<p>The Rosenzweig-MacArthur model, which only considers classical diffusion, has long been pointed out as not having the conditions to generate Turing patterns. However, spatial patterns are not limited to one type of Turing pattern. Under appropriate parameters and initial conditions, the model can still observe other spatiotemporal patterns, such as concentric wave patterns, spiral patterns, and lattice formations. This paper extends the model that originally only considered integer diffusion to the case of fractional diffusion. Theoretically, we have analyzed the existence and stability of the equilibria within the model. Furthermore, we have engaged in comprehensive discussions regarding potential transcritical bifurcation, Hopf bifurcation, and stable limit cycle in the absence of diffusion. Notably, the Hopf bifurcation observed in the spatial fractional model lays the groundwork for our subsequent identification of spatiotemporal patterns. Our analysis reveals that fractional diffusion does not change the conclusion that the model does not produce Turing patterns. Still, it plays an important role in the evolution of spatiotemporal patterns. Specifically, in two-dimensional (2D) space, fractional order controls switch between spiral patterns and concentric waves. In addition, lattice formations have been captured within this spatial fractional model, with the fractional order and initial values impacting the spatial distribution of the population. Unlike previously observed, as the predator’s diffusion rate increases, despite the parameter values meeting the conditions for model instability, the population density within the region transitions from its original periodic oscillation to a stable state. Intriguingly, spiral patterns persist in three-dimensional (3D) space, where fractional orders govern the formation of 3D lattices. These findings contribute to the enrichment of existing research.</p>