Analysis and simulation of the effect of fractional parameters on dynamic behavior for a fractional-in-time three-species reaction-diffusion model

Hao Lu Zhang; Xiao Yu Li; Zhi Yuan Li; Hao Lu Zhang; Xiao Yu Li; Zhi Yuan Li

doi:10.3934/nhm.2026020

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 426-445. doi: 10.3934/nhm.2026020

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Research article

Analysis and simulation of the effect of fractional parameters on dynamic behavior for a fractional-in-time three-species reaction-diffusion model

Hao Lu Zhang ^{1,2
,
,},
Xiao Yu Li ^{3
,
,},
Zhi Yuan Li ^{3
,
,}

1.
College of Civil Engineering, Inner Mongolia University of Technology, Hohhot 010051, China
2.
Production and Operation Center, CCCC Comprehensive Planning and Design Institute Co. Ltd., Beijing 100024, China
3.
College of Intelligent Science and Technology, Inner Mongolia University of Technology, Hohhot 010080, China

Received: 24 January 2026 Revised: 03 March 2026 Accepted: 16 March 2026 Published: 25 March 2026

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.
- a numerical method,
- fractional reaction-diffusion,
- pattern formation,
- short-memory principle,
- stability analysis
Citation: Hao Lu Zhang, Xiao Yu Li, Zhi Yuan Li. Analysis and simulation of the effect of fractional parameters on dynamic behavior for a fractional-in-time three-species reaction-diffusion model[J]. Networks and Heterogeneous Media, 2026, 21(2): 426-445. doi: 10.3934/nhm.2026020

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Abstract

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.

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