mechanism vs. phenomenology in choosing functional forms: neighborhood analyses of tree competition...
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Mechanism vs. phenomenology in choosing functional forms:
Neighborhood analyses of tree competition
Case Study 3
Likelihood Methods in EcologyApril 25 - 29, 2011
Granada, Spain
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 maximum-likelihood, 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…
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”?
Site) Size, n,Competitio Growth, Potential f(Maximum Growth Actual
Effect of Tree Size (DBH) on Potential Growth
221 0
bX
)X/DBHln(/
e Multiplier Size
Lognormal function, where:
•X0 = DBH at maximum potential growth
•Xb = variance parameter
0
0.2
0.4
0.6
0.8
1
1.2
0 25 50 75
DBH (cm)
Fra
ctio
n o
f M
axim
um
Gro
wth
or
Su
rviv
al
X0 = 10
X0 = 20
X0 = 40
X0 = 80
Xb = 0.75
Why use this function?
Recourse to macroecology?The power function
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)
Enquist et al. (1999) have argued from basic principles (assumptions) that
3
1
DdtdD
But trees don’t appear to fit the theory…
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”?
Crowding)Shading,Size, Growth, f(Pot. Growth Actual
Shading of Target Trees by Neighbors(as a function of distance and DBH)
30 cm DBH Target Tree
Neighbor Tree DBH (cm)0 20 40 60 80 100 120
Fra
ctio
n of
Sky
Obs
tru
cted
0.0
0.1
0.2
0.3
0.4
0.5
4 m
6 m
8 m
10 m
20 m
Crowding “Effect”:A Neighborhood Competition Index
(NCI)
n
j ij
ijs
i dist
DBHiNCI
1
)
1 )(
(
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)
A simple size and distance dependent index of 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)
22 yx
n
n
ix
1
sin
n
n
iy
1
cos
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
)1(1
)
1 )(
(
n
j ij
ijs
s
i dist
DBHNCI
= 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
DNCI*Ce Multiplier nCompetitio
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
Neighborhood Competition Index
Fra
ctio
n o
f P
ote
nti
al G
row
th
or
Su
rviv
al
Effect of target tree size on sensitivity to competition
DBHDNCICe Multiplier nCompetitio **
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
NCI
Fra
cti
on
of
Po
ten
tia
l Gro
wth
DBH = 10
DBH = 20
DBH = 30
DBH = 40
DBH = 50
DBH = 60
DBH = 70
c = 250d = 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.
What do these look like?
0.00.10.20.30.40.50.60.70.80.91.01.1
-2 -1 0 1 2 3
Environmental Gradient
Fra
ctio
n o
f M
axim
um
Po
ten
tial
Gro
wth
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5
Environmental Gradient
Max
imu
m P
ote
nti
al G
row
th
Whittaker
Keddy
)(
][
2)/ln(
2/11 )(
)(
1*t
n
j ijdist
ijDBHs
s
ib
ot DBHCShadingSX
XDBH
eeeMaxRG
RG
Radial growth = Maximum growth * size effect * shading*crowding
The full model (for any given species)...
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 coefficients with a simpler λ
model in which all are fixed at a value of 1 λ
PDF and Error Distribution
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)...
),,0( 2
N
f(x)y
But with a larger dataset and more higher R2, residuals were normally distributed with a constant variance…
Hemlock
y = 0.9943x
R2 = 0.2324
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5
ObservedP
red
icte
d
Neutral vs. Niche Theory: are neighbors equivalent in their
competitive effects?
Species n # parameters9 Species of competitors
Intra vs Interspecific
Equivalent Competitors
Shading Only
Size Only R2
Hemlock 245 19 454.22 454.31 475.34 522.86 694.50 76.2%Cedar 192 19 275.12 327.08 364.18 412.93 541.75 79.6%
Amabilis 91 10 160.82 145.03 137.97 154.54 235.83 89.5%Subalpine 95 9 227.39 223.64 218.72 238.79 282.94 56.2%
Spruce 196 18 508.75 524.24 519.99 524.29 640.37 68.7%Pine 93 9 213.34 215.64 210.58 210.72 265.64 73.3%
Aspen 101 10 177.36 171.05 166.21 172.35 186.86 31.2%Cottonwood 39 9 153.94 122.93 114.98 115.99 121.00 61.6%
Birch 245 19 288.41 304.63 299.93 336.69 438.05 79.9%
AICcorr of alternate neighborhood competition models for growth of 9 tree species in the interior cedar-hemlock forests
of north central British Columbia
How do neighbor size and distance affect degree of crowding?
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)
n
j ij
ijs
s
i dist
DBHNCI
1
)
1 )(
(
Should and vary, in principle, depending on the identity of the neighbor?
Does the size of the target tree affect its sensitivity to crowding?
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!
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
NCI
Fra
cti
on
of
Po
ten
tia
l Gro
wth
DBH = 10
DBH = 20
DBH = 30
DBH = 40
DBH = 50
DBH = 60
DBH = 70
c = 250d = 1
DBHDNCICe Multiplier nCompetitio **
Are positive values of biologically realistic?
Are the parameter estimates “robust”?
Astrup et al. 2008, Forest Ecol. Management 10:1659-1665.
Fra
ctio
n of
Pot
entia
l Gro
wth
Fagus grandifolia
-2 -1 0 1 2 3
Rel
ativ
e A
bund
ance
0.0
0.2
0.4
0.6
0.8
1.0
Acer saccharum
-2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
Low Fertility High FertilityDCA Axis 2
dots = relative abundance in each of the plotsline = estimated potential growth (in absence of competition)
Shade tolerant species – fertility gradient
Do species grow best in the sites where they are most abundant?
Note: similar pattern for shade tolerant species along the moisture
gradient (Axis 1)
Fertility Gradient:Shade intolerant species
Quercus rubra
DCA Axis 2
-2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
Fra
ctio
n o
f Po
ten
tial G
row
th
Pinus strobus
-2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
Low Fertility High Fertility
Acer rubrum
-2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
Fraxinus americana
-2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
DCA Axis 2
Rel
ativ
e A
bund
ance