Mean Field Games for Diel Vertical Migration with Diffusion

Area of Science:

• Theoretical Ecology
• Mathematical Biology
• Oceanography

Background:

• Diel vertical migration is a key ecological process in marine ecosystems.
• Understanding the collective behavior of migrating organisms is crucial.
• Previous models often simplify the complex decision-making of individual organisms.

Purpose of the Study:

• To develop a theoretical framework for diel vertical migration using differential mean field games.
• To model the collective behavior of a continuum of agents in a dynamic environment.
• To establish an equivalence between game-theoretic equilibria and optimization problems.

Main Methods:

• Formulation of a differential mean field game for vertical migration.
• Derivation of a mean field system of partial differential equations (PDEs).
• Establishing and solving a PDE-constrained optimization problem.

Main Results:

• The study presents a novel game-theoretic approach to diel vertical migration.
• Finding Nash equilibria in the game is shown to be equivalent to solving a PDE-constrained optimization problem.
• Numerical solutions were obtained for both formulations.

Conclusions:

• The differential mean field game framework provides a robust method for studying collective vertical migration.
• The established equivalence offers alternative analytical and numerical approaches.
• The findings contribute to a deeper understanding of marine ecosystem dynamics.