  This work is devoted to the study of a stochastic logistic growth model with
and without the Allee effect. Such a model describes the evolution of a
population under environmental stochastic fluctuations and is in the form of a
stochastic differential equation driven by multiplicative Gaussian noise. With
the help of the associated Fokker-Planck equation, we analyze the population
extinction probability and the probability of reaching a large population size
before reaching a small one. We further study the impact of the harvest rate,
noise intensity, and the Allee effect on population evolution. The analysis and
numerical experiments show that if the noise intensity and harvest rate are
small, the population grows exponentially, and upon reaching the carrying
capacity, the population size fluctuates around it. In the stochastic
logistic-harvest model without the Allee effect, when noise intensity becomes
small (or goes to zero), the stationary probability density becomes more acute
and its maximum point approaches one. However, for large noise intensity and
harvest rate, the population size fluctuates wildly and does not grow
exponentially to the carrying capacity. So as far as biological meanings are
concerned, we must catch at small values of noise intensity and harvest rate.
Finally, we discuss the biological implications of our results.
