  We propose a stochastic logistic model with mate limitation and stochastic
immigration. Incorporating stochastic immigration into a continuous time Markov
chain model, we derive and analyze the associated master equation. By a
standard result, there exists a unique globally stable positive stationary
distribution. We show that such stationary distribution admits a bimodal
profile which implies that a strong Allee effect exists in the stochastic
model. Such strong Allee effect disappears and threshold phenomenon emerges as
the total population size goes to infinity. Stochasticity vanishes and the
model becomes deterministic as the total population size goes to infinity. This
implies that there is only one possible fate (either to die out or survive) for
a species constrained to a specific community and whether population eventually
goes extinct or persists does not depend on initial population density but on a
critical inherent constant determined by birth, death and mate limitation. Such
a conclusion interprets differently from the classical ordinary differential
equation model and thus a paradox on strong Allee effect occurs. Such paradox
illustrates the diffusion theory's dilemma.
