Download models - sortie-nd

Seminar 1
Neighborhood Analyses of Forest
Ecosystems
using Likelihood Methods and Modeling
Likelihood Methods in Forest Ecology
October 9th – 20th , 2006
Discrete Patch Models of
Community Dynamics

Theory of gap phase dynamics in mesic forests
Logging
gaps at
Date
Creek,
British
Columbia
Limitations of the Traditional Patch
Dynamics Models

The models generally ignore
- Heterogeneity within patches
- Spatial interactions between patches
- Interactions between disturbed patches and the surrounding
undisturbed matrix
More generally, discrete
patches are the exception rather
than the rule…
Arguments for a Spatially-Explicit, Neighborhood
Theory of Forest Ecosystem Dynamics

Local neighborhoods rather than an arbitrary plot size
(or a watershed) as the fundamental units of forest
ecosystems
strong vertical integration
relatively weak horizontal integration
Examples of Neighborhood Phenomena
in Forests

Localized effects of the spatial distribution of tree species on:
-
Seed rain and seedling establishment
-
Spatial variation in understory light levels
-
Soil resource availability and nutrient cycling
-
Abundance and activity of small mammals
-
Competitive interactions between trees
-
Dynamics and effects of pests and pathogens
Foraging patterns of large herbivores
Themes...

Shifting Focus... from simply estimating the mean of a
process in a plot to developing models to understand the
processes that produce spatial and temporal variation in
ecosystem properties.

Model formulation... how do you choose a functional
form to describe a neighborhood process?
But how do you integrate all of this
detail?...

SORTIE: a spatially-explicit model of forest dynamics...
SORTIE
Canham et al. 1994 (CJFR)
(1990 – 1996)
Light
Pacala et al. 1993 (CJFR)
Pacala et al. 1996 (Ecol. Monogr.)
Recruitment
Growth
2
Growth
Seedling
Density
6
1
0
0
10
20
30
4
2
0
40
0
Distance from Parent
25
50
75
100
Light
Ribbens et al. 1994 (Ecology)
Mortality
Mortality
Pacala et al. 1994 (CJFR)
1
0.5
0
0
1
2
Growth
Kobe et al. 1995 (Ecol. Appl.)
What have we added since…

Canopy tree – soil interactions and niche differentiation
along soil nutrient gradients
(Adrien Finzi, Feike Dijkstra, Seth Bigelow)

Effects of herbivores (deer and small mammals)
(Chris Tripler, Jackie Schnurr)

Competition, growth and mortality of adult trees
(Mike Papaik and Maria Uriarte)

Revisiting seed dispersal and seedling dispersion
Did we get it right 10 years ago?

Succession still appears to be largely driven by competition
for light…

But, even very fine-scale variation in soil nutrient availability
can dramatically alter competitive hierarchies (leading to
different successional patterns and dominants)

Predictions of forest structure and biomass require explicit
consideration of adult tree competition

Herbivores can change everything…
SORTIE/ND
A “Neighborhood” Model of Forest Dynamics

The Neighborhood Model approach in SORTIE
- Individual-tree based and spatially-explicit
-
The spatial scale of the effective “neighborhood” varies for
any given property or process, as needed
-
Canopy gaps recognized as heterogeneous entities that emerge
as a result of the process of tree mortality
-
Canopy gaps “perceived” differently by different tree species
because of differences in light requirements and shade
tolerance
Phase 2 (1996 - ): SORTIE-ND

Completely re-design the model with a more open
architecture to provide a flexible modeling platform for
neighborhood dynamics of forests:
Programming: Lora Murphy
(based on earlier work by
Mike Papaik)
http://www.sortie-nd.org
Parameterization of SORTIE-ND
Biome
Forest Type
Field Sites
Temperate
deciduous
forest
oak – northern
hardwood forests
Great Mountain
Forest
(Connecticut)
Temperate
coniferous
forest
Temperate
evergreen
rain forest
Tropical
evergreen
rain forest
interior cedar –
hemlock forests
Date Creek Exp.
Forest
(British Columbia)
Waitutu Forest
(New Zealand)
mixed beech –
podocarp forests
Tabonuco forest
Luquillo Forest
(Puerto Rico)
#
Species
9
9
Focus
plant-animal
interactions;
ecosystem processes;
invasive species;
sustainable forestry
12
effects of introduced
herbivores
12
hurricane disturbance
Parameterization also underway or recently completed in:
• sub-boreal spruce forests of British Columbia (K. D. Coates)
• boreal aspen spruce forests of Quebec (C. Messier and collaborators at UQAM).
A Likelihood Framework for Analysis of
Neighborhood Phenomena in Forests

Specification of alternate models
(as a form of hypothesis testing)

Parameter estimation
(using ML methods)

Model comparison
(using AIC)

Model evaluation
(using a variety of metrics)
s
Effect  
i 1
n
 f (dist , size
j 1
ij
ij
)
 higher order terms...
Examples...

Neighborhood approaches to the prediction of
- Seed predation by small mammals
- Leaf litterfall and nutrient return via litterfall
- Defining the “footprint” of ecosystem transformation by
invasive tree species (Lorena Gomez Aparicio)
Effects of Canopy Tree Neighborhoods on Spatial
Distribution and Activity of Small Mammals

Spatial variation in occurrence of small mammals is strongly
influenced by the spatial distribution of large-seeded tree
species that represent important food resources

Spatial variation in rates of seed and seedling predation vary
accordingly...
Schnurr, J. L., C. D. Canham, R. S. Ostfeld, and R. S. Inouye.
2004. Neighborhood analyses of small-mammal
dynamics: Impacts on seed predation and seedling
establishment. Ecology 85:741-755.
Characterizing the neighborhood…

Use an
ordination to
synthesize the
effects of
neighboring
canopy trees…
Source: Schnurr et al. (2004)
1995
(year after mast)
1.6
1.4
1.2
1
Nonlinear Poisson
regression of small
mammal capture
data
mice
voles
chipmunks
0.8
0.6
Y  A* B
0.4
0.2
Note: 1994 was a mast year for red
oak seed production
0
-5 -4 -3 -2 -1 0 1 2
3 4 5
Oak
Maple
Canopy Tree Neighborhood Ordination Axis
Changes in average capture
rates of small mammals as
a function of local canopy
tree composition and seed
production at GMF
1
0.8
1996
0.6
0.4
0.2
Maple
-5
Source: Schnurr et al. (2004)
( X C )2
-4
Oak
0
-3
-2
-1
0
1
2
3
4
Canopy Tree Neighborhood Ordination Axis
5
Example: Predation by rodents on Rimu
seeds
Deb Wilson and nested
exclosures for deer and
small mammals at Waitutu
Forest, South Island, NZ
Probability of Predation
1
Probability of predation of
Rimu seed as a function of
local canopy tree abundance...
0.8
0.6
0.4
0.2
0
-0.5
Rimu
0
0.5
1
Silver Beech
Canopy Tree Neighborhood Ordination Axis
Leaf Litterfall: It’s easy to collect, but
can we predict it...

Use maximum likelihood methods to estimate spatiallyexplicit leaf litterfall “functions”

Assume litterfall (g/m2) from a source tree is a function
of:
- Species
- Tree size (DBH)
- Distance from the tree (m)
- Direction from the tree (anisotropy)
Leaf Litter Dispersal Functions
Weibull (Exponential) Function: leaf litter declines monotonically
with distance (Ferrari and Sugita 1996):
n
litterfall (g/0.5 m )  TLP 
2
i 1

 dbhi  1  disti 
e


 30.0  
Lognormal Function: leaf litter reaches a peak at some distance
away from the tree (Greene and Johnson 1996):
n
litterfall (g/0.5 m )  TLP 
2
i 1
 dist i 
ln(
)


1
 


2




 dbhi  1
e


30
.
0



2
Anisotropy: Does Direction Matter?
Lognormal Litter Dispersal Function:
n
litterfall (g/0.5 m )  TLP 
2
i 1
 dist i 
ln(
)


1
 

2 




 dbhi  1
e


 30.0  
2
Incorporate Effect of Direction from Source Tree on Modal Disperal
Distance1:
  X 0  X p cos( anglei   
1Staelens,
J., L. Nachtergale, S. Luyssaert, and N. Lust. 2003. A model of wind-influenced
leaf litterfall in a mixed hardwood forest. Canadian Journal of Forest Research.
What is being ignored?

Tree height

Local topography

Temporal variation in timing of leaf fall

...?
What is being simplified?

Assumes anisotropy is a smooth (cosine) function of
direction

Assumes a tree is a point source
Field Methods

Collect leaf litterfall for 1 season (Sept. – Dec.) using two 0.5 m2
littertraps at each of 36 sites at Great Mountain Forest, in
northwest Connecticut1

Collect a subsample of litter from a rain-free period for analysis
of concentrations of calcium (Ca), magnesium (Mg) and
potassium (K) concentrations in fresh leaf litter

Map the distribution of all trees within 25 m of the littertraps
1Note:
3 litter traps could not be used because of damage
Maximum Likelihood Estimation

Assumed that the data were normally distributed

Used numerical integration to estimate the normalizer

Used simulated annealing to find the 4-6 ML parameter estimates,
with a moderately high initial temperature and a slow annealing
schedule (250,000 iterations)
n
litterfall (g/0.5 m )  TLP 
2
i 1
 dist i 
ln(
)


1
 

2 




 dbhi  1
e


 30.0  
2
Comparison of Alternate Leaf Litter Dispersal
Functions…
Species
n
Weibull (4 parameters)
Maximum likelihood
AICcorr
R2
Acer
rubrum
60
4
-229.3
467.4
0.86
Acer
saccharum
57
4
-248.4
505.6
0.82
Fagus
grandifolia
63
4
-243.9
496.4
0.82
Fraxinus
americana
45
4
-174.7
358.4
0.84
Quercus
rubra
45
4
-201.5
412.0
0.78
Tsuga
canadensis
51
4
-179.0
366.8
0.83
Lognormal (4)
Maximum likelihood
AICcorr
R2
4.00
-228.9
466.6
0.86
4.00
-247.5
503.7
0.82
4.00
-243.6
495.9
0.82
4.00
-175.0
359.0
0.84
4.00
-197.5
404.0
0.81
4.00
-177.1
363.0
0.84
Anisotropic Lognormal (6)
Maximum likelihood
AICcorr
R2
6.00
-218.9
451.4
0.90
6.00
-246.3
506.3
0.83
6.00
-239.0
491.5
0.85
6.00
-155.5
325.2
0.93
6.00
-194.3
402.9
0.83
6.00
-151.5
316.9
0.94
4.5
33.8
1.1
46.1
 AICcorr*
15.1
* Strength of evidence for anisotropy (i.e. difference in AICcorr between anisotropic and isotropic lognormal models,
when anisotropic had lower AIC)
100
Downwind
ACRU
ACSA
FAGR
FRAM
QURU
TSCA
75
2
Predicted Leaf
Litter Dispersal
Functions
Leaf Litterfall (g / 0.5 m )
50
25
0
0
5
10
15
20
25
100
Upwind
75
50
25
0
0
5
10
15
Distance (m)
20
25
Model Evaluation
Goodness of fit:
- R2: 0.83 – 0.94 (It can’t get much higher...)
- Bias: slopes of regression of observed on predicted
-
(forced through the origin) = 0.998 – 1.001 (unbiased)
Prediction error: (RMSE – standard deviation of
residuals):
140
» hemlock: 4.74
» red oak: 18.2
120
Red maple
100
Observed

80
60
40
y = 1.0009x
R2 = 0.8979
20
0
0
20
40
60
Predicted
80
100
120
150
125
A
B
115
105
100
N
95
75
ACRU
ACSA
FAGR
FRAM
QURU
TSCA
85
75
-
0
25
50
75
100
125
2.00
C
2.00
105
2.00
2.25
1.50
1.75
2.252.00
1.50
1.50
2.00 2.25
1.75
2.00
1.75
S
1.75
2.752.75
3.25
3.00
1.75
2.00
2.25
3.50
2.50
3.25
3.00
2.75
1.50
75
25
1.50
1.75
1.75
2.00
2.25
2.00
35
350
300
55
65
75
1.75
1.25
1.50
1.50
1.00
1.00
1.25
1.00
0.4
0.4 0.5
1.00
1.50
1.25
0.3
0.75
1.25
1.50
1.25
1.50
Ca (g/m2)
1.25
55
E
65
0.3
0.3
0.4
0.3
0.50.4
0.3
0.4
0.3
0.5 0.4
0.5 0.5
0.4 0.6
0.6
0.5
0.3 0.7
0.3
0.3
0.2
0.3
0.2
0.3
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
75
75
0.3
0.3
Mg (g/m2)
0.3
0.3
0.4
25
-
0.2
0.2
1.00
1.00
0.2
0.2
0.3
0.5 0.5
0.5
0.3
1.50
45
0.3
85
1.00
1.50
0.4
95
0.4
0.4
0.5
0.3
0.4
0.5
0.4
0.4
1.25
2.00
1.75
1.50
1.25
1.50
0.5
0.4
0.75
1.25
0.4
0.3
0.3
0.3
0.5
0.5
105
0.2
0.4
0.4
0.5
1.25
1.00
1.50
500
450
0.3
0.5
115
1.25
1.50
85
D
0.5
1.25
1.75
2.75
2.50
2.25
2.00
1.50
1.25
1.50
600
400
550
450
500
300350
300
400
45
0.5 0.5
0.5
1.25
2.25
2.25
1.502.00
35
0.5
2.00
2.00
95
2.00
1.75
2.25 2.00
2.25
1.75
2.25
300 300
450
500
350400
350 400 350
350
400
350
350
300
450
500
400
450
350 350
750
700
650
600
550
350
500
350
550
400
500
450
450
500
400
500
300
400
550
550
550
350
400
400 450
500
450 600 450
250
300
450
400
700
350
700
600
250
900
850
800
750
400
500
300
450
250250
350
500
400
250
950
350
650
900
850
800
750
700
550 400
200
300
350 300
300
250
350
350
200
350
300
300
250
350
250
400
300
300
300
300
250
350
400
300
200
350 300
450350
400
250
300
450600
550
500
0.5
2.25
2.00
1.75
1.50
1.25
1.75
2.25
2.00 1.75
2.00
400
500 450
500
500
550
450 500
500
600
500
550
550
450 350400
450 400
450
400 400
25
1.251.75
1.50
1.25
1.25
1.50
2.25
2.25
400
400
125
2.50
115
500
450
450
Litterfall (g/m2)
50
2.25
500 500
450 400
125
Predicted
spatial
variation
in total
litterfall,
and
deposition
via
litterfall
of calcium
and
magnesium
550
450
35
45
W
55
65
75
The “Footprint” of an Invasive Species
(research by Lorena Gomez Aparicio)

Invasive tree species alter environmental conditions and
ecosystem processes as they invade a stand

Do these ecosystem effects create feedbacks that either
accelerate or retard the rate of subsequent invasion?

How do these changes alter competitive balances among
the native tree species?
The cast of characters

Two important invasive tree species locally:
- Norway maple (Acer platanoides)
- Tree of heaven (Ailanthus altissima)

Both species are still most abundant along roadsides and
forest edges, but are beginning to move into forest
interiors…
Characterizing the neighborhood
effects of trees
30
Acer platanoides
25
20
What defines a “footprint”
- Leaf litterfall
- Rooting patterns
- Shading
- …?
15
10
Leaf literfall (g/m2)

5
0
0
5
10
15
20
25
30
Ailanthus altissima
25
Upwind
Downwind
20
15
10
5
0
0
5
10
15
Distance (m)
20
25
Modeling “Footprints”

Two approaches:
- Contrast effect of the invasive with the average effect of
all native species
- Fit more complex models that fit individual effects of
both invasive and native species
- In both cases, test for site-specific effects:
does the effect
of an invasive or native species depend on underlying site
conditions?
A simple linear, additive model for
overlapping footprints
Y=a+bX
n

X   DBH i exp   distancei



i 1
Where:
Y = ecosystem state
X = summed effect of the overlapping footprints of
i = 1..n invasive trees
DBH, distance = size of and distance to the invasive trees
0.1
0.35
pH
Tree of
heaven
Relative impact of a 30-cm DBH Ailanthus altissima
0.08
0.3
Ca
0.25
0.06
0.2
0.04
0.15
0.1
0.02
0.05
0
0
0
5
10
15
20
25
0.1
0
5
10
15
20
25
0.35
0.3
K
0.08
Mass N
0.25
0.06
0.2
0.04
0.15
0.1
0.02
0.05
0
0
0
5
10
15
20
25
0.35
0
5
10
15
20
25
0.1
0.3
NO3-
0.25
Nitrification
0.08
0.2
0.06
0.15
0.04
0.1
0.02
0.05
0
0
0
5
10
15
20
25
0
Distance (m)
5
10
15
20
25
0.35
0.1
pH
0.3
Norway maple
Ca
0.08
0.25
0.06
0.2
0.15
0.04
0.1
0.02
0.05
Relative impact of a 30-cm DBH Acer platanoides
0
0
0
5
10
15
20
25
0.1
0
5
10
15
20
25
0.35
Mg
0.08
K
0.3
0.25
0.06
0.2
0.04
0.15
0.1
0.02
0.05
0
0
0
5
10
15
20
25
0.35
0
5
10
15
20
25
0.35
FF depth
0.3
NO3-
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
5
10
15
20
25
0.1
0
5
10
15
20
25
0.1
N mineralization
0.08
Nitrification
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
0
5
10
15
20
25
0
5
10
15
20
0.1
mineralization
Distance fromCa
the
invasive tree (m)
0.08
0.06
25
1 .2
1 .0
Norway maple
0 .8
0 .6
0 .4
0 .0
1 .2
Tree of heaven
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
1 .2
1 .0
Sugar maple
0 .8
0 .6
0 .4
0 .2
0 .0
1 .2
1 .0
White ash
0 .8
0 .6
0 .4
0 .2
0 .0
1 .2
1 .0
Red oak
0 .8
0 .6
0 .4
0 .2
t io
n
al
ra
er
pi
C
a
M
in
es
R
iza
t io
t io
ca
if i
i tr
N
er
al
ob
N
M
in
icr
M
n
t io
lN
ia
iza
lC
ia
H4
ob
N
ic r
M
3
O
N
:N
s
as
M
C
N
C
s
K
as
M
g
M
a
C
pH
pt
h
de
ep
FF
rd
t te
de
lk
Li
ns
i ty
th
n
n
0 .0
Bu
Relative
speciesspecific
effects
0 .2