{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T18:18:06Z","timestamp":1758478686669,"version":"3.37.3"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T00:00:00Z","timestamp":1685577600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T00:00:00Z","timestamp":1685577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,8]]},"DOI":"10.1007\/s12190-023-01881-9","type":"journal-article","created":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T16:02:17Z","timestamp":1685635337000},"page":"3261-3288","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Dynamics of a mutualistic model with advection and a free boundary in heterogeneous environment"],"prefix":"10.1007","volume":"69","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1163-2444","authenticated-orcid":false,"given":"Linfei","family":"Shi","sequence":"first","affiliation":[]},{"given":"Tianzhou","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Jinjin","family":"Mao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,1]]},"reference":[{"key":"1881_CR1","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1137\/090771089","volume":"42","author":"Y Du","year":"2010","unstructured":"Du, Y., Lin, Z.: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377\u2013405 (2010). https:\/\/doi.org\/10.1137\/090771089","journal-title":"SIAM J. Math. Anal."},{"issue":"4","key":"1881_CR2","doi-asserted-by":"publisher","first-page":"583","DOI":"10.3934\/nhm.2012.7.583","volume":"7","author":"G Bunting","year":"2012","unstructured":"Bunting, G., Du, Y., Krakowski, K.: Spreading speed revisited: analysis of a free boundary model. Netw. Heterog. Media. 7(4), 583\u2013603 (2012). https:\/\/doi.org\/10.3934\/nhm.2012.7.583","journal-title":"Netw. Heterog. Media."},{"issue":"2","key":"1881_CR3","first-page":"467","volume":"21","author":"Y Kaneko","year":"2011","unstructured":"Kaneko, Y., Yamada, Y.: A free boundary problem for a reaction-diffusion equation appearing in ecology. Adv. Math. Sci. Appl. 21(2), 467\u2013492 (2011)","journal-title":"Adv. Math. Sci. Appl."},{"issue":"5","key":"1881_CR4","doi-asserted-by":"publisher","first-page":"2007","DOI":"10.3934\/dcds.2013.33.2007","volume":"33","author":"R Peng","year":"2013","unstructured":"Peng, R., Zhao, X.: The diffusive logistic model with a free boundary and seasonal succession. Discrete Contin. Dyn. Syst. 33(5), 2007\u20132031 (2013). https:\/\/doi.org\/10.3934\/dcds.2013.33.2007","journal-title":"Discrete Contin. Dyn. Syst."},{"issue":"10","key":"1881_CR5","doi-asserted-by":"publisher","first-page":"2673","DOI":"10.4171\/JEMS\/568","volume":"17","author":"Y Du","year":"2015","unstructured":"Du, Y., Lou, B.: Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17(10), 2673\u20132724 (2015). https:\/\/doi.org\/10.4171\/JEMS\/568","journal-title":"J. Eur. Math. Soc."},{"key":"1881_CR6","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/j.jtbi.2008.12.032","volume":"258","author":"N Maidana","year":"2009","unstructured":"Maidana, N., Yang, H.: Spatial spreading of West Nile Virus described by traveling waves. J. Theoret. Biol. 258, 403\u2013417 (2009). https:\/\/doi.org\/10.1016\/j.jtbi.2008.12.032","journal-title":"J. Theoret. Biol."},{"key":"1881_CR7","volume-title":"The Effect of Intermediate Advection on Two Competing Species","author":"I Averill","year":"2012","unstructured":"Averill, I.: The Effect of Intermediate Advection on Two Competing Species. Ohio State University, Mathematics, Doctor of Philosophy (2012)"},{"key":"1881_CR8","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.aml.2014.05.015","volume":"37","author":"H Gu","year":"2014","unstructured":"Gu, H., Lin, Z., Lou, B.: Long time behavior of solutions of a diffusion-advection logistic model with free boundaries. Appl. Math. Lett. 37, 49\u201353 (2014). https:\/\/doi.org\/10.1016\/j.aml.2014.05.015","journal-title":"Appl. Math. Lett."},{"issue":"6","key":"1881_CR9","doi-asserted-by":"publisher","first-page":"1714","DOI":"10.1016\/j.jfa.2015.07.002","volume":"269","author":"H Gu","year":"2015","unstructured":"Gu, H., Lou, B., Zhou, M.: Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries. J. Funct. Anal. 269(6), 1714\u20131768 (2015). https:\/\/doi.org\/10.1016\/j.jfa.2015.07.002","journal-title":"J. Funct. Anal."},{"issue":"1","key":"1881_CR10","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/j.jmaa.2015.02.051","volume":"428","author":"Y Kaneko","year":"2015","unstructured":"Kaneko, Y., Matsuzawa, H.: Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations. J. Math. Anal. Appl. 428(1), 43\u201376 (2015). https:\/\/doi.org\/10.1016\/j.jmaa.2015.02.051","journal-title":"J. Math. Anal. Appl."},{"key":"1881_CR11","doi-asserted-by":"publisher","first-page":"3991","DOI":"10.1016\/j.jde.2015.11.002","volume":"260","author":"H Gu","year":"2016","unstructured":"Gu, H., Lou, B.: Spreading in advective environment modeled by a reaction diffusion equation with free boundaries. J. Differ. Eqn. 260, 3991\u20134015 (2016). https:\/\/doi.org\/10.1016\/j.jde.2015.11.002","journal-title":"J. Differ. Eqn."},{"issue":"2","key":"1881_CR12","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1016\/j.jfa.2018.03.006","volume":"275","author":"P Zhou","year":"2018","unstructured":"Zhou, P., Xiao, D.: Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system. J. Funct. Anal. 275(2), 356\u2013380 (2018). https:\/\/doi.org\/10.1016\/j.jfa.2018.03.006","journal-title":"J. Funct. Anal."},{"issue":"2","key":"1881_CR13","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1007\/s10884-017-9571-9","volume":"30","author":"Y Zhao","year":"2018","unstructured":"Zhao, Y., Wang, M.: A reaction-diffusion-advection equation with mixed and free boundary conditions. J. Dyn. Differ. Eqn. 30(2), 743\u2013777 (2018). https:\/\/doi.org\/10.1007\/s10884-017-9571-9","journal-title":"J. Dyn. Differ. Eqn."},{"issue":"7","key":"1881_CR14","doi-asserted-by":"publisher","first-page":"2089","DOI":"10.3934\/dcdsb.2015.20.2089","volume":"20","author":"M Li","year":"2015","unstructured":"Li, M., Lin, Z.: The spreading fronts in a mutualistic model with advection. Discrete Contin. Dyn. Syst. Ser. B 20(7), 2089\u20132105 (2015). https:\/\/doi.org\/10.3934\/dcdsb.2015.20.2089","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"issue":"2","key":"1881_CR15","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1093\/imamat\/hxv035","volume":"81","author":"Y Zhao","year":"2016","unstructured":"Zhao, Y., Wang, M.: Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients. IMA J. Appl. Math. 81(2), 255\u2013280 (2016). https:\/\/doi.org\/10.1093\/imamat\/hxv035","journal-title":"IMA J. Appl. Math."},{"key":"1881_CR16","doi-asserted-by":"publisher","first-page":"4336","DOI":"10.1016\/j.jde.2011.02.011","volume":"250","author":"Y Du","year":"2011","unstructured":"Du, Y., Guo, Z.: Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II. J. Differ. Eqn. 250, 4336\u20134366 (2011). https:\/\/doi.org\/10.1016\/j.jde.2011.02.011","journal-title":"J. Differ. Eqn."},{"key":"1881_CR17","doi-asserted-by":"publisher","unstructured":"Du Y., Liang X.: Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model. Ann. Inst. H. Poincar\u00e9 Anal. Non lin\u00e9aire 32(2), 279\u2013305 (2015). https:\/\/doi.org\/10.1016\/j.anihpc.2013.11.004","DOI":"10.1016\/j.anihpc.2013.11.004"},{"issue":"5","key":"1881_CR18","doi-asserted-by":"publisher","first-page":"2736","DOI":"10.1016\/j.jde.2017.04.013","volume":"263","author":"W Ding","year":"2017","unstructured":"Ding, W., Peng, R., Wei, L.: The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment. J. Differ. Eqn. 263(5), 2736\u20132779 (2017). https:\/\/doi.org\/10.1016\/j.jde.2017.04.013","journal-title":"J. Differ. Eqn."},{"issue":"9","key":"1881_CR19","doi-asserted-by":"publisher","first-page":"2089","DOI":"10.1016\/j.jfa.2013.07.016","volume":"265","author":"Y Du","year":"2013","unstructured":"Du, Y., Guo, Z., Peng, R.: A diffusive logistic model with a free boundary in time-periodic environment. J. Funct. Anal. 265(9), 2089\u20132142 (2013). https:\/\/doi.org\/10.1016\/j.jfa.2013.07.016","journal-title":"J. Funct. Anal."},{"issue":"2","key":"1881_CR20","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1016\/j.jfa.2015.10.014","volume":"270","author":"M Wang","year":"2016","unstructured":"Wang, M.: A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J. Funct. Anal. 270(2), 483\u2013508 (2016). https:\/\/doi.org\/10.1016\/j.jfa.2015.10.014","journal-title":"J. Funct. Anal."},{"issue":"6","key":"1881_CR21","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1016\/j.jde.2013.12.008","volume":"256","author":"P Zhou","year":"2014","unstructured":"Zhou, P., Xiao, D.: The diffusive logistic model with a free boundary in heterogeneous environment. J. Differ. Eqn. 256(6), 927\u20131954 (2014). https:\/\/doi.org\/10.1016\/j.jde.2013.12.008","journal-title":"J. Differ. Eqn."},{"issue":"1","key":"1881_CR22","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1016\/j.jmaa.2014.09.055","volume":"423","author":"J Wang","year":"2015","unstructured":"Wang, J., Zhang, L.: Invasion by an inferior or superior competitor: a diffusive competition model with a free boundary in a heterogeneous environment. J. Math. Anal. Appl. 423(1), 377\u2013398 (2015). https:\/\/doi.org\/10.1016\/j.jmaa.2014.09.055","journal-title":"J. Math. Anal. Appl."},{"key":"1881_CR23","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1016\/j.nonrwa.2013.10.003","volume":"16","author":"J Zhao","year":"2014","unstructured":"Zhao, J., Wang, M.: A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment. Nonlinear Anal. Real World Appl. 16, 250\u2013263 (2014). https:\/\/doi.org\/10.1016\/j.nonrwa.2013.10.003","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"2","key":"1881_CR24","doi-asserted-by":"publisher","first-page":"415","DOI":"10.3934\/dcdsb.2018179","volume":"24","author":"M Wang","year":"2019","unstructured":"Wang, M.: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete Contin. Dyn. Syst. Ser. B 24(2), 415\u2013421 (2019). https:\/\/doi.org\/10.3934\/dcdsb.2018179","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"1881_CR25","doi-asserted-by":"publisher","first-page":"6144","DOI":"10.1016\/j.jde.2016.08.033","volume":"261","author":"H Monobe","year":"2016","unstructured":"Monobe, H., Wu, C.: On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment. J. Differ. Eqn. 261, 6144\u20136177 (2016). https:\/\/doi.org\/10.1016\/j.jde.2016.08.033","journal-title":"J. Differ. Eqn."},{"issue":"4","key":"1881_CR26","doi-asserted-by":"publisher","first-page":"1570","DOI":"10.1016\/j.jde.2019.09.003","volume":"268","author":"D Tang","year":"2020","unstructured":"Tang, D., Zhou, P.: On a Lotka-Volterra competition-diffusion-advection System: homogeneity vs heterogeneity. J. Differ. Eqn. 268(4), 1570\u20131599 (2020). https:\/\/doi.org\/10.1016\/j.jde.2019.09.003","journal-title":"J. Differ. Eqn."},{"issue":"2","key":"1881_CR27","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1093\/imamat\/hxw059","volume":"82","author":"Q Chen","year":"2017","unstructured":"Chen, Q., Li, F., Wang, F.: A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment. IMA J. Appl. Math. 82(2), 445\u2013470 (2017). https:\/\/doi.org\/10.1093\/imamat\/hxw059","journal-title":"IMA J. Appl. Math."},{"key":"1881_CR28","doi-asserted-by":"publisher","first-page":"837","DOI":"10.3934\/DCDSB.2021067","volume":"27","author":"B Duan","year":"2022","unstructured":"Duan, B., Zhang, Z.: A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete Contin. Dyn. Syst. Ser. B 27, 837\u2013861 (2022). https:\/\/doi.org\/10.3934\/DCDSB.2021067","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"1881_CR29","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1038\/261459a0","volume":"261","author":"M May","year":"1976","unstructured":"May, M.: Simple mathematical models with very complicated dynamics. Nature 261, 459\u2013467 (1976)","journal-title":"Nature"},{"key":"1881_CR30","doi-asserted-by":"publisher","first-page":"1512","DOI":"10.1016\/j.jmaa.2019.02.037","volume":"472","author":"Q Zhang","year":"2019","unstructured":"Zhang, Q., Wang, M.: Dynamics for the diffusive mutualist model with advection and different free boundaries. J. Math. Anal. Appl. 472, 1512\u20131535 (2019). https:\/\/doi.org\/10.1016\/j.jmaa.2019.02.037","journal-title":"J. Math. Anal. Appl."},{"issue":"4","key":"1881_CR31","doi-asserted-by":"publisher","first-page":"1252","DOI":"10.1016\/j.jde.2014.10.022","volume":"258","author":"M Wang","year":"2015","unstructured":"Wang, M.: The diffusive logistic equation with a free boundary and sign-changing coefficient. J. Differ. Eqn. 258(4), 1252\u20131266 (2015). https:\/\/doi.org\/10.1016\/j.jde.2014.10.022","journal-title":"J. Differ. Eqn."},{"issue":"3","key":"1881_CR32","doi-asserted-by":"publisher","first-page":"996","DOI":"10.1016\/j.jde.2012.04.014","volume":"253","author":"Y Du","year":"2012","unstructured":"Du, Y., Guo, Z.: The Stefan problem for the Fisher-KPP equation. J. Differ. Eqn. 253(3), 996\u20131035 (2012). https:\/\/doi.org\/10.1016\/j.jde.2012.04.014","journal-title":"J. Differ. Eqn."},{"issue":"3","key":"1881_CR33","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1007\/s10884-015-9503-5","volume":"29","author":"M Wang","year":"2017","unstructured":"Wang, M., Zhao, J.: A free boundary problem for the predator-prey model with double free boundaries. J. Dyn. Differ. Eqn. 29(3), 957\u2013979 (2017). https:\/\/doi.org\/10.1007\/s10884-015-9503-5","journal-title":"J. Dyn. Differ. Eqn."},{"issue":"10","key":"1881_CR34","doi-asserted-by":"publisher","first-page":"3365","DOI":"10.1016\/j.jde.2014.02.013","volume":"256","author":"M Wang","year":"2014","unstructured":"Wang, M.: On some free boundary problems of the prey-predator model. J. Differ. Eqn. 256(10), 3365\u20133394 (2014). https:\/\/doi.org\/10.1016\/j.jde.2014.02.013","journal-title":"J. Differ. Eqn."},{"issue":"3","key":"1881_CR35","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1090\/S0002-9939-2014-12214-3","volume":"143","author":"H Gu","year":"2015","unstructured":"Gu, H., Lin, Z., Lou, B.: Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries. Proc. Am. Math. Soc. 143(3), 1109\u20131117 (2015). https:\/\/doi.org\/10.1090\/S0002-9939-2014-12214-3","journal-title":"Proc. Am. Math. Soc."},{"issue":"2","key":"1881_CR36","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.mbs.2006.09.003","volume":"204","author":"R Cantrell","year":"2006","unstructured":"Cantrell, R., Cosner, C., Lou, Y.: Movement toward better environments and the evolution of rapid diffusion. Math. Biosci. 204(2), 199\u2013214 (2006). https:\/\/doi.org\/10.1016\/j.mbs.2006.09.003","journal-title":"Math. Biosci."},{"key":"1881_CR37","doi-asserted-by":"publisher","first-page":"91","DOI":"10.5206\/mase\/10644","volume":"1","author":"K Lam","year":"2020","unstructured":"Lam, K., Liu, S., Lou, Y.: Selected topics on reaction-diffusion-advection models from spatial ecology. Math. Appl. Sci. Eng. 1, 91\u2013206 (2020). https:\/\/doi.org\/10.5206\/mase\/10644","journal-title":"Math. Appl. Sci. Eng."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01881-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01881-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01881-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,19]],"date-time":"2023-07-19T19:25:44Z","timestamp":1689794744000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01881-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,1]]},"references-count":37,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["1881"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01881-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2023,6,1]]},"assertion":[{"value":"3 January 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 June 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}