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

Strong Allee Effect in A Stochastic Logistic Model with Mate Limitation and Stochastic Immigration