<p>This paper is concerned with the spatiotemporal dynamics of a three species competition–competition–predation Lotka–Volterra system with cross-diffusion. Firstly, by applying cross-diffusion coefficients as the bifurcation parameter, we demonstrate that our model undergoes a spatially inhomogeneous Hopf bifurcation around positive constant steady state when the bifurcation parameter is varied through a critical value. In addition, we prove that the obtained spatially inhomogeneous bifurcating periodic solutions are stable when cross-diffusion coefficients lie in suitable ranges. Secondly, we establish the sufficient conditions for the global asymptotical stability of positive constant steady state. Finally, we prove that the system admits only the positive constant steady state when cross-diffusion coefficients is small enough, which implies that the system does not create stationary patterns when cross-diffusions are small enough. A common feature in the most existing research papers is that the bifurcation parameter that induces Hopf bifurcation appears in the reaction terms rather than diffusion terms. We demonstrate that the spatially inhomogeneous Hopf bifurcation can be induced by the effect of cross-diffusion terms. Our results show that cross-diffusion plays a crucial role in the formation of spatially inhomogeneous periodic oscillatory patterns.</p>