Ecological examples of nonstationarity, nonlinearity, and statistical interactions in dynamic structural equation models

Ecologists are adapting structural causal modelling for spatial, phylogenetic, and time-series analysis. However, ecological extensions of path analysis and structural equation models (SEM) typically assume that interactions (“path coefficients”) are stationary, linear, and additive, while ecological and evolutionary dynamics are often nonstationary, nonlinear, and include statistical interactions. Here, we combine moderated SEM (estimating path coefficients as model variables) with dynamic SEM (estimating both simultaneous and lagged interactions among variables), develop a new “path-lag-slope” notation to specify this combination, and demonstrate it using a simulation experiment and three ecological case studies. The simulation experiment confirms that an autocorrelated “random-slope” model can estimate the nonstationary impact of one variable on another, but that the random slope is shrunk towards a constant value as data become less informative. The first case study then demonstrates nonstationarity by estimating an autoregressive slope linking a regional climate index to local ocean temperature near Vancouver Island. The second demonstrates nonlinearity by approximating Lotka-Volterra dynamics for two predator-prey systems, which closely match estimates of interactions and carrying capacity from traditional ordinary-differential equation methods. The third demonstrates statistical interactions by using monthly plankton samples (1962-1994) to show that resource-consumer-predator interactions in Lake Washington have a dome-shaped response to temperature. We envision several uses in causal analysis: (1) testing whether path coefficients are nonstationary; (2) estimating nonlinear responses given missing data; and (3) linking ecological parameters to hypothesized drivers in applied modelling.