In this work, we consider a mathematical model of syntrophic relationship between two microbial species of the anaerobic digestion process including mortality (or decay) terms. We focus on the acetogenesis and hydrogenotrophic methanogenesis phases. Our study gives a quite comprehensive analysis of a syntrophic model by analyzing the joined effects of syntrophy relationship, mortality, substrate inhibition and input concentrations that were neglected in previous studies. Using a general class of growth rates, the necessary and sufficient conditions of existence and local stability of all steady states of the four-dimensional system are determined according to the operating parameters. This general model exhibits a rich behavior with the coexistence of two microbial species, the bi-stability, the multiplicity of coexistence steady states, and the existence of two steady states of extinction of the first species. The operating diagram shows how the model behaves by varying the control parameters and illustrates the effect of the inhibition and the new input substrate concentration (hydrogen) on the reduction of the coexistence region and the emergence of a bi-stability region. Similarly to the classical chemostat model, including the substrate inhibition can destabilize a two-tiered microbial 'food chain' where the stability depends on the initial condition.