Abstract Over the last 250 years, anthropogenic activity has increased atmospheric carbon dioxide by nearly 40%. This increase is mainly caused by human fossil fuel combustion and deforestation, which are the main causes of global warming. Phytoplankton of the world’s oceans synthesizes half of the carbon dioxide of the total Earth’s photosynthetic activity. Thus, phytoplankton plays a crucial role in controlling Earth’s climate. To study this scenario, we propose and analyze a mathematical model for the carbon-phytoplankton-zooplankton interaction dynamics. Positivity, boundedness, existence, and stability of biologically possible equilibrium points are studied. The system exhibits Hopf bifurcation with respect to the carbon capture coefficient and the criteria of Hopf bifurcation is established around the coexisting equilibrium. Complex spatiotemporal dynamics and patchy pattern formation are observed in the spatially explicit model. The proposed carbon-phytoplankton-zooplankton system incorporates the effect of global warming, and our simulation shows shifts in plankton seasonal dynamics.