Case Study 3

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

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”, “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”?

4. Effect of Tree Size (DBH) on Potential Growth • Lognormal function, where: • X0 = DBH at maximum potential growth • Xb = variance parameter Why use this function?

5. Recourse to macroecology?The power function Enquist et al. (1999) have argued from basic principles (assumptions) that 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)

6. 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…

7. 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”?

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

9. Crowding “Effect”:A Neighborhood Competition Index (NCI) A simple size and distance dependent index of competitive effect: 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

10. target tree 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

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… What happens if you use trees near the edge of the plot as “targets” (observations)? Potential neighborhood “Target” tree

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

17. 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.

18. What do these look like? Whittaker Keddy

19. The basic model (for any given species, without site effects)... 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 Xbare estimated parameters • Shading is the calculated reduction in incident radiation by neighbors, and S is an estimated parameter • DBHijand distij are the size and distance to neighboring tree j of species group i, and C, liand g are estimated parameters

20. 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)?

21. 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

22. PDF and Error Distribution In an 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…

23. 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

24. How do neighbor size and distance affect degree of crowding? • Both α and b varied widely depending on target tree species • a 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: a = 0; Aspen), to only the very large trees having an effect (a = 3.4, Subalpine fir) Should a and b vary, in principle, depending on the identity of the neighbor?

25. Does the size of the target tree affect its sensitivity to crowding? • Models including g 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 g biologically realistic? • Are the g parameter estimates “robust”? Astrup et al. 2008, Forest Ecol. Management 10:1659-1665.

26. Shade tolerant species – fertility gradient Do species grow best in the sites where they are most abundant? dots = relative abundance in each of the plots line = estimated potential growth (in absence of competition) Note: similar pattern for shade tolerant species along the moisture gradient (Axis 1)

27. Fertility Gradient:Shade intolerant species

28. More recent analyses*(using data for the 50 most common tree species in the eastern US) • Does the strength of a species’ competitive effect or response vary along climate gradients? • Yes… • How do the strength of competitive effects and responses vary as a function of a suite of widely used functional traits? • not very consistently with commonly used traits like wood density, tree height, mycorrhizal association… • Is the strength of either competitive effect or response related to the abundance of a species? • not really… *ongoing work with Suzanne Boyden of Clarion University