Spatiotemporal dynamics and bifurcation analysis of a time-delayed water-vegetation model: insights into oscillation patterns

In arid ecosystems, water supply significantly influences vegetation growth, but this influence is not instantaneous. Biogeochemical processes, including delayed precipitation infiltration into deep soil layers and slow water uptake by plant roots, significantly affect vegetation spatial distribution and resilience. Although existing vegetation-water interaction models can describe the pattern formation induced by diffusion mechanisms, they rarely explore the crucial role of time-delay effects in system destabilization and oscillatory behaviors. To clarify how delayed water supply drives critical transitions in arid ecosystems, we introduce a discrete time-delay term into a classical reaction-diffusion vegetation model, constructing a vegetation-water dynamical system with time-delay coupling. Through mathematical analysis, we prove the existence, uniqueness, and boundedness of the system’s solutions. Stability analysis shows that neither diffusion nor time delay has a significant impact on the stability of the bare-soil equilibrium point, while the stability of the vegetation-water coexistence equilibrium point is solely regulated by time delay. By choosing the time delay \(τ \) as the bifurcation parameter, we theoretically derive the critical threshold for Hopf bifurcation and analyze the direction and stability of bifurcating periodic solutions via center manifold reduction. Finally, numerical simulations are performed to validate the theoretical analysis and support the predictions.