Turing Instability of a Closed Nutrient-Phytoplankton-Zooplankton Model with Nutrient Recycling

Abstract

We investigate Turing instability in a closed Nutrient–Phytoplankton–Zooplankton (NPZ) ecosystem that incorporates delayed nutrient recycling, formulated as a reaction–diffusion system. Although spatial diffusion typically enhances system stability, our study focuses on how differing diffusion rates among species can destabilize steady states and lead to the emergence of spatial patterns. To explore this, we first perform a linear stability analysis to identify the conditions under which Turing instability arises. These theoretical predictions are then validated through numerical simulations. Our study progresses systematically: beginning with a two-species model, extending to a threespecies system, and finally to a four species NPZD model. This stepwise framework provides both conceptual insight and quantitative understanding of how diffusion influences instabilities, offering a comprehensive perspective on pattern formation in multi-species plankton ecosystems.