<p>The spatiotemporal dynamics of nonlinear oscillations plays a significant important role in understanding complex behaviors and functions of biological and electronic systems. While the spatiotemporal patterns in a memristive Josephson junction array are still unclear. In this paper, a discrete memristive Josephson junction model (DMJJ) is proposed. Furthermore, we construct a DMJJ array, considering the resistive coupling and magnetic flux coupling generated by the phase difference. Based on the stability analysis, an excitable transition from resting state into periodical firing is triggered by the Neimark-Sacker bifurcation in the DMJJ model. Meanwhile, the chaotic behaviors associated with the period-doubling bifurcation are triggered with increasing the discrete-time parameter in the DMJJ model with external excitation. The DMJJ array can display regular, irregular, and chaotic spatiotemporal patterns. It is found that the transitions of spatiotemporal patterns are dependent upon the firing modes and the external excitation. A transition between spiral wave and target wave is established numerically when the controllable energy is injected. Obtained result is helpful to understand and predict the spatiotemporal dynamics of complex firing behaviors induced in the brain-inspired biophysical systems.</p>