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The need for reliable models that take into account the nonlinear kinetics of dopants is nowadays of paramount importance, particularly with the physical dimensions of electron devices shrinking to the deep nanoscale range and the development of emerging nanoionic systems such as the memristor. In this paper, we present a novel nonlinear dopant drift model that resolves the boundary issues existing in previously reported models that can be easily adjusted to match the dynamics of distinct memristive elements. With the aid of this model, we examine switching mechanisms, current-voltage characteristics, and the collective ion transport in two terminal memristive devices, providing new insights on memristive behavior.