Stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises

Area of Science:

• Mathematical Biology
• Stochastic Partial Differential Equations
• Computational Science

Background:

• Lotka-Volterra models are foundational for predator-prey dynamics.
• Incorporating prey-taxis, diffusion, and noise adds realism to ecological models.
• Stochasticity is crucial for understanding population fluctuations and system stability.

Purpose of the Study:

• To develop a novel stochastic system of nonlinear partial differential equations for predator-prey interactions with prey-taxis.
• To derive a macroscopic model from stochastic kinetic equations.
• To establish the existence of weak martingale solutions and develop numerical approximations.

Main Methods:

• Micro-macro decomposition method for model derivation.
• Faedo-Galerkin method for proving the existence of weak martingale solutions.
• One- and two-dimensional finite volume approximations for kinetic and macroscopic models.

Main Results:

• A new stochastic macroscopic model is derived from kinetic equations.
• The existence of weak martingale solutions is established.
• A stable one-dimensional finite volume scheme is developed, demonstrating convergence and system features.

Conclusions:

• The study successfully models complex predator-prey dynamics with stochasticity and prey-taxis.
• The developed numerical methods provide a robust tool for analyzing such systems.
• The findings offer insights into the behavior of ecological systems under varying conditions.