5
Appendix 2 Goodness-of-fit of Gaussian curves and comparisons with alternative population-front shapes.
Our Poisson regression analysis showed that the average density of Myrcianthes decreased significantly with the squared distance to the nearest riparian forest (see Table 1 and Fig. 2a in main text). These Gaussian curves fitted the observed densities of Myrcianthes with substantial dispersion but with no noticeable systematic error (Fig. A2.1).
To examine further the appropriateness of Gaussian curves, we fitted alternative shapes of the population front to our Myrcianthes data. We framed this analysis within integro-difference models of population expansion (Kot et al. 1996). One important prediction of these models is that the shape of the population front is approximately proportional to that of the dispersal kernel when reproduction is density-independent (Kot et al. 1996). More specifically,
(A2.1)
where λ(x,t) is the density at time t and distance x from the origin of the expansion, λ(0,0) is the population density at the expansion origin, R0 is the net reproductive rate, and k(x) is the dispersal kernel (Kot et al. 1996). Dispersal kernels can be described using a one-dimensional distance-dependent decay function of the form
(A2.2)
where k(x) is the probability of propagules landing at distance x from the release location, Γ( ) is the gamma function, and b and c are distance and shape parameters, respectively (Clark 1998; see also Clark et al. 1998). From the moments of k(x), the mean dispersal distance of propagules moving to the right or left from the source is
(A2.3)
(Clark 1998; Clark et al. 1998). Note that the mean dispersal distance is proportional to the square root of the diffusion coefficient used in the main text, i.e. E[|x|]=2√(Dt/π). Introducing dispersal kernel (A2.2) into model (A2.1) and then taking logs results in a population front of the following form
(A2.4)
Fitting models such as (A2.4) is complicated by the fact that parameters b and c tend to trade off with each other, making their estimation unstable when attempted simultaneously (Ribbens et al. 1994; Clark et al. 1998). This limitation can be bypassed by setting a fixed value of c and then fitting b (Ribbens et al. 1994; Clark et al. 1998). We followed this approach and used a Poisson regression model to fit three different front shapes to Myrcianthes data, each corresponding to one of three fixed values of parameter c: 0.5 (the square root front), 1 (the exponential front), and 2 (the Gaussian front). The Poisson regression model was
(A2.5)
where c is the shape parameter [cf. equation (1) in main text]. We used two alternative measures to evaluate the relative suitability of the resulting two-parameter Poisson regression models (cf. Ribbens et al. 1994; Clark et al. 1998). The residual deviance provides an absolute (inverse) measure of model goodness-of-fit, although it cannot be used to make a goodness-of-fit test in Poisson models corrected for overdispersion, such as ours, because the scaled deviance does not necessarily follow a Chi-squared distribution (see Agresti 2002). We thus calculated the Pearson’s correlation between observed and predicted densities, which is a measure of predictive power useful for model selection that can be formally tested (Agresti 2002). In addition, to evaluate the consistency across front shapes of the results reported in the main text, we calculated the attained spread, mean dispersal distance, and local population growth from the parameters estimated using model (A2.5). Following the same approach as in the main text, our estimation of attained population range for the ij-th environmental condition (and for each fixed shape parameter) was,
(A2.6)
Following equation (A2.3), estimations of mean dispersal distance were calculated as
(A2.7)
while local population growth (r) was visualized through
(A2.8)
Confidence intervals for attained spread, mean dispersal distance, and rij t + ln[λ(0,0)] were calculated following the approaches and methods described in the main text.
Results show that the Gaussian front fitted the data slightly better than both the exponential and the square-root fronts (Table A2.1). As reported by others (e.g. Clark 1998), our results show that the smaller the shape parameter, the longer (and faster) the population spread estimated (Fig. A2.2a). Despite these quantitative results, differences in attained spread among environmental conditions were largely consistent across front shapes (Fig. A2.2a). Palm density and fire history interacted to control Myrcianthes spread in all three front shapes. Attained spread was significantly shorter on burned than unburned savannahs, although this negative effect of fire was smaller on sparse than dense savannahs and even not significant for the Exponential and the Gaussian fronts (Fig. A2.2b). Thus, the results extracted from all three front shapes point out that dense palm groves that remained unaffected by fire were the preferential avenue for Myrcianthes population expansion across the landscape of El Palmar National Park (Fig. A2.2b). Regarding estimations of mean dispersal distance, no significant differences among the environmental conditions were detected within any front shape (Fig. A2.2b). Differences in population growth rate among environmental conditions were qualitatively similar for all three front shapes (Fig. A2.2c). However, confidence intervals for parameter rij t + ln[λ(0,0)] were smaller for the Gaussian front, and more significant differences among environmental conditions were detected relative to the other front shapes (Fig. A2.2c). Overall, the results of this exercise indicate that (i) the assumption Gaussian dispersal may be reasonable for the relatively small spatial scale considered in our study (< 2km), that (ii) the estimated values of population spread of Myrcianthes extracted from the Gaussian front would be conservative with respect to those extracted from models that assume dispersal kernels with fatter tails, and that, however, (iii) the main conclusions about the effects of palm density and fire history on Myrcianthes spread are largely unaffected by the assumed shape of the advancing front.
References
Agresti A (2002) Categorical data analysis, 2nd edition. John Wiley & Sons, Hoboken
Clark JS (1998) Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord. American Naturalist 152:204-224
Clark JS, Macklin E, Wood L (1998) Stages and spatial scales of recruitment limitation in southern Appalachian forests. Ecological Monographs 68:213-235
Kot M, Lewis MA, van den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77:2027-2042
Ribbens E, Silander JA, Pacala SW (1994) Seedling recruitment in forests: Calibrating models to predict patterns of tree seedling dispersion. Ecology 75:1794-1806
Table A2.1. Measures of goodness-of-fit and predictive power of three models characterized by different values of shape parameter c. The smaller the residual deviance (i.e. the unexplained variation), the better the model’s fit to data; total (or null) deviance in Myrcianthes data was 6194. Pearson’s correlation between observed and predicted densities (Correlation O-P) gives a measure of model’s predictive power.
Model c Res. deviance Correlation O-P p value
Square root (fat-tailed) 0.5 801 0.539 <0.001
Exponential 1 767 0.558 <0.001
Gaussian 2 751 0.582 <0.001
Figure captions
Fig. A2.1. Deviance residuals of the Poisson regression model corresponding to a Gaussian-shaped front (summarized in Table 1 in main text). Deviance residuals are defined as the square root of the deviance contribution for the observation, with sign equal to the sign of the raw residual (Agresti 2002). Other types of residuals (e.g. Pearson residuals) show a similar pattern (not shown). Different symbols represent four environmental conditions resulting from the combination of two levels of density of adult palms and two fire histories. D: high palm density (dense palm savannahs); S: low palm density (sparse palm savannahs); B: time since last fire <15 years (burned); UB: time since last fire >25 years (unburned). There is no significant correlation between deviance residuals and distance (Pearson’s correlation = 0.089, p = 0.59).
Fig. A2.2. Estimations of attained spread (a), mean dispersal distance (b), and parameter r t + ln[λ(0,0)] (c) of Myrcianthes populations in four environmental conditions for three models characterized by different values of shape parameter c. Parameter r t + ln[λ(0,0)] is used to visualize differences in density-independent population growth rate r. Dispersion bars are 95% confidence intervals and different lowercase letters above bars indicate significant differences (P<0.05) within each front shape. D: high palm density (dense palm savannahs); S: low palm density (sparse palm savannahs); B: time since last fire <15 years (burned); UB: time since last fire >25 years (unburned).
Fig. A2.1.
Fig. A2.2.