This paper considers the problem of tracking an acoustic sources in three dimensional (3-D) space by using a vertical acoustic vector sensor (AVS) array in a shallow ocean environment. The innovations of this work are double fold: 1) a particle filtering (PF) approach is developed to track the position of an acoustic source; and 2) based on the source motion and wave propagation models, the posterior Cramér-Rao bound (PCRB) is derived to provide a lower performance bound of 3-D position tracking in shallow ocean. The PF approach uses a number of samples to approximate the posterior distribution of interested parameters, by which a complex 3-D search can be avoided for 3-D position estimation. Also, due to incorporating both the source dynamic and measurement information, the tracking approach is able to provide a lower performance bound than the traditional localization approach. The tracking performance is further demonstrated by numerical experiments.