Three-dimensional pattern dynamics of a fractional predator-prey model with cross-diffusion and herd behavior.

• We introduced three-dimensional fractional diffusion into the reaction-diffusion system. • It is maybe the first time to discuss three-dimensional pattern dynamics with fractional cross-diffusion. • The increase of the cross-fractional diffusion coefficient causes the bistable state of the three-dimensional pattern. • The fractional-order α affects the stability of the 3D pattern. • Proper protection of prey refuge is beneficial to the stability of the ecosystem. In this paper, we study the pattern dynamics in a spatial fractional predator-prey model with cross fractional diffusion, herd behavior and prey refuge. In this model, herd behavior exists in the population of predators and the prey. The spatial dynamics of the system are obtained through appropriate threshold parameters, and a series of three-dimensional patterns are observed, such as tubes, planar lamellae and spherical droplets. Specifically, linear stability analysis is applied to obtain the conditions of Hopf bifurcation and Turing instability. Then, by utilizing the central manifold reduction theory analysis, the amplitude equation near the critical point of Turing bifurcation is deduced to study the selection and stability of pattern formation. The theoretical results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]