Time delays play an important role in plant growth and development and the formation of spatiotemporal patterns. Based on the nutrient-vegetation model established by Rietkerk et al.[1], in this paper, we consider the effects of spatial diffusion and two time delays existing between nutrient cycling and vegetation growth. If the two delays are not considered, we find that spatial diffusion can not induce the emergence of Turing instability. Furthermore, we consider the effects of the two delays on the dynamical behaviors. By using the geometric analysis method, we demonstrate that time delays can induce the transition between the stable state and the temporal periodic oscillations, and both time delays and spatial diffusion can drive the complex spatiotemporal pattern formation. Our study provides theoretical insights into the spatiotemporal dynamics of vegetation ecosystems.
