Different Mechanisms of Pattern Formation in a Plankton Model with Cross-Diffusion and Fitness

Abstract

Pattern formation is a ubiquitous phenomenon encountered in various nonequilibrium physical, chemical and biological systems. The resulting spatiotemporal patterns as well as their characteristics are often determined by the type of instability. However, when different instabilities occur simultaneously, the generated pattern formation cannot be expected to be a simple superposition of patterns. To address this issue, we study spatiotemporal dynamics driven by different mechanisms in a reaction–advection–diffusion plankton model. Linear stability analysis is performed upon the uniform steady state to identify conditions for the predator–prey interaction driven, taxis-diffusion driven and cross-diffusion-driven instabilities. For the cross-diffusion-driven instability, we employ weakly nonlinear analysis to derive amplitude equations, which helps to predict the type of patterns turning out to emerge with parameters that are varying. Theoretical results are verified by numerical simulations, and some interesting patterns including spiral and target waves are also numerically observed.