Analysis of taxis-driven instability of a predator–prey system through the plankton community model

The paper deals with a prey–predator model, which describes the spatiotemporal dynamics of plankton community and the nutrients. The system is described by reaction–diffusion–advection equations in a one-dimensional vertical column of water in the surface layer. Advective term of the predator equation represents the vertical movements of zooplankton with velocity, which is assumed to be proportional to the gradient of phytoplankton density. This study aimed to determine the conditions under which these movements (taxis) lead to the spatially heterogeneous structures generated by the system. Assuming diffusion coefficients of all model components to be equal the instability of the system in the vicinity of stationary homogeneous state with respect to small inhomogeneous perturbations is analyzed.
Necessary conditions for the flow-induced instability were obtained through linear stability analysis. Depending on the local kinetics parameters, increasing the taxis rate leads to Turing or wave instability. This fact is in good agreement with conditions for the emergence of spatial and spatiotemporal patterns in a minimal phyto-plankton-zooplankton model after flow-induced instabilities derived by other authors. This mechanism of generating patchiness is more general than the Turing mechanism, which depends on strong conditions on the diffusion coefficients.
While the taxis exceeding a certain critical value, the wave number corresponding to the fastest growing mode remains unchanged. This value determines the type of spatial structure. In support of obtained results, the paper presents the spatiotemporal dynamics of the model components demonstrating Turing-type pattern and standing wave pattern.