### Mathematical Biosciences and Engineering

2013, Issue 4: 979-996. doi: 10.3934/mbe.2013.10.979

# Stability and Hopf bifurcation in a diffusivepredator-prey system incorporating a prey refuge

• Received: 01 February 2013 Accepted: 29 June 2018 Published: 01 June 2013
• MSC : Primary: 35Q92, 35B32; Secondary: 35B35.

• A diffusive predator-prey model with Holling type II functionalresponse and the no-flux boundary condition incorporating aconstant prey refuge is considered. Globally asymptoticallystability of the positive equilibrium is obtained. Regarding theconstant number of prey refuge $m$ as a bifurcation parameter, byanalyzing the distribution of the eigenvalues, the existence ofHopf bifurcation is given. Employing the center manifold theoryand normal form method, an algorithm for determining theproperties of the Hopf bifurcation is derived. Some numericalsimulations for illustrating the analysis results are carried out.

Citation: Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusivepredator-prey system incorporating a prey refuge[J]. Mathematical Biosciences and Engineering, 2013, 10(4): 979-996. doi: 10.3934/mbe.2013.10.979

### Related Papers:

• A diffusive predator-prey model with Holling type II functionalresponse and the no-flux boundary condition incorporating aconstant prey refuge is considered. Globally asymptoticallystability of the positive equilibrium is obtained. Regarding theconstant number of prey refuge $m$ as a bifurcation parameter, byanalyzing the distribution of the eigenvalues, the existence ofHopf bifurcation is given. Employing the center manifold theoryand normal form method, an algorithm for determining theproperties of the Hopf bifurcation is derived. Some numericalsimulations for illustrating the analysis results are carried out.
 [1] Nonlinear Anal-Real, 11 (2010), 246-252. [2] Bull. Math. Biol., 57 (1995), 63-76. [3] J. Differ. Equations, 203 (2004), 331-364. [4] Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349. [5] Trans. Amer. Math. Soc., 349 (1997), 2443-2475. [6] J. Differ. Equations, 229 (2006), 63-91. [7] Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 95-135. [8] Ecol. Model., 166 (2003), 135-146. [9] Nonlinear Anal-Real, 12 (2011), 2385-2395. [10] Cambridge University Press, Cambridge, 1981. [11] Princeton University Press, Princeton, 1978. [12] J. Anim. Ecol., 42 (1973), 693-726. [13] Am. Nat., 122 (1983), 521-541. [14] Taiwan. J. Math., 9 (2005), 151-173. [15] J. Math. Biol., 42 (2001), 489-506. [16] Appl. Math. Comput., 182 (2006), 672-683. [17] Yale Univ. Press, New Haven, Connecticut, 1976. [18] Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681-691. [19] Nonlinear Anal-Real, 10 (2009), 2558-2573. [20] J. Differ. Equations, 231 (2006), 534-550. [21] Theor. Popul. Biol., 53 (1998), 131-142. [22] J. Math. Biol., 36 (1998), 389-406. [23] J. Math. Biol., 88 (1988), 67-84. [24] Nonlinear Anal.-Real, 14 (2013), 1806-1816. [25] J. Math. Anal. Appl., 371 (2010), 323-340. [26] Math. Biosci., 218 (2009), 73-79. [27] Princeton University Press, Princeton, 1974. [28] Theor. Popul. Biol., 29 (1986), 38-63. [29] J. Differ. Equations, 200 (2004), 245-273. [30] J. Differ. Equations, 247 (2009), 866-886. [31] Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164. [32] Theor. Popul. Biol., 47 (1995), 1-17. [33] Ecology, 76 (1995), 2270-2277. [34] Theor. Popul. Biol., 31 (1987), 1-12. [35] Cambridge University Press, Cambridge, 1974. [36] Chapman $&$ Hall, New York, 1984. [37] J. Differ. Equations, 251 (2011), 1276-1304. [38] J. Math. Biol., 62 (2011), 291-331. [39] J. Math. Biol., 43 (2001), 268-290. [40] J. Differ. Equations, 246 (2009), 1944-1977.

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