Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality

We investigate the effects of diffusion on the spatial dynamics of a predator-prey model with hyperbolic mortality in predator population. More precisely, we aim to study the formation of some elementary two-dimensional patterns such as hexagonal spots and stripe patterns. Based on the linear stability analysis, we first identify the region of parameters in which Turing instability occurs. When control parameter is in the Turing space, we analyse the existence of stable patterns for the excited model by the amplitude equations. Then, for control parameter away from the Turing space, we numerically investigate the initial value-controlled patterns. Our results will enrich the pattern dynamics in predator-prey models and provide a deep insight into the dynamics of predator-prey interactions.