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A model is presented for readout-power heating in kinetic inductance detectors. It is shown that the power dissipated by the readout signal can cause the temperature of the quasiparticle system in the superconducting resonator to switch between well-defined states. At low readout powers, only a single solution to the heat balance equation exists, and the resonance curve merely distorts as the readout power is increased. At high readout powers, three states exist, two of which are stable, and the resonance curve shows hysteretic switching. The power threshold for switching depends on the geometry and material used but is typically around -70 dBm for Aluminum resonators. A comprehensive set of simulations is reported, and a detailed account of the switching process is given. Experimental results are also shown, which are in strong qualitative agreement with the simulations. The general features of the model are independent of the precise cooling function, and are even applicable for resonators on suspended, thermally isolated, dielectric membranes, where an increase in quasiparticle lifetime is expected. We discuss various extensions to the technique, including the possibility of recovering the cooling function from large-signal measurements of the resonance curve.