Based on the nonlinear three-wave interaction model including driving and dissipation, we construct a system to describe the nonlinear interaction between energetic particle induced geodesic acoustic mode (EGAM) and the drift waves turbulence in the Dimits shift region, and study both analytically and numerically the linear growth and nonlinear oscillation phases of the system, respectively. Further numerical results show that, without the contribution of EGAM, the system goes through limited cycle oscillation to period doubling, and finally route to chaos with the change of the linear drive/dissipation rate. On this basis, the nonlinear saturated "Dimits region" of the system is constructed, which is then used to study the influence of EGAM on the drift wave in the Dimits region. The results show that for EGAM with different amplitude and frequency, the modulated drift wave can be either excited or suppressed, partly reproduces the results from large scale simulation. Finally, We use the method of phase space analysis to give the corresponding explanation.