Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge

Yunshyong  Chow; Sophia  Jang

doi:10.3934/dcdsb.2016019

Article Contents

2016, Volume 21, Issue 6: 1713-1728. Doi: 10.3934/dcdsb.2016019

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Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge

Yunshyong Chow^1, and
Sophia Jang^2,

1.
Institute of Mathematics, Academia Sinica, Taipei 10617
2.
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042

Received: June 2015

Revised: April 2016

Published: August 2016

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Abstract

In this work, the effect of a host refuge on a host-parasitoid inter- action is investigated. The model is built upon a modied Nicholson-Bailey system by assuming that in each generation a constant proportion of the host is free from parasitism. We derive a sucient condition based on the model parameters for both populations to coexist. We prove that it is possible for the system to undergo a supercritical and then a subcritical Neimark-Sacker bifurcation or for the system only to exhibit a supercritical Neimark-Sacker bifurcation. It is illustrated numerically that a constant proportion of host refuge can stabilize the host-parasitoid interaction.

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Mathematics Subject Classification: Primary: 92D25; Secondary: 39A30.

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References

[1]	L. J. S. Allen, An Introduction to Mathematical Biology, Prentice-Hall, New Jersey, 2006.
[2]	M. Begon, J. Harper and C. Townsend, Ecology: Individuals, Populations and Communities, Blackwell Science Ltd, New York, 1996.
[3]	F. Berezovskaya, B. Song and C. Castillo-Chavez, Role of prey dispersal and refuges on predator-prey dynamics, SIAM J. Appl. Math., 70 (2010), 1821-1839.doi: 10.1137/080730603.
[4]	G. Butler and P. Waltman, Persistence in dynamical systems, J. Diff. Equ., 63 (1986), 255-263.doi: 10.1016/0022-0396(86)90049-5.
[5]	E. Gonzalez-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135-146.doi: 10.1016/S0304-3800(03)00131-5.
[6]	J. K. Hale and H. Koçak, Dynamics and Bifurcations, Springer, New York, 1991.doi: 10.1007/978-1-4612-4426-4.
[7]	M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Princeton Univeristy Press, Princeton, New Jersey, 1978.
[8]	A. Hines and J. Pearse, Abalones, shells, and sea otters: Dynamics of prey populations in central California, Ecol., 63 (1982), 1547-1560.doi: 10.2307/1938879.
[9]	J. Hofbauer and J. So, Uniform persistence and repellors for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142.doi: 10.1090/S0002-9939-1989-0984816-4.
[10]	Y. Huang, F. Chen and Z. Li, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, App. Math. Comput., 182 (2006), 672-683.doi: 10.1016/j.amc.2006.04.030.
[11]	V. Hutson, A theorem on average Liapunov functions, Monash. Math., 98 (1984), 267-275.doi: 10.1007/BF01540776.
[12]	S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model, J. Difference Equ. Appl., 12 (2006), 165-181.doi: 10.1080/10236190500539238.
[13]	L. Ji and C. Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonl. Ana.: RWA, 11 (2010), 2285-2295.doi: 10.1016/j.nonrwa.2009.07.003.
[14]	V. Krivan, Behavioral refuges and predator-prey coexistence, J. Theo. Biol., 339 (2013), 112-121.doi: 10.1016/j.jtbi.2012.12.016.
[15]	Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang and Z. Li, Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges, Math. Biosci., 218 (2009), 73-79.doi: 10.1016/j.mbs.2008.12.008.
[16]	J. Maynard Smith, Models in Ecology, Cambridge Univ. Press, London, 1974.
[17]	J. McNair, The effects of refuges on predator-prey interactions: A reconsideration, Theo. Pop. Biol., 29 (1986), 38-63.doi: 10.1016/0040-5809(86)90004-3.
[18]	J. McNair, Stability effects of prey refuges with entry-exit dynamics, J. Theor. Biol., 125 (1987), 449-464.doi: 10.1016/S0022-5193(87)80213-8.
[19]	W. Murdoch and A. Oaten, Predation and population stability, Adv. Ecol. Res., 9 (1975), 1-131.doi: 10.1016/S0065-2504(08)60288-3.
[20]	A. Sih, Prey refuges and predator-prey stability, Theo. Pop. Biol., 31 (1987), 1-12.doi: 10.1016/0040-5809(87)90019-0.
[21]	R. Taylor, Predation, Chapman and Hall, New York, 1984.doi: 10.1007/978-94-009-5554-7.
[22]	S. Woodin, Refuges, disturbance, and community structure: A marine soft-bottom example, Ecol., 59 (1978), 274-284.doi: 10.2307/1936373.
[23]	S. Woodin, Disturbance and community structure in a shallow water sand flat, Ecol., 62 (1981), 1052-1066.doi: 10.2307/1937004.