Savanna models that are based on recurrent disturbances such as fire result in nonequilibrium savannas, but these models rarely incorporate vegetation feedbacks on fire frequency or include more than two states (grasses and trees). We develop a disturbance model that includes vegetation-fire feedbacks, using a system of differential equations to represent three main components of savannas: grasses, fire-tolerant savanna trees, and fire-intolerant forest trees. We investigate the stability of savannas in the presence of positive feedbacks of fire frequency with (1) grasses, (2) savanna trees, and (3) grasses and savanna trees together while also allowing for negative feedbacks of forest trees on fire frequency. We find that positive feedbacks between fire frequency and savanna trees, alone or together with grasses, can stabilize savannas, blocking the conversion of savannas to forests. Negative feedbacks of forest trees on fire frequency shift the range of parameter space that supports savannas, but they do not generally alter our results. We propose that pyrogenic trees that modify characteristics of fire regimes are ecosystem engineers that facilitate the persistence of savannas, generating both threshold fire frequencies with rapid changes in community composition when these thresholds are crossed and hystereses with bistable community states.