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