A note on stability of stochastic logistic equation

This report is concerned with a famous stochastic logistic equation d x ( t ) = x ( t ) ( 1 − x ( t ) / K ) [ r ( t ) d t + σ ( t ) d B ( t ) ] , where B ( t ) is a standard Brownian motion. Under a simple assumption, sufficient conditions that are close to the necessary conditions for global asymptotical stability of the zero solution and the positive equilibrium are established. Numerical simulations are introduced to support the results. The results show that the noise is unfavorable for the stability of the positive equilibrium.