1. Mechanism vs. phenomenology in choosing functional forms: Neighborhood analyses of tree competition Case Study 3

2. 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.

3. The general approach… where “Size” and “Competition” are multipliers (0-1) that reduce “Maximum Potential Growth”… Should these terms be additive or multiplicative? Why use 0-1 scalars as multipliers? What is “maximum potential growth”? Should we have included a “site” term?

4. Effect of Tree Size (DBH) on Potential Growth Lognormal function, where: X 0 = DBH at maximum potential growth X b = variance parameter Why use this function?

5. Recourse to macroecology? 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 But trees don’t appear to fit the theory…

6. Separating competition into effects and responses… l 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…

7. Separating shading from crowding l Most neighborhood competition studies can not isolate the effects of aboveground vs. belowground competition l 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)

8. Shading of Target Trees by Neighbors (as a function of distance and DBH)

9. Crowding “Effect”: A Neighborhood Competition Index (NCI) 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

10. What if all the neighbors are on one side of the target tree? l The “Sweep” Index: - The fraction of the effective neighborhood circumference obstructed by neighbors rooted within the neighborhood l Zar’s (1974) Index of Angular Dispersion target tree

11. Index of Angular Dispersion (Zar 1974) 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.

12. Basic Model plus Effects of Angular Dispersion  = 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)

13. Competitive “Response”: Relationship Between NCI and Growth

14. Effect of target tree size on sensitivity to competition

15. 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)?

16. The importance of stratifying sampling across a range of neighborhood conditions

17. Effect of Site Quality on Potential Growth l 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.

18. Radial growth = Maximum growth * size effect * shading*crowding The full model (for any given species)... Where: MaxRG is the estimated, maximum potential radial growth DBH t is the size of the target tree, and X o and X b are estimated parameters Shading is the calculated reduction in incident radiation by neighbors, and S is an estimated parameter DBH ij and dist ij are the size and distance to neighboring tree j of species group i, and C, i and  are estimated parameters

19. A sample of basic questions addressed by the analyses l Do different species of competitors have distinctly different effects? l How do neighbor size and distance affect degree of crowding? l Are there thresholds in the effects of competition? l Does sensitivity to competition vary with target tree size? l What is the underlying relationship between potential growth and tree size (i.e. in the absence of competition)?

20. Parameter Estimation and Comparison of Alternate Models l Maximum likelihood parameters estimated using simulated annealing (a global optimization procedure) l Start with a “full” model, then successively simplify the model by dropping terms l Compare alternate models using Akaike’s Information Criterion, corrected for small sample size (AIC corr ), 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

21. 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)... But with a larger dataset and more higher R2, residuals were normally distributed with a constant variance…

22. Neutral vs. Niche Theory: are neighbors equivalent in their competitive effects? AIC corr of alternate neighborhood competition models for growth of 9 tree species in the interior cedar-hemlock forests of north central British Columbia

23. How do neighbor size and distance affect degree of crowding? l Both α and  varied widely depending on target tree species l  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?

24. Does the size of the target tree affect its sensitivity to crowding? l Models including  were more likely for 5 of the 9 species: l 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.