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Fractional Gierer-Meinhardt system with cross-diffusion: Pattern analysis in three-dimensional space

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  • He, Haoming
  • Xiao, Min
  • Cao, Jinde
  • Zhou, Ying
  • Park, Ju H.
  • Rutkowski, Leszek

Abstract

As a kind of classical reaction-diffusion system to explain the formation of biological patterns, the Gierer-Meinhardt system has received significant attention. Works related to the Turing instability and two-dimensional patterns has also revealed its rich dynamic properties. However, considering its background of depicting organ formation, the two-dimensional Turing pattern seems to have a slight lack of realistic information that can be reflected. Therefore, in this paper, we further extend the diffusion space to three dimension in an attempt to reveal a more complete dynamic behavior of the Gierer-Meinhardt system. Meanwhile, we introduce the concept of fractional order Laplacian operator to refine the model. We choose the cross diffusion coefficient d12 as the threshold for inducing Turing instability and calculate the three-dimensional amplitude equation under the body-centered cubic (BCC) spatial lattice structure. Based on the distribution of characteristic roots of the amplitude equation, the stable domain of the three-dimensional patterns is established, and the rich evolution processes of the patterns are demonstrated in the simulation.

Suggested Citation

  • He, Haoming & Xiao, Min & Cao, Jinde & Zhou, Ying & Park, Ju H. & Rutkowski, Leszek, 2026. "Fractional Gierer-Meinhardt system with cross-diffusion: Pattern analysis in three-dimensional space," Applied Mathematics and Computation, Elsevier, vol. 513(C).
    • Handle: RePEc:eee:apmaco:v:513:y:2026:i:c:s0096300325005314
    DOI: 10.1016/j.amc.2025.129806
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