Fractional Vegetation-Water Model in Arid and Semi-Arid Environments: Pattern Formation and Numerical Simulations

Abstract

In this paper, we present a new fractional mathematical model to describe the dynamics and the interaction between plants and water in arid and semi-arid environments with and without slope. By the Caputo fractional operator, the model allows for simulating the phenomena related to the vegetation migration, which occur in domains with different slopes. By assuming the fractional parameter linked to the slope of the domain, the new fractional model represents a connection between the Klausmeier model, where water advection occurs, to the Klausmeier-Gray-Scott model, where water diffuses. The proposed model describes an anomalous physical phenomenon that changes as the fractional parameter changes, modelling an anomalous water advection. An analytical study of the stability of the Hopf bifurcation demonstrates that the migration speed results to be a function of the fractional parameter, confirming the connection between the fractional parameter and the slope of the domain. The oscillatory solutions and the vegetation pattern formation, obtained numerically, validate the analytical results and confirm the reliability and efficiency of the fractional formulation of the considered model.

Figures (11)

• Figure 3: Test 1. Numerical solutions of the reduced models: Top frames: numerical solution $U_j$. Bottom frames: numerical solution $W_j$. Left frames: the solution $U_j$ of the reduced KL model. Right frames: the solution $W_j$ of reduced KL--GS model.
• Figure 4: Test 1. Numerical solutions: Top frames: numerical solution $u_j^n$. Bottom frames: numerical solution $w_j^n$. Left frames: numerical solutions $u_j^n$ and $w_j^n$ of KL model. Right frames: numerical solutions $u_j^n$ and $w_j^n$ of KL--GS model.
• Figure 5: Migration speed $c^{HB}_\alpha$, depending on $\alpha$, given by (\ref{['condHBF']}) for $\nu=460$. Right frame: zoom of the left frame.
• Figure 6: Test 1. The numerical solutions $U_j$ of $U(z)$ and $u_j^n$ of $u(x,t)$ obtained for $a=2$, $m=0.45$, $\nu = 460$. Top: $\alpha=0.33$ and $c_\alpha= 2.895$. Bottom: $\alpha=0.66$ and $c_\alpha= 0.804$.
• Figure 7: Test 2. Numerical solutions $U_j$ of the concentration of plant $U(z_j)$. Left frame: reduced KL model. Right frame: reduced KL--GS model.