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

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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 Schrödinger 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.

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  1. Department of Mathematical Sciences, Northern Illinois University, 1425 Lincoln Highway, DeKalb, IL, USA

    Paramahansa Pramanik

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Pramanik, P. Effects of water currents on fish migration through a Feynman-type path integral approach under \(\sqrt{8/3}\) Liouville-like quantum gravity surfaces. Theory Biosci. 140, 205–223 (2021). https://doi.org/10.1007/s12064-021-00345-7

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  • DOI: https://doi.org/10.1007/s12064-021-00345-7

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