The effect of grazing intensity on pattern dynamics of the vegetation system

The evolution of vegetation system in arid and semi-arid grazing areas is a complex dynamical system which depends not only on the rainfall, but also grazing intensity. Currently, most of the research focuses on the influence of rainfall, but the effects of grazing have not been fully understood. Simultaneously, the intraspecific competition delay widely exists in vegetation system. In this project, we develop a vegetation model coupled with the intraspecific competition delay, discuss the vegetation pattern caused by the combination of grazing intensity, rainfall and competition delay, by analyzing conditions of the occurrence of Turing instability. In particular, we propose a new theoretical method to deal with the Turing instability of delayed diffusion system when the system exists a zero eigenvalue and the corresponding transversity condition is not zero. Using the theory of dynamics, we can show: (i) a bistable region in which vegetation-existence and vegetation-extinction states coexist for relatively large grazing rate, in the corresponding ordinary differential system, while the vegetation-existence state is replaced by vegetation pattern state for non-delayed diffusion system, and the range of bistable region is extended as the increase of rainfall; (ii) spatial distribution structure and spatially mean density of vegetation patterns reveal that degradation of the vegetation becomes more and more obvious with the increase of grazing intensity; (iii) the bistable region corresponding to the delayed diffusion system is narrowed, and vegetation system undergoes regime shift from the pattern state to bare-soil state before reaching the threshold of grazing rate. Overall, this study yields a new theoretical perspective for pattern dynamics of delayed reaction–diffusion equation, and provides valuable insights into the study of vegetation system in grazing ecosystem.