{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T15:54:05Z","timestamp":1769356445714,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,12]],"date-time":"2024-10-12T00:00:00Z","timestamp":1728691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Fujian Province","award":["2021J01613"],"award-info":[{"award-number":["2021J01613"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["2021J011032"],"award-info":[{"award-number":["2021J011032"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["MJY22027"],"award-info":[{"award-number":["MJY22027"]}]},{"name":"Scientific Research Foundation of Minjiang University","award":["2021J01613"],"award-info":[{"award-number":["2021J01613"]}]},{"name":"Scientific Research Foundation of Minjiang University","award":["2021J011032"],"award-info":[{"award-number":["2021J011032"]}]},{"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>A Leslie\u2013Gower predator\u2013prey model with nonlinear harvesting and a generalist predator is considered in this paper. It is shown that the degenerate positive equilibrium of the system is a cusp of codimension up to 4, and the system admits the cusp-type degenerate Bogdanov\u2013Takens bifurcation of codimension 4. Moreover, the system has a weak focus of at least order 3 and can undergo degenerate Hopf bifurcation of codimension 3. We verify, through numerical simulations, that the system admits three different stable states, such as a stable fixed point and three limit cycles (the middle one is unstable), or two stable fixed points and two limit cycles. Our results reveal that nonlinear harvesting and a generalist predator can lead to richer dynamics and bifurcations (such as three limit cycles or tristability); specifically, harvesting can cause the extinction of prey, but a generalist predator provides some protection for the predator in the absence of prey.<\/jats:p>","DOI":"10.3390\/axioms13100704","type":"journal-article","created":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T10:11:04Z","timestamp":1729073464000},"page":"704","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Bifurcation of a Leslie\u2013Gower Predator\u2013Prey Model with Nonlinear Harvesting and a Generalist Predator"],"prefix":"10.3390","volume":"13","author":[{"given":"Mengxin","family":"He","sequence":"first","affiliation":[{"name":"School of Computer and Big Data, Minjiang University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,12]]},"reference":[{"key":"ref_1","first-page":"241","article-title":"A natural population norm","volume":"3","author":"Lotka","year":"1913","journal-title":"J. Wash. Acad. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1038\/118558a0","article-title":"Fluctuations in the abundance of a species considered mathematically","volume":"118","author":"Volterra","year":"1926","journal-title":"Nature"},{"key":"ref_3","first-page":"1","article-title":"The functional response of predators to prey density and its role in mimicry and population regulation","volume":"45","author":"Holling","year":"1965","journal-title":"Mem. Entomol. Soc. Can."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"108198","DOI":"10.1016\/j.cnsns.2024.108198","article-title":"Dynamics of a predator\u2013prey system with foraging facilitation andgroup defense","volume":"138","author":"Yao","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_5","first-page":"328","article-title":"Impact of the fear effect in a prey-predator model incorporating a prey refuge","volume":"356","author":"Zhang","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jde.2023.11.017","article-title":"Hydra effect and global dynamics of predation with strong Allee effect in prey and intraspecific competition in predator","volume":"384","author":"Bai","year":"2024","journal-title":"J. Differ. Equations"},{"key":"ref_7","first-page":"321","article-title":"Stability analysis of a diffusive predator\u2013prey model with Hunting cooperation","volume":"3","author":"Wu","year":"2021","journal-title":"J. Nonlinear Model. Anal."},{"key":"ref_8","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. Equations"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1093\/biomet\/35.3-4.213","article-title":"Some further notes on the use of matrices in population mathematics","volume":"35","author":"Leslie","year":"1948","journal-title":"Biometrika"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1093\/biomet\/45.1-2.16","article-title":"A stochastic model for studying the properties of certain biological systems by numerical methods","volume":"45","author":"Leslie","year":"1958","journal-title":"Biometrika"},{"key":"ref_11","unstructured":"Pielou, E.C. (1977). Mathematical Ecology, John Wiley & Sons. [2nd ed.]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/S0893-9659(01)80029-X","article-title":"A Lyapunov function for Leslie\u2013Gower predator\u2013prey models","volume":"14","author":"Korobeinikov","year":"2001","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"855","DOI":"10.2307\/1936296","article-title":"The stability and the intrinsic growth rates of prey and predator populations","volume":"56","author":"Tanner","year":"1975","journal-title":"Ecology"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1137\/S0036139993253201","article-title":"Global stability for a class of predator\u2013prey system","volume":"55","author":"Hsu","year":"1995","journal-title":"SIAM J. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1721","DOI":"10.1016\/j.jde.2014.04.024","article-title":"Bifurcations in a predator\u2013prey system of Leslie type with generalized Holling type III functional response","volume":"257","author":"Huang","year":"2014","journal-title":"J. Differ. Equations"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.nonrwa.2019.04.003","article-title":"Four limit cycles in a predator\u2013prey system of Leslie type with generalized Holling type III functional response","volume":"50","author":"Dai","year":"2019","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"108561","DOI":"10.1016\/j.aml.2022.108561","article-title":"Global dynamics of a Leslie\u2013Gower predator\u2013prey model with square root response function","volume":"140","author":"He","year":"2023","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1016\/S0893-9659(03)90096-6","article-title":"Boundedness and global stability for a predator\u2013prey model with modified Leslie\u2013Gower and Holling-type II schemes","volume":"16","author":"Okiye","year":"2003","journal-title":"Appl. Math. Lett."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1137\/22M1488466","article-title":"An organizing center of codimension four in a predator\u2013prey 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_21","doi-asserted-by":"crossref","first-page":"109193","DOI":"10.1016\/j.aml.2024.109193","article-title":"Dynamics of a Lesile-Gower predator\u2013prey model with square root response function and generalist predator","volume":"157","author":"He","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107109","DOI":"10.1016\/j.cnsns.2023.107109","article-title":"Dynamic complexity of a modified Leslie\u2013Gower predator\u2013prey system with fear effect","volume":"119","author":"Chen","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"113794","DOI":"10.1016\/j.chaos.2023.113794","article-title":"Stability and Hopf bifurcation of a modified Leslie\u2013Gower predator\u2013prey model with Smith growth rate and B-D functional response","volume":"174","author":"Feng","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","unstructured":"Clark, C.W. (1990). Mathematical Bioeconomics, The Optimal Management of Renewable Resources, John Wiley and Sons. [2nd ed.]."},{"key":"ref_25","first-page":"2101","article-title":"Bifurcations analysis in a predator\u2013prey model with constant-yield predator harvesting","volume":"18","author":"Huang","year":"2013","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/j.jde.2022.01.016","article-title":"Degenerate Bogdanov\u2013Takens bifurcation of codimension 4 in Holling-Tanner model with harvesting","volume":"314","author":"Xiang","year":"2022","journal-title":"J. Differ. Equations"},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"103995","DOI":"10.1016\/j.nonrwa.2023.103995","article-title":"Degenerate codimension-2 cusp of limit cycles in a Holling-Tanner model with harvesting and anti-predator behavior","volume":"76","author":"Xu","year":"2024","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"106800","DOI":"10.1016\/j.cnsns.2022.106800","article-title":"Bifurcations on a discontinuous Leslie-Grower model with harvesting and alternative food for predators and Holling II functional response","volume":"116","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_30","first-page":"289","article-title":"Phase portraits, Hopf bifurcations and limit cycles of Leslie\u2013Gower predator\u2013prey systems with harvesting rates","volume":"14","author":"Zhu","year":"2010","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s12591-012-0142-6","article-title":"Bifurcation analysis and control of Leslie\u2013Gower predator\u2013prey model with Michaelis-Menten type prey-harvesting","volume":"20","author":"Gupta","year":"2012","journal-title":"Differ. Equations Dyn. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"6715","DOI":"10.1002\/mma.4484","article-title":"Bogdanov\u2013Takens bifurcations of codimensions 2 and 3 in a Leslie\u2013Gower predator\u2013prey model with Michaelis\u2013Menten\u2013type prey harvesting","volume":"40","author":"Kong","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1721","DOI":"10.1007\/s10884-022-10242-2","article-title":"Cyclicity of the limit periodic sets for a singularly perturbed Leslie\u2013Gower predator\u2013prey model with prey harvesting","volume":"36","author":"Yao","year":"2024","journal-title":"J. Dyn. Differ. Equations"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7708","DOI":"10.3934\/mbe.2020392","article-title":"Global stability in a modified Leslie\u2013Gower type predation model assuming mutual interference among generalist predators","volume":"17","year":"2020","journal-title":"Math. Biosci. Eng."},{"key":"ref_35","unstructured":"Zhang, Z.F., Ding, T.R., Huang, W.Z., and Dong, Z.X. (1992). Qualitative Theory of Diffrential Equations, American Mathematical Society. Translation of Mathematical Monographs."},{"key":"ref_36","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."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/0022-0396(89)90117-4","article-title":"A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4","volume":"79","author":"Li","year":"1989","journal-title":"J. Differ. Equations"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/j.jde.2022.10.018","article-title":"Bifurcations in Holling-Tanner model with generalist predator and prey refuge","volume":"343","author":"Xiang","year":"2023","journal-title":"J. Differ. Equations"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/j.cam.2009.06.029","article-title":"Decomposition of algebraic sets and applications to weak centers of cubic systems","volume":"23","author":"Chen","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"100939","DOI":"10.1016\/j.ecocom.2021.100939","article-title":"Bubbling and Hydra Effect in a Population System with Allee Effect","volume":"47","author":"Garain","year":"2021","journal-title":"Ecol. Complex."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/704\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:12:16Z","timestamp":1760112736000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/704"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,12]]},"references-count":40,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["axioms13100704"],"URL":"https:\/\/doi.org\/10.3390\/axioms13100704","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,12]]}}}