Delay-driven spatial patterns in a network-organized semiarid vegetation model

Spatiotemporal dynamics of vegetation models are traditionally investigated on a spatially continuous domain. The increasingly fragmented agricultural landscape necessitates network-organized models. In this paper, we develop a semiarid vegetation model to describe the spatiotemporal dynamics between plant and water on a network accounting for fragmented habitats which are connected by dispersal of seeds. Time delay is introduced to account for time lag in water uptake. By linear stability analysis we show that the coexistence equilibrium is asymptotically stable in the absence of time delay, but loses its stability via Hopf bifurcation when time delay is beyond a threshold. Applying the center manifold theory, we derive the explicit formulas that determine the stability and direction of the Hopf bifurcation. Numerical simulations demonstrate the emergence of spatial patterns on a network. Comparing our network-organized model to other model variants, we find that increasing landscape fragmentation is more likely to generate the variation of plant density among different habitats.