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The dynamic complexity of a Holling type-IV predator-prey system with stage structure and double delays. (English) Zbl 1213.37128

Summary: We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
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References:

[1] DOI: 10.1142/S021833900600191X · Zbl 1116.92065
[2] Ecology 73 (5) pp 1530– (1992)
[3] DOI: 10.3934/dcdsb.2008.10.857 · Zbl 1160.34046
[4] Qualitative analysis of mathematical models of populations 25 pp 100– (1972)
[5] DOI: 10.1007/s002850100097 · Zbl 1007.34031
[6] Monographs and Textbooks in Pure and Applied Mathematics 57 pp x+254– (1980)
[7] (1973)
[8] Biomathematics 19 pp xiv+767– (1989)
[9] Biotechnology and Bioengineering 10 pp 707– (1968)
[10] Monographs in Population Biology 13 pp vii+237– (1978)
[11] (2001)
[12] Journal of Animal Ecology 43 pp 747– (1974)
[13] DOI: 10.1016/0022-247X(90)90289-R · Zbl 0711.34091
[14] DOI: 10.1016/S0092-8240(86)90003-0
[15] Discrete and Continuous Dynamical Systems. Series B 9 (2) pp 397– (2008)
[16] Mathematics in Science and Engineering 191 pp xii+398– (1993)
[17] Cambridge Texts in Applied Mathematics pp xiv+402– (1997)
[18] DOI: 10.1016/j.cam.2005.12.035 · Zbl 1117.34070
[19] DOI: 10.1016/j.cam.2005.08.017 · Zbl 1110.34051
[20] DOI: 10.1016/j.nonrwa.2004.04.002 · Zbl 1088.34070
[21] DOI: 10.1007/s00285-004-0278-2 · Zbl 1055.92043
[22] DOI: 10.1016/j.jmaa.2004.07.017 · Zbl 1085.34062
[23] DOI: 10.1016/0025-5564(90)90019-U · Zbl 0719.92017
[24] DOI: 10.1016/j.jmaa.2007.12.038 · Zbl 1146.34057
[25] DOI: 10.1016/j.cam.2008.11.014 · Zbl 1189.34158
[26] DOI: 10.1016/S0022-247X(02)00329-3 · Zbl 1039.34068
[27] DOI: 10.1016/S0022-247X(02)00103-8 · Zbl 1022.34039
[28] DOI: 10.1137/0520025 · Zbl 0692.34053
[29] DOI: 10.1137/0524026 · Zbl 0774.34030
[30] DOI: 10.1016/j.nonrwa.2006.01.004 · Zbl 1152.34374
[31] DOI: 10.1137/S0036141000376086 · Zbl 1013.92034
[32] DOI: 10.1007/s11538-006-9121-9 · Zbl 1296.92102
[33] DOI: 10.1007/s11071-006-9133-x · Zbl 1176.92056
[34] (1993)
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