Mathematical Biosciences and Engineering, 2013, 10(4): 979-996. doi: 10.3934/mbe.2013.10.979.

Primary: 35Q92, 35B32; Secondary: 35B35.

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

1. Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001

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.
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Keywords Hopf bifurcation; prey refuge; holling II functional response; diffusion.

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

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