  A stochastic differential game theoretic model has been proposed to determine
optimal behavior of a fish while migrating against water currents both in
rivers and oceans. Then, a dynamic objective function is maximized subject to
two stochastic dynamics, one represents its location and another its relative
velocity against water currents. In relative velocity stochastic dynamics, a
Cucker-Smale type stochastic differential equation is introduced under white
noise. As the information regarding hydrodynamic environment is incomplete and
imperfect, a Feynman type path integral under $\sqrt{8/3}$ Liouville-like
quantum gravity surface has been introduced to obtain a Wick-rotated
Schr\"odinger type equation to determine an optimal strategy of a fish during
its migration. The advantage of having Feynman type path integral is that, it
can be used in more generalized nonlinear stochastic differential equations
where constructing a Hamiltonian-Jacobi-Bellman (HJB) equation is impossible.
The mathematical analytic results show exact expression of an optimal strategy
of a fish under imperfect information and uncertainty.
