In this paper, under homogeneous Neumann boundary conditions, the complex dynamical behaviors of a diffusive Leslie-Gower predator-prey model with a ratio-dependent Holling type III functional response and nonlinear prey harvesting is carefully studied. By scrupulously analyzing and comprehending the distribution of the eigenvalues, the existence and stability (balance) of the extinction and coexistence equilibrium states are determined, and the bifurcations exhibited by the system are investigated by a mathematical analysis. Additionally, based on the theoretical analysis and numerical simulation, (Harvesting rate-induced, Delay-induced), Turing-Hopf bifurcations points are derived. Our results show that delay and nonlinear prey harvesting rates can create spatially inhomogeneous periodic solutions.