Self-organization of plankton patchiness across multiple scales: An explanation coupling reaction–diffusion model and environmental heterogeneity

¹ Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing 102206, PR China
² Powerchina Beijing Engineering Corporation Limited, Beijing 100024, PR China

^* Corresponding author: tous This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 28 May 2025
Accepted: 3 April 2026

Abstract

The nonlinear mechanisms underlying the emergence of plankton patchiness have attracted considerable attention in recent decades. In this research, a coupled reaction–diffusion model for phytoplankton-zooplankton interactions is developed by incorporating plankton migration across water layers. Conditions for pattern self-organization are derived through stability analysis, and robustness of multiscale pattern regimes is quantified through local and global sensitivity analyses. When the dispersion relation has multiple peaks, both pure Turing and Hopf–Turing instabilities can generate patterns with two or three coexisting patch sizes. Variance spectrum analysis on the multiscale patterns shows a power-law distribution of plankton variance across spatial scales. Using 1 km as a dividing scale, pure Turing instability mainly shapes smaller-scale patchiness, whereas Hopf–Turing instability has a stronger influence at larger scales. By introducing spatially heterogeneous carrying capacity, the self-organized patterns show variance spectra with slopes comparable to those reported in the literature. Additional tests with nutrient limitation and turbulence demonstrate that multiscale patchiness persists under more realistic forcing, and comparison with chlorophyll-a spectra from the Bohai and Yellow Seas supports the model ability to reproduce scaling behavior. These findings suggest that Turing instability and environmental heterogeneity can jointly provide a plausible mechanism for multiscale plankton patchiness observed in natural ecosystems.