APPENDIX 1. Model parameterization and analysis, and background on matrix models.

To construct stage-based population projection matrices, it is first necessary to estimate the average “vital rates” for each stage class. The vital rates are: a) Permanence, the probability an individual will survive one year and remain in its starting stage class; b) Transition, the probability that an individual will survive and grow into another stage class during the course of the year; and c) Fecundity, the annual reproductive output of an individual (Caswell 2001).

The heights and diameters of all juveniles and adult trees within the transects were measured in June 1999 and remeasured in 2000 and 2001. We determined that an adult or juvenile had transitioned to another size class if its diameter increased (or decreased) enough to qualify for that size class. For seedling transition and survival, from June 1999 to 2001 we followed seedlings in 24 plots (3.14 m2) randomly located within 15 m of the main transects. We measured the height of a maximum of 20 seedlings per plot. At the end of the year, if a seedling had grown taller than 70 cm, we considered that it had transitioned to the juvenile class. Juvenile and seedling survivorship was determined by presence\absence at the last census. Adults were determined to be dead if they were uprooted and/or the great majority of their leaves were dry. Using the aforementioned data, we determined the mean permanence and transition rates for each size class.

We used the annual rate of seedling establishment in the plots to estimate fecundity, where fecundity was the average number of seedlings per hectare per individual per adult size class that established within one year. Fecundity was apportioned among adult size classes as follows: We estimated a linear relationship between tree crown size and trunk diameter (y = 4.349x - 39.496, R2 = 0.91, p = 0.01) and assumed that an individual tree’s fecundity was proportional to its crown size. We then determined the mean fecundity per size class.

Population projection matrices have the form: n(t + 1) = A x n(t), where n(t), represents the stage structure (n) at time t, n(t + 1) is the stage structure at the next time interval (the same time the next year), and A is a matrix containing the vital rate averages for each stage class. We used the bootstrap resampling procedure recommended by Caswell (2001) to estimate λ (the finite population growth rate) and confidence intervals (CIs) for λ and the matrix elasticities. In this method, each observation in the bootstrap is one individual and its corresponding history (i.e. vital rates). In each run of the bootstrap, one observation per size class was randomly sampled with replacement for a total of five total observations resampled per run. The remaining observations were used to construct matrix A. We then used standard, numerical, iterative techniques for estimating λ (Caswell 2001). Each value for λ and the elasticities was obtained after 256 iterations. This was repeated 2000 times for a bootstrap distribution of λ and elasticity values. We used the normal theory parametric method to estimate λ and 95% CIs (Caswell 2001) because it gave similar estimates for λ but larger, more conservative 95% CIs than the bootstrap percentile intervals method (Caswell 2001).