Role of instant nutrient replenishment on plankton dynamics with diffusion in a closed system: A pattern formation

A mathematical model is proposed to study the role of instantaneous nutrient recycling on the plankton ecosystem. In this model, we include three state variables namely, nutrient biomass, phytoplankton and zooplankton population with Holling type II response function for the population density transformation from phytoplankton to zooplankton. It is obtained that the local stability of different equilibrium depends on the nutrient supply rate to the phytoplankton for the temporal system and also existence of the oscillatory behavior of the temporal system is established by using Bendixson–Dulac criteria. In the spatiotemporal model, we also determine the diffusion-driven instability condition, with the numerical support for the effect of diffusivity coefficients on chaotic behavior of the system. Furthermore, we obtained the instability condition for linear and no-linear system from the higher order stability analysis. Finally, we analyze the time evaluation pattern formation of the spatial system. This shows that it is useful to use the reaction–diffusion system to reveal the spatial dynamics in the real world.