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.