{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:34:36Z","timestamp":1760240076106,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,6]],"date-time":"2019-03-06T00:00:00Z","timestamp":1551830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Nature Science Foundation of China","award":["61772063"],"award-info":[{"award-number":["61772063"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we researched some dynamical behaviors of a stochastic predator\u2013prey system, which is considered under the combination of Crowley\u2013Martin functional response and stage structure. First, we obtained the existence and uniqueness of the global positive solution of the system. Then, we studied the stochastically ultimate boundedness of the solution. Furthermore, we established two sufficient conditions, which are separately given to ensure the stochastic extinction of the prey and predator populations. In the end, we carried out the numerical simulations to explain some cases.<\/jats:p>","DOI":"10.3390\/e21030252","type":"journal-article","created":{"date-parts":[[2019,3,7]],"date-time":"2019-03-07T10:52:22Z","timestamp":1551955942000},"page":"252","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Extinction Analysis of Stochastic Predator\u2013Prey System with Stage Structure and Crowley\u2013Martin Functional Response"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2024-0418","authenticated-orcid":false,"given":"Conghui","family":"Xu","sequence":"first","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guojian","family":"Ren","sequence":"additional","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9336-0376","authenticated-orcid":false,"given":"Yongguang","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,6]]},"reference":[{"key":"ref_1","unstructured":"Lotka, A. (1956). Elements of Mathematical Biology, Dover Publications."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1051\/m2an:2003029","article-title":"Persistence and bifurcation analysis on a predator-prey system of Holling type","volume":"37","author":"Mukherjee","year":"2003","journal-title":"Esaim Math. Model. Numer. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3002","DOI":"10.1016\/j.amc.2010.04.012","article-title":"Permanence and extinction analysis for a delayed periodic predator-prey system with Holling type II response function and diffusion","volume":"216","author":"Liu","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/s12190-013-0696-7","article-title":"Hopf bifurcation in a predator-prey system with Holling type III functional response and time delays","volume":"44","author":"Zhang","year":"2014","journal-title":"J. Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3083","DOI":"10.1890\/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2","article-title":"Functional responses with predator interference: Viable alternative to Holling type II model","volume":"82","author":"Sklaski","year":"2001","journal-title":"Ecology"},{"key":"ref_6","first-page":"1","article-title":"Permanence and global asymptotic stability of a delayed predator-prey model with Hassell-Varley type functional response","volume":"94","author":"Wang","year":"2013","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.jmaa.2005.10.011","article-title":"Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response","volume":"317","author":"Cui","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1007\/s11071-016-2728-y","article-title":"Dynamics of a switching diffusion modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response","volume":"85","author":"Lahrouz","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s11071-014-1859-2","article-title":"Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge","volume":"80","author":"Tripathi","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.cnsns.2015.06.008","article-title":"Global analysis of a delayed density dependent predator\u2013prey model with Crowley-Martin functional response","volume":"30","author":"Tripathi","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1016\/j.chaos.2009.03.020","article-title":"Dynamics of a three species food chain model with Crowley\u2013Martin type functional response","volume":"42","author":"Upadhyay","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/S0898-1221(97)00056-4","article-title":"A predator-prey system with stage-structure for predator","volume":"33","author":"Wang","year":"1997","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1016\/j.chaos.2007.01.035","article-title":"Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay","volume":"38","author":"Chen","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","first-page":"503","article-title":"Permanence of periodic predator-prey system with two predators and stage structure for prey","volume":"2018","author":"Huang","year":"2014","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/j.mcm.2007.11.003","article-title":"Permanence of periodic Holling type-IV predator-prey system with stage structure for prey","volume":"48","author":"Yang","year":"2008","journal-title":"Math. Comput. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1016\/j.amc.2014.01.139","article-title":"Stability in a predator\u2013prey model with Crowley-Martin function and stage structure for prey","volume":"232","author":"Meng","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s11071-009-9495-y","article-title":"Bifurcation and stability analysis in predator-prey model with a stage-structure for predator","volume":"58","author":"Sun","year":"2009","journal-title":"Nonlinear Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2151","DOI":"10.1016\/j.nonrwa.2010.12.029","article-title":"Global dynamics of a predator-prey model with time delay and stage structure for the prey","volume":"12","author":"Xu","year":"2011","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.amc.2016.10.035","article-title":"A stage-structured predator-prey model with predation over juvenile prey","volume":"297","author":"Lu","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2014\/618587","article-title":"Global stability of a stage-structured predator-prey model with stochastic perturbation","volume":"2014","author":"Yang","year":"2014","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3792","DOI":"10.1016\/j.cnsns.2010.12.026","article-title":"Global stability of stage-structured predator\u2013prey models with Beddington-DeAngelis functional response","volume":"16","author":"Liu","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_22","first-page":"1","article-title":"A stochastic predator-prey system with stage structure for predator","volume":"2014","author":"Zhao","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1007\/s00332-018-9444-3","article-title":"Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type II functional response","volume":"28","author":"Liu","year":"2018","journal-title":"J. Nonlinear Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.nonrwa.2006.09.009","article-title":"Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey","volume":"9","author":"Chen","year":"2008","journal-title":"Nonlinear Anal.-Real World Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s12190-013-0674-0","article-title":"Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response","volume":"43","author":"Liu","year":"2013","journal-title":"J. Appl. Math. Comput."},{"key":"ref_26","unstructured":"Mao, X.R. (1997). Stochastic Differential Equations and Applications, Horwood Publishing."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"May, R.M. (2001). Stability and Complexity in Model Ecosystems, Princeton University Press.","DOI":"10.1515\/9780691206912"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.jmaa.2010.10.053","article-title":"Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching","volume":"376","author":"Li","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1137\/S0036144500378302","article-title":"An algorithmic introduction to numerical simulation of stochastic differential equations","volume":"43","author":"Higham","year":"2001","journal-title":"SIAM Rev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.amc.2017.09.030","article-title":"Stationary distribution and extinction of a stochastic predator\u2013prey model with additional food and nonlinear perturbation","volume":"320","author":"Liu","year":"2018","journal-title":"Appl. Math. Computat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1016\/j.amc.2013.12.026","article-title":"On a stochastic delayed predator-prey model with L\u00e9vy jumps","volume":"228","author":"Liu","year":"2014","journal-title":"Appl. Math. Computat."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1016\/j.nonrwa.2012.07.032","article-title":"Stochastic predator-prey model with Allee effect on prey","volume":"14","author":"Aguirre","year":"2013","journal-title":"Nonlinear Anal. Real World Appl"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.ecolmodel.2013.02.029","article-title":"Study of chaotic behavior in predator\u2013prey interactions in a chemostat","volume":"259","author":"Ali","year":"2013","journal-title":"Ecol. Model."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/3\/252\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:36:41Z","timestamp":1760186201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/3\/252"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,6]]},"references-count":33,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["e21030252"],"URL":"https:\/\/doi.org\/10.3390\/e21030252","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,3,6]]}}}