Transcript Case_Study_3_Neighborhood_Competition - sortie-nd

Likelihood Methods in Ecology
April 25 - 29, 2011
Granada, Spain
Case Study 3
Mechanism vs. phenomenology in choosing
functional forms:
Neighborhood analyses of tree competition
Key References
Canham, C. D., P. T. LePage, and K. D. Coates. 2004. A neighborhood analysis of
canopy tree competition: effects of shading versus crowding. Canadian Journal of
Forest Research 34:778-787.
Uriarte, M, C. D. Canham, J. Thompson, and J. K. Zimmerman. 2004. A maximumlikelihood, neighborhood analysis of tree growth and survival in a tropical forest.
Ecological Monographs 74:591-614.
Canham, C. D., M. Papaik, M. Uriarte, W. McWilliams, J. C. Jenkins, and M. Twery.
2006. Neighborhood analyses of canopy tree competition along environmental
gradients in New England forests. Ecological Applications 16:540-554.
Coates, K. D., C. D. Canham, and P. T. LePage. 2009. Above versus belowground
competitive effects and responses of a guild of temperate tree species. Journal of
Ecology 97:118-130.
The general approach…
Actual Growth  f(Maximum Potential Growth, Competitio n, Size, Site)
where “Size”, “Competition”, and “Site” are multipliers (0-1) that
reduce “Maximum Potential Growth”…
Should these terms be additive or multiplicative?
Why use 0-1 scalars as multipliers?
Just what is “maximum potential growth”?
Effect of Tree Size (DBH) on Potential Growth
Lognormal function, where:
•X0 = DBH at maximum
potential growth
•Xb = variance parameter
Why use this function?
Fraction of Maximum Growth or
Survival
Size Multiplier  e
 ln( DBH / X 0
1 / 2 
Xb

)


2
1.2
1
0.8
X0 = 10
0.6
X0 = 20
X0 = 40
0.4
X0 = 80
0.2
Xb = 0.75
0
0
25
50
DBH (cm)
75
Recourse to macroecology?
The power function
Enquist et al. (1999) have argued from basic principles (assumptions) that
dD
dt
 D
1
3
But trees don’t appear to fit the theory…
Russo, S. E., S. K. Wiser, and D. A. Coomes. 2007. Growth-size scaling relationships of woody
plant species differ from predictions of the Metabolic Ecology Model. Ecology Letters 10:
889-901.
Corrigendum: Ecology Letters 11:311-312 (deals with support intervals)
Separating competition into effects and
responses…

In operational terms, it is common to separate competition
into (sensu Deborah Goldberg)
-
Competitive “effects” : some measure of the aggregate “effect”
of neighbors (i.e. degree of reduction in resource availability,
amount of shade cast)
-
Competitive “responses”: the degree to which performance
of the target tree is reduced given the competitive effects of
neighbors…
Separating shading from crowding

Most neighborhood competition studies cannot isolate the
effects of aboveground vs. belowground competition

The study in BC was an exception
-
Shading by canopy trees is very predictable given the locations,
sizes, and species of neighbors (Canham et al. 1999)
After removing the shading effect, can I call the rest of the
crowding effect “belowground competition”?
Actual Growth  f(Pot. Growth, Size, Shading, Crowding)
Shading of Target Trees by Neighbors
(as a function of distance and DBH)
30 cm DBH Target Tree
Fraction of Sky Obstructed
0.5
4m
0.4
6m
8m
0.3
10 m
0.2
20 m
0.1
0.0
0
20
40
60
80
Neighbor Tree DBH (cm)
100
120
Crowding “Effect”:
A Neighborhood Competition Index (NCI)
A simple size and distance dependent index of competitive
effect:
s
NCI  
i 1
n


j 1
i
( DBH ij )
( distij ) 
For j = 1 to n individuals of i = 1 to s species
within a fixed search radius allowed by the plot size
i= per capita competition coefficient for species i
(scaled to = 1 for the species with strongest competitive effect)
NOTE: NCI is scaled to = 1 for the most crowded neighborhood observed
for a given target tree species
What if all the neighbors are on one side
of the target tree?


The “Sweep” Index:
-
The fraction of the effective
neighborhood circumference
obstructed by neighbors
rooted within the
neighborhood
Zar’s (1974) Index of Angular
Dispersion
target tree
Index of Angular Dispersion (Zar 1974)
  x y
2
2
n
x
 sin
i 1
n
n
y
 cos
i 1
n
where  is the angle from the target tree to the ith
neighbor.
 ranges from 0 when the neighbors are uniformly
distributed to 1 when they are tightly clumped.
Basic Model plus Effects of Angular Dispersion
s
NCI  
 i 1
n
s

j 1
( DBH ij ) 
(1   )

( distij ) 

 = index of angular dispersion of competitors around
the target tree
Bottom line: angular dispersion didn’t improve fit in early
tests, so was abandoned (too much computation time)
Competitive “Response”:
Relationship Between NCI and Growth
D
C
*
NCI
Competition Multiplier  e
Fraction of Potential Growth
or Survival

1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Neighborhood Competition Index
1
Effect of target tree size on sensitivity
to competition
D *DBH 
C
*
NCI
Competition Multiplier  e

Fraction of Potential Growth
1

c = 250
d=1
0.8
DBH = 10
DBH = 20
0.6
DBH = 30
DBH = 40
0.4
DBH = 50
DBH = 60
0.2
DBH = 70
0
0
0.2
0.4
0.6
NCI
0.8
1
Sampling Considerations:
Avoiding A Censored Sample…
Potential
neighborhood
“Target” tree
What happens if you use trees near
the edge of the plot as “targets”
(observations)?
The importance of stratifying sampling across a
range of neighborhood conditions
Effect of Site Quality on Potential Growth

Alternate hypotheses from niche theory:
-
Fundmental niche differentiation (Gleason, Curtis, and Whittaker):
species have optimal growth (fundamental niches) at different
locations along environmental gradients
-
Shifting competitive hierarchy (Keddy): all species have optimal
growth at the resource-rich end of a gradient, their realized
niches reflect competitive displacement to sub-optimal ends of
the gradient
Canham, C. D., M. Papaik, M. Uriarte, W. McWilliams, J. C. Jenkins, and M. Twery. 2006.
Neighborhood analyses of canopy tree competition along environmental gradients in New
England forests. Ecological Applications 16:540-554.
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Whittaker
-2
-1
0
1
2
3
Environmental Gradient
Keddy
1.0
Maximum Potential Growth
Fraction of Maximum
Potential Growth
What do these look like?
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
Environmental Gradient
4
5
The full model
(for any given species)...
RG  MaxRG * e
 ln( DBHt / X o ) 
1 / 2 

Xb


2
e
 s n ( DBH ij ) 

C    s
 ( DBHt )

 S [ Shading]
 i 1 j 1 ( distij ) 
e
Radial growth = Maximum growth * size effect * shading*crowding
Where:
• MaxRG is the estimated, maximum potential radial growth
• DBHt is the size of the target tree, and Xo and Xb are estimated
parameters
• Shading is the calculated reduction in incident radiation by
neighbors, and S is an estimated parameter
•DBHij and distij are the size and distance to neighboring tree j of
species group i, and C, i and  are estimated parameters
A sample of basic questions addressed
by the analyses

Do different species of competitors have distinctly different
effects?

How do neighbor size and distance affect degree of crowding?

Are there thresholds in the effects of competition?

Does sensitivity to competition vary with target tree size?

What is the underlying relationship between potential growth
and tree size (i.e. in the absence of competition)?
Parameter Estimation and Comparison of
Alternate Models

Maximum likelihood parameters estimated using simulated
annealing (a global optimization procedure)

Start with a “full” model, then successively simplify the model by
dropping terms

Compare alternate models using Akaike’s Information Criterion,
corrected for small sample size (AICcorr), and accept simpler
models if they don’t produce a significant drop in information.
-
i.e. do species differ in competitive effects?
» compare a model with separate λ coefficients with a simpler
model in which all λ are fixed at a value of 1
PDF and Error Distribution
3.5
y = 0.9943x
R2 = 0.2324
3
Predicted
In our earlier study (Canham et al.
2004), residuals were
approximately normal, but
variance was not homogeneous (it
appeared to increase as a function
of the mean predicted growth)...
Hemlock
2.5
2
1.5
1
0.5
0
But with a larger dataset and more
higher R2, residuals were normally
distributed with a constant variance…
y  f(x)  
  N (0, 2 ),
0
0.5
1
Observed
1.5
Neutral vs. Niche Theory: are neighbors
equivalent in their competitive effects?
AICcorr of alternate neighborhood competition models for growth of 9
tree species in the interior cedar-hemlock forests
of north central British Columbia
Species
Hemlock
Cedar
Amabilis
Subalpine
Spruce
Pine
Aspen
Cottonwood
Birch
n
245
192
91
95
196
93
101
39
245
9 Species of
Intra vs
Equivalent Shading
Only
# parameters competitors Interspecific Competitors
19
454.22
454.31
475.34
522.86
19
275.12
327.08
364.18
412.93
10
160.82
145.03
137.97
154.54
9
227.39
223.64
218.72
238.79
18
508.75
524.24
519.99
524.29
9
213.34
215.64
210.58
210.72
10
177.36
171.05
166.21
172.35
9
153.94
122.93
114.98
115.99
19
288.41
304.63
299.93
336.69
Size
Only
694.50
541.75
235.83
282.94
640.37
265.64
186.86
121.00
438.05
R2
76.2%
79.6%
89.5%
56.2%
68.7%
73.3%
31.2%
61.6%
79.9%
How do neighbor size and distance affect degree
of crowding?
s
NCI  
i 1
n
s

j 1
( DBH ij )
( distij ) 

Both α and  varied widely depending on target tree species

 ranged from near zero to > 3
-
So, depending on the species of target tree, crowding effects of
neighbors ranged from proportional to simply the density of
neighbors (regardless of size:  = 0; Aspen), to only the very large
trees having an effect ( = 3.4, Subalpine fir)
Should  and vary, in principle, depending on the
identity of the neighbor?
Does the size of the target tree affect
its sensitivity to crowding?
D *DBH 
C
*
NCI
Competition Multiplier  e


Models including were more likely for 5 of the 9 species:
Values for conifers were negative (larger trees less sensitive to
crowding), but values for 2 of the deciduous trees were positive!
Are positive values of 
biologically realistic?
Are the parameter estimates
“robust”?
Astrup et al. 2008, Forest Ecol. Management
10:1659-1665.
1
Fraction of Potential Growth


c = 250
d=1
0.8
DBH = 10
DBH = 20
0.6
DBH = 30
DBH = 40
0.4
DBH = 50
DBH = 60
0.2
DBH = 70
0
0
0.2
0.4
0.6
NCI
0.8
1
Relative Abundance
Shade tolerant species – fertility gradient
1.0
1.0
Acer saccharum
Fagus grandifolia
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-2
-1
0
1
2
3 -2
-1
Low Fertility
DCA Axis 2
dots = relative abundance in each of the plots
line = estimated potential growth (in absence of
competition)
0
1
2
3
Fraction of Potential Growth
Do species grow best in the sites where
they are most abundant?
High Fertility
Note: similar pattern for shade
tolerant species along the moisture
gradient (Axis 1)
Fertility Gradient:Shade intolerant species
Acer rubrum
Relative Abundance
0.8
0.8
Quercus rubra
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-2
-1
0
1
2
3
-2
-1
0
1
2
3
1.0
1.0
0.8
Fraxinus americana
0.8
Pinus strobus
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-2
-1
Low Fertility
0
1
2
3 -2
-1
DCA Axis 2
0
1
2
3
High Fertility
Fraction of Potential Growth
1.0
1.0