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Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge
1. | Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China |
References:
[1] |
L. Chen, F. Chen and L. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal-Real, 11 (2010), 246-252.
doi: 10.1016/j.nonrwa.2008.10.056. |
[2] |
J. Collings, Bifurcation and stability analysis of a temperature-depent mite predator-prey interaction model incroporating a prey refuge, Bull. Math. Biol., 57 (1995), 63-76. |
[3] |
Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differ. Equations, 203 (2004), 331-364.
doi: 10.1016/j.jde.2004.05.010. |
[4] |
Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.
doi: 10.1017/S0308210500000895. |
[5] |
Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.
doi: 10.1090/S0002-9947-97-01842-4. |
[6] |
Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differ. Equations, 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013. |
[7] |
Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 95-135. |
[8] |
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. |
[9] |
X. Guan, W. Wang and Y. Cai, Spatiotemporal dynamics of a Lieslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal-Real, 12 (2011), 2385-2395.
doi: 10.1016/j.nonrwa.2011.02.011. |
[10] |
B. Hassard, N. Kazarinoff and Y-H. Wan, "Theory and Applications of Hopf Bifurcation," Cambridge University Press, Cambridge, 1981. |
[11] |
M. Hassel, "The Dynamics of Arthropod Predator-Prey Systems," Princeton University Press, Princeton, 1978. |
[12] |
M. Hassel and R. May, Stability in insect host-parasite models, J. Anim. Ecol., 42 (1973), 693-726.
doi: 10.2307/3133. |
[13] |
R. Holt, Optimal foraging and the form of the predator isoclin, Am. Nat., 122 (1983), 521-541.
doi: 10.1086/284153. |
[14] |
S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwan. J. Math., 9 (2005), 151-173. |
[15] |
S.-B. Hsu, T.-W. Huang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.
doi: 10.1007/s002850100079. |
[16] |
Y. Huang, F. Chen and L. Zhong, Stability analysis of a predator-prey model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683.
doi: 10.1016/j.amc.2006.04.030. |
[17] |
G. Hutchinson, "The Ecological Theater and the Evolutionary Play," Yale Univ. Press, New Haven, Connecticut, 1976. |
[18] |
T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681-691.
doi: 10.1016/j.cnsns.2003.08.006. |
[19] |
W. Ko and K. Ryu, A qualitative on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal-Real, 10 (2009), 2558-2573.
doi: 10.1016/j.nonrwa.2008.05.012. |
[20] |
W. Ko and K. Ryu, Qualitative analysis of a prey-predator model with Holling type II functional response incorporating a prey refuge, J. Differ. Equations, 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[21] |
V. Krivan, Effects of optimal antipredator behavior of prey on predator-prey dynamics: The role of refuges, Theor. Popul. Biol., 53 (1998), 131-142. |
[22] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[23] |
Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, J. Math. Biol., 88 (1988), 67-84.
doi: 10.1016/0025-5564(88)90049-1. |
[24] |
Y. Li and M. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal.-Real, 14 (2013), 1806-1816.
doi: 10.1016/j.nonrwa.2012.11.012. |
[25] |
X. Liu and Y. Lou, Global dynamics of a predator-prey model, J. Math. Anal. Appl., 371 (2010), 323-340.
doi: 10.1016/j.jmaa.2010.05.037. |
[26] |
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. |
[27] |
R. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1974. |
[28] |
J. M. McNair, The effects of refuges on predator-prey ineractions: A reconsideration, Theor. Popul. Biol., 29 (1986), 38-63.
doi: 10.1016/0040-5809(86)90004-3. |
[29] |
P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differ. Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[30] |
R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differ. Equations, 247 (2009), 866-886.
doi: 10.1016/j.jde.2009.03.008. |
[31] |
R. Peng and M. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.
doi: 10.1017/S0308210500003814. |
[32] |
G. D. Ruxton, Short term refuge use and stability of predator-prey models, Theor. Popul. Biol., 47 (1995), 1-17.
doi: 10.1006/tpbi.1995.1001. |
[33] |
M. Scheffer and R. J. De Boer., Implications of spatial heterogeneity for the paradox of enrichment, Ecology, 76 (1995), 2270-2277.
doi: 10.2307/1941701. |
[34] |
A. Sih, Prey refuges and predator-prey stability, Theor. Popul. Biol., 31 (1987), 1-12.
doi: 10.1016/0040-5809(87)90019-0. |
[35] |
L. Smith, "Models in Ecology," Cambridge University Press, Cambridge, 1974. |
[36] | |
[37] |
J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differ. Equations, 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[38] |
J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
[39] |
D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.
doi: 10.1007/s002850100097. |
[40] |
F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024. |
show all references
References:
[1] |
L. Chen, F. Chen and L. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal-Real, 11 (2010), 246-252.
doi: 10.1016/j.nonrwa.2008.10.056. |
[2] |
J. Collings, Bifurcation and stability analysis of a temperature-depent mite predator-prey interaction model incroporating a prey refuge, Bull. Math. Biol., 57 (1995), 63-76. |
[3] |
Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differ. Equations, 203 (2004), 331-364.
doi: 10.1016/j.jde.2004.05.010. |
[4] |
Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.
doi: 10.1017/S0308210500000895. |
[5] |
Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.
doi: 10.1090/S0002-9947-97-01842-4. |
[6] |
Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differ. Equations, 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013. |
[7] |
Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 95-135. |
[8] |
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. |
[9] |
X. Guan, W. Wang and Y. Cai, Spatiotemporal dynamics of a Lieslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal-Real, 12 (2011), 2385-2395.
doi: 10.1016/j.nonrwa.2011.02.011. |
[10] |
B. Hassard, N. Kazarinoff and Y-H. Wan, "Theory and Applications of Hopf Bifurcation," Cambridge University Press, Cambridge, 1981. |
[11] |
M. Hassel, "The Dynamics of Arthropod Predator-Prey Systems," Princeton University Press, Princeton, 1978. |
[12] |
M. Hassel and R. May, Stability in insect host-parasite models, J. Anim. Ecol., 42 (1973), 693-726.
doi: 10.2307/3133. |
[13] |
R. Holt, Optimal foraging and the form of the predator isoclin, Am. Nat., 122 (1983), 521-541.
doi: 10.1086/284153. |
[14] |
S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwan. J. Math., 9 (2005), 151-173. |
[15] |
S.-B. Hsu, T.-W. Huang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.
doi: 10.1007/s002850100079. |
[16] |
Y. Huang, F. Chen and L. Zhong, Stability analysis of a predator-prey model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683.
doi: 10.1016/j.amc.2006.04.030. |
[17] |
G. Hutchinson, "The Ecological Theater and the Evolutionary Play," Yale Univ. Press, New Haven, Connecticut, 1976. |
[18] |
T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681-691.
doi: 10.1016/j.cnsns.2003.08.006. |
[19] |
W. Ko and K. Ryu, A qualitative on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal-Real, 10 (2009), 2558-2573.
doi: 10.1016/j.nonrwa.2008.05.012. |
[20] |
W. Ko and K. Ryu, Qualitative analysis of a prey-predator model with Holling type II functional response incorporating a prey refuge, J. Differ. Equations, 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[21] |
V. Krivan, Effects of optimal antipredator behavior of prey on predator-prey dynamics: The role of refuges, Theor. Popul. Biol., 53 (1998), 131-142. |
[22] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[23] |
Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, J. Math. Biol., 88 (1988), 67-84.
doi: 10.1016/0025-5564(88)90049-1. |
[24] |
Y. Li and M. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal.-Real, 14 (2013), 1806-1816.
doi: 10.1016/j.nonrwa.2012.11.012. |
[25] |
X. Liu and Y. Lou, Global dynamics of a predator-prey model, J. Math. Anal. Appl., 371 (2010), 323-340.
doi: 10.1016/j.jmaa.2010.05.037. |
[26] |
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. |
[27] |
R. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1974. |
[28] |
J. M. McNair, The effects of refuges on predator-prey ineractions: A reconsideration, Theor. Popul. Biol., 29 (1986), 38-63.
doi: 10.1016/0040-5809(86)90004-3. |
[29] |
P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differ. Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[30] |
R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differ. Equations, 247 (2009), 866-886.
doi: 10.1016/j.jde.2009.03.008. |
[31] |
R. Peng and M. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.
doi: 10.1017/S0308210500003814. |
[32] |
G. D. Ruxton, Short term refuge use and stability of predator-prey models, Theor. Popul. Biol., 47 (1995), 1-17.
doi: 10.1006/tpbi.1995.1001. |
[33] |
M. Scheffer and R. J. De Boer., Implications of spatial heterogeneity for the paradox of enrichment, Ecology, 76 (1995), 2270-2277.
doi: 10.2307/1941701. |
[34] |
A. Sih, Prey refuges and predator-prey stability, Theor. Popul. Biol., 31 (1987), 1-12.
doi: 10.1016/0040-5809(87)90019-0. |
[35] |
L. Smith, "Models in Ecology," Cambridge University Press, Cambridge, 1974. |
[36] | |
[37] |
J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differ. Equations, 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[38] |
J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
[39] |
D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.
doi: 10.1007/s002850100097. |
[40] |
F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024. |
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