In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order $\alpha$, and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, $\beta$, changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns.

This paper investigates a three-species reaction-diffusion model with fractional-order derivatives and proposes an innovative numerical method for its simulation. The method integrates an optimized Grünwald-Letnikov discretization scheme, enhanced by a short-memory principle, with a high-order accurate nine-point compact difference scheme, enabling an efficient and stable solution of fractional operators. Rigorous convergence and stability analyses confirm the theoretical reliability of the algorithm. Through stability and Turing bifurcation analyses, the study systematically reveals the regulatory mechanism of the fractional-order exponent on the dynamic behavior of the system. The numerical results demonstrate that the present method accurately captures the effect of fractional derivatives on the formation process of spatial patterns.

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown.

This paper presents a novel high-order numerical method. The proposed scheme utilizes polynomial generating functions to achieve p order accuracy in time for the Grünwald–Letnikov fractional derivatives, while maintaining second-order spatial accuracy. By incorporating a short-memory principle, the method remains computationally efficient for long-time simulations. The authors rigorously analyze the stability of equilibrium points for the fractional vegetation–water model and perform a weakly nonlinear analysis to derive amplitude equations. Convergence analysis confirms the scheme’s consistency, stability, and convergence. Numerical simulations demonstrate the method’s effectiveness in exploring how different fractional derivative orders influence system dynamics and pattern formation, providing a robust tool for studying complex fractional systems in theoretical ecology.