Appendix D. Among population growth comparison.
To compare growth trends between populations, irrespective of actual time, we standardized each curve to years from initiation of population growth. We define the decade of initiation as the decade in which the second individual was established in the population, since the ratio of establishments per reproductively mature individuals remains at zero while only one individual is present.
Growth models were compared among populations. The standardized-scaled-growth data set from each population was fit to each of the other population’s models and the residual was calculated. To determine if populations were significantly different from each other, we calculated the Aikake Information Criteria (AIC) score for each population fitted to each of the other population’s models. We also calculated the AIC score of the lower 95% CI for each of the populations against its own fitted growth curve. AIC scores were then compared to determine the relative degree of difference between pairs of populations, where a lower AIC score indicates a better model fit (Burnham and Anderson 2002). If the AIC score for a given population fitted to another population’s model was lower than the 95% CI AIC score for the population it was being compared to, we considered those two populations to not be significantly different from each other.
Finally, we investigated the impact of adding individuals to the establishment gap in the early period of Grass Creek’s history. Grass Creek is unique among the four populations in that there is an extended period, spanning more than a century, of no further establishment following the recruitment of the first individual (Fig.2d). We tested whether the addition of even a single tree during this period would affect the modeled growth rate, and make the growth rate more similar to the other populations. To do this we systematically added one tree to the population in each decade between 1621 and 1741. For each of these addition scenarios the revised Grass Creek data were then fitted to the other population’s logistic regression models, and the AIC score was calculated to assess how the additional tree affected the similarity of the Grass Creek growth model to each of the other population’s models.
Population growth curves standardized to the year of growth initiation show similar growth trends at Castle Garden, Cottonwood Creek, and Anchor Dam (Fig. D1). All three populations, irrespective of their true size, appear to have reached a saturation point approximately 300 years following initiation (Fig. D1). Unlike the other populations, the Grass Creek population increased rapidly following the second recruitment event, reaching a saturation point after only approximately 100 years. AIC scores showed that Cottonwood Creek and Castle Garden have the most similar growth curves (Table D1). The data from both Castle Garden and Cottonwood Creek fitted each other’s model within the 95% CI (shown by lower AIC scores, Table D1). Both the Grass Creek and Anchor Dam data fitted poorly with all of the other population’s models. However, Grass Creek consistently shows the worst fit (Table D1).
A major reason for the difference between the Grass Creek population and the other three populations is the long gap between the initial establishment event and the second establishment event at Grass Creek (Fig.2d). Secondary establishment events occurred much more quickly at the other sites (Fig. 2c,e,f), which decreased the slope of the growth curve relative to Grass Creek. When a single individual was added to any of the decades between 1611 and 1731, the Grass Creek growth curve became more similar to the other populations. In all cases, the fit of the Grass Creek data to the other population’s models was better with an added tree than without (Table D1, Table D2); and the fit of the model generally improved when an individual was added nearer to Grass Creek’s initial establishment event (decade of 1601). AIC scores for the Grass Creek data fitted to both the Cottonwood Creek and Castle Garden models improved substantially, and began to approach the lower 95% CI when a tree was added early in the population’s history. AIC scores using the Anchor Dam model also improved, but not as much as with the other two populations (Table D2).
TABLE D1. AIC scores for the observed cumulative growth data of each of the study populations fitted to the logistic regression model of each population. Numbers in bold are the scores for each population fitted to its own model. Numbers in parenthesis are the increase in the AIC score based on the 95% confidence interval for that model. Lower AIC scores indicate a better fit of the data to the model.
TABLE D2. AIC scores for Grass Creek data, with an additional tree added to the data set for each decade between 1611 and 1731, fitted to the other three population’s models. Lower AIC scores indicate a better fit of the data to the model.
|FIG. D1. Modeled cumulative population growth curves for four ponderosa pine populations (from Fig. 3a–d). The x-axis plots years from initiation of population growth for each population, irrespective of the actual calendar year the population was initiated. Hatched areas around each line represent the 95% CI for that model.|
Burnham, K. P., and D. R. Anderson. 2002. Model selection and multimodel inference A practical information-theoretical approach. Springer Science+Business Media. USA. 497pp.