Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system

This paper is concerned with a stochastic generalized logistic equation d x = x [ r − a x θ ] d t + ∑ i = 1 n α i x d B i ( t ) + ∑ i = 1 n β i x 1 + θ d B i ( t ) , where B i ( t ) ( i = 1 , … , n ) are independent Brownian motions. We show that if the intensities of the white noises are sufficiently small, then there is a stationary distribution to this equation and it has an ergodic property. If the intensities of the white noises are sufficiently large, then the equation is extinctive. Some numerical simulations are introduced to support the main results at the end.