{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T11:56:59Z","timestamp":1771329419277,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T00:00:00Z","timestamp":1625011200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T00:00:00Z","timestamp":1625011200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Supercomput"],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s11227-021-03956-5","type":"journal-article","created":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T12:03:08Z","timestamp":1625054588000},"page":"2477-2502","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Application of homotopy perturbation and variational iteration methods for nonlinear imprecise prey\u2013predator model with stability analysis"],"prefix":"10.1007","volume":"78","author":[{"given":"T.","family":"Vijayalakshmi","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4853-5912","authenticated-orcid":false,"given":"R.","family":"Senthamarai","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,30]]},"reference":[{"key":"3956_CR1","first-page":"82","volume-title":"Elements of physical biology.","author":"AJ Lotka","year":"1925","unstructured":"Lotka AJ (1925) Elements of physical biology., vol 21. Williams and Wilkins, Baltimore, p 82"},{"key":"3956_CR2","unstructured":"Volterra V (1926) Variazioni e fluttuazioni del numero di\u02c6aindividui in specie animali conviventi,\u201d Mem. R. Accad. Naz. Dei Lincei, vol 2"},{"key":"3956_CR3","volume-title":"Elements of mathematical biology","author":"M Kot","year":"2001","unstructured":"Kot M (2001) Elements of mathematical biology. Cambridge University Press, Cambridge, UK"},{"key":"3956_CR4","unstructured":"Smith JM, Models in ecology. CUP Arch (1978)"},{"key":"3956_CR5","volume-title":"M","author":"R May","year":"2001","unstructured":"May R (2001) M, 2nd edn. Stability and complexity in model ecosystems, Princeton University Press","edition":"2"},{"issue":"2","key":"3956_CR6","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1137\/0144026","volume":"44","author":"JM Cushing","year":"1984","unstructured":"Cushing JM (1984) Periodic two-predator, one-prey interactions and the time sharing of a resource niche. SIAM J Appl Math 44(2):392\u2013410. https:\/\/doi.org\/10.1137\/0144026","journal-title":"SIAM J Appl Math"},{"issue":"6","key":"3956_CR7","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1007\/BF00276947","volume":"27","author":"KP Hadeler","year":"1989","unstructured":"Hadeler KP, Freedman HI (1989) Predator-prey populations with parasitic infection. J Math Biol 27(6):609\u2013631. https:\/\/doi.org\/10.1007\/BF00276947","journal-title":"J Math Biol"},{"issue":"5","key":"3956_CR8","doi-asserted-by":"publisher","first-page":"2905","DOI":"10.1016\/j.nonrwa.2008.09.009","volume":"10","author":"F Chen","year":"2009","unstructured":"Chen F, Chen L, Xie X (2009) On a Leslie\u2013Gower predator-prey model incorporating a prey refuge. Nonlinear Anal Real World Appl 10(5):2905\u20132908. https:\/\/doi.org\/10.1016\/j.nonrwa.2008.09.009","journal-title":"Nonlinear Anal Real World Appl"},{"key":"3956_CR9","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/S0895-7177(03)90099-9","volume":"38","author":"TK Kar","year":"2003","unstructured":"Kar TK (2003) Selective harvesting in a prey-predator fishery with time delay. Math Comput Model 38:449\u2013458. https:\/\/doi.org\/10.1016\/S0895-7177(03)90099-9","journal-title":"Math Comput Model"},{"issue":"1","key":"3956_CR10","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.cam.2005.01.035","volume":"185","author":"TK Kar","year":"2006","unstructured":"Kar TK (2006) Modelling and analysis of a harvested prey-predator system incorporating a prey refuge. J Comput Appl Math 185(1):19\u201333. https:\/\/doi.org\/10.1016\/j.cam.2005.01.035","journal-title":"J Comput Appl Math"},{"key":"3956_CR11","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.jtbi.2012.06.032","volume":"310","author":"TK Kar","year":"2012","unstructured":"Kar TK, Ghorai A, Jana S (2012) Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide. J Theor Biol 310:187\u2013198. https:\/\/doi.org\/10.1016\/j.jtbi.2012.06.032","journal-title":"J Theor Biol"},{"issue":"3","key":"3956_CR12","doi-asserted-by":"publisher","first-page":"1805","DOI":"10.1007\/s11071-012-0575-z","volume":"70","author":"K Chakraborty","year":"2012","unstructured":"Chakraborty K, Jana S, Kar TK (2012) Effort dynamics of a delay-induced prey-predator system with reserve. Nonlinear Dyn 70(3):1805\u20131829. https:\/\/doi.org\/10.1007\/s11071-012-0575-z","journal-title":"Nonlinear Dyn"},{"issue":"3","key":"3956_CR13","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1007\/s002850100095","volume":"43","author":"A Martin","year":"2001","unstructured":"Martin A, Ruan S (2001) Predator-prey models with delay and prey harvesting. J Math Biol 43(3):247\u2013267. https:\/\/doi.org\/10.1007\/s002850100095","journal-title":"J Math Biol"},{"issue":"4","key":"3956_CR14","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1016\/j.cnsns.2004.12.011","volume":"11","author":"TK Kar","year":"2006","unstructured":"Kar TK, Pahari UK (2006) Non-selective harvesting in prey predator models with delay. Commun Nonlinear Sci Numer Simul 11(4):499\u2013509. https:\/\/doi.org\/10.1016\/j.cnsns.2004.12.011","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1\u20132","key":"3956_CR15","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s11071-010-9923-z","volume":"66","author":"Y Zhang","year":"2011","unstructured":"Zhang Y, Zhang Q (2011) Dynamical behavior in a delayed stage-structure population model with stochastic fluctuation and harvesting. Nonlinear Dyn 66(1\u20132):231\u2013245. https:\/\/doi.org\/10.1007\/s11071-010-9923-z","journal-title":"Nonlinear Dyn"},{"key":"3956_CR16","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.matcom.2012.10.003","volume":"85","author":"S Jana","year":"2012","unstructured":"Jana S, Chakraborty M, Chakraborty K, Kar TK (2012) Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge. Math Comput Simul 85:57\u201377. https:\/\/doi.org\/10.1016\/j.matcom.2012.10.003","journal-title":"Math Comput Simul"},{"key":"3956_CR17","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/j.amc.2014.10.097","volume":"250","author":"S Jana","year":"2015","unstructured":"Jana S, Ghorai A, Guria S, Kar TK (2015) Global dynamics of a predator, weaker prey and stronger prey system. Appl Math Comput 250:235\u2013248. https:\/\/doi.org\/10.1016\/j.amc.2014.10.097","journal-title":"Appl Math Comput"},{"issue":"7","key":"3956_CR18","doi-asserted-by":"publisher","first-page":"1493","DOI":"10.1007\/s11538-016-0192-y","volume":"78","author":"D Pal","year":"2016","unstructured":"Pal D, Mahapatra GS, Samanta GP (2016) Stability and bionomic analysis of fuzzy prey-predator harvesting model in presence of toxicity: a dynamic approach. Bull Math Biol 78(7):1493\u20131519. https:\/\/doi.org\/10.1007\/s11538-016-0192-y","journal-title":"Bull Math Biol"},{"key":"3956_CR19","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1016\/j.ecolmodel.2016.07.014","volume":"337","author":"C Walters","year":"2016","unstructured":"Walters C, Christensen V, Fulton B, Smith ADM, Hilborn R (2016) Predictions from simple predator-prey theory about impacts of harvesting forage fishes. Ecol Model 337:272\u2013280. https:\/\/doi.org\/10.1016\/j.ecolmodel.2016.07.014","journal-title":"Ecol Model"},{"key":"3956_CR20","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/9512879","author":"A Das","year":"2019","unstructured":"Das A, Pal M (2019) Theoretical analysis of an imprecise prey-predator model with harvesting and optimal control\u2019. J Optim. https:\/\/doi.org\/10.1155\/2019\/9512879","journal-title":"J Optim"},{"key":"3956_CR21","doi-asserted-by":"publisher","DOI":"10.4172\/2168-9679.1000358","author":"PK Jha","year":"2017","unstructured":"Jha PK, Ghorai S (2017) \u2018Stability of prey-predator model with Holling type response function and selective Harvesting. J Appl Comput Math. https:\/\/doi.org\/10.4172\/2168-9679.1000358","journal-title":"J Appl Comput Math"},{"key":"3956_CR22","doi-asserted-by":"publisher","first-page":"4044","DOI":"10.1016\/.apm.2011.11.029","volume":"36","author":"S Charaborty","year":"2012","unstructured":"Charaborty S, Pal S (2012) Bairagi N \u2018\u2018Predator-Prey interaction with harvesting mathematical study with biological ramifications. Appl Math Model 36:4044\u20134059. https:\/\/doi.org\/10.1016\/.apm.2011.11.029","journal-title":"Appl Math Model"},{"key":"3956_CR23","first-page":"541","volume":"12","author":"KS Chen","year":"1981","unstructured":"Chen KS (1981) Uniqueness of a limit cycle of a predator\u2013prey system, SIAM. J Math Anal 12:541\u2013548","journal-title":"J Math Anal"},{"issue":"1","key":"3956_CR24","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/0092-8240(94)00024-7","volume":"57","author":"JB Collings","year":"1995","unstructured":"Collings JB (1995) Bifurcation and stability analysis of a temperature dependent mite predator\u2013prey interaction model incorporating a prey refuge. Bull Math Biol 57(1):63\u201376","journal-title":"Bull Math Biol"},{"key":"3956_CR25","volume-title":"The dynamics of anthropoid predator-prey systems","author":"MP Hassel","year":"1978","unstructured":"Hassel MP (1978) The dynamics of anthropoid predator-prey systems. Princeton University Press, Princeton, NJ"},{"key":"3956_CR26","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/0025-5564(90)90139-P","volume":"99","author":"J Hofbauer","year":"1990","unstructured":"Hofbauer J, So JWH (1990) Multiple limit cycles for predator\u2013prey models. Math Biosci 99:71\u201375","journal-title":"Math Biosci"},{"key":"3956_CR27","first-page":"3","volume":"45","author":"CS Holling","year":"1965","unstructured":"Holling CS (1965) The functional response of predators to prey density and its role in mimicry and population regulation. Mem Entomol Soc Can 45:3\u201360","journal-title":"Mem Entomol Soc Can"},{"key":"3956_CR28","doi-asserted-by":"publisher","first-page":"012139","DOI":"10.1088\/1742-6596\/1000\/1\/012139","volume":"1000","author":"R Senthamarai","year":"2018","unstructured":"Senthamarai R, Vijayalakshmi T (2018) An analytical approach to top predator interference on the dynamics of a food chain model. J Phys Conf Ser 1000:012139. https:\/\/doi.org\/10.1088\/1742-6596\/1000\/1\/012139","journal-title":"J Phys Conf Ser"},{"key":"3956_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2005.03.006","author":"J-H He","year":"2005","unstructured":"He J-H (2005) Application of Homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals. https:\/\/doi.org\/10.1016\/j.chaos.2005.03.006","journal-title":"Chaos Solitons Fractals."},{"key":"3956_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7462(98)00085-7","author":"J-H He","year":"1999","unstructured":"He J-H (1999) Homotopy perturbation technique. Comput Method Appl Mech Eng. https:\/\/doi.org\/10.1016\/S0020-7462(98)00085-7","journal-title":"Comput Method Appl Mech Eng"},{"key":"3956_CR31","doi-asserted-by":"publisher","first-page":"020077","DOI":"10.1063\/1.5112262","volume":"2112","author":"T Vijayalakshmi","year":"2019","unstructured":"Vijayalakshmi T, Senthamarai R (2019) An analytical approach to the density dependent prey-predator system with Beddington\u2013DeAngelies functional response. AIP Conf Proc 2112:020077. https:\/\/doi.org\/10.1063\/1.5112262","journal-title":"AIP Conf Proc"},{"key":"3956_CR32","doi-asserted-by":"publisher","first-page":"903","DOI":"10.1016\/j.camwa.2006.12.058","volume":"54","author":"B Batiha","year":"2007","unstructured":"Batiha B, Noorani MSM, Hashim I (2007) Variational iteration method for solving multispecies Lotka\u2013Volterra equations. Comput Math Appl 54:903. https:\/\/doi.org\/10.1016\/j.camwa.2006.12.058","journal-title":"Comput Math Appl"},{"key":"3956_CR33","doi-asserted-by":"publisher","first-page":"1701","DOI":"10.1016\/j.amc.2006.08.077","volume":"186","author":"M Rafei","year":"2007","unstructured":"Rafei M, Daniali H, Ganji DD (2007) Variational iteration method for solving the epidemic model and the prey and predator problem. Appl Math Comput 186:1701. https:\/\/doi.org\/10.1016\/j.amc.2006.08.077","journal-title":"Appl Math Comput"},{"key":"3956_CR34","doi-asserted-by":"publisher","first-page":"1197","DOI":"10.1016\/j.camwa.2006.12.076","volume":"54","author":"F Shakeri","year":"2007","unstructured":"Shakeri F, Dehghan M (2007) Numerical solution of a biological population model using He\u2019s variational iteration method. Comput Math Appl 54:1197. https:\/\/doi.org\/10.1016\/j.camwa.2006.12.076","journal-title":"Comput Math Appl"}],"container-title":["The Journal of Supercomputing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-021-03956-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11227-021-03956-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-021-03956-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,24]],"date-time":"2022-01-24T11:28:06Z","timestamp":1643023686000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11227-021-03956-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,30]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["3956"],"URL":"https:\/\/doi.org\/10.1007\/s11227-021-03956-5","relation":{},"ISSN":["0920-8542","1573-0484"],"issn-type":[{"value":"0920-8542","type":"print"},{"value":"1573-0484","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,30]]},"assertion":[{"value":"17 June 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 June 2021","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors state that no conflicts of interest exist.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}