<p>Marine dynamics is known for having a strong impact on the atmosphere, aquatic food web, and global climate change. In this paper, we consider and study a two-species phytoplankton-zooplankton interaction model. We consider the Holling type-III and Holling type-IV interactions among the species. The impact of self and cross-diffusion have been studied on the density distribution of the species via a reaction-diffusion model. A detailed analytical study has been performed for proposed model system in the absence as well as presence of diffusion. For the temporal system, numerically, we have investigated the detailed bifurcation analysis and mathematically, the boundedness and positivity of solutions, the existence of equilibria, stability of the proposed model system, and Hopf-bifurcation of the interior equilibrium point. By analyzing the linear stability of the spatial homogeneous equilibrium state of this model, sufficient condition of Turing instability and amplitude equation are obtained. Numerical simulations have been performed to validate the theoretical results. Under consideration, the results obtained appear to enrich the findings of the proposed system. We identify that the half-saturation constant acts as a control parameter for dynamical behavior of the temporal system through a bifurcation diagram and system dynamics shows oscillatory to stable behavior. Time iteration, fish predation rate of zooplankton, and half-saturation constant have a great impact on the spatial distribution of the species. The spots and stripes like patterns is obtained for spatial model system through numerical simulation. From the spatial patterns, it is clear that the dynamics of the model system are significantly more affected by cross-diffusion compared to self-diffusion.</p>