Hao Lu Zhang

This paper investigates the effect of fractional parameters on chaos and pattern formation in the fractional-in-time complex Swift–Hohenberg model. A novel higher-order numerical scheme is developed, combining a [ ]th-order Grünwald–Letnikov discretization for the Caputo time-fractional derivative with a nine-point finite difference approximation for spatial operators, achieving fourth-order spatial accuracy and incorporating a short-memory principle for computational efficiency. Linear stability and Turing bifurcation analysis reveal that the fractional order [ ] nonmonotonically modulates both the complexity measures and the Turing instability threshold, enabling multiple stability switches and transitions to globally unstable regimes. Numerical simulations demonstrate that [ ] acts as a bifurcation parameter, inducing complex pattern selection and chaotic dynamics, with the proposed scheme effectively capturing rich spatiotemporal behaviors across various parameter regimes.

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.

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.

This paper investigates the dynamic behavior of a fractional-order reaction-diffusion system for vegetation pattern formation, incorporating interactions between plant biomass, soil water, and salt concentration. The model utilizes Grünwald–Letnikov fractional derivatives to capture memory effects and anomalous diffusion. A novel high-order numerical scheme is developed, featuring a high-accuracy, short-memory time discretization with a nine-point finite difference method in space to enhance stability. Bifurcation analysis is performed to determine equilibrium stability and identify Turing instability thresholds. Numerical simulations illustrate the emergence of diverse vegetation patterns, highlighting how different fractional orders influence spatiotemporal dynamics. Overall, the proposed framework provides an effective computational tool for analyzing complex fractional ecological systems.

This paper focuses on the dynamical behavior of the fractional-order Hastings–Powell (HP) model. First, a high-precision numerical method is given and the limitations of the method are analyzed. Then, the stability of the equilibrium point of the model is analyzed, numerical simulations of the model with different orders and initial values are carried out, and a comparison is made between the high-precision numerical method and other methods. Finally, the Turing instability analysis and the weak nonlinear analysis of the HP model with diffusion terms are carried out and numerical simulations are performed in a nine-point difference format. The consistency between the numerical results and the numerical simulations is also verified.

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.