Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration

Chuang  Xu

doi:10.3934/dcdsb.2016049

Article Contents

2016, Volume 21, Issue 7: 2321-2336. Doi: 10.3934/dcdsb.2016049

This issue Previous Article Kolmogorov-type systems with regime-switching jump diffusion perturbations Next Article Stochastic dynamics: Markov chains and random transformations

Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration

Chuang Xu^1,

1.
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001

Received: January 31, 2015

Revised: October 31, 2015

Published: August 2016

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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.

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Mathematics Subject Classification: Primary: 60J27, 92D25, 92D40; Secondary: 60J80.

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