Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations

Area of Science:

• Mathematical Biology
• Ecology
• Dynamical Systems

Background:

• Integro-differential equations model complex population interactions.
• These models generalize Lotka-Volterra equations and apply to phenomena like cancer drug resistance.
• Understanding species persistence is crucial in ecological and evolutionary studies.

Purpose of the Study:

• To analyze the asymptotic behavior of N-population integro-differential equations.
• To establish conditions for the persistence of all interacting species.
• To identify traits that are asymptotically selected under specific interaction conditions.

Main Methods:

• Asymptotic analysis of integro-differential equations.
• Utilizing a Lyapunov function to derive conditions for stability.
• Investigating both general interaction matrices and specific cases of mutualistic interactions.

Main Results:

• A general condition on the interaction matrix ensuring species persistence and providing a convergence rate.
• A distinct condition for persistence in mutualistic interaction scenarios.
• Characterization of asymptotically selected traits when persistence conditions are met.

Conclusions:

• The study provides robust mathematical conditions for ensuring the long-term survival of interacting populations.
• The findings offer a framework for predicting evolutionary trait selection in ecological models.
• This work contributes to a deeper understanding of population dynamics and evolutionary stability.