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

TL;DR

This paper analyzes a stochastic logistic model with mate limitation and immigration, revealing a strong Allee effect characterized by a bimodal stationary distribution, which disappears as population size grows large, transitioning to deterministic behavior.

Contribution

It introduces a stochastic model incorporating immigration and mate limitation, demonstrating the existence of a strong Allee effect and its transition to deterministic dynamics at large populations.

Findings

Existence of a bimodal stationary distribution indicating a strong Allee effect.

Disappearance of the Allee effect as population size approaches infinity.

Transition from stochastic to deterministic dynamics with increasing population size.

Abstract

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.