The Lyapunov-Foster criteria is one of the well known methods for verifying ergodicity of Markov processes. The "Standard Lyapunov function” has played a significant role in the analysis of deterministic reaction networks, especially in regards to deficiency 0 networks. I will provide conditions on reaction networks that guarantee the Standard Lyapunov function satisfies the Lyapunov-Foster criteria. The analysis utilizes hierarchies of intensities and rate functions (Tier structures). I will also provide some special classes of reaction networks satisfying the conditions, thus proving networks in those classes are positive recurrent. I will also provide some results pertaining to the mixing times of reaction networks.