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Population games with instantaneous behavior and the Rosenzweig-MacArthur model.

Emil F Frølich1, Uffe H Thygesen2

  • 1Department of Applied Mathematics and Computer Science - DTU Compute, Technical University of Denmark, Building 303B, Matematiktorvet, Kgs. Lyngby, Denmark. jaem@dtu.dk.

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This study introduces a new general method for determining animal spatial distributions and population dynamics using mean-field games and variational inequalities. The approach provides unique solutions for habitat distributions and population dynamics in various ecosystems.

Keywords:
Game theoryHabitat choicePopulation dynamicsPopulation game

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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.