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

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0 25 50 75

DBH (cm)

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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

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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

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Neighborhood Competition Index

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Effect of target tree size on sensitivity to competition

DBHDNCICe Multiplier nCompetitio **

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NCI

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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

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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

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ObservedP

red

icte

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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!

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NCI

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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.

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Fagus grandifolia

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Acer saccharum

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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

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Pinus strobus

-2 -1 0 1 2 30.0

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Low Fertility High Fertility

Acer rubrum

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Fraxinus americana

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DCA Axis 2

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