{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T16:05:35Z","timestamp":1760889935334,"version":"3.37.3"},"reference-count":71,"publisher":"World Scientific Pub Co Pte Ltd","issue":"11","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61533011"],"award-info":[{"award-number":["61533011"]}],"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":["61703251"],"award-info":[{"award-number":["61703251"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2019,10]]},"abstract":"<jats:p> In this paper, a diffusive predator\u2013prey model is considered in which the predator and prey populations both exhibit schooling behavior. The system\u2019s spatial dynamics are captured via a suitable threshold parameter, and a sequence of spatiotemporal patterns such as hexagons, stripes and a mixture of the two are observed. Specifically, the linear stability analysis is applied to obtain the conditions for Hopf bifurcation and Turing instability. Then, employing the multiple-scale analysis, the amplitude equations near the critical point of Turing bifurcation are derived, through which the selection and stability of pattern formations are investigated. The theoretical results are verified by numerical simulations. <\/jats:p>","DOI":"10.1142\/s0218127419501463","type":"journal-article","created":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T06:46:35Z","timestamp":1571208395000},"page":"1950146","source":"Crossref","is-referenced-by-count":9,"title":["Pattern Dynamics in a Predator\u2013Prey Model with Schooling Behavior and Cross-Diffusion"],"prefix":"10.1142","volume":"29","author":[{"given":"Wen","family":"Wang","sequence":"first","affiliation":[{"name":"School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2281-9378","authenticated-orcid":false,"given":"Shutang","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhibin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Da","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Management Science and Engineering, Shandong Normal University, Jinan 250358, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2019,10,16]]},"reference":[{"key":"S0218127419501463BIB001","doi-asserted-by":"publisher","DOI":"10.1137\/070705210"},{"key":"S0218127419501463BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5193(05)80379-0"},{"key":"S0218127419501463BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5193(89)80211-5"},{"key":"S0218127419501463BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-016-2844-8"},{"key":"S0218127419501463BIB005","doi-asserted-by":"publisher","DOI":"10.1142\/2284"},{"key":"S0218127419501463BIB006","doi-asserted-by":"publisher","DOI":"10.2307\/3866"},{"key":"S0218127419501463BIB007","first-page":"31","volume":"29","author":"Brattey J.","year":"1997","journal-title":"NAFO Sci. Coun. Stud."},{"key":"S0218127419501463BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2011.12.014"},{"key":"S0218127419501463BIB009","doi-asserted-by":"publisher","DOI":"10.1006\/jmsc.2002.1282"},{"key":"S0218127419501463BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2017.03.028"},{"key":"S0218127419501463BIB011","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127412500617"},{"key":"S0218127419501463BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2012.04.012"},{"key":"S0218127419501463BIB013","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2013.12.481"},{"key":"S0218127419501463BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2016.02.007"},{"key":"S0218127419501463BIB015","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2017.11.014"},{"key":"S0218127419501463BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/j.pbiomolbio.2010.05.008"},{"key":"S0218127419501463BIB017","doi-asserted-by":"publisher","DOI":"10.1007\/s002850050095"},{"key":"S0218127419501463BIB018","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.65.851"},{"key":"S0218127419501463BIB019","doi-asserted-by":"publisher","DOI":"10.2307\/1467324"},{"key":"S0218127419501463BIB020","doi-asserted-by":"publisher","DOI":"10.2307\/1936298"},{"key":"S0218127419501463BIB021","doi-asserted-by":"publisher","DOI":"10.1111\/j.1439-0485.2009.00328.x"},{"key":"S0218127419501463BIB022","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.54.261"},{"key":"S0218127419501463BIB023","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.78.061917"},{"key":"S0218127419501463BIB024","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2012.11.009"},{"key":"S0218127419501463BIB025","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2009.06.012"},{"key":"S0218127419501463BIB026","doi-asserted-by":"publisher","DOI":"10.1038\/2231133a0"},{"key":"S0218127419501463BIB027","doi-asserted-by":"publisher","DOI":"10.4039\/Ent91293-5"},{"key":"S0218127419501463BIB028","doi-asserted-by":"publisher","DOI":"10.4039\/entm9745fv"},{"key":"S0218127419501463BIB029","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.64.026219"},{"key":"S0218127419501463BIB030","doi-asserted-by":"publisher","DOI":"10.1126\/science.284.5421.1826"},{"key":"S0218127419501463BIB031","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/47.3-4.219"},{"volume-title":"Elements of Physical Biology","year":"1925","author":"Lotka A. J.","key":"S0218127419501463BIB032"},{"key":"S0218127419501463BIB033","doi-asserted-by":"publisher","DOI":"10.1007\/s11431-015-5961-6"},{"key":"S0218127419501463BIB034","doi-asserted-by":"publisher","DOI":"10.1142\/S0217979214502397"},{"key":"S0218127419501463BIB035","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2016.06.075"},{"key":"S0218127419501463BIB036","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-017-3565-3"},{"key":"S0218127419501463BIB037","first-page":"35","volume":"317","author":"Manna D.","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"S0218127419501463BIB038","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(97)00164-4"},{"key":"S0218127419501463BIB039","doi-asserted-by":"publisher","DOI":"10.1063\/1.165931"},{"volume-title":"Nonlinear Science and the Pattern Dynamics Introduction","year":"2010","author":"Ouyang Q.","key":"S0218127419501463BIB040"},{"key":"S0218127419501463BIB041","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(99)00070-9"},{"key":"S0218127419501463BIB042","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2014.02.039"},{"key":"S0218127419501463BIB043","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127418300045"},{"key":"S0218127419501463BIB044","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(72)90090-2"},{"key":"S0218127419501463BIB045","doi-asserted-by":"publisher","DOI":"10.1137\/120899510"},{"key":"S0218127419501463BIB046","doi-asserted-by":"publisher","DOI":"10.3390\/mi9010035"},{"key":"S0218127419501463BIB047","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2013.08.030"},{"volume-title":"Models in Ecology","year":"1974","author":"Smith J. M.","key":"S0218127419501463BIB048"},{"key":"S0218127419501463BIB049","doi-asserted-by":"publisher","DOI":"10.1111\/sapm.12165"},{"key":"S0218127419501463BIB050","first-page":"021005","volume":"3","author":"Steyn-Ross M. L.","year":"2013","journal-title":"Phys. Rev. X"},{"key":"S0218127419501463BIB051","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-009-9594-9"},{"key":"S0218127419501463BIB052","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-012-0330-5"},{"key":"S0218127419501463BIB053","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-013-0935-3"},{"key":"S0218127419501463BIB054","doi-asserted-by":"publisher","DOI":"10.1016\/j.plrev.2016.08.002"},{"key":"S0218127419501463BIB055","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2018.04.010"},{"key":"S0218127419501463BIB056","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2014.12.006"},{"key":"S0218127419501463BIB057","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-016-2873-3"},{"key":"S0218127419501463BIB058","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2011.05.022"},{"key":"S0218127419501463BIB059","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2017.01.087"},{"key":"S0218127419501463BIB060","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.81.026213"},{"key":"S0218127419501463BIB061","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.1952.0012"},{"key":"S0218127419501463BIB062","doi-asserted-by":"publisher","DOI":"10.1051\/mmnp\/20116105"},{"key":"S0218127419501463BIB063","doi-asserted-by":"publisher","DOI":"10.1039\/B813825G"},{"key":"S0218127419501463BIB064","first-page":"31","volume":"2","author":"Volterra V.","year":"1926","journal-title":"Mem. R. Accad. Naz. dei Lincei"},{"key":"S0218127419501463BIB065","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2015.10.036"},{"key":"S0218127419501463BIB066","doi-asserted-by":"publisher","DOI":"10.1631\/jzus.A1800334"},{"key":"S0218127419501463BIB067","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.10.024"},{"key":"S0218127419501463BIB068","first-page":"647","volume":"22","author":"Yi F.","year":"2017","journal-title":"Discr. Contin. Dyn. Syst. Ser. B"},{"key":"S0218127419501463BIB069","doi-asserted-by":"publisher","DOI":"10.1063\/1.4812724"},{"key":"S0218127419501463BIB070","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.90.052908"},{"key":"S0218127419501463BIB071","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-014-1438-6"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127419501463","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T06:47:05Z","timestamp":1571208425000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127419501463"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10]]},"references-count":71,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2019,10]]}},"alternative-id":["10.1142\/S0218127419501463"],"URL":"https:\/\/doi.org\/10.1142\/s0218127419501463","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"type":"print","value":"0218-1274"},{"type":"electronic","value":"1793-6551"}],"subject":[],"published":{"date-parts":[[2019,10]]}}}