This paper proposes and studies a predator-prey model incorporating distributed memory and gestation delay to more accurately describe animal movement. First, the stability conditions of the positive equilibrium in the absence of delays are analyzed. Second, the conditions for the occurrence of Turing and Hopf bifurcation without gestation delay are derived. Subsequently, the combined effects of memory delay and gestation delay on the stability of the positive equilibrium are investigated, revealing that their interaction can generate more complex spatiotemporal patterns. Furthermore, normal form theory is employed to determine the direction and stability of the Hopf bifurcation induced solely by memory delay in the absence of gestation delay. Finally, numerical simulations are conducted to validate the theoretical results. In addition, variations in the memory-based diffusion coefficient, memory delay, and gestation delay are shown to trigger transitions among spatially homogeneous/nonhomogeneous steady states and spatially homogeneous/nonhomogeneous periodic patterns.