Bifurcation and pattern dynamic behavior in a fractional-order ecosystem model

Hai Yan Zhang; Wei Zhang; Hao Lu Zhang; Hai Yan Zhang; Wei Zhang; Hao Lu Zhang

doi:10.3934/nhm.2026008

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 183-197. doi: 10.3934/nhm.2026008

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Bifurcation and pattern dynamic behavior in a fractional-order ecosystem model

1.
School of Physics and Electronic Information Engineering, Jining Normal University, Ulanqab, Inner Mongolia 012000, China
2.
Institute of Economics and Management, Jining Normal University, Ulanqab, Inner Mongolia 012000, China
3.
College of Civil Engineering, Inner Mongolia University of Technology, Hohhot 010051, China

Received: 24 November 2025 Revised: 15 January 2026 Accepted: 22 January 2026 Published: 27 January 2026

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.
- high-precision numerical method,
- stability analysis,
- bifurcation analysis
Citation: Hai Yan Zhang, Wei Zhang, Hao Lu Zhang. Bifurcation and pattern dynamic behavior in a fractional-order ecosystem model[J]. Networks and Heterogeneous Media, 2026, 21(1): 183-197. doi: 10.3934/nhm.2026008

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Abstract

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.

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