Evolution of the light-induced absorption in a Bi₁₂TiO₂₀ crystal is investigated at a wavelength of 633 nm. It is found that the absorption coefficient evolution consists of fast and slow components with different characteristic response time. The light absorption coefficient decreases from 0.9 to 0.16 cm^-1 after keeping the sample in darkness for six months. We propose a new charge-transport model that includes the deep donors in three valence states and the shallow traps. It is shown that the theory developed provides an explanation for the observed features of the increase and dark decay of the light-induced absorption. The material parameters of the Bi₁₂TiO₂₀ crystal are estimated from the best fit of the theoretical model with the experimental data.