Maximum Entropy Models for Unimodal Time Series: Case Studies of Universe 25 and St. Matthew Island

Abstract

We present a maximum entropy modeling framework for unimodal time series: signals that begin at a reference level, rise to a single peak, and return. Such patterns are commonly observed in ecological collapse, population dynamics, and resource depletion. Traditional dynamical models are often inapplicable in these settings due to limited or sparse data, frequently consisting of only a single historical trajectory. In addition, standard fitting approaches can introduce structural bias, particularly near the mode, where most interpretive focus lies. Using the maximum entropy principle, we derive a least-biased functional form constrained only by minimal prior knowledge, such as the starting point and estimated end. This leads to analytically tractable and interpretable models. We apply this method to the collapse of the Universe 25 mouse population and the reindeer crash on St. Matthew Island. These case studies demonstrate the robustness and flexibility of the approach in fitting diverse unimodal time series with minimal assumptions. We also conduct a cross-comparison against established models, including the Richards, Skewnormal, and Generalized Gamma functions. While models typically fit their own generated data best, the maximum entropy models consistently achieve the lowest off-diagonal root-mean-square losses, indicating superior generalization. These results suggest that maximum entropy methods provide a unifying and efficient alternative to mechanistic models when data is limited and generalization is essential.