JARABE DE PANELA

I used the Gompertz population model to estimate the population growth parameters of density dependence and density independent growth. The Gompertz model is closely related to the familiar logistic model of population growth. Like the logistic, the Gompertz model contains two parameters one for density dependent growth and one for density independent growth. However, the parameters in the Gompertz model are easier to estimate empirically than those in the logistic (parameters). As a result the Gompertz model has been frequently used to estimate density dependence of populations (e.g., (Roy, McIntire, & Cumming, 2016; Thorson et al., 2015)).

The equation for population growth (F(N) ) is expressed in the Gompertz form as: 𝐹(𝑁_𝑡) = 𝑁𝑡+1= 𝑁𝑡𝑒𝑎+(𝑏−1)log (𝑁𝑡)

where Nt is population density at time t, a is density independent growth of the population, and b is the degree of density dependence.

I can simplify this equation to make it straightforward to model and interpret. Taking the natural log of both sides of this equation for F(N) results in the Gompertz model in the following form:

log(𝑁_𝑡+1) = 𝑎 + 𝑏 𝑙𝑜𝑔(𝑁_𝑡) + 𝜖_𝑡 (Equation 4.1)

A value of b greater than 1 represents exponential growth, a value of b = 1 implies no density

dependence, a value of b less than but close to 1 represents weak density dependence, and a value of b

close to 0 represents strong density dependence. The model includes an error term too, εt, which represents environmental stochasticity. This model can be fit using linear regression of log(Nt) against

log(Nt+1),with the intercept serving as an estimate for a and the slope serving as an estimate for b. Since the estimates of both a and b were used to calculate the STT, I examined the strength of these values and how they varied between and across the invasion sites.

In the interests of simplifying the models and focusing the scope of this study, I decided to only examine the population growth process known as infilling. Infilling can be defined as when the spreading population colonises areas within its existing range, and it occurs because of local dispersal (L. A. V. Taylor, Hasenkopf, & Cruzan, 2015; Warren, Ursell, Keiser, & Bradford, 2013). Because most of the population growth of these tree invasions occurs at short distances (Buckley et al., 2005; Nuñez & Paritsis, 2018; K. T. Taylor et al., 2016), I will limit my scope to infilling.

To account for spatial autocorrelation due to dispersal at short to intermediate distances and correlated environmental conditions in adjacent areas, I added a spatial random effects term to the Gompertz equation. I used a Conditional Autoregressive (CAR) model for the spatial random effects term since this model was straight forward to set up and is known to perform well with spatial data (Duncan, White, & Mengersen, 2017). This auto-regressive model works by accounting for the local neighbourhood and effectively smooths over adjacent areas.

Equal weight (w) was given to each neighbouring plot, so that:

𝑤_𝑖,𝑗= {_{0 ; 𝑖𝑓 𝑎𝑟𝑒𝑎𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑠}1 ; 𝑖𝑓 𝑎𝑟𝑒𝑎𝑠 𝑖 𝑎𝑛𝑑 𝑖 𝑎𝑟𝑒 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑠

Since each neighbouring plot had the same weight, the CAR model assumed the datasets were

isotropic. Therefore, this model must be interpreted with caution because the datasets were potentially anisotropic. For each site and set of time-steps examined, I defined a new neighbourhood matrix based on the occupied areas in the site.

I ran the models in a Bayesian framework. I used informative priors based on estimates of density dependence and density independent growth for similar Gompertz models in the literature (Dennis, Ponciano, Lele, Taper, & Staples, 2006; Thorson et al., 2015). The following prior distributions were used:

𝑎 ~ 𝑁(0, 0.2) 𝑏 ~ 𝑁(0, 1) 𝜏 ~ 𝐺(01. , 0.1)

I used a CAR normal distribution for the prior of the CAR model, with a precision term τsp with a gamma distribution.

I set up and ran a model for each time interval and site. Time intervals were defined as the difference between one time step and another, and there were different lengths of time between time steps due to which imagery data was available. I ran the models with Markov Chain Monte Carlo (MCMC) sampling, implemented in the R package Nimble (de Valpine et al., 2017). I tested whether the spatial model improved fit over an aspatial model without the spatial random effects term. To compare the spatial and aspatial models, I calculated the Watanabe-Akaike Information Criterion (WAIC) values (Watanabe, 2010). WAIC values are recommended to be used instead of Deviance Information

Criterion (DIC) values according to Vehtari et al. (2016). Each model used 3 chains and ran for 10,000 iterations with the first 5,000 samples discarded as the burn-in. The initial conditions of each chain were randomly generated within the expected parameter range. MCMC samples were not thinned as

thinning has been shown to decrease the precision of estimates from MCMC sampling (Link & Eaton, 2012).

To check for model convergence, I used the trace plots, posterior densities, and R̂ values from the R package shinystan (Gabry, 2018), similar to methods by (Elderd & Miller, 2016). I also examined the goodness of fit of the models by plotting their predictive intervals with the observed data to determine whether the model could predict all of the observed values (Gelman, Meng, & Stern, 1996). All of the models were coded and run in R (R Core Team, 2017), and the model code is shown in Appendix G.1.