Multiple timescales in collective motion: daily and intraday upstream fish migration focusing on Feller condition

Fish migration is a collective phenomenon that has multiple timescales, ranging from daily to intraday (hourly or even finer). We propose a unified mathematical approach using diffusion bridges, nonlinear stochastic differential equations with pinned initial and terminal conditions, to model both daily and intraday fish migration phenomena. Drift and diffusion coefficients of these bridges are determined based on time-dependent parameterized average and variance curves fitted against fish count data, with which the unique existence of their solutions is rigorously guaranteed. We show that sample paths of the diffusion bridges have qualitatively distinctive properties depending on the Feller condition, namely, the ratio between the sizes of diffusion and drift. Our application study about the juvenile upstream migration of Plecoglossus altivelis altivelis (Ayu) in Japan clarifies similarities and differences between daily and intraday migration phenomena. Particularly, we discuss that the daily and intraday fish count data correspond to distinctive Feller indices, showing that the former is qualitatively less randomized and intermittent. The results obtained in this study suggest that the Feller condition potentially serves as an effective tool for evaluating fish migration phenomena of Ayu across different timescales.