Noise-induced vegetation transitions in the Grazing Ecosystem

The Grazing Ecosystem is a special case of predator-prey systems, which has attracted widespread attention since a long time ago. Because of the ubiquity of noise, there is a growing need to research the influences of noise on the Grazing Ecosystem. This paper is devoted to investigating the transition behaviors of the high vegetation biomass in the Grazing Ecosystem subjected to Gaussian noise and Lévy noise, respectively. Firstly, the original system is translated into the Itô stochastic differential equation, which is utilized to derive the analytical expression of the escape probability through the Dirichlet boundary value problems. Then the transitions between the two vegetation potential wells are explored by calculating the size of the stochastic basin of attraction based on the escape probability. The comparison between the analytical results and the ones through Monte Carlo simulations shows that the proposed method works very well. It turns out that the Gaussian white noise intensity, Lévy noise stability parameter and herbivore density have different impact mechanisms on the basin stability of high density vegetation in the stochastic Grazing Ecosystem.