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We perform and detail the steady-state and Hopf bifurcation to present the formation of nonhomogeneous stationary or time-periodic solutions from the homogeneous one. In particular, the steady-state reduces to that of a two-component PDE system, independent of the initial condition and the system parameters. Moreover, we analyze the stability of these nontrivial patterns and our results find a wave mode selection mechanism, which in turn suggests that a large interval length admits time-periodic solutions, while a small interval length most likely drives spatially monotonous nonhomogeneous stationary solutions. Finally, we provide numerical experiments to illustrate and support our theoretical findings. <\/jats:p>","DOI":"10.1142\/s0218127424500974","type":"journal-article","created":{"date-parts":[[2024,6,28]],"date-time":"2024-06-28T08:07:50Z","timestamp":1719562070000},"source":"Crossref","is-referenced-by-count":0,"title":["Bifurcation Analysis and Synapse Pattern Formation with Bidirectional Motor-Mediated Transports"],"prefix":"10.1142","volume":"34","author":[{"given":"Qin","family":"Gu","sequence":"first","affiliation":[{"name":"Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, P. R. China"}]},{"given":"Siyao","family":"Jiang","sequence":"additional","affiliation":[{"name":"Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, P. R. 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