The hysteretic conduction characteristics and fatigue profile of La_1/3Ca_2/3MnO₃ (LCMO)-based memristive devices were investigated. The oxide films were grown by pulsed laser deposition (PLD) and sandwiched between Ag and Pt electrodes. The devices exhibit bipolar resistive switching (RS) effect with well-defined intermediate conduction states that arise from partial SET and RESET events. The current-voltage curves are modeled and simulated using a compact memristive approach. Two equations are considered: one for the electron transport based on the double-diode equation and the other for the memory state of the device driven by the play operator with logistic ridge functions. An expression that accounts for the remnant resistance of the device is obtained after simplifying the model equations in the low-voltage limit. The role played by the power dissipation in the LCMO reset dynamics as well as the asymmetrical reduction of the resistance window caused by long trains of switching pulses are discussed.