Bifurcation analysis of an extended Klausmeier-Gray-Scott model with infiltration delay

Given that the precipitation infiltrates into the soil at the cost of time, we establish an extended vegetation-water model with infiltration delay based on generalized Klausmeier-Gray-Scott model. Moreover, considering that the precipitation in a certain region is basically balanced, the vegetation mortality is selected as the key parameter. By dynamical theory analysis, the number of vegetation-existence steady states varies from two to one, and then to none with the increase of vegetation mortality, but the bare-soil state always exists. The infiltration delay of water and spatial factor have no effect on stable feature of the bare-soil state, but it induces occurrence of the periodic oscillation pattern of nonspatial and spatial systems at the vegetation-existence steady state compared with ODE system. Furthermore, the introduction of spatial factors makes the saddle node stable. The increase of relevant spatial factors can enlarge the bistable region, namely, coexistence of the vegetation-existence and bare-soil steady states, which increases the possibility of sustainable existence of the vegetation. The results obtained are helpful to understand significant influence of vegetation mortality on vegetation system, and then provide some theoretical basis for the study of interaction between the vegetation and water and evolution development of the vegetation system.