Summary: This paper deals with a very general case of predator-prey model called Gauss type model, in the presence of cross-diffusion, where the existence of global solution has been shown and some priori bound of solution. The dynamic near Turing-Hopf bifurcation has been studied, where the existence of Hopf and Turing-Hopf bifurcation has been shown, and for the stability of the periodic solution the normal form on the center of the manifold of Turing-Hopf bifurcation has been calculated. Further, an application of the considered model has been applied in treated models and non treated ones, also an extend numerical simulation has been used to illustrate the theoretical results.