American Institute of Mathematical Sciences

Advanced
2013, 10(4): 979-996. doi: 10.3934/mbe.2013.10.979

Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge

Xiaoyuan Chang 1, and  Junjie Wei 1,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

Received  February 2013 Revised  April 2013 Published  June 2013

A diffusive predator-prey model with Holling type II functional response and the no-flux boundary condition incorporating a constant prey refuge is considered. Globally asymptotically stability of the positive equilibrium is obtained. Regarding the constant number of prey refuge $m$ as a bifurcation parameter, by analyzing the distribution of the eigenvalues, the existence of Hopf bifurcation is given. Employing the center manifold theory and normal form method, an algorithm for determining the properties of the Hopf bifurcation is derived. Some numerical simulations for illustrating the analysis results are carried out.
Keywords: Hopf bifurcation, prey refuge, holling II functional response, diffusion..
Mathematics Subject Classification: Primary: 35Q92, 35B32; Secondary: 35B3.
Citation: Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Hassard, N. Kazarinoff and Y-H. Wan, "Theory and Applications of Hopf Bifurcation," Cambridge University Press, Cambridge, 1981.  Google Scholar

[11]

M. Hassel, "The Dynamics of Arthropod Predator-Prey Systems," Princeton University Press, Princeton, 1978.  Google Scholar

[12]

M. Hassel and R. May, Stability in insect host-parasite models, J. Anim. Ecol., 42 (1973), 693-726. doi: 10.2307/3133.  Google Scholar

[13]

R. Holt, Optimal foraging and the form of the predator isoclin, Am. Nat., 122 (1983), 521-541. doi: 10.1086/284153.  Google Scholar

[14]

S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwan. J. Math., 9 (2005), 151-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. Hutchinson, "The Ecological Theater and the Evolutionary Play," Yale Univ. Press, New Haven, Connecticut, 1976. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1974. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

A. Sih, Prey refuges and predator-prey stability, Theor. Popul. Biol., 31 (1987), 1-12. doi: 10.1016/0040-5809(87)90019-0.  Google Scholar

[35]

L. Smith, "Models in Ecology," Cambridge University Press, Cambridge, 1974. Google Scholar

[36]

R. J. Taylor, "Predation," Chapman $&$ Hall, New York, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Hassard, N. Kazarinoff and Y-H. Wan, "Theory and Applications of Hopf Bifurcation," Cambridge University Press, Cambridge, 1981.  Google Scholar

[11]

M. Hassel, "The Dynamics of Arthropod Predator-Prey Systems," Princeton University Press, Princeton, 1978.  Google Scholar

[12]

M. Hassel and R. May, Stability in insect host-parasite models, J. Anim. Ecol., 42 (1973), 693-726. doi: 10.2307/3133.  Google Scholar

[13]

R. Holt, Optimal foraging and the form of the predator isoclin, Am. Nat., 122 (1983), 521-541. doi: 10.1086/284153.  Google Scholar

[14]

S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwan. J. Math., 9 (2005), 151-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. Hutchinson, "The Ecological Theater and the Evolutionary Play," Yale Univ. Press, New Haven, Connecticut, 1976. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1974. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

A. Sih, Prey refuges and predator-prey stability, Theor. Popul. Biol., 31 (1987), 1-12. doi: 10.1016/0040-5809(87)90019-0.  Google Scholar

[35]

L. Smith, "Models in Ecology," Cambridge University Press, Cambridge, 1974. Google Scholar

[36]

R. J. Taylor, "Predation," Chapman $&$ Hall, New York, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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