels
Local – among individual plants
Regional – among communities / sites
We know from manipulated experiments that drought increase the
competitive effect of Deschampsia on Calluna (Ransijn et al. in prep)
Negative spatial covariation at the site level and significant effects of
precipitation on the observed cover of Deschampsia and Calluna
It is suggested that the oceanic–continental climatic gradient
determine the relative abundance of the two species
Effects of soil may be confounded with precipitation gradient?
The cover ratio of the two species may be used to monitor the effect
of climate change on dry heathlands
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