Convergence of a structured metapopulation model to Levins's model

Area of Science:

• Ecology
• Mathematical Biology
• Population Dynamics

Background:

• Metapopulation models are crucial for understanding species distribution across fragmented habitats.
• The Levins model provides a foundational framework for unstructured metapopulation dynamics.
• Investigating structured metapopulation models is essential for ecological realism.

Purpose of the Study:

• To determine conditions where structured metapopulation dynamics can be approximated by unstructured models.
• To analyze the equilibrium distribution of a structured metapopulation model.
• To develop simplified formulae for equilibrium distributions in limiting regimes.

Main Methods:

• Analysis of a structured metapopulation model with mean-field migration.
• Asymptotic analysis as carrying capacity and other parameters scale.
• Application of Stein's method for probabilistic approximations.
• Numerical examples to illustrate theoretical findings.

Main Results:

• The equilibrium distribution of the structured model converges to a bimodal distribution under specific scaling conditions.
• This bimodal distribution includes a point mass at zero and a positive part approximated by a shifted Poisson distribution.
• Simplified approximate formulae for the equilibrium distribution were derived for the limiting regime.
• Methods for computing persistence regions in parameter space were established.

Conclusions:

• Structured metapopulation models can be effectively approximated by unstructured models under certain conditions.
• The derived limiting distribution provides insights into species persistence and distribution patterns.
• The study offers practical tools for analyzing metapopulation dynamics and predicting species persistence.