{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T23:02:06Z","timestamp":1780700526123,"version":"3.54.1"},"reference-count":35,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,1,23]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Turing models of pattern formation provide insight into an intriguing question in developmental biology, like how nature exhibits various structures, shapes, and organized patterns. These natural patterns include the pattern and texture on a desert dune, spots and stripes on the skin of various animals, the growth of a body from a single cell, and the formation of fingerprints. The current work emphasized stability analysis and parameter settings to obtain diverse patterns in nature. As growth is an inevitable continuous process, it is responsible for producing different structures and patterns in living beings. The proposed numerical framework and simulation exhibit realistic natural patterns using reaction-diffusion (RD) models driven by Turing-type instability in the Gray-Scott model. The study proposes a parameter space for Turing patterns using stability analysis. The implemented mathematical model discretizes the continuous partial differential equations into their discrete counterpart by employing a finite-difference scheme. The presented framework combines state-of-the-art spatial and temporal discretization techniques together with stability analysis to mirror stable Turing-type patterns. The proposed numerical scheme is robust, efficient, accurate, and capable of exhibiting diverse biological patterns for the set of parameters in the Turing space, which are validated through stability analysis.<\/jats:p>","DOI":"10.1515\/comp-2025-0052","type":"journal-article","created":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T22:24:46Z","timestamp":1780698286000},"source":"Crossref","is-referenced-by-count":0,"title":["Numerical framework and\u00a0stability analysis of\u00a0the Gray-Scott model"],"prefix":"10.1515","volume":"16","author":[{"given":"Ramzan","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of General Education , University of Doha for Science and Technology , 24449 , Doha , Qatar"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Attique","family":"Ahmed","sequence":"additional","affiliation":[{"name":"School of Arts and Sciences , University of Central Asia , Naryn Campus, 722918 , Bishkek , Kyrgyz Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2026,6,8]]},"reference":[{"key":"2026060522244340015_j_comp-2025-0052_ref_001","doi-asserted-by":"crossref","unstructured":"A. M. Turing, \u201cThe chemical basis of morphogenesis,\u201d Philos. Trans. R. Soc. Lond. B, Biol. Sci., vol. 237, pp. 37\u201372, 1952.","DOI":"10.1098\/rstb.1952.0012"},{"key":"2026060522244340015_j_comp-2025-0052_ref_003","doi-asserted-by":"crossref","unstructured":"M. D. G. Maliheiros, H. Fensterseifer, and M. Walter, \u201cThe leopard never changes its spots: Realistic pigmentation pattern formation by coupling tissue growth with reaction-diffusion,\u201d ACM Trans. Graphics, vol.\u00a039, no.\u00a04, 2020.","DOI":"10.1145\/3386569.3392478"},{"key":"2026060522244340015_j_comp-2025-0052_ref_002","unstructured":"A. Tschentscher, \u201cAfrican leopard near Okevi waterhole, Estoha, Namibia,\u201d 2019."},{"key":"2026060522244340015_j_comp-2025-0052_ref_004","doi-asserted-by":"crossref","unstructured":"Q. Ouyang and H. L. Swinney, \u201cTransition from a uniform state to hexagonal and striped Turing patterns,\u201d Nature, vol. 352, no. 6336, pp. 610\u2013612, 1991. https:\/\/doi.org\/10.1038\/352610a0.","DOI":"10.1038\/352610a0"},{"key":"2026060522244340015_j_comp-2025-0052_ref_005","doi-asserted-by":"crossref","unstructured":"A. Gierer and H. Meinhardt, \u201cA theory of biological pattern formation,\u201d Kybernetik, vol.\u00a012, no.\u00a01, pp.\u00a030\u201339, 1972, https:\/\/doi.org\/10.1007\/BF00289234.","DOI":"10.1007\/BF00289234"},{"key":"2026060522244340015_j_comp-2025-0052_ref_006","doi-asserted-by":"crossref","unstructured":"J. E. Pearson, \u201cComplex patterns in a simple system,\u201d Science, vol. 261, pp. 189\u2013192, 1993.","DOI":"10.1126\/science.261.5118.189"},{"key":"2026060522244340015_j_comp-2025-0052_ref_007","unstructured":"R. P. Munafo, \u201cStable localized moving patterns in the 2-D Gray-Scott model,\u201d 2014. Available at: https:\/\/arxiv.org\/abs\/1501.01990v1."},{"key":"2026060522244340015_j_comp-2025-0052_ref_008","unstructured":"Muhammad, A. Yau, M. U. Adehi, and M. Garba, \u201cSpot patterns in Gray Scott model with application to epidemic control,\u201d Int. J. Math. Model. Comput., vol.\u00a04, no.\u00a04, pp.\u00a0389\u2013400, 2014."},{"key":"2026060522244340015_j_comp-2025-0052_ref_009","doi-asserted-by":"crossref","unstructured":"S. Hasnain, S. Bashir, P. Linker, and M. Saqib, \u201cEfficiency of numerical schemes for two-dimensional Gray Scott model,\u201d AIP Adv., vol.\u00a09, no.\u00a010, 2019, https:\/\/doi.org\/10.1063\/1.5095517.","DOI":"10.1063\/1.5095517"},{"key":"2026060522244340015_j_comp-2025-0052_ref_010","doi-asserted-by":"crossref","unstructured":"A. A. A. Amin and D. S. Mashat, \u201cAnalysis of Gray Scott\u2019s model numerically,\u201d Am. J. Comput. Math., vol. 11, no. 4, pp. 273\u2013288, 2021. https:\/\/doi.org\/10.4236\/ajcm.2021.114018.","DOI":"10.4236\/ajcm.2021.114018"},{"key":"2026060522244340015_j_comp-2025-0052_ref_011","doi-asserted-by":"crossref","unstructured":"U. Hayat, R. Ali, S. Shaiq, and A. Shahzad, \u201cA numerical study on thin film flow and heat transfer enhancement for copper nanoparticles dispersed in ethylene glycol,\u201d Rev. Adv. Mater. Sci., vol. 62, no. 1, 2023, Art. no. 20220320, https:\/\/doi.org\/10.1515\/rams-2022-0320.","DOI":"10.1515\/rams-2022-0320"},{"key":"2026060522244340015_j_comp-2025-0052_ref_012","unstructured":"R. Ali, \u201cNumerical techniques for the simulation of PDEs on surfaces for biomathematical problems,\u201d Dissertation, Dortmund, Technische Universit\u00e4t, 2016."},{"key":"2026060522244340015_j_comp-2025-0052_ref_013","doi-asserted-by":"crossref","unstructured":"K. U. Rehman, Q. M. Al-Mdallal, R. Mahmood, M. Y. Malik, and R. Ali, \u201cOn inclined heated square obstacle in a liquid stream carried by partially heated channel: Finite element analysis,\u201d Case Stud. Therm. Eng., vol. 15, 2019, Art. no. 100532. https:\/\/doi.org\/10.1016\/j.csite.2019.100532.","DOI":"10.1016\/j.csite.2019.100532"},{"key":"2026060522244340015_j_comp-2025-0052_ref_014","doi-asserted-by":"crossref","unstructured":"A. Shahzad et al.., \u201cNumerical study of axisymmetric flow and heat transfer in a liquid film over an unsteady radially stretching surface,\u201d Math. Probl. Eng., vol. 2020, no. 1, 2020, Art. no. 6737243.","DOI":"10.1155\/2020\/6737243"},{"key":"2026060522244340015_j_comp-2025-0052_ref_015","doi-asserted-by":"crossref","unstructured":"A. Sokolov, R. Ali, and S. Turek, \u201cAn AFC-stabilized implicit finite element method for partial differential equations on evolving-in-time surfaces,\u201d J. Comput. Appl. Math., vol. 289, no. 1, pp. 101\u2013115, 2015. https:\/\/doi.org\/10.1016\/j.cam.2015.03.002.","DOI":"10.1016\/j.cam.2015.03.002"},{"key":"2026060522244340015_j_comp-2025-0052_ref_016","unstructured":"L. Ringham, \u201cModelling natural phenomenon with reaction-diffusion,\u201d Canada, Uni of Calgary, 2020."},{"key":"2026060522244340015_j_comp-2025-0052_ref_017","unstructured":"T. Leppanen, \u201cComputational studies of pattern formation in Turing systems,\u201d Finland, Helsinki University of Technology Laboratory of Computational Engineering Publications, 2004."},{"key":"2026060522244340015_j_comp-2025-0052_ref_018","unstructured":"F. Buric, \u201cPattern formation and chemical evolution in extended Gray-Scott models,\u201d Gothenburg, Chalmers University of Technology, 2014."},{"key":"2026060522244340015_j_comp-2025-0052_ref_019","doi-asserted-by":"crossref","unstructured":"A. Al Bayati, S. A. Manaa, and A. M. Al-Rozbayani, \u201cStability analysis of Gray-Scott model in one-dimension,\u201d Rafidain J. Comput. Sci. Math., vol. 5, no. 2, 2008. https:\/\/doi.org\/10.33899\/csmj.2008.163972.","DOI":"10.33899\/csmj.2008.163972"},{"key":"2026060522244340015_j_comp-2025-0052_ref_020","doi-asserted-by":"crossref","unstructured":"J. Mcgough and K. Riley, \u201cPattern formation in the Gray-Scott model,\u201d Nonlinear Anal. Real World Appl., vol. 5, no. 1, pp. 105\u2013121, 2003. https:\/\/doi.org\/10.1016\/s1468-1218(03)00020-8.","DOI":"10.1016\/S1468-1218(03)00020-8"},{"key":"2026060522244340015_j_comp-2025-0052_ref_021","doi-asserted-by":"crossref","unstructured":"J. Murray, Mathematical Biology II. Spatial Models and Biomedical Applications, New York, USA, Springer-Verlag New York, 2003.","DOI":"10.1007\/b98869"},{"key":"2026060522244340015_j_comp-2025-0052_ref_022","doi-asserted-by":"crossref","unstructured":"V. Orlov and M. Gasanov, \u201cExistence and uniqueness theorem for a solution to a class of a third-order nonlinear differential equation in the domain of analyticity,\u201d Axioms, vol. 11, pp. 1\u201314, 2022.","DOI":"10.3390\/axioms11050203"},{"key":"2026060522244340015_j_comp-2025-0052_ref_023","doi-asserted-by":"crossref","unstructured":"V. Orlov and M. Gasanov, \u201cAnalytic approximate solution in the neighborhood of a moving singular point of a class of nonlinear equations,\u201d Axioms, vol. 11, pp. 1\u201318, 2022.","DOI":"10.3390\/axioms11110637"},{"key":"2026060522244340015_j_comp-2025-0052_ref_024","doi-asserted-by":"crossref","unstructured":"H. Meinhardt, \u201cPigment patterns on sea shells \u2013 a beautiful case of biological pattern formation,\u201d in Growth, Dissolution and Pattern Formation in Geosystems, B. Jamtveit and P. Meakin, Eds., Dordrecht, Springer, 1999.","DOI":"10.1007\/978-94-015-9179-9_10"},{"key":"2026060522244340015_j_comp-2025-0052_ref_025","doi-asserted-by":"crossref","unstructured":"A. Goswami, \u201cNon equilibrium dynamics and crystallization pattern formation in mollusk shells,\u201d 2022. Available at: https:\/\/ssrn.com\/abstract=5015663.","DOI":"10.2139\/ssrn.4310000"},{"key":"2026060522244340015_j_comp-2025-0052_ref_026","doi-asserted-by":"crossref","unstructured":"R. Chirat, D. E. Moulton, and A. Goriely, \u201cMechanical basis of morphogenesis and convergent evolution of spiny seashells,\u201d Proc. Natl. Acad. Sci. U. S. A., vol.\u00a0110, no.\u00a015, pp.\u00a06015\u20136020, 2013, https:\/\/doi.org\/10.1073\/pnas.1220443110.","DOI":"10.1073\/pnas.1220443110"},{"key":"2026060522244340015_j_comp-2025-0052_ref_027","doi-asserted-by":"crossref","unstructured":"E. Siero and E. E. Deinum, \u201cThe Turing heritage for plant biology: All spots and stripes?\u201d Quant. Plant Biol., vol.\u00a06, p.\u00a0e1, 2025, https:\/\/doi.org\/10.1017\/qpb.2024.16.","DOI":"10.1017\/qpb.2024.16"},{"key":"2026060522244340015_j_comp-2025-0052_ref_028","doi-asserted-by":"crossref","unstructured":"E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak, and E. Meron, \u201cEcosystem engineers: From pattern formation to habitat creation,\u201d Phys. Rev. Lett., vol.\u00a093, pp.\u00a01\u20134, 2004, https:\/\/doi.org\/10.1103\/PhysRevLett.93.098105.","DOI":"10.1103\/PhysRevLett.93.098105"},{"key":"2026060522244340015_j_comp-2025-0052_ref_029","doi-asserted-by":"crossref","unstructured":"J. D. Murray, P. K. Maini, and R. T. Tranquillo, \u201cMechanochemical models for generating biological pattern and form in development,\u201d Phys. Rep., vol.\u00a0171, no.\u00a02, pp.\u00a059\u201384, 1988, https:\/\/doi.org\/10.1016\/0370-1573(88)90003-8.","DOI":"10.1016\/0370-1573(88)90003-8"},{"key":"2026060522244340015_j_comp-2025-0052_ref_030","doi-asserted-by":"crossref","unstructured":"Y. Nagashima, S. Tsugawa, A. Mochizuki, T. Sasaki, H. Fukuda, and Y. Oda, \u201cA Rho-based reaction-diffusion systems governs cell wall patterning in metaxylem vessels,\u201d Sci. Rep., vol.\u00a08, no.\u00a01, p.\u00a011542, 2018, https:\/\/doi.org\/10.1038\/s41598-018-29543-y.","DOI":"10.1038\/s41598-018-29543-y"},{"key":"2026060522244340015_j_comp-2025-0052_ref_031","doi-asserted-by":"crossref","unstructured":"R. E. Goldstein, D. J. Muraki, and D. M. Petrich, \u201cInterface proliferation and the growth of labyrinths in a reaction-diffusion systems,\u201d Phys. Rev. E, vol. 53, no. 4, pp. 3933\u20133957, 1996.","DOI":"10.1103\/PhysRevE.53.3933"},{"key":"2026060522244340015_j_comp-2025-0052_ref_032","doi-asserted-by":"crossref","unstructured":"T. Y.-C. Tsai and D. Pinheiro, \u201cCoping with uncertainty: Challenges for robust pattern formation in dynamical tissues,\u201d Semin. Cell Dev. Biol., vol. 175, no. 1, 2025. https:\/\/doi.org\/10.1016\/j.semcdb.2025.103629.","DOI":"10.1016\/j.semcdb.2025.103629"},{"key":"2026060522244340015_j_comp-2025-0052_ref_033","doi-asserted-by":"crossref","unstructured":"A. Kicheva and J. Briscoe, \u201cControl of tissue development by morphogens,\u201d Annu. Rev. Cell Dev. Biol., vol. 39, no. 1, pp. 91\u2013121, 2023. https:\/\/doi.org\/10.1146\/annurev-cellbio-020823-011522.","DOI":"10.1146\/annurev-cellbio-020823-011522"},{"key":"2026060522244340015_j_comp-2025-0052_ref_034","doi-asserted-by":"crossref","unstructured":"T. Fulton et al.., \u201cAxis specification in zebrafish is robust to cell mixing and reveals a regulation of pattern formation by morphogenesis,\u201d Curr. Biol., vol. 30, no. 15, pp. 2984\u20132994, 2020. https:\/\/doi.org\/10.1016\/j.cub.2020.07.022.","DOI":"10.1016\/j.cub.2020.05.048"},{"key":"2026060522244340015_j_comp-2025-0052_ref_035","doi-asserted-by":"crossref","unstructured":"J. Buceta and L. Guitou, \u201cDevelopmental pattern formation: Spanish contributions from a biophysical perspective,\u201d Biophysica, vol.\u00a03, no.\u00a02, pp.\u00a0335\u2013347, 2023, https:\/\/doi.org\/10.3390\/biophysica3020022.","DOI":"10.3390\/biophysica3020022"}],"container-title":["Open Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/comp-2025-0052\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/comp-2025-0052\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T22:25:01Z","timestamp":1780698301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/comp-2025-0052\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,1]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,6,8]]},"published-print":{"date-parts":[[2026,1,23]]}},"alternative-id":["10.1515\/comp-2025-0052"],"URL":"https:\/\/doi.org\/10.1515\/comp-2025-0052","relation":{},"ISSN":["2299-1093"],"issn-type":[{"value":"2299-1093","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,1]]},"article-number":"20250052"}}