Ocean waste is a serious environmental problem affecting marine ecosystems and marine life. A large portion of this waste consists of recyclable materials. If managed correctly, it provides both economic and environmental benefits. By using mathematical models, it is possible to predict the spread of these wastes and develop strategies. In this paper, a discrete-time compartmental model is introduced for the cycle of processing the waste plastic management in the ocean. The model comprises three compartments plastic waste, marine debris, and recycled materials. A system of difference equations is formulated for this process according to the transmissions of materials between compartments. Then two equilibria for the model are obtained; the marine debris-free equilibrium point and the marine debris-included equilibrium point. Also, the basic reproduction number for the model is given, then its stability is investigated and stated in terms of this quantity. Furthermore, the bifurcations of the model including transcritical, period- doubling, and Neimark-Sacker bifurcation, are studied and the conditions for their occurrence are given. The obtained theoretical results are verified via several examples and simulations.