研究一类具有Allee效应及非线性扰动的随机单种群模型的平稳分布及灭绝性. 首先证明模型全局正解的存在唯一性, 接着通过构造合适的Lyapunov函数给出 模型存在唯一平稳分布的充分条件, 其次讨论Allee效应及噪声强度如何影响种群动力学并给出种群灭绝的充分条件. 最后给出数值模拟来例证理论结果.

This paper investigates the stationary distribution and extinction of a stochastic population model with Allee effect and nonlinear perturbation. The author first proves the existence of global positive solution of the model. Then by constructing a suitable stochastic Lyapunov function, the author establishes sufficient conditions for the existence of an ergodic stationary distribution of the model. Then the author obtains sufficient conditions for extinction of the population in two cases. One is how Allee effect affects extermination of the population and the other is how noises affect extermination of the population. At last, some examples together with numerical simulations are provided to illustrate the analytical results. }