Mean Field Games for Diel Vertical Migration with Diffusion

We present a theoretical framework, based on differential mean field games, for expressing diel vertical migration in the ocean as a game with a continuum of players. In such a game, each agent partially controls its own state by adjusting its vertical velocity but the vertical position in a water column is also subject to random fluctuations. A representative player has to make decisions based on aggregated information about the states of the other players. For this vertical differential game, we derive a mean field system of partial differential equations for finding a Nash equilibrium for the whole population. It turns out that finding Nash equilibria in the game is equivalent to solving a PDE-constrained optimization problem. We detail this equivalence when the expected fitness of the representative player can be approximated with a constant and solve both formulations numerically. We illustrate the results on simple numerical examples and construct several test cases to compare the two analytical approaches.