If there is media for the bacteria to colonize, food for them, and appropriate temp, then they will do so. The volume of water is pertinent in several ways, but not to the presence of bacteria.
Small tanks are not as stable over the long run as a larger tank, but they will cycle. The instability crops up in two ways--First, pollutants will increase in concentration quicker than in a large colume of water. Picture 2 drops of food coloring in a tea cup versus a gallon jar. In the tea cup, the dispersion is fairly fast, and the resulting color is noticable. The same two drops disperse slowly to all corners of the gallon jar, and result in only a faint tinge of color. Same thing for wastes. In a one gallon tank, the wastes build up quickly.
Second, because the volume of water is small, water parameters (hardness, pH) can shift dynamically from normal bio-processes. In large tanks, the volume of water means these parameters are stable under normal maintenance. Insmall tanks, more frequent water changes are needed to prevent pH crashes.
So, what this means is that frequent, small water changes are needed to prevent toxins from building up (talking about things other than testable nitrogens), and to maintain an appropriate buffer to stabilize pH.