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Chapter 4 Model full

In this chapter, I fitted the selected model with full data.

4.1 Data

I focused on trees at 20 meters from any plot edges for neighbourhood effect. I used only recruited trees in the censuses with at least 10 measurements of diameter at breast height (DBH, cm). I used only species with at least 10 trees following previous requirements (Tab. 4.1 & Fig. 4.1).

Table 4.1: Metrics on inventory data used to fit the full model including sample size (N), memdian, minimum and maximum values for families, genera, species, individuals, observations, cenusus, recruitment year (year0), last censused year (yearmax), recruitment diameter (dbh0) and last censused diameter (dbhmax).

	N	Median	Minimum	Maximum
families	38
genera	95
species	138
individuals	7 961
observations	117 688
census		13	11	30
year0		1 995	1 985	2 010
yearmax		2 019	1 995	2 021
dbh0		11	5	15
dbhmax		15	6	77

Figure 4.1: Tree diameter trajectories in reduced data. Color represent individuals.

4.2 Model

I used a Gompertz model (Hérault et al. 2011), were the diameter of individual \(i\) at year \(t\) is the sum of annual growth from \(t0\) to \(t\):

\[ DBH_{t,i,s} \sim \mathcal N (10 + Gmax_i \times \sum _{y=1|DBH_{t=0}} ^{y=t} exp(-\frac12.[\frac{log(\frac{DBH_{t,i}}{100.Dopt_i})}{Ks_i}]^2)), \sigma) \\| Dopt_i \sim \mathcal N(Dopt_s,\sigma_D), Ks_i \sim \mathcal N(Ks_s,\sigma_K) \]

The annual growth rate for individual \(i\) at year \(y\) with a diameter of \(DBH_{y,i}\) is defined following a Gompertz model (Gompertz 1825) already identified as the best model for growth-trajectories in Paracou (Hérault et al. 2011), where \(Gmax_i\) is the fixed maximum growth potential of every individual, \(Dopt_i\) is the optimal diameter at which the individual reaches its maximum growth potential, and \(Ks_i\) is the kurtosis defining the width of the bell-shaped growth-trajectory (see figure 1 in Hérault et al. 2011). \(Dopt_i\) and \(Ks_i\) are random effects centered on species parameters \(Dopt_s\) and \(Ks_s\) with associated variances \(\sigma_D\) and \(\sigma_K\).

4.3 Fit

The model correctly converged (\(\hat R < 1.1\)) for the majority of \(Gmax_i\). All \(\sigma\) have a small posterior but difficulties to converge. The correlation between \(Dopt\) and \(Ks\) is acceptable but marked. \(Gmax_i\) posteriors have logical uncertainty but are varying widely among individuals.

References

Gompertz, B. (1825). On the nature of the function expressive of the law of humanmortality, and on a newmode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513–583. Retrieved from https://www.tandfonline.com/doi/full/10.1080/14786445908642737

Hérault, B., Bachelot, B., Poorter, L., Rossi, V., Bongers, F., Chave, J., Paine, C.E.T., Wagner, F. & Baraloto, C. (2011). Functional traits shape ontogenetic growth trajectories of rain forest tree species. Journal of Ecology, 99, 1431–1440.