Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to alternative food sources. It has long been known that sigmoid (Holling's Type III) functional responses may stabilize an otherwise unstable equilibrium of prey and predators in Lotka-Volterra models. This poses the question of under what conditions such switching-mediated stability is likely to occur. A more complete understanding of the effect of predator switching would therefore require the analysis of one-predator/two-prey models, but these are difficult to analyze. We studied a model based on the simplifying assumption that the alternative food source has a fixed density. A well-known result from optimal foraging theory is that when prey density drops below a threshold density, optimally foraging predators will switch to alternative food, either by including the alternative food in their diet (in a fine-grained environment) or by moving to the alternative food source (in a coarse-grained environment). Analyzing the population dynamical consequences of such stepwise switches, we found that equilibria will not be stable at all. For suboptimal predators, a more gradual change will occur, resulting in stable equilibria for a limited range of alternative food types. This range is notably narrow in a fine-grained environment. Yet, even if switching to alternative food does not stabilize the equilibrium, it may prevent unbounded oscillations and thus promote persistence. These dynamics can well be understood from the occurrence of an abrupt (or at least steep) change in the prey isocline. Whereas local stability is favored only by specific types of alternative food, persistence of prey and predators is promoted by a much wider range of food types.