Analysis of Reaction-Diffusion Predator-Prey System under Random Switching

TL;DR

This paper studies a reaction-diffusion predator-prey model with environmental randomness modeled by Markovian switching, analyzing long-term dynamics, species persistence, and extinction thresholds through rigorous mathematical and numerical methods.

Contribution

It introduces the first rigorous analysis of a spatially diffusive predator-prey system under Markovian switching, linking spatial ecology with stochastic hybrid PDE models.

Findings

Derived a critical threshold for species extinction or persistence.

Characterized the asymptotic behavior and omega-limit sets of solutions.

Validated theoretical results with numerical simulations showing regime transitions.

Abstract

This paper investigates the long-term dynamics of a reaction-diffusion predator-prey system subject to random environmental fluctuations modeled by Markovian switching. The model is formulated as a hybrid system of partial differential equations (PDEs), where the switching between different ecological regimes captures the randomness in environmental conditions. We derive a critical threshold parameter that determines whether the predator species will eventually go extinct or persist. We further characterize the system's asymptotic behavior by providing a detailed pathwise description of the omega-limit set of solutions. This analysis reveals how the effects of random switching shape the distribution and long-term coexistence of the species. Numerical simulations are provided to validate and illustrate the theoretical findings, highlighting transitions between different dynamical regimes. To the best of our knowledge, this is the first work that rigorously analyzes a spatially diffusive predator-prey model under Markovian switching, thereby bridging the gap between spatial ecology and stochastic hybrid PDE systems.