A minimization principle behind the diffusion bridge of diurnal fish migration

Fish migration is a mass movement that affects the hydrosphere and ecosystems. While it occurs on multiple temporal scales, including daily and intraday fluctuations, the latter remains less studied. In this study, for a stochastic differential equation model of the intraday unit-time fish count at a fixed observation point, we demonstrate that the model can be derived from a minimization problem in the form of a stochastic control problem. The control problem assumes the form of the Schrödinger Bridge but differs from classical formulations by involving a degenerate diffusion process and an objective function with a novel time-dependent weight coefficient. The well-posedness of the control problem and its solution are discussed in detail by using a penalized formulation. The proposed theory is applied to juvenile upstream migration events of the diadromous fish species Plecoglossus altivelis altivelis commonly called Ayu in Japan. We also conduct sensitivity analysis of the models identified from real data.

💡 Research Summary

This paper develops a rigorous variational formulation for the stochastic diffusion‑bridge model that was previously introduced to describe the diurnal migration of the Japanese ayu (Plecoglossus altivelis altivelis). The authors start from a one‑dimensional stochastic differential equation (SDE) for the unit‑time fish count X(t) observed at a fixed point in a river. The dynamics contain a constant source term a, a time‑dependent reversion coefficient h(t), and a degenerate diffusion term σ√X(t) that guarantees non‑negativity of the process. The SDE reads

 dX(t) =