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Likelihood Methods in Forest Ecology October 9 th – 20 th , 2005

Mechanism vs. phenomenology in choosing functional forms: neighborhood analyses of tree competition. Seminar 4. Likelihood Methods in Forest Ecology October 9 th – 20 th , 2005. Key References.

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Likelihood Methods in Forest Ecology October 9 th – 20 th , 2005

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  1. Mechanism vs. phenomenology in choosing functional forms: neighborhood analyses of tree competition Seminar 4 Likelihood Methods in Forest Ecology October 9th – 20th , 2005

  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. Papaik, M. J. and C. D. Canham. Tree competition along environmental gradients in southern New England forests: An application of multi-model inference. Ecological Applications, in press.

  3. Spatially-explicit, neighborhood analysis of tree growth where Competition, Size and Site are “Multipliers” (0-1) that reduce Maximum Potential Growth…

  4. Forest Inventory and Analysis (FIA) Plots in Vermont and New Hampshire Old 1/5 or 1/6 acre plot (49’ or 52.7’ radius) “Observations” = trees in new plot that were present in previous census Censused 1982-83 neighborhood search radius = 24’ New 24’ radius plot (census all stems > 5”) Censused 1996-98 Selected plots in “forest land” land-use classes that were not logged during the census period (n = 802 plots)

  5. Competitive “Effect”:A Neighborhood Competition Index (NCI) A simple size and distance dependent model: For j = 1 to n individuals of i = 1 to s species within a fixed search radius allowed by the plot size • = species-specific competition coefficient (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

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

  7. Effect of target tree size on sensitivity to competition

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

  9. 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

  10. Whittaker(fundamental niche differentiation) • Normal function • NOTE: Bivariate normal function can be used to test for effects of two environmental factors simultaneously (x is the axis score, and Xo and Xb are estimated)

  11. Keddy(shifting competitive hierarchy) Logistic function (x is the axis score, and Xo and Xb are estimated) Bivariate form used to test for response to 2 axes simultaneously

  12. What can we use as measures of site quality? • Direct measures of site conditions are limited • i.e., physiographic class data • Instead, use multivariate analyses of vegetation composition to produce an ordination that captures the effects of environmental gradients on trees in a given plot… • detrended correspondence analysis (DECORANA)

  13. Detrended Correspondence Analysis of FIA Plots in VT and NH red crosses = species black circles = plots no obvious clustering into a few, discrete community types… DCA AXIS 2 DCA1 DCA2 Eigenvalues 0.69 0.54 Axis lengths 4.02 4.52 DCA AXIS 1

  14. DCA AXIS 1 HIGH Soil Fertility DCA AXIS 2 LOW MESIC XERIC Soil Moisture

  15. The full model (for any given species)... Radial growth = Maximum growth * site effect * size effect * competition effect • Where: • MaxRG is the estimated, maximum potential growth • g = ordination axis score for the plot containing tree t, and Go and Gb are estimated parameters • DBHt is the size of the target tree, and Xo and Xbare estimated parameters • DBHijand distij are the size and distance to neighboring tree j of species group i, and C, D, lsand g are estimated parameters

  16. Relative Abundance – Live Trees target tree species

  17. 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? • How does potential growth vary along environmental gradients? • Are species most abundant in the sites where they perform the best in the absence of competition? • What is the underlying relationship between potential growth and tree size (i.e. in the absence of competition)?

  18. 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

  19. PDF and Error Distribution Residuals were approximately normal, but variance was not homogeneous (it appeared to increase as a function of the mean predicted growth)... So, I fitted the models using the assumption that the residuals were normally distributed, but with a variance that was a linear function of the mean. Maximum likelihood estimates of a and b were determined as part of the optimization

  20. Analysis Summary • Sufficient sample sizes for analysis of the 14 most common tree species (n = 53 – 930 target trees per species) • Overall model fits generally poor (growth is noisy and unpredictable) (R2 = 0.125 – 0.351) • 8 of the 14 species showed significant variation in potential growth along at least 1 of the 2 ordination axes

  21. Comparison of Models AICcorr of models with alternate forms for the effects of Axis 1 and Axis 2 (Gaussian vs. Logistic)

  22. Comparison of Models (continued) AICcorr of models with and without gamma (effect of target tree DBH on sensitivity to competition)

  23. How do neighbor size and distance affect degree of crowding? • α mean value:1.8 (range : 0.6 – 4.0) • So, effects of neighbors on growth are roughly proportional to the neighbors’ basal area and biomass • β mean value: 0.3 (range : 0 – 0.6) • So, effects of neighbors on growth decline slowly with distance

  24. Are there thresholds in the effects of competition on growth? • Basically NO: Simple negative exponential model was the most parsimonious fit for all of the species except black birch (i.e. D parameter = 1) D = 5 D = 3 D =1

  25. Do target tree species differ in their response to competition?

  26. Does the size of the target tree affect its sensitivity to crowding? • Models including g were more likely for 6 of the 14 species: ABBA, ACRU, BEAL, PIST, POTR, and QURU • mean value: –1.6 (range –1.1 to –1.9) White Pine • So, smaller trees of these 6 mid- to early successional species are much more sensitive to competition than are larger trees…

  27. How does potential growth vary with target tree size (DBH)?

  28. Does potential growth vary along the environmental gradients represented in the ordination? • Compare alternate models (for each of the 14 target species) with: • Gaussian or logistic variation along both axes • Gaussian or logistic variation along the axis with the strongest effect • No axes • Choose the model with the lowest AICcorr (i.e. the least loss of information using the fewest parameters)

  29. Results: Variation in Potential Growth along the Moisture Gradient

  30. Variation in Potential Growth along the Fertility Gradient

  31. 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)

  32. Fertility Gradient:Shade intolerant species

  33. Moisture Gradient – Shade tolerant Species

  34. Moisture Gradient Shade intolerant Species

  35. Summary of niche patterns • Fundamental niches: The individualistic hypothesis survives: species show optimal growth at different points along resource gradients... • Realized niches: • Shade tolerant species appear to reach greatest abundance in sites where they have highest potential growth • Less tolerant species consistently displaced to lower ends of the 2 gradients

  36. Are different species of competitors equivalent in their effects on a given tree species? • Compare 4 alternate models: • “Full” model: separate competition coefficients (l) for all common species of neighbors (grouping rare species into an “other” species category) • “Mixed” model: group interspecific competitors into weak, intermediate, and strong competitors, based on results of “full” model • “Intra- vs. Interspecific” model: lump all interspecific competitors into 1 group, with a separate l for intraspecific competitors • “Equivalent” model: set l = 1 for all species of neighbors

  37. Model Comparisons

  38. The matrix of estimated competition coefficients (l)

  39. What species have the strongest and weakest per capita competitive effect on sugar maple? • Weak competitors (l < 0.2): red maple and eastern white pine • Strong Competitors (l > 0.8): balsam fir and beech • Strongest Competitor (l = 1.0): other sugar maples

  40. Are species that co-occur in the same environments strong competitors with each other? • Compared the average distance between pairs of species “centroids” in the ordination with the strength of competition between the two species • Bottom line: no obvious pattern... • Strong competitors were not necessarily either close to or far away from each other in ordination space... • Ditto for weak competitors...

  41. Limitations • Analysis ignores • effects of crowding by trees smaller than 5” DBH • changes in neighborhood during the census interval • Scarcity of large trees and truly all-aged stands in the landscape

  42. Results from temperate coniferous forests of British Columbia AICcorr of alternate neighborhood competition models for growth of 9 tree species in the interior cedar-hemlock forests of north central British Columbia Coates and Canham, in preparation

  43. Next Steps… • Extend analysis to entire NE U. S. (~100,000 plots) • Finish development of the software (maximum likelihood estimation and SORTIE modeling) and hardware (clusters of workstations) needed for the expanded analyses • Incorporate regional climatic gradients in the analyses • Use the results to explore a wide range of partial harvesting scenarios

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