{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T05:45:30Z","timestamp":1780638330203,"version":"3.54.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,11,5]],"date-time":"2016-11-05T00:00:00Z","timestamp":1478304000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100002790","name":"Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002790","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["PolyU 253004\/14P"],"award-info":[{"award-number":["PolyU 253004\/14P"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11301442"],"award-info":[{"award-number":["11301442"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2017,4]]},"DOI":"10.1007\/s00332-016-9344-3","type":"journal-article","created":{"date-parts":[[2016,11,5]],"date-time":"2016-11-05T09:11:18Z","timestamp":1478337078000},"page":"573-603","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":83,"title":["A Theoretical Approach to Understanding Population Dynamics with Seasonal Developmental Durations"],"prefix":"10.1007","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3864-2001","authenticated-orcid":false,"given":"Yijun","family":"Lou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao-Qiang","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,11,5]]},"reference":[{"key":"9344_CR1","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/0025-5564(90)90019-U","volume":"101","author":"WG Aiello","year":"1990","unstructured":"Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139\u2013153 (1990)","journal-title":"Math. Biosci."},{"key":"9344_CR2","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1111\/j.1461-0248.2005.00879.x","volume":"9","author":"S Altizer","year":"2006","unstructured":"Altizer, S., Dobson, A., Hosseini, P., et al.: Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467\u2013484 (2006)","journal-title":"Ecol. Lett."},{"key":"9344_CR3","doi-asserted-by":"crossref","first-page":"219","DOI":"10.2307\/3933","volume":"47","author":"RM Anderson","year":"1978","unstructured":"Anderson, R.M., May, R.M.: Regulation and stability of host-parasite population interactions: I. Regulatory processes. J. Anim. Ecol. 47, 219\u2013247 (1978)","journal-title":"J. Anim. Ecol."},{"key":"9344_CR4","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1007\/s11538-006-9166-9","volume":"69","author":"N Baca\u00ebr","year":"2007","unstructured":"Baca\u00ebr, N.: Approximation of the basic reproduction number $$R_0$$ R 0 for vector-borne diseases with a periodic vector population. Bull. Math. Biol. 69, 1067\u20131091 (2007)","journal-title":"Bull. Math. Biol."},{"key":"9344_CR5","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s00285-006-0015-0","volume":"53","author":"N Baca\u00ebr","year":"2006","unstructured":"Baca\u00ebr, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco. J. Math. Biol. 53, 421\u2013436 (2006)","journal-title":"J. Math. Biol."},{"key":"9344_CR6","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s00285-011-0479-4","volume":"65","author":"N Baca\u00ebr","year":"2012","unstructured":"Baca\u00ebr, N., Ait Dads, A.H.: On the biological interpretation of a definition for the parameter $$R_0$$ R 0 in periodic population models. J. Math. Biol. 65, 601\u2013621 (2012)","journal-title":"J. Math. Biol."},{"key":"9344_CR7","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.mbs.2015.09.005","volume":"269","author":"Z Bai","year":"2015","unstructured":"Bai, Z.: Threshold dynamics of a time-delayed SEIRS model with pulse vaccination. Math. Biosci. 269, 178\u2013185 (2015)","journal-title":"Math. Biosci."},{"key":"9344_CR8","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1007\/s00285-014-0821-8","volume":"69","author":"MV Barbarossa","year":"2014","unstructured":"Barbarossa, M.V., Hadeler, K.P., Kuttler, C.: State-dependent neutral delay equations from population dynamics. J. Math. Biol. 69, 1027\u20131056 (2014)","journal-title":"J. Math. Biol."},{"key":"9344_CR9","doi-asserted-by":"crossref","unstructured":"Cushing, J.M.: An Introduction to Structured Population Dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1998)","DOI":"10.1137\/1.9781611970005"},{"key":"9344_CR10","doi-asserted-by":"crossref","first-page":"487","DOI":"10.2307\/5339","volume":"61","author":"AP Dobson","year":"1992","unstructured":"Dobson, A.P., Hudson, P.J.: Regulation and stability of a free-living host-parasite system: Trichostrongylus tenuis in red grouse. II. Population models. J. Anim. Ecol. 61, 487\u2013498 (1992)","journal-title":"J. Anim. Ecol."},{"key":"9344_CR11","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4342-7","volume-title":"Introduction to Functional Differential Equations","author":"JK Hale","year":"1993","unstructured":"Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)"},{"key":"9344_CR12","volume-title":"Periodic-Parabolic Boundary Value Problems and Positivity, , Pitman Research Notes in Mathematics","author":"P Hess","year":"1991","unstructured":"Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, vol. 247. Longman Scientific and Technical, Harlow (1991)"},{"key":"9344_CR13","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s00285-011-0463-z","volume":"65","author":"H Inaba","year":"2012","unstructured":"Inaba, H.: On a new perspective of the basic reproduction number in heterogeneous environments. J. Math. Biol. 65, 309\u2013348 (2012)","journal-title":"J. Math. Biol."},{"key":"9344_CR14","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1017\/S0031182099006095","volume":"121","author":"RR Kao","year":"2000","unstructured":"Kao, R.R., Leathwick, D.M., Roberts, M.G., Sutherland, I.A.: Nematode parasites of sheep: a survey of epidemiological parameters and their application in a simple model. Parasitology 121, 85\u2013103 (2000)","journal-title":"Parasitology"},{"key":"9344_CR15","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1137\/15M1021271","volume":"76","author":"M Kloosterman","year":"2016","unstructured":"Kloosterman, M., Campbell, S.A., Poulin, F.J.: An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton. SIAM J. Appl. Math. 76, 551\u2013577 (2016)","journal-title":"SIAM J. Appl. Math."},{"key":"9344_CR16","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1890\/12-1407.1","volume":"23","author":"SJ Leroux","year":"2013","unstructured":"Leroux, S.J., Larriv\u00e9e, M., Boucher-Lalonde, V., et al.: Mechanistic models for the spatial spread of species under climate change. Ecol. Appl. 23, 815\u2013828 (2013)","journal-title":"Ecol. Appl."},{"key":"9344_CR17","doi-asserted-by":"crossref","first-page":"169","DOI":"10.3934\/dcdsb.2009.12.169","volume":"12","author":"Y Lou","year":"2009","unstructured":"Lou, Y., Zhao, X.-Q.: Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete Contin. Dyn. Syst. Ser. B 12, 169\u2013186 (2009)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"9344_CR18","doi-asserted-by":"crossref","first-page":"479","DOI":"10.2307\/2963491","volume":"66","author":"E McCauley","year":"1996","unstructured":"McCauley, E., Nisbet, R.M., De Roos, A.M., et al.: Structured population models of herbivorous zooplankton. Ecol. Monogr. 66, 479\u2013501 (1996)","journal-title":"Ecol. Monogr."},{"key":"9344_CR19","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1111\/ele.12022","volume":"16","author":"PK Moln\u00e1r","year":"2013","unstructured":"Moln\u00e1r, P.K., Kutz, S.J., Hoar, B.M., Dobson, A.P.: Metabolic approaches to understanding climate change impacts on seasonal host-macroparasite dynamics. Ecol. Lett. 16, 9\u201321 (2013)","journal-title":"Ecol. Lett."},{"key":"9344_CR20","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.ijpara.2004.12.013","volume":"35","author":"NH Ogden","year":"2005","unstructured":"Ogden, N.H., Bigras-Poulin, M., O\u2019Callaghan, C.J., et al.: A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis. Int. J. Parasitol. 35, 375\u2013389 (2005)","journal-title":"Int. J. Parasitol."},{"key":"9344_CR21","doi-asserted-by":"crossref","first-page":"13844","DOI":"10.1073\/pnas.0903423106","volume":"106","author":"KP Paaijmans","year":"2009","unstructured":"Paaijmans, K.P., Read, A.F., Thomas, M.B.: Understanding the link between malaria risk and climate. PNAS 106, 13844\u201313849 (2009)","journal-title":"PNAS"},{"key":"9344_CR22","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/j.amc.2014.05.079","volume":"242","author":"D Posny","year":"2014","unstructured":"Posny, D., Wang, J.: Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments. Appl. Math. Comput. 242, 473\u2013490 (2014)","journal-title":"Appl. Math. Comput."},{"key":"9344_CR23","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.3934\/dcdsb.2014.19.1155","volume":"19","author":"C Rebelo","year":"2014","unstructured":"Rebelo, C., Margheri, A., Baca\u00ebr, N.: Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete Contin. Dyn. Syst. Ser. B 19, 1155\u20131170 (2014)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"9344_CR24","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/0169-4758(95)80150-2","volume":"11","author":"MG Roberts","year":"1995","unstructured":"Roberts, M.G.: A pocket guide to host-parasite models. Parasitol. Today 11, 172\u2013177 (1995)","journal-title":"Parasitol. Today"},{"key":"9344_CR25","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/S0169-4758(99)01430-1","volume":"15","author":"MG Roberts","year":"1999","unstructured":"Roberts, M.G.: The immunoepidemiology of nematode parasites of farmed animals: a mathematical approach. Parasitol. Today 15, 246\u2013251 (1999)","journal-title":"Parasitol. Today"},{"key":"9344_CR26","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1093\/imammb\/8.2.83","volume":"8","author":"MG Roberts","year":"1991","unstructured":"Roberts, M.G., Grenfell, B.T.: The population dynamics of nematode infections of ruminants: periodic perturbations as a model for management. IMA J. Math. Appl. Med. Biol. 8, 83\u201393 (1991)","journal-title":"IMA J. Math. Appl. Med. Biol."},{"key":"9344_CR27","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1017\/S0031182000064830","volume":"110","author":"MG Roberts","year":"1995","unstructured":"Roberts, M.G., Heesterbeek, J.A.: The dynamics of nematode infections of farmed ruminants. Parasitol 110, 493\u2013502 (1995)","journal-title":"Parasitol"},{"key":"9344_CR28","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/978-4-431-36025-4_17","volume-title":"Micromammals and Macroparasites from Evolutionary Study to Management, Morand","author":"R Ros\u00e0","year":"2006","unstructured":"Ros\u00e0, R., Rizzoli, A., Ferrari, N., Pugliese, A.: Models for host-macroparasite interactions in micromammals. In: Krasnov, B.R., Poulin, R. (eds.) Micromammals and Macroparasites from Evolutionary Study to Management, Morand, pp. 319\u2013348. Springer, Tokyo (2006)"},{"key":"9344_CR29","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/1-4020-3647-7_11","volume-title":"Delay Differential Equations and Applications","author":"S Ruan","year":"2006","unstructured":"Ruan, S.: Delay differential equations in single species dynamics. In: Arino, O., Hbid, M.L., Ait Dads, E. (eds.) Delay Differential Equations and Applications, pp. 477\u2013517. Springer, Berlin (2006)"},{"key":"9344_CR30","unstructured":"Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. American Mathematical Society (1995)"},{"key":"9344_CR31","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.ecolmodel.2014.11.001","volume":"297","author":"L Stancampiano","year":"2015","unstructured":"Stancampiano, L., Usai, F.: The role of density-dependent arrested larval stages on parasite dynamics and stability: lessons from nematodes and donkeys. Ecol. Model. 297, 69\u201379 (2015)","journal-title":"Ecol. Model."},{"key":"9344_CR32","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1098\/rspb.2015.1453","volume":"282","author":"TJ Stevenson","year":"2015","unstructured":"Stevenson, T.J., Visser, M.E., Arnold, W., et al.: Disrupted seasonal biology impacts health, food security and ecosystems. Proc. R. Soc. B 282, 1453 (2015)","journal-title":"Proc. R. Soc. B"},{"key":"9344_CR33","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1137\/080732870","volume":"70","author":"HR Thieme","year":"2009","unstructured":"Thieme, H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188\u2013211 (2009)","journal-title":"SIAM J. Appl. Math."},{"key":"9344_CR34","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1603\/ME10117","volume":"48","author":"J Wang","year":"2011","unstructured":"Wang, J., Ogden, N.H., Zhu, H.: The impact of weather conditions on Culex pipiens and Culex restuans (Diptera: Culicidae) abundance: a case study in Peel region. J. Med. Entomol. 48, 468\u2013475 (2011)","journal-title":"J. Med. Entomol."},{"key":"9344_CR35","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1007\/s10884-008-9111-8","volume":"20","author":"W Wang","year":"2008","unstructured":"Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699\u2013717 (2008)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9344_CR36","volume-title":"Theory of Nonlinear Age-Dependent Population Dynamics","author":"GF Webb","year":"1985","unstructured":"Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1985)"},{"key":"9344_CR37","doi-asserted-by":"crossref","first-page":"775","DOI":"10.2307\/3275531","volume":"46","author":"IB Wood","year":"1960","unstructured":"Wood, I.B., Hansen, M.F.: Experimental transmission of ruminant nematodes of the genera Cooperia, Ostertagia and Haenronchus to laboratory rabbits. J. Parasitol. 46, 775\u2013776 (1960)","journal-title":"J. Parasitol."},{"key":"9344_CR38","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.jtbi.2012.11.014","volume":"319","author":"X Wu","year":"2013","unstructured":"Wu, X., Duvvuri, V.R., Lou, Y., et al.: Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada. J. Theor. Biol. 319, 50\u201361 (2013)","journal-title":"J. Theor. Biol."},{"key":"9344_CR39","doi-asserted-by":"crossref","first-page":"3464","DOI":"10.1002\/mma.3424","volume":"38","author":"X Wu","year":"2015","unstructured":"Wu, X., Magpantay, F.M.G., Wu, J., Zou, X.: Stage-structured population systems with temporally periodic delay. Math. Methods Appl. Sci. 38, 3464\u20133481 (2015)","journal-title":"Math. Methods Appl. Sci."},{"key":"9344_CR40","doi-asserted-by":"crossref","first-page":"3011","DOI":"10.1016\/j.jde.2014.12.032","volume":"258","author":"L Zhang","year":"2015","unstructured":"Zhang, L., Wang, Z.-C., Zhao, X.-Q.: Threshold dynamics of a time periodic reaction\u2013diffusion epidemic model with latent period. J. Differ. Equ. 258, 3011\u20133036 (2015)","journal-title":"J. Differ. Equ."},{"key":"9344_CR41","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21761-1","volume-title":"Dynamical Systems in Population Biology","author":"X-Q Zhao","year":"2003","unstructured":"Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer-Verlag, New York (2003)"},{"key":"9344_CR42","doi-asserted-by":"publisher","unstructured":"Zhao, X.-Q.: Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. (2015). doi: 10.1007\/s10884-015-9425-2","DOI":"10.1007\/s10884-015-9425-2"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-016-9344-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-016-9344-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-016-9344-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,27]],"date-time":"2020-09-27T04:32:26Z","timestamp":1601181146000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-016-9344-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,5]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,4]]}},"alternative-id":["9344"],"URL":"https:\/\/doi.org\/10.1007\/s00332-016-9344-3","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,11,5]]}}}