Hierarchical permutation entropy regression identifies decadal decline in the intrinsic predictability of butterfly populations

Abstract

Short-term population dynamics are shaped by both external environmental stochasticity and endogenous ecological dynamics. The extent to which population trajectories maintain predictable temporal structure reflects underlying determinism and sets limits on inherent forecastability. Permutation entropy quantifies this intrinsic predictability using the relative frequency of short fluctuation motifs. However, requirements for long timeseries have restricted ecological applications. Here, I introduce a Bayesian hierarchical regression framework, incorporating entropy-neutral priors, that estimates expected permutation entropy from shorter, spatially-distributed ecological datasets. I validate the approach's capacity to detect the impact of increasing environmental variance through simulations. Applied to four decades of UK butterfly data, the approach identified increasing permutation entropy across 35 species, consistent with an increasing role of stochastic weather events. Outlier weather events did not have a consistent relationship with permutation entropy, highlighting the potential for environmental forcing to both disrupt and constrain trajectories. With this flexible approach, permutation entropy offers complementary perspectives on longstanding questions around forecasting and population dynamics.