The Stability of a Predator–Prey Model with Cross-Dispersal in a Multi-Patch Environment

This paper investigates the stability of predator–prey models within multi-patch environments, with a particular focus on the influence of cross-dispersion across patches. We apply Kirchhoff’s matrix tree theorem and Liapunov’s method to derive criteria related to the cross-dispersion topology, thus solving the challenge of determining global asymptotic stability conditions. The method incorporates realistic ecological interactions and spatial heterogeneity, offering a framework for stability analysis. Our findings demonstrate that an appropriate level of cross-dispersion can effectively mitigate oscillations and foster convergence toward equilibrium. Two numerical examples validate these theoretical results and demonstrate the feasibility and effectiveness of the model across multiple patches.