August  2016, 21(6): 1713-1728. doi: 10.3934/dcdsb.2016019

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

1. 

Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan

2. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States

Received  June 2015 Revised  April 2016 Published  June 2016

In this work, the effect of a host refuge on a host-parasitoid inter- action is investigated. The model is built upon a modi ed 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.
Citation: Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019
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.

show all references

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.

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