A Study on the Effects of Riesz Fractional Diffusion on Pattern Formation in a Toxic-Phytoplankton–Zooplankton Model

In this paper, we investigate Turing instability and spatiotemporal pattern formation in a toxic-phytoplankton–zooplankton model incorporating Riesz fractional diffusion. By replacing classical Laplacians with fractional-order operators (−Δ)ϕ/2 and (−Δ)ψ/2, we establish a framework to capture species-specific anomalous diffusion in aquatic ecosystems. Theoretical analysis demonstrates that the fractional orders significantly modulate the Turing bifurcation threshold and the range of unstable modes. Numerical simulations, implemented via a DCT-based fractional finite difference scheme, reveal that: (1) fractional diffusion triggers pattern formation below the classical critical threshold, generating complex mixed structures unattainable in integer-order models; (2) the two fractional orders exert opposing influences on pattern morphology—decreasing the phytoplankton order ϕ promotes a transition from stripes to hexagonal spots. Our results suggest that anomalous diffusion is a key mechanism for ecological pattern selection, providing new insights into the spatiotemporal complexity of plankton populations in heterogeneous environments.