Regularized Latent Trajectory Models for Spatio-temporal Population Dynamics

Abstract

Climate change impacts ecosystems variably in space and time. Landscape features may confer resistance against environmental stressors, whose intensity and frequency also depend on local weather patterns. Characterizing spatio-temporal variation in population responses to these stressors improves our understanding of what constitutes climate change refugia. We developed a Bayesian hierarchical framework that allowed us to differentiate population responses to seasonal weather patterns depending on their “sensitive” or “resilient” states. The framework inferred these sensitivity states based on latent trajectories delineating dynamic state probabilities. The latent trajectories are composed of linear initial conditions, functional regression models, and additive random effects representing ecological mechanisms such as topological buffering and effects of legacy weather conditions. Further, we developed a Bayesian regularization strategy that promoted temporal coherence in the inferred states. We demonstrated our hierarchical framework and regularization strategy using simulated examples and a case study of native brook trout (Salvelinus fontinalis) count data from the Great Smoky Mountains National Park, southeastern USA. Our study provided insights into ecological processes influencing brook trout sensitivity. Our framework can also be applied to other species and ecosystems to facilitate management and conservation.

Appendix A: Simulation Study

We generated count data from $n = 20$ sites for $T = 10$ years. For $i = 1, \dots , 20$ and $t = 1, \dots , 10$, we simulated covariates $\varvec{x}_i = \left( 1, x_{i, 1}, x_{i, 2}\right) '$ and $\varvec{h}_{i, t} = \left( h_{i, t, 1}, h_{i, t, 2}\right) '$ from independent standard normal distributions. To specify the latent trajectory, we let $\varvec{m}_i = \varvec{x}_i$ for the initial conditions. We simulated a continuous driver covariate, $w_{i, 1}(\tau )$, by sampling from independent standard normal distributions, and an indicator driver covariate, $w_{i, 2}(\tau )$, by sampling from independent Bernoulli distributions with probability 0.1. Further, we generated spatially correlated random effects, $\varvec{\epsilon }_{t, q} = \left( \epsilon _{1, t, q}, \dots , \epsilon _{10, t, q}\right) ', q = 1, 2$, using Equation 6. We calculated state probabilities, $\rho _{i, t, q}$, using Equation 4, and generated $z_{i, t, q} \sim \text {Bern}\left( \rho _{i, t, q}\right)$. We calculated population densities, $\lambda _{i, t}$, using Equation 3, and generated true abundance, $N_{i, t}$, using Equation 2, where we let $A_i = 1$ for all sites. Finally, we generated observed counts, $y_{i, t, j}, j = 1, 2, 3$, using Equation 1.

In terms of model fitting, we first selected the optimal tuning parameter from $\varvec{c} = (0, 0.5, 1, 1.5, 2)'$. The array was concluded from the iterative selection process described in Section 2.2. We conducted a three-fold cross-validation, where at each non-overlapping fold, we randomly designated two-thirds of the simulated counts as the training set and the remaining one-third as the test set. We fit the models under different penalties to the training sets and evaluated their predictive performance on the test sets using the procedure described in Section 3.2. We then fit the best predictive model to all simulated data and summarized marginal posterior distributions. We repeated the above data simulation and model fitting processes twenty times to derive the empirical coverage rate of the inferred 95% credible intervals (Table 2).