{
  "id": "2412.20097",
  "version": "v1",
  "published": "2024-12-28T09:39:04.000Z",
  "updated": "2024-12-28T09:39:04.000Z",
  "title": "Parameter spaces for cross-diffusive-driven instability in a reaction-diffusion system on an annular domain",
  "authors": [
    "Gulsemay Yigit",
    "Wakil Sarfaraz",
    "Raquel Barreira",
    "Anotida Madzvamuse"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this work, the influence of geometry and domain size on spatiotemporal pattern formation is investigated to establish parameter spaces for a cross-diffusive reaction-diffusion model on an annulus. By applying linear stability theory, we derive conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on non-convex geometries e.g. annulus. Hence, the key research methodology and outcomes of our studies include: a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction-diffusion system; a linear stability analysis to characterise the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through lower and upper bounds of the domain size; the full classification of the model parameters, and a demonstration of how cross-diffusion relaxes the general conditions for the reaction-diffusion system to exhibit pattern formation. To validate theoretical findings and predictions, we employ the finite element method to reveal spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction-diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumours. Specifically, the cross-section of an actively invasive region in a hypoxic tumour can be effectively approximated by an annulus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-28T09:39:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reaction-diffusion system",
      "parameter spaces",
      "cross-diffusive-driven instability",
      "domain size",
      "pattern formation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}