Turing pattern selection for a plant-wrack model with cross-diffusion

We investigate the Turing instability and pattern formation mechanism of a plant-wrack model with both self-diffusion and cross-diffusion terms. We first study the effect of self-diffusion on the stability of equilibrium. We then derive the conditions for the occurrence of the Turing patterns induced by cross-diffusion based on self-diffusion stability. Next, we analyze the pattern selection by using the amplitude equation and obtain the exact parameter ranges of different types of patterns, including stripe patterns, hexagonal patterns and mixed states. Finally, numerical simulations confirm the theoretical results.