{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T05:36:08Z","timestamp":1767245768024,"version":"3.48.0"},"reference-count":33,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T00:00:00Z","timestamp":1767052800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2025J011265"],"award-info":[{"award-number":["2025J011265"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific Research Foundation of Minjiang University","award":["MJY22027"],"award-info":[{"award-number":["MJY22027"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we consider a generalist predator\u2013prey model with nonlinear harvesting, which has at most eight non-negative equilibria. We prove that the double positive equilibrium is a cusp of codimension up to 3; therefore, the system exhibits a cusp-type degenerate Bogdanov\u2013Takens bifurcation of the same codimension. The elementary antisaddle equilibrium can act as a weak focus of the order of no more than two, giving rise to a degenerate Hopf bifurcation of codimension up to two. These high-codimension bifurcations identify organizing centers in parameter space, indicating regions where the ecosystem is highly sensitive and prone to abrupt regime shifts. Our results indicate that the generalist predator can induce a richer bifurcation phenomenon and more complex dynamics and can drive the system to certain desired stable states.<\/jats:p>","DOI":"10.3390\/axioms15010031","type":"journal-article","created":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T15:05:01Z","timestamp":1767193501000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bifurcation Analysis of a Generalist Predator-Prey Model with Holling Type II Harvesting"],"prefix":"10.3390","volume":"15","author":[{"given":"Mengxin","family":"He","sequence":"first","affiliation":[{"name":"School of Computer and Data Science, Minjiang University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiqin","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics Research, Fujian Institute of Education, Fuzhou 350025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,30]]},"reference":[{"key":"ref_1","unstructured":"Lotka, A.J. (1925). Elements of Physical Biology, Williams and Wilkins."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1038\/118558a0","article-title":"Fluctuations in the abundance of species considered mathematically","volume":"118","author":"Volterra","year":"1926","journal-title":"Nature"},{"key":"ref_3","unstructured":"Bazykin, A.D. (1976). Structural and Dynamic Stability of Model Predator-Prey Systems, International Institute for Applied Systems Analysis."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kuznetsov, Y. (2004). Elements of Applied Bifurcation Theory, Springer. [3rd ed.].","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.jde.2021.01.025","article-title":"Global analysis in Bazykin\u2019s model with Holling II functional response and predator competition","volume":"280","author":"Lu","year":"2021","journal-title":"J. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s12346-023-00940-7","article-title":"The impact of Allee effect on a Leslie-Gower predator-prey model with hunting cooperation","volume":"23","author":"Liu","year":"2024","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_7","first-page":"2152","article-title":"Bifurcation analysis in a modified Leslie-Gower with nonlocal competition and Beddington-DeAngelis functional response","volume":"15","author":"Ma","year":"2025","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"107936","DOI":"10.1016\/j.cnsns.2024.107936","article-title":"Double Hopf bifurcation induced by spatial memory in a diffusive predator-prey model with Allee effect and maturation delay of predator","volume":"132","author":"Li","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2250086","DOI":"10.1142\/S0218127422500869","article-title":"Modeling Allee effect in the Lesile-Gower predator-prey system incorporating a prey refuge","volume":"32","author":"Yin","year":"2022","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_10","first-page":"267","article-title":"Spatial dynamics of a diffusive predator-prey model with Leslie-Gower functional response and strong Allee effect","volume":"2","author":"Wei","year":"2020","journal-title":"J. Nonlinear Model. Anal."},{"key":"ref_11","first-page":"274","article-title":"Stability analysis of a delayed predator-prey model with generalist predator and fear effect","volume":"228","author":"Guo","year":"2025","journal-title":"Math. Comput. Simul."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3730","DOI":"10.3934\/dcdsb.2025043","article-title":"Bifurcation analysis of a modified Leslie-Gower predator-prey system with fear effect on prey","volume":"30","author":"Zhang","year":"2025","journal-title":"Discret. Contin. Dyn.-Syst.-B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1111\/sapm.12492","article-title":"Linking bifurcation analysis of Holling-Tanner model with generalist predator to a changing environment","volume":"49","author":"Xiang","year":"2022","journal-title":"Stud. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/imammb\/dqm011","article-title":"Control of invasive hosts by generalist parasitoids","volume":"25","author":"Magal","year":"2008","journal-title":"Math. Med. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4618","DOI":"10.1016\/j.jde.2019.10.036","article-title":"Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response","volume":"268","author":"Xiang","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2833","DOI":"10.1007\/s10884-021-10079-1","article-title":"Dynamics complexity of generalist predatory mite and the leafhopper pest in tea plantations","volume":"35","author":"Yuan","year":"2023","journal-title":"J. Dyn. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1016\/j.apm.2020.09.046","article-title":"Allee effect in prey\u2019s growth reduces the dynamical complexity in prey-predator model with generalist predator","volume":"91","author":"Sen","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1137\/22M1488466","article-title":"An organizing center of codimension four in a predator-prey model with generalist predator: From tristability and quadristability to transients in a nonlinear environmental change","volume":"22","author":"Lu","year":"2023","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109193","DOI":"10.1016\/j.aml.2024.109193","article-title":"Dynamics of a Leslie-Gower predator-prey model with square root response function and generalist predator","volume":"157","author":"He","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2450076","DOI":"10.1142\/S0218127424500767","article-title":"Bifurcation analysis of a Holling-Tanner model with generalist predator and constant-yield harvesting","volume":"34","author":"Wu","year":"2024","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"116132","DOI":"10.1016\/j.chaos.2025.116132","article-title":"Cooperation and harvesting-induced delays in a predator-prey model with prey fear response: A crossing curves approach","volume":"194","author":"Singh","year":"2025","journal-title":"Chaos Solitons Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.matcom.2024.09.001","article-title":"Bistability in a predator-prey model characterized by the Crowley-Martin functional response: Effects of fear, hunting cooperation, additional foods and nonlinear harvesting","volume":"228","author":"Roy","year":"2025","journal-title":"Math. Comput. Simul."},{"key":"ref_23","first-page":"550023","article-title":"Dynamics of a harvested Leslie-Gower predator-prey model with simplified Holling type IV functional response","volume":"25","author":"Huangfu","year":"2025","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.nonrwa.2016.05.010","article-title":"Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting","volume":"33","author":"Hu","year":"2017","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"111071","DOI":"10.1016\/j.chaos.2021.111071","article-title":"Dynamic analysis of a predator-prey system with nonlinear prey harvesting and square root functional response","volume":"148","author":"Mortuja","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.jmaa.2012.08.057","article-title":"Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting","volume":"398","author":"Gupta","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"109732","DOI":"10.1016\/j.aml.2025.109732","article-title":"Dynamics of a Holling-Tanner predator-prey model with harvesting and anti-predator behavior","volume":"172","author":"He","year":"2026","journal-title":"Appl. Math. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"He, M.X., and Li, Z. (2024). Bifurcation of a Leslie-Gower predator-prey model with nonlinear harvesting and a generalist predator. Axioms, 13.","DOI":"10.3390\/axioms13100704"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.cam.2004.10.001","article-title":"On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay","volume":"180","author":"Chen","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"ref_30","unstructured":"Zhang, Z.F., Ding, T.R., Huang, W.Z., and Dong, Z.X. (1992). Qualitative Theory of Diffrential Equations, American Mathematical Society."},{"key":"ref_31","first-page":"2101","article-title":"Bifurcations analysis in a predator-prey model with constant-yield predator harvesting","volume":"18","author":"Huang","year":"2013","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s12346-025-01291-1","article-title":"Bifurcation analysis of a Leslie-Gower predator-prey model with Allee effect on predator and simplified Holling type IV functional response","volume":"24","author":"Zhang","year":"2025","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1017\/S0143385700004119","article-title":"Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3","volume":"7","author":"Dumortier","year":"1987","journal-title":"Ergod. Theory Dyn. Syst."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T05:32:58Z","timestamp":1767245578000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,30]]},"references-count":33,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010031"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010031","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,12,30]]}}}