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**On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity.**
*(English)*
Zbl 1221.34192

Summary: We study the effect of the degree of habitat complexity and gestation delay on the stability of a predator–prey model. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. The qualitative dynamical behavior of the model system is verified with the published data of Paramecium aurelia (prey) and Didinium nasutum (predator) interaction. It is observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the gestation period exceeds some critical value. However, the fluctuations in the population levels can be controlled completely by increasing the degree of habitat complexity.

### MSC:

34K20 | Stability theory of functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

92D25 | Population dynamics (general) |

37N25 | Dynamical systems in biology |

### Keywords:

predator; prey interaction; habitat complexity; limit cycle; stability and direction of Hopf-bifurcation
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\textit{N. Bairagi} and \textit{D. Jana}, Appl. Math. Modelling 35, No. 7, 3255--3267 (2011; Zbl 1221.34192)

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### References:

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