Uncovering state-dependent relationships in shallow lakes using Bayesian latent variable regression

Ecosystems sometimes undergo dramatic shifts between contrasting regimes. Shallow lakes, for instance, can transition between two alternative stable states: a clear state dominated by submerged aquatic vegetation and a turbid state dominated by phytoplankton. Theoretical models suggest that critical nutrient thresholds differentiate three lake types: highly resilient clear lakes, lakes that may switch between clear and turbid states following perturbations, and highly resilient turbid lakes. For effective and efficient management of shallow lakes and other systems, managers need tools to identify critical thresholds and state-dependent relationships between driving variables and key system features. Using shallow lakes as a model system for which alternative stable states have been demonstrated, we developed an integrated framework using Bayesian latent variable regression (BLR) to classify lake states, identify critical total phosphorus (TP) thresholds, and estimate steady state relationships between TP and chlorophyll a (chl a) using cross-sectional data. We evaluated the method using data simulated from a stochastic differential equation model and compared its performance to k-means clustering with regression (KMR). We also applied the framework to data comprising 130 shallow lakes. For simulated data sets, BLR had high state classification rates (median/mean accuracy >97%) and accurately estimated TP thresholds and state-dependent TP-chl a relationships. Classification and estimation improved with increasing sample size and decreasing noise levels. Compared to KMR, BLR had higher classification rates and better approximated the TP-chl a steady state relationships and TP thresholds. We fit the BLR model to three different years of empirical shallow lake data, and managers can use the estimated bifurcation diagrams to prioritize lakes for management according to their proximity to thresholds and chance of successful rehabilitation. Our model improves upon previous methods for shallow lakes because it allows classification and regression to occur simultaneously and inform one another, directly estimates TP thresholds and the uncertainty associated with thresholds and state classifications, and enables meaningful constraints to be built into models. The BLR framework is broadly applicable to other ecosystems known to exhibit alternative stable states in which regression can be used to establish relationships between driving variables and state variables.