Dynamic stability of a stochastic vegetation-water model in arid ecosystems

This paper investigates the deterministic and stochastic dynamic stability of a three-variable vegetation-water ecosystem under different stochastic environmental disturbances, considering both multiplicative and additive noises. For the deterministic case, the stability of equilibria is assessed through the Jacobian matrix eigenvalues, initial state trajectories and bifurcation structure. In the stochastic case, the stability of the cube basin is analyzed using the first escape probability, derived from the Itô stochastic differential equation with six boundary conditions. Theoretical results align with Monte Carlo simulations of the original system, confirming the validity of the calculations. Findings reveal that the stability of the vegetation-free equilibrium depends on the existence of vegetation-positive equilibrium. Stochastic disturbances, especially the environmental disturbance of vegetation, enhances the stability of ecosystem. On the contrary, soil water and rainfall disturbances disrupt the stability of the system, providing insights for ecosystem protection.