Landfilled biodegradable materials are subject to biological evolution, namely biodegradation or digestion, under the action of aerobic bacteria and further anaerobic bacteria. In the absence of special provisions, the liquid flowing through the mass of waste produces leachate, being charged with chemical or biological substances. Therefore, the mathematical modeling is of great interest for making decisions in the field of waste management since it makes possible to predict the behavior of the various complex systems underlying the management of waste. The analysis of the dynamics of the model considered in our work describes the process of anaerobic biodegradation with leachate reinjection, and allows us to predict the amount of biogas produced in the long term. The system admits an infinity of non hyperbolic equilibria but we are able to describe the asymptotic behavior of the solutions depending on the initial condition. Thanks to this model, we can highlight the influence of the leachate recirculation on the production of biogas, with the help of numerical simulations. This recirculation could be a part of the sustainable solutions adapted by recent technologies for the treatment of waste, not only to optimize the total production of biogas, but also to obtain a good quality of the outgoing liquid to meet the discharge standards.