The dynamical behavior of a class of stochastic vegetation models

Abstract

We investigate the influence of random fluctuations in environmental parameters (e.g. rainfall and grazing rate) on the dynamical behavior of a class of ecological models for vegetation biomass in semi-arid regions. Firstly, we show the global existence and uniqueness of positive solutions for the number of vegetation biomass B(t). Then some sufficient conditions are presented for extinction and persistence of B(t) by using the stochastic linearization technique, the Hurwitz criterion and the Lyapunov function method. In particular, a threshold value R0s for the stochastic vegetation model is established in order to illustrate extinction and persistence of B(t). We also find that R0s=1 can be a warning signal for the emergence of desertification. In the case of R0s>1, we show that the stochastic vegetation system has a unique stationary distribution. Finally, we present conditions for persistence of B(t), and prove the pth moment stability and almost sure stability of the null solution in a general decay rate sense (including the exponential, polynomial and logarithmic rates) where the parameters in the stochastic vegetation system can also be dependent on time. These results are illustrated by numerical simulations.