Mean Field Games and Ideal Free Distribution

Area of Science:

• Ecology
• Mathematical Biology
• Game Theory

Background:

• The ideal free distribution (IFD) by Fretwell and Lucas describes habitat selection in animal populations.
• Existing models often lack a dynamic framework for habitat selection.
• Mean field game systems offer a powerful tool for modeling large population dynamics.

Purpose of the Study:

• To dynamically model the habitat selection game using a mean field game system.
• To establish the existence of classical solutions for ergodic mean field game systems.
• To demonstrate the convergence of population density to the ideal free distribution.

Main Methods:

• Utilizing a mean field game system with local coupling.
• Establishing existence of classical solutions for ergodic systems.
• Analyzing convergence as control cost approaches zero.

Main Results:

• Existence of classical solutions for the ergodic mean field game system, including heterogeneous diffusion in 1D.
• Population density of agents converges to the ideal free distribution.
• Provides a derivation of IFD in a dynamical context.

Conclusions:

• The study successfully models habitat selection dynamically using mean field games.
• Confirms convergence to the ideal free distribution under specific conditions.
• Offers a novel dynamical derivation of the ideal free distribution concept.