# American Institute of Mathematical Sciences

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

 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.
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. [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] R. J. Taylor, "Predation," Chapman & Hall, New York, 1984. [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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##### 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] R. J. Taylor, "Predation," Chapman & Hall, New York, 1984. [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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