Study On Dynamical Behaviors Of Stochastic Tree-grass Coexistence Model

In this paper,we mainly study the long-time behaviors of tree-grass coexistence model with random perturbations by using the methods of stochastic Lyapunov analysis,construction of auxiliary process,comparison theorem of stochastic differential equations and strong ergodicity theorem etc.Firstly,the existence and uniqueness of the global positive solution of the stochastic tree-grass coexistence model are given,and we prove the moment boundedness of the solution.Then,the sufficient and almost necessary conditions for persistence and extinction of the model are investigated by proposing two thresholds λ1 and λ2.To be more specific,when the thresholds λ1>0 and λ2<0,the trees will be persistent,and the grass will be extinct at an exponential rate.At this time,the vegetation type in this area will gradually evolve into forest.When the thresholds λ1>0 and λ2>0,both trees and grass are persistent,the stochastic tree-grass coexistence model has a unique invariant measure.At this time,the vegetation type in this area will gradually evolve into savanna with coexistence of trees and grass.When the thresholds λ1<0 and λ2<0,both trees and grass will be extinct at exponential rates.At this time,this area will gradually evolve into desert without vegetation.When the thresholdsλ1<0 and λ2>0,the trees will be extinct at an exponential rate,and the grass will be persistent.At this time,the vegetation type in this area will gradually evolve into grassland.Finally,the conclusions are verified by numerical simulations.