{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T05:53:13Z","timestamp":1762062793638,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,21]],"date-time":"2022-08-21T00:00:00Z","timestamp":1661040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["No. 2572020BC06","No.31702289"],"award-info":[{"award-number":["No. 2572020BC06","No.31702289"]}]},{"name":"National Natural Science Foundation of China","award":["No. 2572020BC06","No.31702289"],"award-info":[{"award-number":["No. 2572020BC06","No.31702289"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper considers the time taken for young predators to become adult predators as the delay and constructs a stage-structured predator\u2013prey system with Holling III response and time delay. Using the persistence theory for infinite-dimensional systems and the Hurwitz criterion, the permanent persistence condition of this system and the local stability condition of the system\u2019s coexistence equilibrium are given. Further, it is proven that the system undergoes a Hopf bifurcation at the coexistence equilibrium. By using Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium and the coexistence equilibrium are globally asymptotically stable, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Some numerical simulations are carried out to illustrate the main results.<\/jats:p>","DOI":"10.3390\/axioms11080421","type":"journal-article","created":{"date-parts":[[2022,8,21]],"date-time":"2022-08-21T21:05:51Z","timestamp":1661115951000},"page":"421","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Stability Analysis of a Stage-Structure Predator\u2013Prey Model with Holling III Functional Response and Cannibalism"],"prefix":"10.3390","volume":"11","author":[{"given":"Yufen","family":"Wei","sequence":"first","affiliation":[{"name":"College of Science, Heilongjiang Bayi Agricultural University, Daqing 163319, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4993-7377","authenticated-orcid":false,"given":"Yu","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1007\/s11071-018-4079-3","article-title":"Dynamical behaviour of a delayed three species predator\u2013prey model with cooperation among the prey species","volume":"92","author":"Maitra","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1142\/S1793524511001556","article-title":"Ratio-dependent predator-prey model with stage structure and time delay","volume":"5","author":"Zha","year":"2012","journal-title":"Int. J. Biomath."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1007\/s11071-015-2589-9","article-title":"A delayed prey\u2013predator system with prey subject to the strong Allee effect and disease","volume":"84","author":"Biswas","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/s11071-014-1691-8","article-title":"Stability and bifurcation analysis of a diffusive prey\u2013predator system in Holling type III with a prey refuge","volume":"79","author":"Yang","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1080\/17513758.2020.1850892","article-title":"Dynamical analysis of a delayed diffusive predator\u2013prey model with schooling behaviour and Allee effect","volume":"84","author":"Meng","year":"2020","journal-title":"J. Biol. Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"493","DOI":"10.15388\/NA.2018.4.3","article-title":"Optimal harvesting policy of a prey\u2013predator model with Crowley\u2013Martin-type functional response and stage structure in the predator","volume":"23","author":"Dubey","year":"2018","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1016\/j.apm.2020.08.054","article-title":"Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay","volume":"91","author":"Zhang","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"655","DOI":"10.4028\/www.scientific.net\/AMM.687-691.655","article-title":"Stability and Hopf bifurcation in delayed prey\u2013predator system with ratio dependent","volume":"687\u2013691","author":"Yuan","year":"2014","journal-title":"Appl. Mech. Mater."},{"key":"ref_9","first-page":"527","article-title":"Permanence and global stability for single-species model with three life stages and time delay","volume":"26","author":"Gao","year":"2006","journal-title":"Acta Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1016\/j.nonrwa.2011.11.007","article-title":"Stability and travelling waves for a time-delayed population system with stage structure","volume":"13","author":"Liang","year":"2012","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"88","DOI":"10.4028\/www.scientific.net\/AMR.978.88","article-title":"Data processing for cynamic consequences of prey refuge in a prey\u2013predator system with stage structure and time delay","volume":"978","author":"Han","year":"2014","journal-title":"Adv. Mater. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s40840-014-0033-9","article-title":"Bifurcation analysis of a population dynamics in a critical state","volume":"38","author":"Xia","year":"2015","journal-title":"Bull. Malays. Math. Soc. Ser."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1007\/s11071-021-06549-2","article-title":"Dynamics of a Filippov prey\u2013predator system with stage-specific intermittent harvesting","volume":"105","author":"Bhattacharyya","year":"2021","journal-title":"Nonlinear Dynam."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.jtbi.2016.02.016","article-title":"Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect","volume":"396","author":"Morozov","year":"2016","journal-title":"J. Theor. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.chaos.2018.01.038","article-title":"Traveling wave fronts of a single species model with cannibalism and nonlocal effect","volume":"108","author":"Zhang","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.nonrwa.2003.12.003","article-title":"Global dynamics and Hopf bifurcation of a structured population model","volume":"6","author":"Xu","year":"2005","journal-title":"Nonlinear Anal.-Real."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yu, X., Zhu, Z., and Chen, F. (2020). Dynamic behaviors of a single species stage structure model with Michaelis-Menten-type juvenile population harvesting. Mathematics, 8.","DOI":"10.3390\/math8081281"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Rayungsari, M., Suryanto, A., Kusumawinahyu, W.M., and Darti, I. (2022). Dynamical analysis of a predator-prey model incorporating predator cannibalism and refuge. Axioms, 11.","DOI":"10.3390\/axioms11030116"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"724325","DOI":"10.1155\/2013\/724325","article-title":"Global dynamics of a prey\u2013predator model with stage structure and delayed predator response","volume":"2013","author":"Wang","year":"2013","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_20","first-page":"431671","article-title":"Global stability and Hopf bifurcation of a prey\u2013predator model with time delay and stage structure","volume":"2014","author":"Wang","year":"2014","journal-title":"Chin. J. Eng. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1186\/s13662-015-0548-x","article-title":"Stability and Hopf bifurcation for a ratio-dependent prey\u2013predator system with stage structure and time delay","volume":"2015","author":"Wang","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s12190-014-0762-9","article-title":"Global dynamics of a delayed predator\u2013prey model with stage structure and holling type II functional response","volume":"47","author":"Wang","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3937","DOI":"10.1002\/mma.3328","article-title":"Global dynamics of a delayed prey\u2013predator model with stage structure for the predator and the prey","volume":"38","author":"Wang","year":"2015","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_24","first-page":"693","article-title":"Stability and Hopf bifurcation in a time-delayed predator-prey system with stage structures for both predator and prey","volume":"36","author":"Zhu","year":"2020","journal-title":"Chin. J. Eng. Math."},{"key":"ref_25","first-page":"39","article-title":"Stability and Hopf bifurcation in a predator-prey system with Holling-III functional response and stage structure","volume":"36","author":"Wei","year":"2019","journal-title":"J. Nat. Sci. Heilongjiang Univ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"109919","DOI":"10.1016\/j.chaos.2020.109919","article-title":"A new adaptive synchronization and hyperchaos control of a biological snap oscillator","volume":"138","author":"Sajjadi","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Srivastava, H., and Khader, M. (2021). Numerical simulation for the treatment of nonlinear predator-prey equations by using the finite element optimization method. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5020056"},{"key":"ref_28","first-page":"209","article-title":"Bifurcation analysis of a diffusive predator-prey model with schooling behaviour and cannibalism in prey","volume":"11","author":"Djilali","year":"2021","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"12679","DOI":"10.1002\/ece3.6901","article-title":"Infant cannibalism in wild white-faced capuchin monkeys","volume":"10","author":"Nishikawa","year":"2020","journal-title":"Ecol. Evol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.jmaa.2015.04.079","article-title":"Ecological and evolutionary dynamics of two-stage models of social insects with egg cannibalism","volume":"430","author":"Kang","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.mbs.2018.11.004","article-title":"Dynamical analysis of a stage-structured predator-prey model with cannibalism","volume":"307","author":"Zhang","year":"2019","journal-title":"Math. Biosci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1186\/s13662-019-2289-8","article-title":"Dynamic behaviors of Lotka-Volterra predator-prey model incorporating predator cannibalism","volume":"2019","author":"Deng","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_33","unstructured":"Ma, Z.E., and Zhou, Y.C. (2001). Qualitative and Stable Methods for Ordinary Differential Equations, Science Press."},{"key":"ref_34","first-page":"1265798","article-title":"Global behavior of solutions in a prey\u2013predator cross-diffusion model with cannibalism","volume":"2020","author":"Chen","year":"2020","journal-title":"Complexity"},{"key":"ref_35","first-page":"388","article-title":"Persistence in infinite-dimensional systems","volume":"20","author":"Hale","year":"1989","journal-title":"J. Math. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/421\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:13:09Z","timestamp":1760141589000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/421"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,21]]},"references-count":35,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080421"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080421","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,8,21]]}}}