A model of edge-defined film-fed (EFG) crystal growth is developed to study melt growth of cylinders of the scintillator crystal cesium iodide (CsI). This system is characterized by strongly nonlinear interactions of heat transfer, capillarity, and die geometry that give rise to multiple solution states under a single set of operating conditions. A thermal-capillary instability is identified that stems from meniscus sag due to gravity, which causes an inection point to appear in the meniscus shape. This shape allows two solutions to coexist that correspond to different crystal diameters and gap sizes (axial length of liquid bridge between die and crystal). Also identified is an instability of convective heat transfer characterized by a strong interaction of thermal convection with the geometry of the growth interface.