Nonlinearities have been shown to play an important role in increasing the extracted energy of energy harvesting devices at the macro and micro scales. Vibration-based energy harvesting on the nano scale has also received attention. In this paper, we characterize the nonlinear dynamical behavior of a strained nanostructured graphene for its potential use in energy harvesting applications. A compressed vibrating membrane graphene sheet free from any external excitation is first studied. We present a continuous time dynamical model of the system in the form of a double-well single degree of freedom system. Equilibrium points are obtained and their stability analysis is carried out. Then, random vibrations are considered as the main ambient energy source for the system and its performances in terms of the well occupation zones, RMS value of the position, and the corresponding energy harvested are presented in the steady state non-equilibrium regime when the noise level is considered as a control parameter. From this model, nonlinear analysis is carried out by computing state space trajectories, probability density and FFT spectra under a deterministic excitation. The ultimate goal of this parameter space exploration based upon a behavioral model is to provide design-oriented guidelines for engineering graphene-based mechanical harvesters