Stationary distributions of persistent ecological systems

Area of Science:

• Ecology
• Mathematical Biology
• Environmental Science

Background:

• Ecological systems are often subject to unpredictable environmental changes.
• Understanding species coexistence and long-term population dynamics is crucial in ecology.
• Analytical solutions for complex ecological models with random fluctuations are often intractable.

Purpose of the Study:

• To develop and apply a numerical method for approximating invariant probability measures in ecological systems.
• To investigate the impact of various environmental fluctuation types on ecosystem stability.
• To provide biological insights into population dynamics and species coexistence under stochastic conditions.

Main Methods:

• Analysis of ecological systems using stochastic differential equations.
• Modeling with piecewise deterministic Markov processes incorporating random environmental switches.
• Development of numerical approximation techniques for invariant probability measures.
• Application to Lotka-Volterra, predator-prey, and rock-paper-scissors models.

Main Results:

• General conditions for species coexistence and convergence to a unique stationary distribution were established.
• A novel numerical method for approximating invariant probability measures was developed and validated.
• The influence of different environmental fluctuation types (stochastic differential equations, Markov processes, switching) on ecosystem dynamics was analyzed.
• Parameter impacts on stationary distributions were quantified for specific ecological models.

Conclusions:

• The developed numerical method effectively approximates invariant probability measures, aiding ecological analysis.
• Environmental fluctuations significantly influence species coexistence and long-term population stability.
• The study provides new biological insights into the resilience and dynamics of ecosystems facing environmental uncertainty.