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Inferring microbial community structure based on temporal metagenomics datais an important goal in microbiome studies. The deterministic generalizedLotka-Volterra differential (GLV) equations have been used to model thedynamics of microbial data. However, these approaches fail to take randomenvironmental fluctuations into account, which may negatively impact theestimates. We propose a new stochastic GLV (SGLV) differential equation model,where the random perturbations of Brownian motion in the model can naturallyaccount for the external environmental effects on the microbial community. Weestablish new conditions and show various mathematical properties of thesolutions including general existence and uniqueness, stationary distribution,and ergodicity. We further develop approximate maximum likelihood estimatorsbased on discrete observations and systematically investigate the consistencyand asymptotic normality of the proposed estimators. Our method is demonstratedthrough simulation studies and an application to the well-known "movingpicture" temporal microbial dataset.

Stochastic Generalized Lotka-Volterra Model with An Application to Learning Microbial Community Structures