{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T12:05:57Z","timestamp":1784635557738,"version":"3.55.0"},"reference-count":47,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T00:00:00Z","timestamp":1748563200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Program for Middle-aged and Young Scholars of Fujian Provincial Department of Education Fujian Province","award":["JAT220511"],"award-info":[{"award-number":["JAT220511"]}]},{"name":"Science and Technology Program for Middle-aged and Young Scholars of Fujian Provincial Department of Education Fujian Province","award":["2024J01273"],"award-info":[{"award-number":["2024J01273"]}]},{"name":"Science and Technology Program for Middle-aged and Young Scholars of Fujian Provincial Department of Education Fujian Province","award":["2023J01163"],"award-info":[{"award-number":["2023J01163"]}]},{"name":"Science and Technology Program for Middle-aged and Young Scholars of Fujian Provincial Department of Education Fujian Province","award":["ZCKC23005"],"award-info":[{"award-number":["ZCKC23005"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["JAT220511"],"award-info":[{"award-number":["JAT220511"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["2024J01273"],"award-info":[{"award-number":["2024J01273"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["2023J01163"],"award-info":[{"award-number":["2023J01163"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["ZCKC23005"],"award-info":[{"award-number":["ZCKC23005"]}]},{"name":"Science and Education Innovation Group Cultivation Project of Fuzhou University Zhicheng College","award":["JAT220511"],"award-info":[{"award-number":["JAT220511"]}]},{"name":"Science and Education Innovation Group Cultivation Project of Fuzhou University Zhicheng College","award":["2024J01273"],"award-info":[{"award-number":["2024J01273"]}]},{"name":"Science and Education Innovation Group Cultivation Project of Fuzhou University Zhicheng College","award":["2023J01163"],"award-info":[{"award-number":["2023J01163"]}]},{"name":"Science and Education Innovation Group Cultivation Project of Fuzhou University Zhicheng College","award":["ZCKC23005"],"award-info":[{"award-number":["ZCKC23005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study investigates a modified Lotka\u2013Volterra commensalism system that incorporates the weak Allee effect in prey and symmetric (equal harvesting effort for both species) non-selective harvesting, addressing a critical gap in ecological modeling. Unlike previous work, we rigorously examine how the interaction between the Allee effect and harvesting disrupts system stability, giving rise to novel bifurcation phenomena and population collapse thresholds. Using eigenvalue analysis and the Dulac\u2013Bendixson criterion, we derive sufficient conditions for the existence and stability of equilibria. We find that harvesting destabilizes the system by inducing two saddle-node bifurcations. Notably, prey abundance can increase with greater Allee intensity under controlled harvesting\u2014a rare and counterintuitive ecological outcome. Moreover, exceeding a critical harvesting threshold drives both species to extinction, while controlled harvesting allows sustainable coexistence. Numerical simulations support these analytical findings and identify critical parameter ranges for species coexistence. These results contribute to theoretical ecology and offer insights for designing sustainable harvesting strategies that balance exploitation with conservation.<\/jats:p>","DOI":"10.3390\/sym17060852","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T09:22:41Z","timestamp":1748596961000},"page":"852","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Dynamics of a Modified Lotka\u2013Volterra Commensalism System Incorporating Allee Effect and Symmetric Non-Selective Harvest"],"prefix":"10.3390","volume":"17","author":[{"given":"Kan","family":"Fang","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Fuzhou University Zhicheng College, Fuzhou 350002, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yiqin","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics Research, Fujian Institute of Education, Fuzhou 350001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3617-5550","authenticated-orcid":false,"given":"Fengde","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoying","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mathematics Research, Fujian Institute of Education, Fuzhou 350001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s40435-024-01551-z","article-title":"Exploring periodic behavior and dynamical analysis in a harvested discrete-time commensalism system","volume":"13","author":"Ditta","year":"2025","journal-title":"Int. J. Dyn. Control."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108381","DOI":"10.1016\/j.aml.2022.108381","article-title":"Note on the persistence and stability property of a commensalism model with Michaelis-Menten harvesting and Holling type II commensalistic benefit","volume":"134","author":"Chen","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"212","DOI":"10.17268\/sel.mat.2024.02.01","article-title":"A seasonal commensalism model with a weak Allee effect to describe climate-mediated shifts","volume":"11","author":"Osuna","year":"2024","journal-title":"Geiser. Sel. Mat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4111","DOI":"10.1007\/s12190-023-01913-4","article-title":"Dynamical analysis of a Beddington-DeAngelis commensalism system with two time delays","volume":"69","author":"Qu","year":"2023","journal-title":"J. Appl. Math. Comput."},{"key":"ref_5","first-page":"1","article-title":"Stability and Bifurcation Analysis of a Commensal Model with Allee Effect and Herd Behavior","volume":"32","author":"Cai","year":"2022","journal-title":"Open Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2369","DOI":"10.1007\/s40435-023-01367-3","article-title":"Dynamics of stability, bifurcation and control for a commensal symbiosis model","volume":"12","author":"Patra","year":"2024","journal-title":"Int. J. Dyn. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"18793","DOI":"10.3934\/math.20221034","article-title":"Stability property of the boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in commensal populations","volume":"7","author":"Liu","year":"2022","journal-title":"AIMS Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Xu, L.L., Xue, Y.L., Xie, X.D., and Lin, Q.F. (2022). Dynamic Behaviors of an Obligate Commensal Symbiosis Model with Crowley\u2013Martin Functional Responses. Axioms, 11.","DOI":"10.3390\/axioms11060298"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"515","DOI":"10.37394\/23206.2022.21.57","article-title":"Positive periodic solution of a discrete Lotka-Volterra commensal symbiosis model with Michaelis\u2013Menten type harvesting","volume":"21","author":"Xu","year":"2022","journal-title":"WSEAS Trans. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2150204","DOI":"10.1142\/S0218127421502047","article-title":"Stability and bifurcation analysis of a commensal model with additive Allee effect and nonlinear growth rate","volume":"13","author":"Wei","year":"2021","journal-title":"Internat. J. Bifur. Chaos."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1515\/math-2019-0134","article-title":"Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales","volume":"17","author":"Xue","year":"2019","journal-title":"Open Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1186\/s13662-019-1989-4","article-title":"The influence of commensalism on a Lotka-Volterra commensal symbiosis model with Michaelis-Menten type harvesting","volume":"2019","author":"Chen","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","unstructured":"Deng, H., and Huang, X. (2018). The influence of partial closure for the populations to a harvesting Lotka-Volterra commensalism model. Commun. Math. Biol. Neurosci., 2018."},{"key":"ref_14","unstructured":"Han, R.Y., and Chen, F.D. (2015). Global stability of a commensal symbiosis model with feedback controls. Commun. Math. Biol. Neurosci., 2015."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Gao, W.J., Jia, X., and Shi, R.Q. (2024). Dynamics and optimal harvesting for fishery models with reserved areas. Symmetry, 16.","DOI":"10.3390\/sym16070800"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"53","DOI":"10.3390\/axioms14010053","article-title":"Dynamics of a Fractional-Order Eco-Epidemiological Model with Two Disease Strains in a Predator Population Incorporating Harvesting","volume":"14","author":"Moustafa","year":"2025","journal-title":"Axioms"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"He, M.X., and Li, Z. (2024). Bifurcation of a Leslie\u2013Gower Predator\u2013Prey Model with Nonlinear Harvesting and a Generalist Predator. Axioms, 13.","DOI":"10.3390\/axioms13100704"},{"key":"ref_18","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_19","doi-asserted-by":"crossref","unstructured":"Kashyap, A.J., Zhu, Q.X., Sarmah, H.K., and Bhattacharjee, D. (2023). Dynamical study of a predator\u2013prey system with Michaelis\u2013Menten type predator-harvesting. Int. J. Biomath., 16.","DOI":"10.1142\/S1793524522501352"},{"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":"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_22","doi-asserted-by":"crossref","unstructured":"Allee, W.C. (1932). Animal Aggregations: A Study in General Sociology, University of Chicago Press.","DOI":"10.5962\/bhl.title.7313"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"384","DOI":"10.2307\/3546980","article-title":"Allee effect and population dynamics in the glanville fritillary butterfly","volume":"82","author":"Kuussaari","year":"1998","journal-title":"Oikos"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/S0169-5347(99)01684-5","article-title":"Consequences of the Allee effect for behaviour, ecology and conservation","volume":"14","author":"Stephens","year":"1999","journal-title":"Trends Ecol. Evol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1098\/rspb.1999.0672","article-title":"Population dynamics of obligate cooperators","volume":"266","author":"Courchamp","year":"1999","journal-title":"Proc. R. Soc. Lond. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1552","DOI":"10.3390\/sym16111552","article-title":"Dynamical complexity of modified Leslie\u2013Gower predator\u2014Prey model incorporating double Allee effect and fear effect","volume":"16","author":"Manoj","year":"2024","journal-title":"Symmetry"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"115994","DOI":"10.1016\/j.chaos.2025.115994","article-title":"Bifurcations in a Leslie\u2013Gower predator\u2014Prey model with strong Allee effects and constant prey refuges","volume":"192","author":"Chen","year":"2025","journal-title":"Chaos Solitons Fract."},{"key":"ref_28","first-page":"2350077","article-title":"Dynamical analysis of a Lotka-Volterra competition model with both Allee and fear effects","volume":"17","author":"Chen","year":"2024","journal-title":"Int. J. Bifurc. Chaos."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Wang, F., and Yang, R. (2023). Dynamics of a delayed reaction\u2013diffusion predator\u2013prey model with nonlocal competition and double Allee effect in prey. IInt. J. Biomath., 1.","DOI":"10.1142\/S1793524523500973"},{"key":"ref_30","first-page":"302","article-title":"Influence of the strong Allee effect on prey and the competition among predators in Leslie-Gower type predation models","volume":"7","year":"2020","journal-title":"Int. J. Bifurcat. Chaos."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"24","DOI":"10.17268\/sel.mat.2022.01.02","article-title":"Stability of a Leslie-Gower type predator-prey model with a strong Allee effect with delay","volume":"9","author":"Cruz","year":"2022","journal-title":"Sel. Mat."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.matcom.2024.01.015","article-title":"Turing patterns in a predator\u2013prey model with double Allee effect","volume":"220","author":"Wang","year":"2024","journal-title":"Math. Comput. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2150050","DOI":"10.1142\/S0218127421500504","article-title":"Global dynamics of a Holling-II amensalism system with nonlinear growth rate and Allee effect on the first species","volume":"31","author":"Luo","year":"2021","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_34","unstructured":"Guan, X.Y., Liu, Y., and Xie, X.D. (2018). Stability analysis of a Lotka-Volterra type predator-prey system with Allee effect on the predator species. Commun. Math. Biol. Neurosci., 2018."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s12346-024-01000-4","article-title":"Impact of fear on searching efficiency of first species: A two species Lotka-Volterra competition model with weak Allee effect","volume":"23","author":"Zhu","year":"2024","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/s12346-022-00591-0","article-title":"Qualitative and Bifurcation Analysis in a Leslie-Gower Model with Allee Effect","volume":"21","author":"Fang","year":"2022","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_37","first-page":"220201","article-title":"Qualitative and Bifurcation Analysis of a Single Species Logistic Model with Allee Effect and Feedback Control","volume":"52","author":"Fang","year":"2022","journal-title":"IAENG Int. J. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2050169","DOI":"10.1142\/S0218127420501692","article-title":"Spatiotemporal dynamics of a modified Leslie-Gower model with weak Allee effect","volume":"30","author":"Liu","year":"2020","journal-title":"Int. J. Bifurcat. Chaos."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"8857","DOI":"10.3934\/mbe.2021437","article-title":"Dynamical analysis of an aquatic amensalism model with non-selective harvesting and Allee effect","volume":"18","author":"Liu","year":"2021","journal-title":"Min. Math. Biosci. Eng."},{"key":"ref_40","first-page":"1","article-title":"Stability and bifurcation analysis of an Amensalism model with weak Allee effect","volume":"23","author":"Wei","year":"2020","journal-title":"Qual. Theor. Dyn. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1186\/s13662-018-1647-2","article-title":"Stability analysis of a single species logistic model with Allee effect and feedback control","volume":"2018","author":"Lin","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Mustafa, N., Rahman, J.U., Ishtiaq, U., and Popa, I.L. (2025). Artificial Neural Network-Based Approach for Dynamic Analysis and Modeling of Marburg Virus Epidemics for Health Care. Symmetry, 17.","DOI":"10.3390\/sym17040578"},{"key":"ref_43","first-page":"33","article-title":"On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay","volume":"180","author":"Chen","year":"2005","journal-title":"Commun. Math. Biol. Neurosci."},{"key":"ref_44","unstructured":"Zhang, Z.F., Ding, T.R., Huang, W.Z., and Dong, Z.X. (1992). Qualitative Theory of Differential Equation, American Mathematical Society. Translation of Mathematical Monographs."},{"key":"ref_45","unstructured":"Chen, L.S. (1988). Mathematical Models and Methods in Ecology, Scienec Press."},{"key":"ref_46","unstructured":"Perko, L. (2013). Differential Equations and Dynamical Systems, Springer."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"7337","DOI":"10.1007\/s40808-024-02169-y","article-title":"Modelling of Marburg virus transmission dynamics: A deep learning-driven approah with the efect of quarantine and health awareness interventions","volume":"10","author":"Mustafa","year":"2024","journal-title":"Model. Earth Syst. Environ."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/852\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:43:46Z","timestamp":1760031826000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/852"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,30]]},"references-count":47,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["sym17060852"],"URL":"https:\/\/doi.org\/10.3390\/sym17060852","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,30]]}}}