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Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction\u2013diffusion\u2013advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator\u2013prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations. <\/jats:p>","DOI":"10.1142\/s0218127424501554","type":"journal-article","created":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T04:06:22Z","timestamp":1725854782000},"source":"Crossref","is-referenced-by-count":1,"title":["Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction\u2013Diffusion\u2013Advection System"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-8990-5214","authenticated-orcid":false,"given":"Hongfan","family":"Lu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong 264209, P.\u00a0R.\u00a0China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9293-4238","authenticated-orcid":false,"given":"Yuxiao","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong 264209, P.\u00a0R.\u00a0China"}]}],"member":"219","published-online":{"date-parts":[[2024,9,6]]},"reference":[{"key":"S0218127424501554BIB001","doi-asserted-by":"publisher","DOI":"10.1142\/S021812741950189X"},{"key":"S0218127424501554BIB002","doi-asserted-by":"publisher","DOI":"10.3390\/math10091583"},{"key":"S0218127424501554BIB003","doi-asserted-by":"publisher","DOI":"10.5206\/mase\/14112"},{"key":"S0218127424501554BIB004","doi-asserted-by":"publisher","DOI":"10.3934\/mbe.2023543"},{"key":"S0218127424501554BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2024.04.002"},{"key":"S0218127424501554BIB006","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127424500792"},{"key":"S0218127424501554BIB007","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021220117746"},{"key":"S0218127424501554BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2012.07.002"},{"key":"S0218127424501554BIB009","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127413500144"},{"key":"S0218127424501554BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106976"},{"key":"S0218127424501554BIB011","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127419500895"},{"key":"S0218127424501554BIB012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-00-02280-7"},{"key":"S0218127424501554BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-023-08502-x"},{"key":"S0218127424501554BIB014","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127414300304"},{"key":"S0218127424501554BIB015","doi-asserted-by":"publisher","DOI":"10.3934\/era.2023126"},{"key":"S0218127424501554BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2023.128504"},{"key":"S0218127424501554BIB017","doi-asserted-by":"publisher","DOI":"10.1007\/s00033-023-02077-8"},{"key":"S0218127424501554BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2023.128100"},{"key":"S0218127424501554BIB019","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3968-0"},{"key":"S0218127424501554BIB020","volume-title":"Introduction to Non-Linear Mechanics","author":"Krylov N. 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