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 & 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, (2006). Google Scholar

[2]

M. Begon, J. Harper and C. Townsend, Ecology: Individuals, Populations and Communities,, Blackwell Science Ltd, (1996). Google Scholar

[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. doi: 10.1137/080730603. Google Scholar

[4]

G. Butler and P. Waltman, Persistence in dynamical systems,, J. Diff. Equ., 63 (1986), 255. doi: 10.1016/0022-0396(86)90049-5. Google Scholar

[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. doi: 10.1016/S0304-3800(03)00131-5. Google Scholar

[6]

J. K. Hale and H. Koçak, Dynamics and Bifurcations,, Springer, (1991). doi: 10.1007/978-1-4612-4426-4. Google Scholar

[7]

M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems,, Princeton Univeristy Press, (1978). Google Scholar

[8]

A. Hines and J. Pearse, Abalones, shells, and sea otters: Dynamics of prey populations in central California,, Ecol., 63 (1982), 1547. doi: 10.2307/1938879. Google Scholar

[9]

J. Hofbauer and J. So, Uniform persistence and repellors for maps,, Proc. Am. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar

[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. doi: 10.1016/j.amc.2006.04.030. Google Scholar

[11]

V. Hutson, A theorem on average Liapunov functions,, Monash. Math., 98 (1984), 267. doi: 10.1007/BF01540776. Google Scholar

[12]

S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model,, J. Difference Equ. Appl., 12 (2006), 165. doi: 10.1080/10236190500539238. Google Scholar

[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. doi: 10.1016/j.nonrwa.2009.07.003. Google Scholar

[14]

V. Krivan, Behavioral refuges and predator-prey coexistence,, J. Theo. Biol., 339 (2013), 112. doi: 10.1016/j.jtbi.2012.12.016. Google Scholar

[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. doi: 10.1016/j.mbs.2008.12.008. Google Scholar

[16]

J. Maynard Smith, Models in Ecology,, Cambridge Univ. Press, (1974). Google Scholar

[17]

J. McNair, The effects of refuges on predator-prey interactions: A reconsideration,, Theo. Pop. Biol., 29 (1986), 38. doi: 10.1016/0040-5809(86)90004-3. Google Scholar

[18]

J. McNair, Stability effects of prey refuges with entry-exit dynamics,, J. Theor. Biol., 125 (1987), 449. doi: 10.1016/S0022-5193(87)80213-8. Google Scholar

[19]

W. Murdoch and A. Oaten, Predation and population stability,, Adv. Ecol. Res., 9 (1975), 1. doi: 10.1016/S0065-2504(08)60288-3. Google Scholar

[20]

A. Sih, Prey refuges and predator-prey stability,, Theo. Pop. Biol., 31 (1987), 1. doi: 10.1016/0040-5809(87)90019-0. Google Scholar

[21]

R. Taylor, Predation,, Chapman and Hall, (1984). doi: 10.1007/978-94-009-5554-7. Google Scholar

[22]

S. Woodin, Refuges, disturbance, and community structure: A marine soft-bottom example,, Ecol., 59 (1978), 274. doi: 10.2307/1936373. Google Scholar

[23]

S. Woodin, Disturbance and community structure in a shallow water sand flat,, Ecol., 62 (1981), 1052. doi: 10.2307/1937004. Google Scholar

show all references

References:
[1]

L. J. S. Allen, An Introduction to Mathematical Biology,, Prentice-Hall, (2006). Google Scholar

[2]

M. Begon, J. Harper and C. Townsend, Ecology: Individuals, Populations and Communities,, Blackwell Science Ltd, (1996). Google Scholar

[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. doi: 10.1137/080730603. Google Scholar

[4]

G. Butler and P. Waltman, Persistence in dynamical systems,, J. Diff. Equ., 63 (1986), 255. doi: 10.1016/0022-0396(86)90049-5. Google Scholar

[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. doi: 10.1016/S0304-3800(03)00131-5. Google Scholar

[6]

J. K. Hale and H. Koçak, Dynamics and Bifurcations,, Springer, (1991). doi: 10.1007/978-1-4612-4426-4. Google Scholar

[7]

M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems,, Princeton Univeristy Press, (1978). Google Scholar

[8]

A. Hines and J. Pearse, Abalones, shells, and sea otters: Dynamics of prey populations in central California,, Ecol., 63 (1982), 1547. doi: 10.2307/1938879. Google Scholar

[9]

J. Hofbauer and J. So, Uniform persistence and repellors for maps,, Proc. Am. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar

[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. doi: 10.1016/j.amc.2006.04.030. Google Scholar

[11]

V. Hutson, A theorem on average Liapunov functions,, Monash. Math., 98 (1984), 267. doi: 10.1007/BF01540776. Google Scholar

[12]

S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model,, J. Difference Equ. Appl., 12 (2006), 165. doi: 10.1080/10236190500539238. Google Scholar

[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. doi: 10.1016/j.nonrwa.2009.07.003. Google Scholar

[14]

V. Krivan, Behavioral refuges and predator-prey coexistence,, J. Theo. Biol., 339 (2013), 112. doi: 10.1016/j.jtbi.2012.12.016. Google Scholar

[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. doi: 10.1016/j.mbs.2008.12.008. Google Scholar

[16]

J. Maynard Smith, Models in Ecology,, Cambridge Univ. Press, (1974). Google Scholar

[17]

J. McNair, The effects of refuges on predator-prey interactions: A reconsideration,, Theo. Pop. Biol., 29 (1986), 38. doi: 10.1016/0040-5809(86)90004-3. Google Scholar

[18]

J. McNair, Stability effects of prey refuges with entry-exit dynamics,, J. Theor. Biol., 125 (1987), 449. doi: 10.1016/S0022-5193(87)80213-8. Google Scholar

[19]

W. Murdoch and A. Oaten, Predation and population stability,, Adv. Ecol. Res., 9 (1975), 1. doi: 10.1016/S0065-2504(08)60288-3. Google Scholar

[20]

A. Sih, Prey refuges and predator-prey stability,, Theo. Pop. Biol., 31 (1987), 1. doi: 10.1016/0040-5809(87)90019-0. Google Scholar

[21]

R. Taylor, Predation,, Chapman and Hall, (1984). doi: 10.1007/978-94-009-5554-7. Google Scholar

[22]

S. Woodin, Refuges, disturbance, and community structure: A marine soft-bottom example,, Ecol., 59 (1978), 274. doi: 10.2307/1936373. Google Scholar

[23]

S. Woodin, Disturbance and community structure in a shallow water sand flat,, Ecol., 62 (1981), 1052. doi: 10.2307/1937004. Google Scholar

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