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This paper examines the spatio-temporal dynamics of a marine ecosystem. The system is described by two reaction–diffusion equations. We consider a phytoplankton–zooplankton system with Ivlev-type grazing function. The dynamics of the reaction–diffusion system of phytoplankton–zooplankton interaction has been studied with both constant and variable diffusion coefficients. Periodic oscillations of the phytoplankton and zooplankton populations are shown with constant and variable diffusion coefficients. In order to obtain spatio-temporal patterns, we perform numerical simulations of the coupled system describing phytoplankton–zooplankton dynamics in the presence of diffusive forces. We explain how the concentration of species changes due to local reactions and diffusion. Our results suggest that patchiness is one of the basic characteristics of the functioning of an ecological system. Two-dimensional spatial patterns of phytoplankton–zooplankton dynamics are self-organized and, therefore, can be considered to provide a theoretical framework to understand patchiness in marine environments.