Threshold dynamics of a stochastic single population model with Allee effect

Abstract

A stochastic single species population model with Allee effect is considered in this paper. By using the theory of Random Dynamical System (RDS), the complete classification of the global dynamics of stochastic system is given. The theoretical results show that the dynamics of the system is completely determined by the threshold λ=2r−σ2: if λ≤0, the population will extinct, i.e., the solutions of the model will go to zero; if λ>0, the population is permanent. Moreover, the transition probabilities of the solutions weakly converge to the unique stationary distribution and the explicit density function of the stationary distribution is given.