This work presents a dynamic analysis for an anaerobic digester, supported on the analytical application of the indirect Lyapunov method. The mass-balance model considered is based on two biological reaction pathways and involves both Monod and Haldane representations of the specific biomass growth rates. The dilution rate, the influent concentration of chemical oxygen demand (COD), and the influent concentration of volatile fatty acids (VFA) are considered as stability parameters. Several characteristics are determined analytically for the normal operation equilibrium point: (i) equilibrium coordinates, (ii) parameter conditions that lead to positive values of the equilibrium state variables, (iii) parameter conditions for locally stable nature of the equilibrium, (iv) coordinates for the local bifurcation points—fold and transcritical—, and (v) coordinates of the crossing between bifurcation points. These factors are computed analytically and explicitly as expressions of the dilution rate and the influent concentrations of COD and VFA.