{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T18:09:23Z","timestamp":1780423763712,"version":"3.54.1"},"reference-count":30,"publisher":"World Scientific Pub Co Pte Ltd","issue":"15","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61672021"],"award-info":[{"award-number":["61672021"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61907030"],"award-info":[{"award-number":["61907030"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["1171262"],"award-info":[{"award-number":["1171262"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61872227"],"award-info":[{"award-number":["61872227"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["GK201903088"],"award-info":[{"award-number":["GK201903088"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007128","name":"Natural Science Foundation of Shaanxi Province","doi-asserted-by":"publisher","award":["2018JQ1021"],"award-info":[{"award-number":["2018JQ1021"]}],"id":[{"id":"10.13039\/501100007128","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2020,12,15]]},"abstract":"<jats:p>This paper is dedicated to a study of a diffusive one-prey and two-cooperative-predators model with C\u2013M functional response subject to Dirichlet boundary conditions. We first discuss the existence of positive steady states by the fixed point index theory and the degree theory. In the meantime, we analyze the uniqueness and stability of coexistence states under conditions that one predator\u2019s consumer rate is small and the effect of interference intensity of another predator is large. Then, steady-state bifurcations from two strong semi-trivial steady states (provided that they uniquely exist under some conditions) and from one weak semi-trivial steady state are investigated in detail by the Crandall\u2013Rabinowitz bifurcation theorem, the technique of space decomposition and the implicit function theorem. In addition, we study the asymptotic behaviors including the extinction and permanence of the time-dependent system by the comparison principle, upper-lower solution method and monotone iteration scheme. Finally, numerical simulations are done not only to validate the theoretical conclusions, but also to further clarify the impacts of parameters on the three species.<\/jats:p>","DOI":"10.1142\/s0218127420502247","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T10:44:48Z","timestamp":1607597088000},"page":"2050224","source":"Crossref","is-referenced-by-count":3,"title":["A Diffusive One-Prey and Two-Cooperative-Predators Model with C\u2013M Functional Response"],"prefix":"10.1142","volume":"30","author":[{"given":"Xiao","family":"Yan","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9614-4927","authenticated-orcid":false,"given":"Yanling","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan\u2019e","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2020,12,9]]},"reference":[{"key":"S0218127420502247BIB001","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1016\/j.amc.2012.08.032","volume":"219","author":"Baek S.","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"S0218127420502247BIB002","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0022-1236(71)90015-2","volume":"8","author":"Crandall M. G.","year":"1971","journal-title":"J. Funct. Anal."},{"key":"S0218127420502247BIB003","doi-asserted-by":"crossref","first-page":"211","DOI":"10.2307\/1467324","volume":"8","author":"Crowley P. H.","year":"1989","journal-title":"J. N. Am. Benthol. Soc."},{"key":"S0218127420502247BIB004","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/0022-247X(83)90098-7","volume":"91","author":"Dancer E. N.","year":"1983","journal-title":"J. Math. Anal. Appl."},{"key":"S0218127420502247BIB005","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1090\/S0002-9947-1984-0743741-4","volume":"284","author":"Dancer E. N.","year":"1984","journal-title":"Trans. Amer. Math. Soc."},{"key":"S0218127420502247BIB006","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0362-546X(94)E0063-M","volume":"24","author":"Dancer E. N.","year":"1995","journal-title":"Nonlin. Anal."},{"key":"S0218127420502247BIB007","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127415501102"},{"key":"S0218127420502247BIB008","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s10440-014-9985-x","volume":"139","author":"Dong Y. Y.","year":"2015","journal-title":"Acta Appl. Math."},{"key":"S0218127420502247BIB009","doi-asserted-by":"crossref","first-page":"2443","DOI":"10.1090\/S0002-9947-97-01842-4","volume":"349","author":"Du Y. H.","year":"1997","journal-title":"Trans. Amer. Math. Soc."},{"key":"S0218127420502247BIB010","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1112\/plms\/pdr070","volume":"105","author":"Guo G. H.","year":"2012","journal-title":"Proc. Lond. Math. Soc. (3)"},{"key":"S0218127420502247BIB011","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/j.nonrwa.2006.01.006","volume":"8","author":"Hei L. J.","year":"2007","journal-title":"Nonlin. Anal: Real World Appl."},{"key":"S0218127420502247BIB012","doi-asserted-by":"crossref","first-page":"4185","DOI":"10.1016\/j.na.2009.02.119","volume":"71","author":"Ko W.","year":"2009","journal-title":"Nonlin. Anal."},{"key":"S0218127420502247BIB013","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.jmaa.2012.07.026","volume":"397","author":"Ko W.","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"S0218127420502247BIB014","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.jmaa.2012.07.025","volume":"397","author":"Ko W.","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"S0218127420502247BIB015","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BF00276925","volume":"7","author":"Krikorian N.","year":"1979","journal-title":"J. Math. Biol."},{"key":"S0218127420502247BIB016","first-page":"127","volume":"17","author":"Li H. L.","year":"2012","journal-title":"Discr. Contin. Dyn. Syst. Ser. B"},{"key":"S0218127420502247BIB017","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1016\/j.camwa.2015.03.007","volume":"69","author":"Li S. B.","year":"2015","journal-title":"Comput. Math. Appl."},{"key":"S0218127420502247BIB018","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10440-015-0009-2","volume":"141","author":"Li S. B.","year":"2016","journal-title":"Acta Appl. Math."},{"key":"S0218127420502247BIB019","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.nonrwa.2015.07.010","volume":"27","author":"Li H. X.","year":"2016","journal-title":"Nonlin. Anal. Real World Appl."},{"key":"S0218127420502247BIB020","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1007\/s00526-019-1558-4","volume":"58","author":"Li S. B.","year":"2019","journal-title":"Calc. Var."},{"key":"S0218127420502247BIB021","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.ecocom.2008.02.001","volume":"5","author":"Namba T.","year":"2008","journal-title":"Ecol. Compl."},{"key":"S0218127420502247BIB022","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1016\/0362-546X(95)00058-4","volume":"26","author":"Pao C. V.","year":"1996","journal-title":"Nonlin. Anal."},{"key":"S0218127420502247BIB023","volume-title":"Systems Analysis and Simulations in Ecology II","author":"Patten B. C.","year":"1972"},{"key":"S0218127420502247BIB024","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.cnsns.2015.06.008","volume":"30","author":"Tripathi J. P.","year":"2016","journal-title":"Commun. Nonlin. Sci. Numer. Simul."},{"key":"S0218127420502247BIB025","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1137\/0521018","volume":"21","author":"Yamada Y.","year":"1990","journal-title":"SIAM J. Appl. Math."},{"key":"S0218127420502247BIB026","first-page":"2269","volume":"20","author":"Yang W. B.","year":"2015","journal-title":"Discr. Contin. Dyn. Syst. Ser. B"},{"key":"S0218127420502247BIB027","volume-title":"Introduction of Reaction\u2013Diffusion Equation","author":"Ye Q. X.","year":"1994"},{"key":"S0218127420502247BIB028","first-page":"3755","volume":"23","author":"Zhang G. H.","year":"2018","journal-title":"Discr. Contin. Dyn. Syst. Ser. B"},{"key":"S0218127420502247BIB029","doi-asserted-by":"crossref","first-page":"902","DOI":"10.1016\/j.nonrwa.2010.08.015","volume":"12","author":"Zhou J.","year":"2011","journal-title":"Nonlin. Anal.: Real World Appl."},{"key":"S0218127420502247BIB030","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.3934\/cpaa.2015.14.1127","volume":"14","author":"Zhou J.","year":"2015","journal-title":"Commun. Pure Appl. Anal."}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127420502247","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,17]],"date-time":"2021-04-17T04:50:27Z","timestamp":1618635027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127420502247"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,9]]},"references-count":30,"journal-issue":{"issue":"15","published-print":{"date-parts":[[2020,12,15]]}},"alternative-id":["10.1142\/S0218127420502247"],"URL":"https:\/\/doi.org\/10.1142\/s0218127420502247","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,9]]}}}