Appendix C. Bayesian prior for white spruce seed rain at distance zero from the seed source.

The structure of our model and data led to a strong negative correlation between ω_{1} (mean, or expected, viable seed rain per ground area at distance zero from the source stand) and β_{0} (intercept of the linear establishment function); i.e., a given number of recruits can result from high seed rain and low establishment or vice versa. This trade-off resulted in slow convergence of the Bayesian posteriors for these parameters. To speed up convergence, we specified an informative Gaussian prior for w1 based on seed trap (6–10 traps per year; 0.25 m^{2} or 0.5 m^{2} traps) and germination data from a nearby white spruce-paper birch stand (LTER site UP3A; http://www.lter.uaf.edu/) that originated from the same 1780 fire as our source stand. To specify a prior for ω_{1}, we estimated the mean and error variance (the square of the standard error) of total viable seed rain (seeds m^{-2 }of ground area) in stand UP3A for 1981–1991, which included two mast years, 1982 and 1987. We used the error variance of mean seed rain, rather than the sample variance across seed traps, as the variance of the prior, because ω_{1} is a mean (i.e., the expected seed rain at distance zero from the source). We chose 1991 as the end of the seed rain period. The two years prior to the 1983 fire were included because white spruce seeds may remain viable for up to two years and may have survived the fire (Archibold 1980). For each of the two mast years, we calculated the mean (μ_{M1} and μ_{M2}) and error variance (EV_{M1} and EV_{M2}) of seed rain across traps. Seed trap data were missing for some non-mast years, so we estimated annual seed rain for non-mast years from 22 non-mast years for which data were available from 1969–2003. For each of the 22 years, we calculated the mean within-year seed rain across traps, and then calculated the mean (μ_{NM}) and error variance across years (EV_{NM}). We then calculated the total mean and error variance in seed rain for 1981-1991 as μ = μ_{M1} + μ_{M2} + 9μ_{NM} and EV = EV_{M1} + EV_{M2} + 9EV_{NM}. We use 9 (as opposed to 81) as the multiplier for EV_{NM} because we are assuming independence across the 9 non-mast years. This decision has little impact on our analysis, because the two mast years accounted for > 90% of the estimated total seed rain for 1981–1991. To account for the difference in white spruce basal area in our source stand (11.4 m^{2} ha^{-1}) compared to stand UP3A (where the seed traps were located; 32.0 m^{2} ha^{-1}), we scaled the ω_{1 }prior by 11.4/32. To account for the fact that the seed trap data were collected in the forest interior, while ω_{1} represents seed rain at the forest edge, we scaled the prior by ½. The resulting prior mean is μ_{p} = (11.4/64) × μ = 207.9 seeds m^{-2 }and the prior variance is v_{p} = (11.4/64)2 × EV = 65.2. To determine if our results were robust to our choice of ω_{1} prior, we re-ran the white spruce model with three additional Gaussian priors with the following means and variances: (μ_{p}, 10v_{p}), (0.1μ_{p}, 0.01v_{p}), and (10μ_{p}, 100v_{p}). In all four cases, the posterior mean and variance for ω_{1} were almost identical to its prior mean and variance, and the posterior mean for β_{0} compensated for changes in ω1 (e.g., if we increased the prior mean for ω_{1} , the ω_{1} posterior mean increased, and the β_{0} posterior mean decreased). However, the posterior means and variances for all other parameters were only slightly affected. Therefore, although our analysis is not informative as to the absolute levels of white spruce seed rain and establishment rate, the shape of the distance-dependent seed rain function and our inferences on how establishment depends on predictor variables are robust to our choice of ω_{1} prior. For black spruce, the potential tradeoff between seed rain and β_{0} did not lead to slow convergence, as it did for white spruce. However, we note that this tradeoff is inherent in our analysis for both species. Thus our absolute estimates of seed rain and establishment rate for both species should be interpreted with caution. In contrast, our results on how seed rain and establishment rate vary with predictor variables should not be affected by the parameter tradeoff.

LITERATURE CITED

Archibold, O. W. 1980. Seed input into a post-fire forest site in northern Saskatchewan. Canadian Journal Forestry Research 10:129–134.