Moment expansions in spatial ecological models and moment closure through Gaussian approximation

Area of Science:

• Theoretical Ecology
• Mathematical Biology
• Statistical Physics

Background:

• Ecological models often simplify species interactions.
• Mean-field approximations neglect the impact of fluctuations.
• Understanding fluctuation effects is crucial for accurate ecological dynamics.

Purpose of the Study:

• To model competing species dynamics using spatial moments.
• To incorporate fluctuations via a Gaussian approximation.
• To analyze the limitations of the Gaussian approximation.

Main Methods:

• Describing species dynamics via spatial moment interactions.
• Closing the moment hierarchy using a Gaussian approximation.
• Assuming independent, normally distributed fluctuations.

Main Results:

• The Gaussian approximation systematically corrects mean-field approximations by including fluctuations.
• At low fluctuation levels, Gaussian approximation aligns with mean-field results.
• The Gaussian approximation deviates from mean-field as fluctuations increase.
• Breakdown occurs at high fluctuation levels due to distribution bimodality and skewness.

Conclusions:

• The Gaussian approximation offers a systematic improvement over mean-field theory for ecological dynamics.
• Fluctuation strength significantly impacts the validity of the Gaussian approximation.
• The model's limitations at high fluctuation levels highlight the need for advanced methods.