Effects of water currents on fish migration through a Feynman-type path integral approach under $\sqrt{8/3}$ Liouville-like quantum gravity surfaces

TL;DR

This paper develops a novel stochastic model using Feynman path integrals on Liouville-like quantum gravity surfaces to determine optimal fish migration strategies against water currents under uncertain environmental information.

Contribution

It introduces a new approach combining quantum gravity-inspired path integrals with stochastic differential games for fish migration modeling, enabling solutions in complex nonlinear stochastic systems.

Findings

Exact optimal migration strategy derived under uncertainty.

Path integral approach extends to nonlinear stochastic differential equations.

Provides a mathematical framework for fish behavior in imperfect environments.

Abstract

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.