Patterns robust to disorder in spatially interacting generalized Lotka-Volterra ecosystems.

Area of Science:

• Ecology
• Mathematical Biology
• Statistical Physics

Background:

• Understanding species distribution is key to ecosystem dynamics.
• Generalized Lotka-Volterra models describe interspecies competition and coexistence.
• Nonlocal interactions and diffusion are crucial factors in spatial ecology.

Purpose of the Study:

• To investigate how interspecies interactions and diffusion influence spatial distribution.
• To determine the stability criteria for spatially homogeneous ecosystems.
• To identify transitions in species distribution patterns.

Main Methods:

• Developed a spatially extended generalized Lotka-Volterra model with diffusion and nonlocal interactions.
• Computed the stability criterion for homogeneous states.
• Analyzed the interaction matrix spectrum weighted by species abundances.
• Derived an explicit solution for species density using dynamical mean-field theory and Fisher-Kolmogorov-Petrovski-Piskounov equations.

Main Results:

• Ecosystem stability critically depends on the most abundant species.
• The interplay between reproduction timescale and interaction range dictates stability.
• A Baik-Ben Arous-Péché transition was identified, correlating with pattern changes.
• An explicit solution for species density was obtained under conditions of small disorder.

Conclusions:

• Species interactions and spatial dynamics fundamentally govern ecosystem structure.
• The study provides a theoretical framework for understanding complex spatial patterns in ecosystems.
• Highlights potential for integrating active matter and disordered systems approaches in ecology.