{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T14:20:59Z","timestamp":1766067659005,"version":"3.41.2"},"reference-count":38,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T00:00:00Z","timestamp":1672963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>The complexity of the dynamical behaviors of interaction between prey and its predator is studied. The prey and predator relationship involves the age structure and intraspecific competition on predators and the nonlinear harvesting of prey following the Michaelis\u2013Menten type term. Some biological validities are shown for the constructed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions. Three equilibrium points, namely the origin, axial, and interior points, are found including their global dynamics by employing the Lyapunov function along with the generalized Lassale invariant principle. The changes in dynamical behaviors driven by the harvesting and the memory effect are exhibited, including transcritical, saddle-node, backward, and Hopf bifurcations. The appearance of these interesting phenomena is strengthened by giving numerical simulations consisting of bifurcation diagrams, phase portraits, and their time series.<\/jats:p>","DOI":"10.3389\/fams.2022.1077831","type":"journal-article","created":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T15:34:51Z","timestamp":1673019291000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Bifurcation analysis of a predator\u2013prey model involving age structure, intraspecific competition, Michaelis\u2013Menten type harvesting, and memory effect"],"prefix":"10.3389","volume":"8","author":[{"given":"Hasan S.","family":"Panigoro","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emli","family":"Rahmi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Resmawan","family":"Resmawan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1965","published-online":{"date-parts":[[2023,1,6]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","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 Diff Equat"},{"key":"B2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41598-020-61414-3","article-title":"The impact of microplastic particles on population dynamics of predator and prey: implication of the Lotka-Volterra model","volume":"10","author":"Huang","year":"2020","journal-title":"Sci Rep"},{"key":"B3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41598-018-25436-2","article-title":"Asymptotic stability of a modified Lotka-Volterra model with small immigrations","volume":"8","author":"Tahara","year":"2018","journal-title":"Sci Rep"},{"key":"B4","doi-asserted-by":"publisher","first-page":"1961","DOI":"10.1007\/s10473-020-0622-7","article-title":"Existence and uniqueness of the positive steady state solution for a Lotka-Volterra predator-prey model with a crowding term","volume":"40","author":"Zeng","year":"2020","journal-title":"Acta Math Sci"},{"key":"B5","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/j.ecolmodel.2005.01.052","article-title":"The effect of harvesting on a predator-prey system with stage structure","volume":"187","author":"Gui","year":"2005","journal-title":"Ecol Modell"},{"key":"B6","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/S0025-5564(98)10051-2","article-title":"Destabilizing effect of cannibalism on a structured predator-prey system","volume":"155","author":"Magn\u00fasson","year":"1999","journal-title":"Math Biosci"},{"key":"B7","doi-asserted-by":"publisher","first-page":"329592","DOI":"10.1155\/2013\/329592","article-title":"Stability analysis of a harvested prey-predator model with stage structure and maturation delay","volume":"2013","author":"Liu","year":"2013","journal-title":"Math Problems Eng"},{"key":"B8","doi-asserted-by":"publisher","first-page":"493","DOI":"10.15388\/NA.2018.4.3","article-title":"Optimal harvesting policy of a prey-predator model with Crowley-Martin-type functional response and stage structure in the predator","volume":"23","author":"Dubey","year":"2018","journal-title":"Nonlinear Anal Model Control"},{"key":"B9","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1515\/math-2019-0014","article-title":"Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge","volume":"17","author":"Xiao","year":"2019","journal-title":"Open Math"},{"key":"B10","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1016\/j.physd.2018.10.002","article-title":"Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis-Menten type functional response","volume":"389","author":"Zhang","year":"2019","journal-title":"Physica D"},{"key":"B11","doi-asserted-by":"publisher","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":"B12","doi-asserted-by":"publisher","first-page":"8162","DOI":"10.3934\/math.2022454","article-title":"Periodic solution of a stage-structured predator-prey model with Crowley-Martin type functional response","volume":"7","author":"Lu","year":"2022","journal-title":"AIMS Math"},{"key":"B13","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/j.apm.2021.11.029","article-title":"The impact of role reversal on the dynamics of predator-prey model with stage structure","volume":"104","author":"Li","year":"2022","journal-title":"Appl Math Model"},{"key":"B14","doi-asserted-by":"publisher","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":"B15","doi-asserted-by":"publisher","first-page":"7711","DOI":"10.1002\/mma.5234","article-title":"Local stability analysis on Lotka-Volterra predator-prey models with prey refuge and harvesting","volume":"41","author":"Chow","year":"2018","journal-title":"Math Methods Appl Sci"},{"key":"B16","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1016\/j.mcm.2009.04.018","article-title":"Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator-prey model with linear harvesting rate: analysis and computation","volume":"50","author":"Xiao","year":"2009","journal-title":"Math Comput Model"},{"key":"B17","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1186\/s13662-018-1581-3","article-title":"Bifurcation analysis of a modified Leslie-Gower model with Holling type-IV functional response and nonlinear prey harvesting","volume":"2018","author":"Zhang","year":"2018","journal-title":"Adv Diff Equat"},{"key":"B18","unstructured":"MulheisenM\n            AllenC\n            ParrCS\n          Lycaon Pictus. (On-line), Animal Diversity Web2002"},{"key":"B19","unstructured":"Aepyceros melampus. The IUCN Red List of Threatened Species 2016: eT550A501808282016"},{"key":"B20","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/j.nonrwa.2015.05.014","article-title":"The effect of vaccines on backward bifurcation in a fractional order HIV model","volume":"26","author":"Huo","year":"2015","journal-title":"Nonlinear Anal. Real World Appl"},{"key":"B21","doi-asserted-by":"publisher","first-page":"49","DOI":"10.18576\/pfda\/050106","article-title":"Stage structure and refuge effects in the dynamical analysis of a fractional order Rosenzweig-MacArthur prey-predator model","volume":"5","author":"Moustafa","year":"2019","journal-title":"Progr Fract Different Appl"},{"key":"B22","doi-asserted-by":"publisher","first-page":"012051","DOI":"10.1088\/1742-6596\/1821\/1\/012051","article-title":"Stability analysis of a fractional-order leslie-gower model with allee effect in predator","volume":"1821","author":"Rahmi","year":"2021","journal-title":"J Phys"},{"key":"B23","doi-asserted-by":"publisher","first-page":"823","DOI":"10.3934\/dcdss.2020047","article-title":"Dynamical behaviour of fractional-order predator-prey system of Holling-type","volume":"13","author":"Owolabi","year":"2020","journal-title":"Discrete Contin Dyn Syst S"},{"key":"B24","doi-asserted-by":"publisher","first-page":"101483","DOI":"10.1016\/j.ecoinf.2021.101483","article-title":"Modelling hiding behaviour in a predator-prey system by both integer order and fractional order derivatives","volume":"67","author":"Barman","year":"2022","journal-title":"Ecol Inform"},{"key":"B25","doi-asserted-by":"publisher","first-page":"1736","DOI":"10.1002\/mma.5999","article-title":"Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative","volume":"43","author":"Ghanbari","year":"2020","journal-title":"Math Methods Appl Sci"},{"key":"B26","doi-asserted-by":"publisher","first-page":"110711","DOI":"10.1016\/j.chaos.2021.110711","article-title":"Effects of fear in a fractional-order predator-prey system with predator density-dependent prey mortality","volume":"145","author":"Yousef","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"B27","doi-asserted-by":"publisher","first-page":"110531","DOI":"10.1016\/j.chaos.2020.110531","article-title":"Memory effect on Bazykin's prey-predator model: stability and bifurcation analysis","volume":"143","author":"Ghosh","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"B28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.28919\/cmbn\/6586","article-title":"A discrete-time fractional-order rosenzweig-macarthur predator-prey model involving prey refuge","volume":"2021","author":"Panigoro","year":"2021","journal-title":"Commun Math Biol Neurosci"},{"key":"B29","doi-asserted-by":"publisher","first-page":"355","DOI":"10.34312\/jjom.v4i2.15220","article-title":"A fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey","volume":"4","author":"Panigoro","year":"2022","journal-title":"Jambura J Math"},{"key":"B30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40819-021-01224-x","article-title":"Dynamics of caputo fractional order SEIRV epidemic model with optimal control and stability analysis","volume":"8","author":"Mahata","year":"2022","journal-title":"Int J Appl Comput Math"},{"key":"B31","doi-asserted-by":"publisher","first-page":"35","DOI":"10.3390\/fractalfract4030035","article-title":"Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate","volume":"4","author":"Yavuz","year":"2020","journal-title":"Fractal Fract"},{"key":"B32","doi-asserted-by":"publisher","first-page":"1329","DOI":"10.1007\/s11071-021-07031-9","article-title":"Impact of fear effect in a fractional-order predator-prey system incorporating constant prey refuge","volume":"107","author":"Maji","year":"2022","journal-title":"Nonlinear Dyn"},{"key":"B33","doi-asserted-by":"publisher","first-page":"110420","DOI":"10.1016\/j.chaos.2020.110420","article-title":"Impact of predator incited fear and prey refuge in a fractional order prey predator model","volume":"142","author":"Barman","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"B34","doi-asserted-by":"publisher","first-page":"105","DOI":"10.5614\/cbms.2019.2.2.4","article-title":"Dynamics of a fractional-order predator-prey model with infectious diseases in prey","volume":"2","author":"Panigoro","year":"2019","journal-title":"Commun Biomath Sci"},{"key":"B35","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.cnsns.2014.12.013","article-title":"Volterra-type Lyapunov functions for fractional-order epidemic systems","volume":"24","author":"Vargas-De-Le\u00f3n","year":"2015","journal-title":"Commun Nonlinear Sci Num Simulat"},{"key":"B36","doi-asserted-by":"publisher","first-page":"2951","DOI":"10.1016\/j.cnsns.2014.01.022","article-title":"Lyapunov functions for fractional order systems","volume":"19","author":"Aguila-Camacho","year":"2014","journal-title":"Commun Nonlinear Sci Num Simulat"},{"key":"B37","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1023\/A:1016592219341","article-title":"A predictor-corrector approach for the numerical solution of fractional differential equations","volume":"29","author":"Diethelm","year":"2002","journal-title":"Nonlinear Dyn"},{"key":"B38","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.tpb.2021.05.002","article-title":"Bifurcations and hydra effects in Bazykin's predator-prey model","volume":"140","author":"Adhikary","year":"2021","journal-title":"Theor Populat Biol"}],"container-title":["Frontiers in Applied Mathematics and Statistics"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2022.1077831\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T15:34:54Z","timestamp":1673019294000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2022.1077831\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,6]]},"references-count":38,"alternative-id":["10.3389\/fams.2022.1077831"],"URL":"https:\/\/doi.org\/10.3389\/fams.2022.1077831","relation":{},"ISSN":["2297-4687"],"issn-type":[{"type":"electronic","value":"2297-4687"}],"subject":[],"published":{"date-parts":[[2023,1,6]]},"article-number":"1077831"}}