Spatio‐Temporal Patterns Induced by Cross‐Diffusion and Nonlocal Time Delay in a Circular‐Distributed Nutrient‐Plankton Model

ABSTRACT In this paper, the spatio‐temporal dynamics of a nutrient‐phytoplankton model with nonlocal time delay and cross‐diffusion on a circular domain are studied theoretically and numerically. We first transform such a two‐dimensional model with nonlocal delay into a three‐dimensional non‐delayed model in polar coordinates. Then we pay attention to the stability and the Turing bifurcation analysis. The amplitude equation near the Turing bifurcation boundary is derived by using the multi‐scale analysis method. Through stability analysis of the amplitude equation, the conditions that enable the formation of Turing patterns, including stripes, spots, and a combination of stripes and spots, have been identified. Finally, the numerical simulations are given. The different types of standing waves and target wave patterns are obtained on the circular domain, along with more intricate spatiotemporal patterns that have random initial values across both the circular domain and the rectangular domain. Moreover, there is a certain correspondence between the patterns in the circular domain and the four types of patterns in the conventional rectangular domain. This indicates that in the real world, the nutrient cycle and spatial distribution of phytoplankton, as well as the nonlocal time delay, have a positive impact on the environment.