Spatiotemporal heterogeneity in a diffusive toxin-producing phytoplankton–zooplankton model: The role of spatial memory and cross diffusion

Spatial memory is essential for regulating the spatiotemporal distribution of plankton. In this paper, a spatial memory term is incorporated into a diffusive toxin-producing phytoplankton–zooplankton model with three-dimensional patches. The aim is to examine the combined influence of memory delay and the resulting cross-diffusion coefficient on the spatiotemporal dynamics of the populations. We first prove the existence and uniqueness of solutions for the model and analyze the existence and stability of its homogeneous equilibria. Next, for the model without memory delay, we take the cross-diffusion coefficient as the bifurcation parameter and derive the conditions under which Turing instability arises. When the delay effect is included, the model undergoes an n-mode Hopf bifurcation under the joint action of cross-diffusion and delay. Finally, we support the theoretical analysis with numerical simulations. The results show that toxins play a substantial regulatory role in sustaining ecosystem stability. In addition, the introduction of cross-diffusion leads to spatially heterogeneous steady-state distributions. The inclusion of memory delay further causes the model to exhibit periodic temporal oscillations, giving rise to spatiotemporally coupled non-uniform patterns. Biologically, toxin concentration and spatial memory cycles help promote the coexistence of phytoplankton and zooplankton, thus acting as a key mechanism for preserving ecosystem homeostasis.