Stability and bifurcation analysis of a time-order fractional model for water-plants: Implications for vegetation pattern formation

The water-plant model is a significant tool for studying vegetation patterns. It helps researchers understand the interactions between water availability and plant growth, which are crucial for analyzing ecological dynamics and predicting changes in vegetation due to environmental factors. However, there has been limited research on the memory effect associated with the water-plant model. This paper investigates a fractional-order water-plant model with cross-diffusion, in which the fractional order signifies the memory effect. First, we examine the conditions for the equilibrium point in a spatially homogeneous model, followed by an analysis of the model’s linear stability and the existence of Hopf bifurcation. Subsequently, we analyze the stability of spatiotemporal models incorporating cross-diffusion, along with the presence of Turing bifurcation, Hopf bifurcation, and Turing–Hopf bifurcation. Finally, we present several numerical simulations to validate the theoretical results. The results indicate that the Hopf bifurcation parameters increase with the fractional order $\tau$, leading to a larger parameter space for Hopf instability. As the fractional order $\tau$ increases, it results in a smaller parameter space for Turing instability and a reduced parameter space for stability. This indicates that an increase in the fractional order $\tau$ accelerates the transition of vegetation patterns, thereby affecting the stability of the system.