In this paper we consider a prey-predator model with prey refuge and intraspecific competition between predators using the Crowley-Martin functional response and investigate the dynamic characteristics of spatial and non-spatial prey-predator systems with each analytical and numerical approach. The local stability of non-trivial interior equilibrium, the existence of a Hopf bifurcation, and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. For the spatial system, the Turing and non-Turing patterns are evaluated for some set parametric beliefs, and for prey and predator inhabitants some exciting results are obtained. Numerical simulation demonstrates that both prey and predator populations will not converge to the stationary state at any foreseeable future time when the parametric values are ingested in the Turing domain.