### Steven R. Beissinger, Joseph M. Wunderle, Jr., J. Michael Meyers, Bernt-Erik Sæther, and Steinar Engen. 2008. Anatomy of a bottleneck: diagnosing factors limiting population growth in the Puerto Rican Parrot. Ecological Monographs 78:185–203.

Appendix C. Life-stage simulation analysis.

A life-stage simulation analysis (LSA) was developed to evaluate the relative roles of each matrix model parameter on population growth rates (Wisdom and Mills 1997, Wisdom et al. 2000). As discussed below, in this analysis we used a slight variation of the 3-stage prebreeding matrix model presented above. To implement the LSA, we constructed 1000 matrices by randomly drawing vital rates from specified distributions using @Risk version 4.5 (Palisade 2005) and calculated the population growth rate () for each matrix. We then regressed l against the value of each vital rate. Vital rates that account for the greatest variation in population growth rate have the greatest effects on population dynamics (Cross and Beissinger 2001).

Distributions were chosen for each demographic parameter (Table A3) by fitting a set of potential distributions in @Risk (Palisade 2005) to raw data composed of 25–27 years of annual measures. Chi-square goodness-of-fit tests were used to compare and rank fits. Uniform distributions were used if data were sparse. When required, distributions were truncated to avoid unrealistic or impossible values.

To assess the roles of various limiting factors, the stage-based deterministic matrix model was altered in two ways. First, adult survival P3was shown to be affected by hurricanes (hur) and rainfall (rain) in the following manner (Fig. 6, Table 3) from logistic regression:

 Logit (P3) = -1.814534*hur + 0.000245*rain + 1.055547

Therefore, for each matrix we first determined if it was a hurricane year and then a rainfall (mm) value was chosen. Two hurricanes struck the parrot population during the 27 years of study, yielding an annual probability of 0.074074. Rainfall records during the nesting years were analyzed from 1975–1999 were normally distributed with a mean of 3508 and a standard deviation of 802.

The second way the matrix model was altered was to explicitly parameterize fecundity (m) as a function of the processes on a per egg basis found to be important during the study:

 m = (C*Hf*Hp*F) + (1-s)*r* C*Hf*Hp*F

where C is clutch size, Hf is the probability that an egg is fertile, Hp is the probability that a fertile egg will survive to hatching, F is the probability that a nestling will survive to fledge, s is the percent of nests that fledged at least one young, and r is probability of renesting within a breeding season if a nest fails. All of these parameters were estimated on a per egg basis from analyses presented in the text except s, which was approximated by calculating the likelihood that all eggs in a 3-egg clutch would survive as .

TABLE C1. Distributions for the demographic parameters of the Puerto Rican Parrot matrix population model used in the life-stage simulation analysis (LSA). Distributions and parameters related to those used in @Risk (Palisade 2005).

 Parameter Mean Min Max Distribution Probability of becoming a breeder (G) 0.120 0.00 1.00 Exponential: b = 0.118538 Adult survival (P3) 0.888 0.40 1.00 Logistic regression: hur = -1.814534, rain = 0.000245, constant = 1.055547 Subadult 2 survival (P2) 0.768 P3-G Subadult 1 survival (P1) 0.800 0.75 0.88 10% less than adult survival (P3) Juvenile survival (P0) 0.680 0.52 0.75 Uniform Fecundity (m) 1.548 Calculated from product of next 5 rows Egg survival hatching success (Hf) 0.730 0.18 1.00 Logistic: a = 0.70108, b = 0.11008 Egg loss to predation and other (Hp) 0.930 0.50 1.00 Beta general: a1= 0.4888, a2 = 0.12363 Nestling survival (F) 0.760 0.25 1.00 Logistic: a = 0.75345, b = 0.11868 Probability of renesting following failure (r) 0.188 0.00 0.30 Uniform Clutch size (C) 3.000 2.40 3.40 Lognormal: u = 2.9958, s = 0.26475

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