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It provides necessary and sufficient conditions for the emergence of Hopf, Turing, and Turing\u2013Hopf bifurcations. Utilizing the normal form of the Turing\u2013Hopf bifurcation, the spatiotemporal dynamics close to the bifurcation point are categorized into six types, encompassing spatially inhomogeneous and homogeneous periodic solutions, as well as spatially homogeneous and inhomogeneous steady states, along with their transitions. Notably, it\u2019s observed that the systems may lack stable spatially inhomogeneous periodic solutions under specific parameters, despite the emergence of Turing\u2013Hopf bifurcation. These findings are supported by numerical simulations. <\/jats:p>","DOI":"10.1142\/s0218127424501621","type":"journal-article","created":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T09:55:37Z","timestamp":1726653337000},"source":"Crossref","is-referenced-by-count":2,"title":["Turing\u2013Hopf Bifurcation Coinduced by Diffusion and Delay in Gierer\u2013Meinhardt Systems"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-3313-1854","authenticated-orcid":false,"given":"Xiangyi","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8696-5182","authenticated-orcid":false,"given":"Jinliang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. 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