Population games with instantaneous behavior and the Rosenzweig-MacArthur model

Area of Science:

• Ecology
• Mathematical Biology
• Game Theory

Background:

• Determining animal spatial distribution and population dynamics are fundamental ecological questions.
• Existing methods lack a general approach for coupling instantaneous behavior with population dynamics for spatial distribution analysis.

Purpose of the Study:

• To develop a general method for determining spatial distributions and population dynamics of interacting populations.
• To model instantaneous habitat choice using mean-field games and analyze population dynamics.

Main Methods:

• Utilizing the framework of variational inequalities to establish existence and uniqueness of habitat distributions.
• Applying mean-field games to model interacting populations with instantaneous habitat choice.
• Investigating a Rosenzweig-MacArthur model in a continuous habitat both theoretically and numerically.

Main Results:

• Existence and uniqueness of habitat distributions for interacting populations in continuous and discrete habitats were determined.
• Existence and uniqueness of fixed-points for population dynamics and spatial distributions were shown under specific conditions.
• Efficient numerical methods were applied to analyze emergent dynamics through the variational inequality approach.

Conclusions:

• The proposed method offers a general framework for studying spatial distributions and population dynamics.
• This approach can be applied to complex ecosystems, such as analyzing enrichment impacts in marine environments.
• The integration of mean-field games and variational inequalities provides robust analytical and numerical tools for ecological modeling.