We consider the focusing mass critical Zakharov-Kuznetsov equation in 2D. We will provide a complete classification of the long time behavior of solutions with initial data near the ground and with a suitable decay on the first variable. We will show that only three behaviors are possible: 1. converging to a traveling wave, 2. blowing up in finite time, 3. linear behavior. We also prove the nonexistence of minimal mass blow-up solutions. Our result is an extention of the work of Martel-Merle-Raphael for mass critical gKdV equations. This work is joint with G. Chen and X. Yuan.